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Controllability for nonlinear evolution equations with monotone operators

Yong Han Kang1, Jin-Mun Jeong2* and Hyun-Hee Rho2

Author Affiliations

1 Institute of Liberal Education, Catholic University of Daegu, Daegue, 712-702, Korea

2 Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea

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Journal of Inequalities and Applications 2013, 2013:534  doi:10.1186/1029-242X-2013-534

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/534


Received:7 August 2013
Accepted:22 October 2013
Published:12 November 2013

© 2013 Kang et al.; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we investigate the approximate controllability for nonlinear evolution equations with monotone operators and nonlinear controllers according to monotone operator theory. We also give the regularity for the nonlinear equation. Finally, an example, to which our main result can be applied, is given.

MSC: 35F25, 93C20.

Keywords:
nonlinear evolution equation; monotone operator; approximate controllability; regularity; hemicontinuous

1 Introduction

In this paper, we deal with the approximate controllability for the semilinear equation in a Hilbert space H as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M1">View MathML</a>

(1.1)

In (1.1), the principal operator −A generates an analytic semigroup <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M2">View MathML</a>. Let U be a Hilbert space of control variables, and let B be a linear (or nonlinear) operator from U to H, which is called a controller.

First, we consider the following initial value problem of a semilinear equation:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M3">View MathML</a>

(1.2)

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M4">View MathML</a> is an unbounded operator, Di Blasio et al.[1] proved <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M5">View MathML</a>-regularity for a retarded linear system in Hilbert spaces, and Jeong [2] (also see [3]) considered the control problem for retarded linear systems with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M6">View MathML</a>-valued controller and more general Lipschitz continuity of nonlinear terms.

For the theory of monotone operators, there are many literature works; for example, see Lions [4], Stampacchia [5], Browder [6], and the references cited therein. Kenmochi [7] derived new results on monotone operator equations, and Ouchi [8] proved the analyticity of solutions of semilinear parabolic differential equations with monotone nonlinearity. For the existence of solutions for a class of nonlinear evolution equations with monotone perturbations, one can refer to [9-11]. We refer to Pascali and Sburlan [12], Morosanu [13] to see the applications of nonlinear mapping of monotone type and nonlinear evolution equations. The classical solutions of (1.2) were obtained by Kato [14] under the monotonicity condition on the nonlinear term f as an operator from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M7">View MathML</a> to H.

In the first part of this note, we apply results of [14] to find <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M5">View MathML</a>-regularity of solutions in the wider sense of (1.2) under the more general monotonicity of a nonlinear operator f from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M9">View MathML</a> to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>, which is related to the results of Tanabe [[15], Theorem 6.6.2].

Next, we extend and develop control problems on this topic. In recent years, as for the controllability for semilinear differential equations with Lipschitz continuity of a nonlinear operator f, Naito [16] and [17-19] proved the approximate controllability under the range conditions of the controller B. However, we can find few articles which extend the known general controllability problems to nonlinear evolution equation (1.1) with monotone operators and nonlinear controllers.

In this paper, based on the regularity for solutions of equation (1.2), we obtain the approximate controllability for nonlinear evolution equation (1.1) with monotone operators and nonlinear controllers.

The paper is organized as follows. In Section 2, we explain several notations of this paper and state results about <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M5">View MathML</a>-regularity for linear equations in the sense of [1,15,20]. In Section 3, we give the regularity for nonlinear equation (1.2). In Section 4, we obtain the approximate controllability for nonlinear evolution equation (1.1) with hemicontinuous monotone operators by using the theory of monotone operators. In the end, an example is provided to illustrate the application of the obtained results.

2 Preliminaries

If H is identified with its dual space, we may write <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M12">View MathML</a> densely and the corresponding injections are continuous. The norm on V, H and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a> will be denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M14">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M15">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M16">View MathML</a>, respectively. The duality pairing between the element <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M17">View MathML</a> of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a> and the element <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M19">View MathML</a> of V is denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M20">View MathML</a>, which is the ordinary inner product in H if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M21">View MathML</a>.

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M22">View MathML</a>, we denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M23">View MathML</a> by the value <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M24">View MathML</a> of l at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M25">View MathML</a>. The norm of l as element of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a> is given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M27">View MathML</a>

Therefore, we assume that V has a stronger topology than H and, for brevity, we may regard that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M28">View MathML</a>

(2.1)

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M29">View MathML</a> be a bounded sesquilinear form defined in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M30">View MathML</a> and satisfying Gårding’s inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M31">View MathML</a>

(2.2)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M32">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M33">View MathML</a> is a real number. Let A be an operator associated with this sesquilinear form:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M34">View MathML</a>

Then −A is a bounded linear operator from V to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a> by the Lax-Milgram theorem. The realization of A in H, which is the restriction of A to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M36">View MathML</a>

is also denoted by A. It is well known that A is positive definite and self-adjoint and generates an analytic semigroup <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M2">View MathML</a> in both H and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>. From the following inequalities:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M39">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M40">View MathML</a>

is the graph norm of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M41">View MathML</a>, it follows that there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M42">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M43">View MathML</a>

(2.3)

Thus we have the following sequence:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M44">View MathML</a>

(2.4)

where each space is dense in the next one, which is continuous injection.

Lemma 2.1With notations (2.3), (2.4), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M45">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M46">View MathML</a>denotes the real interpolation space betweenVand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a> (Section 2.4 of[21]or[22]).

If X is a Banach space, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M48">View MathML</a> is the collection of all strongly measurable square integrable functions from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M49">View MathML</a> into X, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M50">View MathML</a> is the set of all absolutely continuous functions on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M51">View MathML</a> such that their derivative belong to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M48">View MathML</a>. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M53">View MathML</a> will denote the set of all continuous functions from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M51">View MathML</a> into X with the supremum norm. If X and Y are two Banach spaces, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M55">View MathML</a> is the collection of all bounded linear operators from X into Y, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M56">View MathML</a> is simply written as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M57">View MathML</a>. Here, we note that by using interpolation theory, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M58">View MathML</a>

(2.5)

First, we consider the following linear system:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M59">View MathML</a>

(2.6)

By virtue of Theorem 3.3 of [1] (or Theorem 3.1 of [2,15]), we have the following result on the corresponding linear equation of (2.6).

Lemma 2.2 (1) For<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M60">View MathML</a> (see Lemma 2.1) and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M61">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M62">View MathML</a>, there exists a unique solutionxof (2.6) belonging to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M63">View MathML</a>

and satisfying

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M64">View MathML</a>

(2.7)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M65">View MathML</a>is a constant depending onT.

(2) Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M67">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M62">View MathML</a>. Then there exists a unique solutionxof (2.6) belonging to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M69">View MathML</a>

and satisfying

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M70">View MathML</a>

(2.8)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M65">View MathML</a>is a constant depending onT.

Throughout this paper, strong convergence is denoted by ‘→’ and weak convergence by ‘⇀’.

Definition 2.1 Let X and Y be Banach spaces and L be a mapping from X into Y. The domain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M72">View MathML</a> of L is assumed to be convex. L is called hemicontinuous if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M73">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M74">View MathML</a> is continuous in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M75">View MathML</a> in weak topology of Y.

The linear operator is obviously hemicontinuous.

Definition 2.2 Let X and Y be Banach spaces and L be a single-valued mapping from X into Y. L is called demicontinuous if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M76">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M77">View MathML</a> imply that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M78">View MathML</a>.

Definition 2.3 Let L be a mapping from a Banach space X into its conjugate space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79">View MathML</a>. L is said to be pseudo-monotone if the following condition is satisfied. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M80">View MathML</a> is a directed family of points, contained in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M72">View MathML</a>, which converges weakly to an element x of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M72">View MathML</a> and if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M83">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M84">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M85">View MathML</a>.

Definition 2.4 Let L be a generally multi-valued mapping from a Hilbert space X into itself. If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M86">View MathML</a>

(2.9)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M87">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M88">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M89">View MathML</a>, then L is called a monotone operator. Sometimes L is also called a monotone operator if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M90">View MathML</a>

holds instead of (2.9).

Definition 2.5 A real-valued continuous function j is called a gauge function defined on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M91">View MathML</a> if it is strictly monotone increasing and satisfies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M92">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M93">View MathML</a>.

Let X be a Banach space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79">View MathML</a> its conjugate. For any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M95">View MathML</a>, we set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M96">View MathML</a>

The multi-valued operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M97">View MathML</a> is called the duality mapping of X with a gauge function j.

Let us denote by Λ the operator determined by an inner product <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M98">View MathML</a> on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M99">View MathML</a>. Then it is immediate that Λ is a duality mapping from V into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a> with a gauge function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M101">View MathML</a>. It is also known that the duality mapping is monotone and hemicontinuous, and hence it is pseudo-monotone.

Lemma 2.3We have briefly explained the theory of monotone operators (see [[15], Section 6.6]).

(1) In Definition 2.3 above, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M102">View MathML</a>is seen by taking<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M103">View MathML</a>.

(2) Hemicontinuous monotone mappings from a Banach spaceXinto<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79">View MathML</a>are pseudo-monotone.

(3) LetXbe a reflexive Banach space, and let bothXand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79">View MathML</a>be strictly convex. Further, let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M106">View MathML</a>be monotone andΛbe a duality mapping fromXinto<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M108">View MathML</a>, thenMis maximal monotone.

(4) LetXbe a reflexive Banach space andLbe a closed monotone linear operator fromXand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79">View MathML</a>. If the dual operator<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M110">View MathML</a>is monotone, thenLis maximal monotone.

(5) LetXbe a reflexive Banach space, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M111">View MathML</a>be maximal monotone andLbe a pseudo-monotone bounded mapping from<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M112">View MathML</a>into<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79">View MathML</a>. If there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M114">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M115">View MathML</a>

then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M116">View MathML</a>, that is, for every<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M117">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M118">View MathML</a>has a solution<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M119">View MathML</a>.

The following inequality is referred to as Young’s inequality.

Lemma 2.4 (Young’s inequality)

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M120">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M121">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M122">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M123">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M124">View MathML</a>. Then, for every<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M125">View MathML</a>, one has

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M126">View MathML</a>

Lemma 2.5LetHbe a Hilbert space and V be a reflexive Banach space. Suppose thatVis a dense subspace ofHand thatVhas a stronger topology thanH. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M12">View MathML</a>. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M62">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M129">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M130">View MathML</a>. Then the operatorLdefined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M131">View MathML</a>

is maximal monotone linear.

Proof Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M132">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M133">View MathML</a>, noting that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M134">View MathML</a> and by Young’s inequality, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M135">View MathML</a>

so that x belongs to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M136">View MathML</a>. Therefore, we find that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M137">View MathML</a> is well defined as an element of H. It is easily shown that L is a closed linear operator from X into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M72">View MathML</a> is dense in X. Further, since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M140">View MathML</a>

we know that L is monotone. It is also easily seen that the adjoint operator of L is given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M141">View MathML</a>

Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M110">View MathML</a> is also monotone. Therefore, from (4) of Lemma 2.3, it is concluded that L is maximal monotone linear. □

3 Nonlinear equations

We consider the following initial value problem of a semilinear equation:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M143">View MathML</a>

(3.1)

The following result is due to Kato [7] (or [[15], Theorem 6.6.1]).

Lemma 3.1Letfbe a demicontinuous bounded mapping from<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M144">View MathML</a>intoH. Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M145">View MathML</a>is monotone for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M146">View MathML</a>:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M147">View MathML</a>

Assume further thatAis a generator of a contraction semigroup<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M2">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M149">View MathML</a>). Then, for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M150">View MathML</a>, there exists a solution<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M151">View MathML</a>of the integral equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M152">View MathML</a>

corresponding to (3.1), and it is unique. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M153">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M154">View MathML</a>be the solutions with initial values<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M155">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M156">View MathML</a>, respectively. Then the estimate

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M157">View MathML</a>

holds on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M158">View MathML</a>. Hence, the mapping which carries the initial value<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M159">View MathML</a>to the solutionxis a continuous mapping fromHinto<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M136">View MathML</a>.

Next, we apply Lemma 3.1 to find a solution in the wider sense of (3.1) under somewhat different assumptions. Concerning the nonlinear mapping f, assume the following hypothesis.

Assumption (F) The mapping f is demicontinuous bounded from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M7">View MathML</a> into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M163">View MathML</a> for each t is monotone as a mapping from V into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>.

The following theorem is a part of Theorem 6.6.2 due to Tanabe [15].

Theorem 3.1Let Assumption (F) be satisfied, and let the assumptions on the principal operatorAstated in Section 2 be satisfied. Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M159">View MathML</a>is an arbitrary element ofHand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M166">View MathML</a>. Then there exists a solution<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M167">View MathML</a>, satisfying<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M168">View MathML</a>, of

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M169">View MathML</a>

(3.2)

and it is unique. Moreover, there exists a constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M170">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M171">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M170">View MathML</a>is a constant depending onTand the mapping

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M173">View MathML</a>

is Lipschitz continuous.

Proof As far as the existence of the solution is concerned, we may put <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M174">View MathML</a>. By Lemma 2.5, the operator defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M175">View MathML</a>

is a maximal monotone linear operator from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M176">View MathML</a> into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177">View MathML</a>. Let us write <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M178">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M179">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M167">View MathML</a>. Then both A and F are monotone operators from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M176">View MathML</a> into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M183">View MathML</a>. Since we assumed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M174">View MathML</a>, equation (3.2) is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M185">View MathML</a>

Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M186">View MathML</a> is monotone, if it is shown to be maximal monotone, assumption (5) of Lemma 2.3 is satisfied with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M187">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M188">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M174">View MathML</a>. Then we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M190">View MathML</a>, which implies the existence of the solution. Thus, from now on, we prove the maximal monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M186">View MathML</a>. Since Λ determined by an inner product <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M98">View MathML</a> on V is a duality mapping from V into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>, as seen under Definition 2.5, to see the maximal monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M186">View MathML</a>, on account of (3) of Lemma 2.3, it is enough to verify that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M195">View MathML</a>.

Let h be an arbitrary element of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M197">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M198">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177">View MathML</a>. Since Λ is positive definite and self-adjoint in both H and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>, the domain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M201">View MathML</a> coincides with V and H, respectively. Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M202">View MathML</a> is a contraction operator in both H and V, and it converges strongly to I as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M203">View MathML</a>. It is also easy to see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M204">View MathML</a> holds for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M205">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M206">View MathML</a>. Let us define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M207">View MathML</a>. Then the mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M208">View MathML</a> satisfies the assumption for f in Lemma 3.1. Hence, Lemma 3.1 can be applied to the initial value problem

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M209">View MathML</a>

(3.3)

Let us denote the semigroup generated by Λ by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M210">View MathML</a>. Then it ensures the existence of a solution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M151">View MathML</a> of the equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M212">View MathML</a>

(3.4)

Multiplying by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M213">View MathML</a> on (3.3) and integrating over <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M214">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M215">View MathML</a>

(3.5)

By using Young’s inequality and the monotonicity of f, the following holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M216">View MathML</a>

So, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M217">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M170">View MathML</a> is a constant, so that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M219">View MathML</a> is bounded in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M220">View MathML</a>. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M221">View MathML</a> is bounded in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177">View MathML</a>. By replacing them by their subsequence, we may assume <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M223">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M176">View MathML</a>. By letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M203">View MathML</a> in (3.4), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M226">View MathML</a>

Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M227">View MathML</a>

from multiplying by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M228">View MathML</a> and the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M208">View MathML</a>, it follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M230">View MathML</a>

Thus, noting that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M231">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M232">View MathML</a> converge strongly to h and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M233">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177">View MathML</a>, respectively, we see that x is a solution of the equation <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M235">View MathML</a>. Finally, to prove the uniqueness of the solution, suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M153">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M154">View MathML</a> are solutions with initial conditions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M155">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M156">View MathML</a> and forcing terms <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M240">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M241">View MathML</a>, respectively. Then it is easy to see that there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M242">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M243">View MathML</a>

This completes the proof of Theorem 3.1. □

Remark 3.1 In a similar way to Theorem 3.1, we also obtain the existence of solutions of (3.2) in the case where f is a demicontinuous bounded mapping from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M244">View MathML</a> into H. Moreover, assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M163">View MathML</a> for each t is monotone as a mapping from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M41">View MathML</a> into H and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M247">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M248">View MathML</a>, then there exists a unique solution x of (3.1) such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M249">View MathML</a>

Moreover, there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M170">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M251">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M170">View MathML</a> is a constant depending on T and the mapping

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M253">View MathML</a>

is Lipschitz continuous.

4 Approximate controllability

In this section, we deal with the approximate controllability for the semilinear equation in H as follows.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M254">View MathML</a>

(4.1)

In (4.1), the principal operator −A generates an analytic semigroup <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M2">View MathML</a> as stated in Section 2. Let U be a Hilbert space of control variables, and let B be a bounded linear operator from U to H, which is called a controller. The mild solution of initial value problem (4.1) is the following form:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M256">View MathML</a>

Let f be a nonlinear mapping satisfying the following.

Assumption (F1) The mapping f is demicontinuous bounded from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M257">View MathML</a> into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>. Assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M259">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M260">View MathML</a> is monotone as a mapping from V into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M262">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M263">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M264">View MathML</a> is monotone as a mapping from U into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>.

For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M266">View MathML</a>, let us define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M267">View MathML</a>. Then from Theorem 3.1 it follows that solution (4.1) exists and is unique in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M268">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M269">View MathML</a> be a state value of system (4.1) at time T corresponding to the nonlinear term f and the control u. We define the reachable sets for system (4.1) as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M270">View MathML</a>

Definition 4.1 System (4.1) is said to be approximately controllable at time T if for every desired final state <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M271">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M272">View MathML</a>, there exists a control function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M273">View MathML</a> such that the solution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M269">View MathML</a> of (4.1) satisfies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M275">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M276">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M277">View MathML</a> is the closure of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M278">View MathML</a> in H.

Definition 4.2 Let L be a mapping from a Banach space X into its conjugate space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79">View MathML</a>. T is called coercive if there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M280">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M281">View MathML</a>

Remark 4.1 [[23], Theorem 1.3]

It is well known that if X is a reflexive Banach space and L is monotone, everywhere defined and hemicontinuous from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M112">View MathML</a> into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M79">View MathML</a>, then L is maximal monotone. If in addition L is coercive monotone, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M284">View MathML</a>.

First, we consider the approximate controllability of system (4.1) in the case where the controller B is the identity operator on H under Assumption (F1) on the nonlinear operator f. So, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M285">View MathML</a> obviously. Consider the linear system given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M286">View MathML</a>

(4.2)

and the following semilinear control system:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M287">View MathML</a>

(4.3)

Lemma 4.1Let Assumption (F1) be satisfied, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M288">View MathML</a>be the solution of (4.2) corresponding to a controlu. Then there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M289">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M290">View MathML</a>

(4.4)

Proof Set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M291">View MathML</a>

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M292">View MathML</a>. Then equation (4.4) is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M293">View MathML</a>

(4.5)

It is easy to see that G is monotone as an operator from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294">View MathML</a> to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177">View MathML</a>, and is a demicontinuous bounded mapping as an operator from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294">View MathML</a> into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177">View MathML</a>. Let the collection of all finite dimensional subspaces of H be denoted by , and when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M299">View MathML</a>, let the orthogonal projection on Y be denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M300">View MathML</a>. For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M301">View MathML</a>, let us define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M302">View MathML</a>; thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M300">View MathML</a> also denotes the orthogonal projection in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294">View MathML</a>. According to Assumption (F1), we have that the operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M305">View MathML</a> is a coercive monotone operator from Y into itself. In general, any demicontinuous operator is hemicontinuous. Therefore, by Remark 4.1, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M306">View MathML</a>, which (4.5) implies the existence of a solution to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M307">View MathML</a>

(4.6)

Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M308">View MathML</a>

we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M309">View MathML</a>

Hence, the solution of (4.6) is bounded on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M136">View MathML</a>. Let w be an arbitrary element of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294">View MathML</a>. Then we can take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M312">View MathML</a> satisfying (4.6) such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M313">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294">View MathML</a>. Since G is monotone as an operator from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294">View MathML</a> to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177">View MathML</a>, and is a demicontinuous bounded mapping, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M317">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M177">View MathML</a>. Hence, we obtain that for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M319">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M320">View MathML</a>

as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M321">View MathML</a>, so that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M322">View MathML</a>

(4.7)

If v is replaced by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M323">View MathML</a> in (4.7), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M324">View MathML</a>

which, by the demicontinuity of G, leads to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M325">View MathML</a>

in the limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M203">View MathML</a>. Since v is arbitrary, we obtain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M327">View MathML</a>. □

Remark 4.2 As seen in [24], we know that if X is a Hilbert space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M328">View MathML</a> is maximal monotone, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M329">View MathML</a>. So, (4.5) is easily obtained if the operator G in Theorem 4.1 is maximal monotone.

Theorem 4.1Under Assumption (F1) and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M330">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M331">View MathML</a>

Therefore, if linear system (4.2) with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M332">View MathML</a>is approximately controllable at timeT, then so is semilinear system (4.3).

Proof Let y, x be the solutions of (4.2) and (4.3), respectively. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M333">View MathML</a> in the sense of Lemma 4.1. Then, since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M334">View MathML</a>

we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M335">View MathML</a>

Acting on both sides of the above equation, by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M336">View MathML</a>, from the monotonicity of f, it follows

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M337">View MathML</a>

which is

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M338">View MathML</a>

By using Gronwall’s inequality, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M103">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M136">View MathML</a>. Noting that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M341">View MathML</a>, every solution of the linear system with control u is also a solution of the semilinear system with control v, that is, we have that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M342">View MathML</a>. □

From now on, we consider the initial value problem for semilinear parabolic equation (4.1). Let U be a Hilbert space, and let the controller operator B be a nonlinear operator from U to H.

Theorem 4.2Let Assumption (F1) and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M343">View MathML</a>be satisfied. Assume that the inverse mapping<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M344">View MathML</a>of the controllerBexists and is monotone. Then the linear system

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M345">View MathML</a>

(4.8)

is approximately controllable at timeT, so is nonlinear system (4.1).

Proof Let y be a solution of (4.8) corresponding to a control u. Consider the following semilinear system:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M346">View MathML</a>

(4.9)

Set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M347">View MathML</a>

(4.10)

We put

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M348">View MathML</a>

Equation (4.10) is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M349">View MathML</a>

(4.11)

Here, similarly to the proof of Lemma 4.1, we have that there exists an element <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M350">View MathML</a> satisfying (4.10), that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M351">View MathML</a>. In a similar way to the proof of Theorem 4.1, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M103">View MathML</a>. Since system (4.1) is equivalent to (4.9), we conclude that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M342">View MathML</a>. □

Now we consider the control problem of (4.1) when the controller B is a nonlinear mapping in the case where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M354">View MathML</a>. In this case, we suppose that Assumption (F) and the next additional assumption are satisfied.

Assumption (F2) Assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M355">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M264">View MathML</a> is maximal monotone as a mapping from U into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>.

The following result is well known from semigroup properties.

Lemma 4.2If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M358">View MathML</a>and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M359">View MathML</a>

then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M360">View MathML</a>for almost all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M361">View MathML</a>.

Theorem 4.3Let Assumption (F2) and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M343">View MathML</a>be satisfied. Assume thatBis a hemicontinuous monotone mapping fromVinto<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>; moreover, if it is coercive, then linear system (4.8) is approximately controllable at timeT, so is semilinear system (4.1).

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M364">View MathML</a>. We define the linear operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M365">View MathML</a> from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M294">View MathML</a> to H by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M367">View MathML</a>

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M368">View MathML</a>. As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M364">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M370">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M371">View MathML</a>

for instance, take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M372">View MathML</a>. By expressing <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M373">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M374">View MathML</a>. By Remark 4.1, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M375">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M376">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M377">View MathML</a>

Since p is an arbitrary element of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M378">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M379">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M380">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M378">View MathML</a>. This implies that linear system (4.8) is approximately controllable.

To prove the approximate controllability of (4.1), we will show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M382">View MathML</a>, i.e., for given <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M383">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M364">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M385">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M386">View MathML</a>

(4.12)

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M387">View MathML</a>. Then we write <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M388">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M389">View MathML</a>. Then we rewrite (4.12) as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M390">View MathML</a>

Thus, in view of Lemma 4.2, it is enough to verify that there exists an arbitrary element u of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M391">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M392">View MathML</a>. By (2) of Lemma 2.3, B is pseudo-monotone and satisfies the condition (5) of Lemma 2.3. Thus, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M393">View MathML</a>. Since p is an arbitrary element of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M378">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M379">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M396">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M378">View MathML</a>. This implies inequality (4.3) and completes the proof of the theorem. □

Remark 4.3 We know that by Assumption (F1) and (4.8), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M398">View MathML</a> is monotone, hemicontinuous and coercive from U into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M10">View MathML</a>. Therefore, as seen in Remark 4.1, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M400">View MathML</a>, that is, system (4.1) is approximately controllable.

5 Example

Let Ω be a bounded region in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M401">View MathML</a> with smooth boundary Ω. We define the following spaces:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M402">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M403">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M404">View MathML</a> are derivatives of u in the distribution sense. The norm of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M405">View MathML</a> is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M406">View MathML</a>

Hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M405">View MathML</a> is a Hilbert space.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M408">View MathML</a>

The norm and inner product of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M409">View MathML</a> are defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M410">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M411">View MathML</a>. We put <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M412">View MathML</a>. Define the operator A by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M413">View MathML</a>

The operator A in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M414">View MathML</a> is defined so that for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M415">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M416">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M417">View MathML</a>

Then, for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M418">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M419">View MathML</a> and A is a positive definite self-adjoint operator.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M420">View MathML</a> be a dual space of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M409">View MathML</a>. For any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M422">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M415">View MathML</a>, the notation <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M23">View MathML</a> denotes the value l at v.

Let u be fixed if we consider the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M425">View MathML</a>, this function is continuous linear. For any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M422">View MathML</a>, it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M427">View MathML</a>. We denote that for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M428">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M429">View MathML</a>

that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M430">View MathML</a>. The operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M431">View MathML</a> is a one-to-one mapping from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M409">View MathML</a> to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M433">View MathML</a>. The relation of operators A and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M431">View MathML</a> satisfy that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M435">View MathML</a>

From now on, both A and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M431">View MathML</a> are denoted simply by A.

We introduce a simple example of the control operator B which satisfies the condition in Theorem 4.2. Consider the case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M437">View MathML</a>, and define the intercept operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M438">View MathML</a> on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M439">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M440">View MathML</a>

Then B is a continuous monotone mapping such that there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M441">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M442">View MathML</a>

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M443">View MathML</a>

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M444">View MathML</a>. Then by Sobolev’s imbedding theorem, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M445">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M446">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M447">View MathML</a>. Assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M448">View MathML</a> is a continuous and increasing function defined on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M449">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M450">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M451">View MathML</a>. If we put <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M452">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M453">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M454">View MathML</a>, so that it is clear that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M455">View MathML</a> is monotone as an operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M456">View MathML</a> into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M457">View MathML</a>. To show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M455">View MathML</a> is a demicontinuous mapping from H into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M457">View MathML</a>, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M460">View MathML</a> in H. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M219">View MathML</a> is bounded in H, so is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M462">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M457">View MathML</a>. Hence, there exists a subsequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M464">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M465">View MathML</a> almost everywhere in Ω and there exists an element <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M466">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M467">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M468">View MathML</a> is a continuous function of real variables λ, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M469">View MathML</a> almost everywhere in Ω. Otherwise, one can find an appropriate convex combination <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M470">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M471">View MathML</a>, which is strongly convergent to g in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M472">View MathML</a>. This says that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M473">View MathML</a> for all y for which <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M474">View MathML</a>. Therefore, we obtain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M475">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/534/mathml/M476">View MathML</a>. Thus, all the conditions stated in Theorem 4.2 are satisfied. Therefore, nonlinear system (4.1) with monotone operators is approximately controllable at time T.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YHK drafted the manuscript and corrected the main results, JMJ carried out the main proof of this paper, and HHR participated in its design and coordination.

Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0007560).

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