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On solving Lipschitz pseudocontractive operator equations
Journal of Inequalities and Applications volume 2014, Article number: 314 (2014)
Abstract
We analyze the convergence of the Mann-type double sequence iteration process to the solution of a Lipschitz pseudocontractive operator equation on a bounded closed convex subset of arbitrary real Banach space into itself. Our results extend the result in (Moore in Comp. Math. Appl. 43: 1585-1589, 2002).
MSC:47H10, 54H25.
1 Introduction
Let E be a real Banach space and be the dual space of E. Let J be the normalized duality mapping from E to defined by
for all where denotes the generalized duality pairing. A single-valued duality map will be denoted by j.
An operator is said to be
-
pseudocontractive if there exists such that
for any ;
-
accretive if for any , there exists satisfying
-
strongly pseudocontractive if there exist and a constant such that
for any ;
-
strongly accretive if for any , there exist and a constant satisfying
for all .
As a consequence of a result of Kato [1], the concept of pseudocontractive operators can equivalently be defined as follows:
T is strongly pseudocontractive if there exists such that the inequality
holds for all and . If in the inequality (1.1), then T is pseudocontractive.
It is easy to see that T is pseudocontractive if and only if is accretive where I denotes the identity mapping on E.
Let C be a compact convex subset of a real Hilbert space and let be a Lipschitz pseudocontraction. It remains as an open question whether the Mann iteration process always converges to a fixed point of T. In [2] it was proved that the Ishikawa iteration process converges strongly to a fixed point of T. In 2001, Mutangadura and Chidume [3] constructed the following example to demonstrate that the Mann iteration process is not guaranteed to converge to a fixed point of a Lipschitz pseudocontraction mapping a compact convex subset of a real Hilbert space H into itself.
Example [3]
Let with the usual Euclidean inner product, and for define . Now, let ; the closed unit ball in H and let , . Define the map by
Observe that T is pseudocontractive, Lipschitz continuous (with Lipschitz constant 5) and has the origin as its unique fixed point; C is compact and convex. However, for any , we have
while for any , we have
and therefore no Mann sequence can converge to , the unique fixed point of T, unless the initial guess is the fixed point itself.
Moore [4] introduced the concept of a Mann-type double sequence iteration process and proved that it converges strongly to a fixed point of a continuous pseudocontraction which maps a bounded closed convex nonempty subset of a real Hilbert space into itself.
Definition 1.1 [4]
Let denote the set of all nonnegative integers (the natural numbers) and let E be a normed linear space. By a double sequence in E is meant a function defined by . A double sequence is said to converge strongly to if given any , there exist such that for all , . If , , we have , then the double sequence is said to be Cauchy. Furthermore, if for each fixed n, as and then as , then as .
Theorem 1.1 [4]
Let C be a bounded closed convex nonempty subset of a (real) Hilbert space H, and let be a continuous pseudocontractive map. Let be real sequences satisfying the following conditions:
-
(i)
,
-
(ii)
, ,
-
(iii)
,
-
(iv)
.
For an arbitrary but fixed , and for each , define by , . Then the double sequence generated from an arbitrary by
converges strongly to a fixed point of T in C.
The following lemma will be useful in the sequel.
Lemma 1.2 [5]
Let and be two sequences of nonnegative real numbers satisfying the inequality
Here . If , then .
It is our purpose in this paper to extend Theorem 1.1 from Hilbert space to an arbitrary real Banach space with no further assumptions on the real sequences , .
2 Main results
Theorem 2.1 Let C be a bounded closed convex subset of a Banach space E and be a Lipschitz pseudocontraction with . Let be real sequences satisfying the following conditions:
-
(i)
,
-
(ii)
.
For an arbitrary but fixed , and for each , define by , . Then the double sequence generated from an arbitrary by
converges strongly to a fixed point of T in C.
Proof Since T is Lipschitzian, there exists such that
Since T is pseudocontractive, for each , we have
Hence, is Lipschitz and strongly pseudocontractive. Also, C is invariant under for all , by convexity. Thus, for each , has a unique fixed point , say, in C.
Now, we proceed in the following steps.
-
(I)
for each , as .
-
(II)
as .
-
(III)
.
Proof of (I). In fact, it follows from (2.1) that
Thus, if is a fixed point of , , then
Using inequality (1.1), it follows that
On the other hand, by (2.1) we obtain
Therefore,
Substituting (2.3) into (2.2), we arrive at
which implies that
and so
Since C is bounded, there exists such that
Hence, it follows from (2.4) that
Since and , it follows from Lemma 1.2 that
i.e., as .
Proof of (II). We prove that converges to some . For this purpose, we need only to prove that is a Cauchy sequence.
In fact, we have
that is,
hence
where . If follows from condition (i) that
This completes step (II) of the proof.
Proof of (III). In order to accomplish step (III), we first have to prove that is an approximate fixed point sequence for T. In fact, from , we have
where . Hence . Since as , T is continuous and using continuity of the norm, we get , i.e., . This completes the proof. □
Corollary 2.2 Let C be a bounded closed convex subset of a Banach space E and be a nonexpansive mapping with . Let be real sequences satisfying conditions (i)-(ii) in Theorem 2.1. For an arbitrary but fixed , and for each , define by , . Then the double sequence generated from an arbitrary by
converges strongly to a fixed point of T in C.
Proof Obvious, observing the fact that every nonexpansive mapping is Lipschitz and pseudocontractive. □
The following corollary follows from Theorem 2.1 on setting .
Corollary 2.3 Let C, E, T, , be as in Theorem 2.1. For an arbitrary but fixed , and for each , define by for all . Then the double sequence generated from an arbitrary by
converges strongly to a fixed point of T in C.
Remark 2.1 Theorem 2.1 improves and extends Theorem 3.1 of Moore [3] in three respects:
-
(1)
It abolishes the condition that .
-
(2)
It abolishes the condition that .
-
(3)
The ambient space is no longer required to be a Hilbert space and is taken to be the more general Banach space instead.
Remark 2.2
-
(1)
Whereas the Ishikawa iteration process was proved to converge to a fixed point of a Lipschitz pseudocontractive mapping in compact convex subsets of a Hilbert space, we imposed no compactness conditions to obtain the strong convergence of the double sequence iteration process to a fixed point of a Lipschitz pseudocontraction.
-
(2)
Our results may easily be extended to the slightly more general classes of Lipschitz hemicontractive and Lipschitz quasi-nonexpansive mappings.
-
(3)
Prototypes of the sequences and are
References
Kato T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn. 1967, 19: 508-520. 10.2969/jmsj/01940508
Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147-150. 10.1090/S0002-9939-1974-0336469-5
Mutangadura SA, Chidume CE: An example on the Mann iteration method for Lipschitz pseudocontractions. Proc. Am. Math. Soc. 2001, 129: 2359-2363. 10.1090/S0002-9939-01-06009-9
Moore C: A double sequence iteration process for fixed points of continuous pseudocontractions. Comput. Math. Appl. 2002, 43: 1585-1589. 10.1016/S0898-1221(02)00121-9
Liu QH: A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings. J. Math. Anal. Appl. 1990, 146: 302-305.
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Abdelhakim, A.A., Gu, F. On solving Lipschitz pseudocontractive operator equations. J Inequal Appl 2014, 314 (2014). https://doi.org/10.1186/1029-242X-2014-314
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DOI: https://doi.org/10.1186/1029-242X-2014-314