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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Mappings of type Orlicz and generalized Cesáro sequence space

Nashat F Mohamed1* and Awad A Bakery12

Author Affiliations

1 Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt

2 Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais, 21589, Saudi Arabia

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Journal of Inequalities and Applications 2013, 2013:186  doi:10.1186/1029-242X-2013-186


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/186


Received:7 January 2013
Accepted:5 April 2013
Published:18 April 2013

© 2013 Mohamed and Bakery; 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

We study the ideal of all bounded linear operators between any arbitrary Banach spaces whose sequence of approximation numbers belong to the generalized Cesáro sequence space and Orlicz sequence space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M1">View MathML</a>, when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M2">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M3">View MathML</a>; our results coincide with that known for the classical sequence space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M4">View MathML</a>.

Keywords:
approximation numbers; operator ideal; generalized Cesáro sequence space; Orlicz sequence space

1 Introduction

By <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M5">View MathML</a>, we denote the space of all bounded linear operators from a normed space X into a normed space Y. The set of natural numbers will denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M6">View MathML</a> and the real numbers by ℝ. By ω, we denote the space of all real sequences. A map which assigns to every operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M7">View MathML</a> a unique sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M8">View MathML</a> is called an s-function and the number <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M9">View MathML</a> is called the nth s-numbers of T if the following conditions are satisfied:

(a) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M10">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M7">View MathML</a>.

(b) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M12">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M13">View MathML</a>.

(c) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M14">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M15">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M16">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M17">View MathML</a>.

(d) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M18">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M7">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M20">View MathML</a>.

(e) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M21">View MathML</a> If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M22">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M7">View MathML</a>.

(f) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M24">View MathML</a> where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M25">View MathML</a> is the identity operator on the Euclidean space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M26">View MathML</a>. Example of s-numbers, we mention approximation number <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M27">View MathML</a>, Gelfand numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M28">View MathML</a>, Kolmogorov numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M29">View MathML</a> and Tichomirov numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M30">View MathML</a> defined by: All of these numbers satisfy the following condition:

(I) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M31">View MathML</a>.

(II) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M32">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M33">View MathML</a> is a metric injection (a metric injection is a one to one operator with closed range and with norm equal one) from the space Y into a higher space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M34">View MathML</a> for suitable index set Λ.

(III) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M35">View MathML</a>.

(IV) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M36">View MathML</a>.

(g) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M37">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M13">View MathML</a>.

An operator ideal U is a subclass of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M39">View MathML</a> such that its components <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M40">View MathML</a> satisfy the following conditions:

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M41">View MathML</a>, where K denotes the 1-dimensional Banach space, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M42">View MathML</a>.

(ii) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M43">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M44">View MathML</a> for any scalars <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M45">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M46">View MathML</a>.

(iii) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M47">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M48">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M17">View MathML</a> then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M50">View MathML</a>. See [1-3].

An Orlicz function is a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M51">View MathML</a> which is continuous, non-decreasing and convex with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M52">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M53">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M54">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M55">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M56">View MathML</a>. See [4,5].

If convexity of Orlicz function M is replaced by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M57">View MathML</a>. Then this function is called modulus function, introduced by Nakano [6]; also, see [7,8] and [9]. An Orlicz function M is said to satisfy <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M58">View MathML</a>-condition for all values of u, if there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M59">View MathML</a>, such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M60">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M61">View MathML</a>). The <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M58">View MathML</a>-condition is equivalent to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M63">View MathML</a> for all values of u and for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M64">View MathML</a>. Lindentrauss and Tzafriri [10] used the idea of Orlicz function to construct Orlicz sequence space

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M65">View MathML</a>

which is a Banach space with respect to the norm

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M66">View MathML</a>

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M2">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M68">View MathML</a> the space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M1">View MathML</a> coincides with the classical sequence space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M4">View MathML</a>. Recently, different classes of sequences have been introduced by using an Orlicz function. See [11] and [12].

Remark 1.1 Let M be an Orlicz function then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M71">View MathML</a> for all λ with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M72">View MathML</a>.

For a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M73">View MathML</a> of positive real numbers with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M74">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M75">View MathML</a> the generalized Cesáro sequence space is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M76">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M77">View MathML</a>

The space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M78">View MathML</a> is a Banach space with the norm

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M79">View MathML</a>

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M80">View MathML</a> is bounded, we can simply write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M81">View MathML</a>

Also, some geometric properties of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M78">View MathML</a> are studied by Sanhan and Suantai [13].

Throughout this paper, the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M83">View MathML</a> is a bounded sequence of positive real numbers, we denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M84">View MathML</a> where 1 appears at ith place for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M85">View MathML</a>. Different classes of paranormed sequence spaces have been introduced and their different properties have been investigated. See [14-18] and [19].

For any bounded sequence of positive numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M86">View MathML</a>, we have the following well-known inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M87">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M88">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M89">View MathML</a>. See [20].

2 Preliminary and notation

Definition 2.1 A class of linear sequence spaces E, called a special space of sequences (sss) having the following conditions:

(1) E is a linear space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M90">View MathML</a>, for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M91">View MathML</a>.

(2) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M92">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M93">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M94">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M96">View MathML</a>i.e.E is solid’,

(3) if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M97">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M98">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M99">View MathML</a> denotes the integral part of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M100">View MathML</a>.

We call such space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M101">View MathML</a> a pre modular special space of sequences if there exists a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M102">View MathML</a>, satisfies the following conditions:

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M103">View MathML</a><a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M104">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M105">View MathML</a>, where θ is the zero element of E,

(ii) there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M106">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M107">View MathML</a> for all values of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M96">View MathML</a> and for any scalar λ,

(iii) for some numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M109">View MathML</a>, we have the inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M110">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M111">View MathML</a>,

(iv) if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M94">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95">View MathML</a> then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M114">View MathML</a>,

(v) for some numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M115">View MathML</a> we have the inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M116">View MathML</a>,

(vi) for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M117">View MathML</a> there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M118">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M119">View MathML</a>. This means the set of all finite sequences is ρ-dense in E.

(vii) for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M120">View MathML</a> there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M121">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M122">View MathML</a>.

It is clear that from condition (ii) that ρ is continuous at θ. The function ρ defines a metrizable topology in E endowed with this topology is denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M101">View MathML</a>.

Example 2.2<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M4">View MathML</a> is a pre-modular special space of sequences for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M125">View MathML</a>, with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M126">View MathML</a>.

Example 2.3<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M127">View MathML</a> is a pre-modular special space of sequences for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M128">View MathML</a>, with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M129">View MathML</a>.

Definition 2.4

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M130">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M131">View MathML</a>

3 Main results

Theorem 3.1<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M132">View MathML</a>is an operator ideal ifEis a special space of sequences (sss).

Proof To prove <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M132">View MathML</a> is an operator ideal:

(i) let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M134">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M135">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M136">View MathML</a>, since E is a linear space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M90">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M139">View MathML</a>; for that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M140">View MathML</a>, which implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M141">View MathML</a>.

(ii) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M142">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M143">View MathML</a> then from Definition 2.1 condition (3) we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M144">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M145">View MathML</a>, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M146">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M147">View MathML</a> is a decreasing sequence and from the definition of approximation numbers we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M148">View MathML</a>

Since E is a linear space and from Definition 2.1 condition (2) we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M149">View MathML</a>, hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M150">View MathML</a>.

(iii) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M47">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M152">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M17">View MathML</a>, then we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M154">View MathML</a> and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M155">View MathML</a>, from Definition 2.1 conditions (1) and (2) we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M156">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M157">View MathML</a>.

 □

Theorem 3.2<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M158">View MathML</a>is an operator ideal, ifMis an Orlicz function satisfying<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M58">View MathML</a>-condition and there exists a constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M106">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M161">View MathML</a>.

Proof

(1-i) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M162">View MathML</a>, since M is non-decreasing, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M163">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M164">View MathML</a>.

(1-ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M20">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M166">View MathML</a> since M satisfies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M58">View MathML</a>-condition, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M168">View MathML</a>, for that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M169">View MathML</a>, then from (1-i) and (1-ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M1">View MathML</a> is a linear space over the field of numbers. Also <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M171">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95">View MathML</a> since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M173">View MathML</a>.

(2) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M94">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M176">View MathML</a>, since M is none decreasing, then we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M177">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M178">View MathML</a>.

(3) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M179">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M180">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M181">View MathML</a>. Hence, from Theorem 3.1, it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M158">View MathML</a> is an operator ideal.

 □

Theorem 3.3<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M183">View MathML</a>is an operator ideal, if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M83">View MathML</a>is an increasing sequence of positive real numbers, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M185">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M186">View MathML</a>.

Proof

(1-i) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M187">View MathML</a> since

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M189">View MathML</a>.

(1-ii) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M20">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M191">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M192">View MathML</a>

we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M193">View MathML</a>, from (1-i) and (1-ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M194">View MathML</a> is a linear space.

To show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M195">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M196">View MathML</a>, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M186">View MathML</a> we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M198">View MathML</a>. Thus, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M199">View MathML</a>

Hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M195">View MathML</a>.

(2) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M94">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M203">View MathML</a>

since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M204">View MathML</a>. Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M205">View MathML</a>.

(3) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M206">View MathML</a>, then we have

Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M208">View MathML</a>. Hence, from Theorem 3.1 it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M183">View MathML</a> is an operator ideal.

 □

Theorem 3.4LetMbe an Orlicz function. Then the linear space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M210">View MathML</a>is dense in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M211">View MathML</a>.

Proof Define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M212">View MathML</a> on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M1">View MathML</a>. First we prove that every finite mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M214">View MathML</a> belongs to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M211">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M216">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M196">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M1">View MathML</a> is a linear space then for every finite mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M214">View MathML</a> the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M220">View MathML</a> contains only finitely many numbers different from zero. To prove that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M221">View MathML</a>, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M222">View MathML</a>, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M223">View MathML</a>, and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M224">View MathML</a>, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M225">View MathML</a> then there exists a natural number <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M226">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M227">View MathML</a>, since ρ is none decreasing and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M228">View MathML</a> is decreasing for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M230">View MathML</a>

then there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M231">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M232">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M233">View MathML</a>, and by using the conditions of M we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M234">View MathML</a>

 □

Corollary 3.5If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M3">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M2">View MathML</a>, we get<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M237">View MathML</a>. See[3].

Theorem 3.6The linear space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M210">View MathML</a>is dense in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M239">View MathML</a>, if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M83">View MathML</a>is an increasing sequence of positive real numbers with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M241">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M186">View MathML</a>.

Proof First we prove that every finite mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M214">View MathML</a> belongs to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M239">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M245">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M196">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M247">View MathML</a> is a linear space, then for every finite mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M214">View MathML</a>i.e. the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M249">View MathML</a> contains only finitely many numbers different from zero. Now we prove that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M250">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M186">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M198">View MathML</a>, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M253">View MathML</a> we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M254">View MathML</a>, and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M255">View MathML</a>, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M256">View MathML</a> then there exists a natural number <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M226">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M258">View MathML</a> for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M259">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M260">View MathML</a>, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M228">View MathML</a> is decreasing for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M95">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M263">View MathML</a>

(1)

then there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M231">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M265">View MathML</a>

(2)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M266">View MathML</a>

(3)

since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M267">View MathML</a>. Then there exists a natural number <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M268">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M269">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M270">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M271">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M272">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M273">View MathML</a>, so we can take

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M274">View MathML</a>

(4)

since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M83">View MathML</a> is an increasing sequence and by using (1), (2), (3) and (4), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M276">View MathML</a>

 □

Theorem 3.7LetXbe a normed space, Ya Banach space and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M101">View MathML</a>be a pre modular special space of sequences (sss), then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M278">View MathML</a>is complete.

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M279">View MathML</a> be a Cauchy sequence in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M280">View MathML</a>, then by using Definition 2.1 condition (vii) and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M281">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M282">View MathML</a>

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M279">View MathML</a> is also Cauchy sequence in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M284">View MathML</a>. Since the space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M284">View MathML</a> is a Banach space, then there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M7">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M287">View MathML</a> and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M288">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M289">View MathML</a>, ρ is continuous at θ and using Definition 2.1(iii), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M290">View MathML</a>

Hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M291">View MathML</a> as such <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M292">View MathML</a>. □

Corollary 3.8LetXbe a normed space, Ya Banach space andMbe an Orlicz function such thatMsatisfies<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M58">View MathML</a>-condition. ThenMis continuous at<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M294">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M211">View MathML</a>is complete.

Corollary 3.9LetXbe a normed space, Ya Banach space and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M83">View MathML</a>be an increasing sequence of positive real numbers with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M241">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M186">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/186/mathml/M239">View MathML</a>is complete.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

NFM gave the idea of the article. AAB carried out the proofs and its application. All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors wish to thank the referees for their careful reading of the paper and for their helpful suggestions.

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