A Pata-type fixed point theorem in modular spaces with application
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RESEARCH
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A Pata-type fixed point theorem in modular spaces with application Mohadeseh Paknazar1* , Madjid Eshaghi2 , Yeol Je Cho3 and Seyed Mansour Vaezpour4 *
Correspondence: [email protected] 1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran Full list of author information is available at the end of the article
Abstract In this paper, we present a Pata-type fixed point theorem in modular spaces which generalizes and improves some old results. As an application, we study the existence of solutions of integral equations in modular function spaces. MSC: Primary 47H10; secondary 46A80; 45G10 Keywords: fixed point; modular spaces; nonlinear integral equations
1 Introduction and preliminaries In Nakano [] introduced the theory of modular spaces in connection with the theory of ordered spaces. Musielak and Orlicz [] in redefined and generalized it to obtain a generalization of the classical function spaces Lp . Khamsi et al. [] investigated the fixed point results in modular function spaces. There exists an extensive literature on the topic of the fixed point theory in modular spaces (see, for instance, [–]) and the papers referenced there. Recently, Pata [] improved the Banach principle. Using the idea of Pata, we prove a fixed point theorem in modular spaces. Then we show how our results generalize old ones. Also, we prepare an application of our main results to the existence of solutions of integral equations in Musielak-Orlicz spaces. In the first place, we recall some basic notions and facts about modular spaces. Definition . Let X be an arbitrary vector space over K (= R or C). (a) A function ρ : X → [, +∞] is called a modular if (i) ρ(x) = if and only if x = ; (ii) ρ(αx) = ρ(x) for every scalar α with |α| = ; (iii) ρ(αx + βy) ≤ ρ(x) + ρ(y) if α + β = and α ≥ , β ≥ for all x, y ∈ X. (b) If (iii) is replaced by (iv) ρ(αx + βy) ≤ αρ(x) + βρ(y) if α + β = and α ≥ , β ≥ , we say that ρ is convex modular. (c) A modular ρ defines a corresponding modular space, i.e., the vector space Xρ given by Xρ = x ∈ X : ρ(λx) → as λ → . Example . Let (X, · ) be a norm space, then · is a convex modular on X. But the converse is not true. © 2013 Paknazar et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Paknazar et al. Fixed Point Theory and Applications 2013, 2013:239 http://www.fixedpointtheoryandapplications.com/content/2013/1/239
In general the modular ρ does not behave as a norm or a distance because it is not subadditive. But one can associate to a modular the F-norm (see []). Definition . The modular space Xρ can be equipped with the F-norm defined by x |x|ρ = inf α > ; ρ ≤α . α Namely, if ρ is convex, then the functional x xρ = inf α > ; ρ ≤ , α is a nor
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