Fixed Points of Multivalued Maps in Modular Function Spaces
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Research Article Fixed Points of Multivalued Maps in Modular Function Spaces Marwan A. Kutbi and Abdul Latif Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia Correspondence should be addressed to Abdul Latif, [email protected] Received 7 February 2009; Accepted 14 April 2009 Recommended by Jerzy Jezierski The purpose of this paper is to study the existence of fixed points for contractive-type and nonexpansive-type multivalued maps in the setting of modular function spaces. We also discuss the concept of w-modular function and prove fixed point results for weakly-modular contractive maps in modular function spaces. These results extend several similar results proved in metric and Banach spaces settings. Copyright q 2009 M. A. Kutbi and A. Latif. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and Preliminaries The well-known Banach fixed point theorem on complete metric spaces specifically, each contraction self-map of a complete metric space has a unique fixed point has been extended and generalized in different directions. For example, see Edelstein 1, 2, Kasahara 3, Rhoades 4, Siddiq and Ansari 5, and others. One of its generalizations is for nonexpansive single-valued maps on certain subsets of a Banach space. Indeed, these fixed points are not necessarily unique. See, for example, Browder 6–8 and Kirk 9. Fixed point theorems for contractive and nonexpansive multivalued maps have also been established by several authors. Let H denote the Hausdorff metric on the space of all bounded nonempty subsets of a metric space X, d. A multivalued map J : X → 2X where 2X denotes the collection of all nonempty subsets of X with bounded subsets as values is called contractive 10 if H Jx, J y ≤ hd x, y
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for all x, y ∈ X and for a fixed number h ∈ 0, 1. If the Lipschitz constant h 1, then J is called a multivalued nonexpansive mapping 11. Nadler 10, Markin 11, Lami-Dozo 12, and others proved fixed point theorems for these maps under certain conditions in the setting of
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Fixed Point Theory and Applications
metric and Banach spaces. Note that an element x ∈ X is called a fixed point of a multivalued map J : X → 2X if x ∈ Jx. Among others, without using the concept of the Hausdorff metric, Husain and Tarafdar 13 introduced the notion of a nonexpansive-type multivalued map and proved a fixed point theorem on compact intervals of the real line. Using such type of notions Husain and Latif 14 extended their result to general Banach space setting. The fixed point results in modular function spaces were given by Khamsi et al. 15. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated in modular spaces. For instance, fixed point theorems are proved in 15, 16 for nonexpansive maps. In this paper, we d
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