Abstract
The existence of fixed points of single-valued mappings in modular function spaces has been studied by many authors. The approximation of fixed points in such spaces via convergence of an iterative process for single-valued mappings has also been attempted very recently by Dehaish and Kozlowski (Fixed Point Theory Appl. 2012:118, 2012). In this paper, we initiate the study of approximating fixed points by the convergence of a Mann iterative process applied on multivalued ρ-nonexpansive mappings in modular function spaces. Our results also generalize the corresponding results of (Dehaish and Kozlowski in Fixed Point Theory Appl. 2012:118, 2012) to the case of multivalued mappings. MSC:47H09, 47H10, 54C60.
Highlights
1 Introduction and preliminaries The theory of modular spaces was initiated by Nakano [ ] in connection with the theory of ordered spaces, which was further generalized by Musielak and Orlicz [ ]
No results were obtained for the approximation of fixed points in modular function spaces until recently Dehaish and Kozlowski [ ] tried to fill this gap using a Mann iterative process for asymptotically pointwise nonexpansive mappings
We make the first ever effort to fill the gap between the existence and the approximation of fixed points of ρ-nonexpansive multivalued mappings in modular function spaces
Summary
Introduction and preliminariesThe theory of modular spaces was initiated by Nakano [ ] in connection with the theory of ordered spaces, which was further generalized by Musielak and Orlicz [ ]. Kumam [ ] obtained some fixed point theorems for nonexpansive mappings in arbitrary modular spaces. The existence of fixed points for multivalued nonexpansive mappings in uniformly convex Banach spaces was proved by Lim [ ].
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