Uniformly Lipschitzian mappings in modular function spaces

By T. Benavides, Mohamed Amine Khamsi, S. Samadi. 2001

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The theory of modular spaces was initiated by Nakano [14] in 1950 in connection with the theory of order spaces and rede8ned and generalized by Musielak and Orlicz [13] in 1959. De8ning a norm, particular Banach spaces of functions can be considered. Metric 8xed theory for these Banach spaces of functions has been widely studied (see, for instance, [15]). Another direction is based on considering an abstractly given functional which controls the growth of the functions. Even though a metric is not de8ned, many problems in 8xed point theory for nonexpansive mappings can be reformulated in modular spaces (see, for instance, [8] and references therein). In this paper, we study the existence of 8xed points for a more general class of mappings: uniformly Lipschitzian mappings. Fixed point theorems for this class of mappings in Banach spaces have been studied in [2,3] and in metric spaces in [11,12] (for further information about this subject, see [1, Chapter VIII] and references therein). The main tool in our approach is the coeAcient of normal structure Ñ(L ). We prove that under suitable conditions a k-uniformly Lipschitzian mapping has a 8xed point if k ¡ ( Ñ(L ))−1=2. In the last section we show a class of modular spaces where Ñ(L )¡ 1 and so, the above theorem can be successfully applied.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4ac8e2b6f5c1143c702dd14bbdb553c741de3bc9

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