Existence and convergence theorems for generalized hybrid mappings in uniformly convex metric spaces

Withun Phuengrattana () and Suthep Suantai ()
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Withun Phuengrattana: Nakhon Pathom Rajabhat University
Suthep Suantai: Chiang Mai University

Indian Journal of Pure and Applied Mathematics, 2014, vol. 45, issue 1, 121-136

Abstract: Abstract In this paper, we show a relationship between strictly convexity of type (I) and (II) defined by Takahashi and Talman, and we prove that any uniformly convex metric space is strictly convex of type (II). Continuity of the convex structure is also shown on a compact domain. Then, we prove the existence of a minimum point of a convex, lower semicontinuous and d-coercive function defined on a nonempty closed convex subset of a complete uniformly convex metric space. By using this property, we prove fixed point theorems for (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Using this result, we also obtain a common fixed point theorem for a countable commutative family of (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Finally, we establish strong convergence of a Mann type iteration to a fixed point of (α, β)-generalized hybrid mapping in a uniformly convex metric space without assuming continuity of convex structure. Our results can be applied to obtain the existence and convergence theorems for (α, β)-generalized hybrid mappings in Hilbert spaces, uniformly convex Banach spaces and CAT(0) spaces.

Keywords: (α; β)-generalized hybrid mappings; convex metric spaces; uniformly convex; CAT(0) spaces; fixed point (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s13226-014-0055-x

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