Iterative approximation to common best proximity points of proximally mean nonexpansive mappings in Banach spaces
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Iterative approximation to common best proximity points of proximally mean nonexpansive mappings in Banach spaces V. Pragadeeswarar1
· R. Gopi1
Received: 5 May 2019 / Accepted: 14 August 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this paper, we introduce the new class of mappings called proximally mean nonexpansive mappings. Further, we construct the Ishikawa iteration scheme associated with two non-self mappings, and we approximate a common best proximity point for a pair of proximally mean nonexpansive mappings via this Ishikawa iteration. As a consequence of our main result, we approximate a best proximity point for proximal mean nonexpansive mappings. Our new results complement and extend recent related results in the literature. Keywords Mean nonexpansive mapping · Ishikawa’s iteration · Best proximity points · Fixed points Mathematics Subject Classification 54C60 · 47H10
1 Introduction and preliminaries Let M and N be two nonempty disjoint subsets of a metric space (X , d). If a mapping Λ from M to N , does not possess a solution for the fixed point equation Λ(w) = w. In this situation, it is natural to determine an approximate solution w such that the error d(w, Λw) is minimum. Best proximity point theorems guarantee the existence and uniqueness of such an optimal approximate solution for the fixed point equation. The point w is called the best proximity point of Λ : M → N , if d(w, Λw) = d(M, N ), where d(M, N ) = in f {w − z : w ∈ M, z ∈ N }. One can note that the best proximity point reduces to a fixed point if Λ is a selfmapping. Recently, many authors studied the existence of a best proximity point under some suitable contraction conditions, for more details; see [2,4,8,13,16] and references therein.
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V. Pragadeeswarar [email protected] R. Gopi [email protected]
1
Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore 641112, TN, India
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V. Pragadeeswarar , R. Gopi
Similarly, when we have two mappings Λ, Γ : M → N , it may happen that the fixed point equations Λ(w) = w and Γ (w) = w do not have a common solution, namely common fixed point of the mappings Λ, Γ . Subsequently, it is attempted to determine an approximate common solution w such that the errors d(w, Λw) and d(w, Γ w) are minimum. The common best proximity point theorems deal the existence of such an optimal approximate solution, namely common best proximity point. The point w is called common best proximity point of the non-self mappings Λ, Γ : M → N , if d(w, Λw) = d(w, Γ w) = d(M, N ), where d(M, N ) = in f {w − z : w ∈ M, z ∈ N }. For existence theorems on common best proximity point, we refer [3,5,10,15] and references therein. Another important and current branch of best proximity point theory is investigating the approximate best proximity point via well known iteration processes, such as Picard, Mann, Ishikawa, etc. In [7,9,11,12,14,17,18], the authors approximated a common fixed point for
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