Approximating best proximity points for Reich type non-self nonexpansive mappings
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Approximating best proximity points for Reich type non-self nonexpansive mappings Rajendra Pant1 · Rahul Shukla1 · Vladimir Rakoˇcevi´c2 Received: 7 February 2020 / Accepted: 27 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In this paper, we consider a general class of non-self non expansive mappings and present best proximity point results in Hilbert spaces. More precisely, we employ a Krasnosel’ski˘ı–Mann type algorithm to approximate best proximity points of non-self mappings. We also propose some hybrid algorithms and obtain certain strong convergence results. Keywords Best proximity point · Non-self mapping · Hilbert space Mathematics Subject Classification Primary 47H10 · 47H09
1 Introduction Let A be a nonempty subset of a metric space (X , d) and S : A → A a self-mapping. A point p ∈ A is said to be a fixed point of S if S( p) = p. The set of fixed points of S will be denoted by F(S). In a wide range of mathematical problems the existence of a solution is equivalent to the existence of a fixed point for a suitable operator. Thus the existence of a fixed point is of paramount importance in several areas of mathematics and other sciences. Fixed point results provide conditions under which mappings have fixed points. If the mapping S is non-self (that is, S : A → B with A ∩ B = ∅, and A and B are two nonempty subsets of X ) then the equation S(u) = u does not have a solution. So, it is contemplated to resolve the problem of finding a point x in A such that x is in proximity to S(x). More precisely, find a point x in A such that the function ϕ(x) = d(x, S(x)) attains its minimum value d(A, B). This point x is a global minimization of function ϕ and the point x is known as best proximity point of
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Vladimir Rakoˇcevi´c [email protected] Rajendra Pant [email protected]; [email protected] Rahul Shukla [email protected]; [email protected]
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Department of Mathematics and Applied Mathematics, University of Johannesburg, Kingsway Campus, Auckland Park 2006, South Africa
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Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš, Niš 18000, Serbia 0123456789().: V,-vol
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S in A, [25]. For initial results on best approximation and best proximity points, we refer to Ky Fan [4]. In the last two decades a number of best proximity point results have been obtained by many mathematicians [3,5,9,11–14,20,21,27–29]. Best proximity point problem has umpteen applications in equilibrium problem, economics and others (cf. [11,19]). Iterative methods for finding fixed points of nonexpansive operators in Hilbert spaces have been described in many publications and especially by Cegielski [2] in his monograph. He shows that the convergence of a big class of iteration methods follows from general properties of some classes of operators and from some general convergence theorems, in particular from Opial’s theorem [17] or from its modifications. In 2017, Jacob et al. [9] considered hybrid methods for approximating best proximity points
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