On best proximity points for set-valued contractions of Nadler type with respect to b -generalized pseudodistances in b
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On best proximity points for set-valued contractions of Nadler type with respect to b-generalized pseudodistances in b-metric spaces Robert Plebaniak* * Correspondence: [email protected] Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łód´z, Banacha 22, Łód´z, 90-238, Poland
Abstract In this paper, in b-metric space, we introduce the concept of b-generalized pseudodistance which is an extension of the b-metric. Next, inspired by the ideas of Nadler (Pac. J. Math. 30:475-488, 1969) and Abkar and Gabeleh (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 107(2):319-325, 2013), we define a new set-valued non-self-mapping contraction of Nadler type with respect to this b-generalized pseudodistance, which is a generalization of Nadler’s contraction. Moreover, we provide the condition guaranteeing the existence of best proximity points for T : A → 2B . A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the error inf{d(x, y) : y ∈ T(x)}, and hence the existence of a consummate approximate solution to the equation T(x) = x. In other words, the best proximity points theorem achieves a global optimal minimum of the map x → inf{d(x; y) : y ∈ T(x)} by stipulating an approximate solution x of the point equation T(x) = x to satisfy the condition that inf{d(x; y) : y ∈ T(x)} = dist(A; B). The examples which illustrate the main result given. The paper includes also the comparison of our results with those existing in the literature. MSC: 47H10; 54C60; 54E40; 54E35; 54E30 Keywords: b-metric spaces; b-generalized pseudodistances; global optimal minimum; best proximity points; Nadler contraction; set-valued maps
1 Introduction A number of authors generalize Banach’s [] and Nadler’s [] result and introduce the new concepts of set-valued contractions (cyclic or non-cyclic) of Banach or Nadler type, and they study the problem concerning the existence of best proximity points for such contractions; see e.g. Abkar and Gabeleh [–], Al-Thagafi and Shahzad [], Suzuki et al. [], Di Bari et al. [], Sankar Raj [], Derafshpour et al. [], Sadiq Basha [], and Włodarczyk et al. []. In , Abkar and Gabeleh [] introduced and established the following interesting and important best proximity points theorem for a set-valued non-self-mapping. First, we recall some definitions and notations. Let A, B be nonempty subsets of a metric space (X, d). Then denote: dist(A, B) = inf{d(x, y) : x ∈ A, y ∈ B}; A = {x ∈ A : d(x, y) = dist(A, B) for some y ∈ B}; B = {y ∈ B : ©2014 Plebaniak; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Plebaniak Fixed Point Theory and Applications 2014, 2014:39 http://www.fixedpointtheoryandappli
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