## On best proximity points for set-valued contractions of Nadler type with respect to b-generalized pseudodistances in b-metric spaces

##### 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.

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