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

摘要

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 → 2 B . 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.