On Best Proximity Points in metric and Banach spaces

Subject Areas : Applied Mathematics

Hamid Mazaheri Tehrani ¹ , Raham Rahmani Jafarbeigi ²

1 - Department of Mathematics, Yazd University, Yazd, Iran
2 - Department of Mathematics, Yazd University, Yazd, Iran

Received: 2019-03-05 Accepted : 2019-08-08 Published : 2021-05-01

Keywords: Best proximity point, Weak ϕ-Contraction map, ϕ-Contraction,

Abstract :

Notice that best proximity point results have been studied to find necessaryconditions such that the minimization problemminx∈A∪Bd(x,Tx)has at least one solution, where T is a cyclic mapping defined on A∪B.A point p ∈ A∪B is a best proximity point for T if and only if thatis a solution of the minimization problem (2.1). Let (A,B) be a nonemptypair in a normed linear space X and S,T : A∪B → A∪B be two cyclicmappings. Let (A,B) be a nonempty pair in a normed linear space X andS,T : A∪B → A∪B be two cyclic mappings. A point p ∈ A∪B is called acommon best proximity point for the cyclic pair (T,S) provided that∥p − Tp∥ = d(A,B) = ∥p − Sp∥In this paper, we survey the existence, uniqueness and convergence of a com-mon best proximity point for a cyclic weak ST − ϕ-contraction map, whichis equivalent to study of a solution for a nonlinear programming problem inthe setting of uniformly convex Banach spaces. We will provide examples toillustrate our results.