On Best Proximity Points in Metric and Banach Spaces

Notice that best proximity point results have been studied to find necessary conditions such that the minimization problem min x∈A ∪ B d(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 that is a solution of the minimization problem (2.1). Let (A,B) be a nonempty pair in a normed linear space X and S, T : A ∪ B → A ∪ B be two cyclic mappings. Let (A,B) be a nonempty pair in a normed linear space X and S, T : A ∪ B → A ∪ B be two cyclic mappings. A point p ∈ A ∪ B is called a common 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 common best proximity point for a cyclic weak ST − ϕ-contraction map, which is equivalent to study of a solution for a nonlinear programming problem in the setting of uniformly convex Banach spaces. Moreover, we provide some examples to illustrate and support the results.