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Consider a self map T defined on the union of two subsets A and B of a metric space and satisfying T (A) ⊆ B and T (B) ⊆ A. We give some contraction type existence results for a best proximity point, that is, a point x such that d(x, T x) = dist (A, B). We also give an algorithm to find a best proximity point for the map T in the setting of a uniformly convex Banach space.

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Abstract. The notion of proximal normal structure is introduced and used to study mappings that are "relatively nonexpansive" in the sense that they are defined on the union of two subsets A and B of a Banach space X and satisfy T x − T y ≤ x − y for all x ∈ A, y ∈ B. It is shown that if A and B are weakly compact and convex, and if the pair (A, B) has proximal normal structure, then a relatively nonexpansive mapping , B). If in addition the norm of X is strictly convex, and if (i) is replaced with (i)′ T (A) ⊆ A and T (B) ⊆ B, then the conclusion is that there exist x 0 ∈ A and y 0 ∈ B such that x 0 and y 0 are fixed points of T and x 0 − y 0 = dist(A, B). Because every bounded closed convex pair in a uniformly convex Banach space has proximal normal structure, these results hold in all uniformly convex spaces. A Krasnosel'skiȋ type iteration method for approximating the fixed points of relatively nonexpansive mappings is also given, and some related Hilbert space results are discussed.1. Introduction. Let X be a normed linear space and D ⊆ X. Recall that a mapping T : D → D is nonexpansive if T x − T y ≤ x − y for all x, y ∈ D. In this paper we consider mappings that are "relatively nonexpansive" in the sense that they are defined on the union of two subsets A and B of X and satisfy T x − T y ≤ x − y for all x ∈ A, y ∈ B. We introduce the notion of "proximal normal structure", and we show that if A and B are weakly compact and convex, and the pair (A, B) has proximal normal structure, then every relatively nonexpansive mapping T : A ∪ B → A ∪ B for which T (A) ⊆ B and T (B) ⊆ A has a best proximity point. This means that there exists x ∈ A∪B such that x−T x = dist(A, B). As a companion result we show that if, in addition, the norm of X is strictly convex, then the assumptions T (A) ⊆ A and T (B) ⊆ B imply the existence of x 0 ∈ A and y 0 ∈ B such that x 0 and y 0 are fixed points of T and x 0 − y 0 = dist(A, B).

The objective of this paper is to deal with a kind of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. Some important and interesting results are obtained which in turn lead to a solution of fuzzy linear programming problems without converting them to crisp linear programming problems.Keywords Fuzzy numbers . Ranking . Fuzzy linear programming AMS subject classification: 90C05 . 90C70 Bellman and Zadeh (1970) proposed the concept of decision making in fuzzy environments. After the pioneering work on fuzzy linear programming by Tanaka et al. (1974Tanaka et al. ( , 1984 and Zimmermann (1974), several kinds of fuzzy linear programming problems have appeared in the literature and different methods have been proposed to solve such problems. Numerous methods for comparison of fuzzy numbers have been suggested in the literature. Maleki, Mashinchi (1996, 2000) used the Rouben's method of comparison of fuzzy numbers and obtained an optimal solution. In this paper, we introduce a new type of fuzzy arithmetic for symmetric trapezoidal fuzzy numbers and propose a method for solving fuzzy linear programming problems without converting them to crisp linear programming problems. This paper is organized as follows: In section 1, we give the definitions of fuzzy linear programming, symmetric trapezoidal fuzzy numbers and some related results of fuzzy arithmetic on symmetric trapezoidal fuzzy numbers. In section 2, we prove fuzzy analogues of some important theorems of linear programming. A numerical example involving symmetric trapezoidal fuzzy numbers is also given to illustrate the theory developed in this paper.