Abstract
The purpose of this paper is to present some definitions and basic concepts of best proximity point in a new class of probabilistic metric spaces and to prove the best proximity point theorems for the contractive mappings and weak contractive mappings. In order to get the best proximity point theorems, some new probabilistic contraction mapping principles have been proved. Meanwhile the error estimate inequalities have been established. Further, a method of the proof is also new and interesting, which is to use the mathematical expectation of the distribution function studying the related problems.
Highlights
Introduction and preliminariesProbabilistic metric spaces were introduced in by Menger [ ]
A mapping T is a probabilistic contraction if T is such that for some constant < k
A probabilistic metric space is a pair (E, F), where E is a nonempty set, F is a mapping from E × E into D+ such that, if Fx,y denotes the value of F at the pair (x, y), the following conditions hold: (PM- ) Fx,y(t) = H(t) if and only if x = y; (PM- ) Fx,y(t) = Fy,x(t) for all x, y ∈ E and t ∈ (–∞, +∞); (PM- ) Fx,z(t) =, Fz,y(s) = implies Fx,y(t + s) = for all x, y, z ∈ E and –∞ < t < +∞
Summary
Introduction and preliminariesProbabilistic metric spaces were introduced in by Menger [ ]. A probabilistic metric space is a pair (E, F), where E is a nonempty set, F is a mapping from E × E into D+ such that, if Fx,y denotes the value of F at the pair (x, y), the following conditions hold: (PM- ) Fx,y(t) = H(t) if and only if x = y; (PM- ) Fx,y(t) = Fy,x(t) for all x, y ∈ E and t ∈ (–∞, +∞); (PM- ) Fx,z(t) = , Fz,y(s) = implies Fx,y(t + s) = for all x, y, z ∈ E and –∞ < t < +∞.
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