Abstract
The purpose of this paper is to present a new class of probabilistic normed spaces and to study fixed point problems in this class of probabilistic normed spaces. This paper includes the following two contents: (1) The definition of a new class of probabilistic normed spaces, the so-called S-probabilistic normed spaces, is given. In order to study the fixed point problems, some relevant properties of S-probabilistic normed spaces are discussed and some basic useful results are obtained; (2) The notion of probabilistic weak convergence is firstly presented in this paper. Therefore the probabilistic weak and strong convergence theorems of fixed points for nonexpansive mappings, asymptotically nonexpansive mappings and strongly pseudocontractive mappings are also proved by using the new methods and techniques.
Highlights
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, s < +∞
A Menger probabilistic metric space is a triple (E, F, ), where E is a nonempty set, is a continuous t-norm and 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: (MPM- ) Fx,y(t) = H(t) if and only if x = y; (MPM- ) Fx,y(t) = Fy,x(t) for all x, y ∈ E and t ∈ (–∞, +∞); (MPM- ) Fx,y(t + s) ≥ (Fx,z(t), Fz,y(s)) for all x, y, z ∈ E and t >, s >
Summary
Introduction and preliminariesProbabilistic metric spaces were introduced in by Menger [ ]. A Menger probabilistic metric space is a triple (E, F, ), where E is a nonempty set, is a continuous t-norm and 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: (MPM- ) Fx,y(t) = H(t) if and only if x = y; (MPM- ) Fx,y(t) = Fy,x(t) for all x, y ∈ E and t ∈ (–∞, +∞); (MPM- ) Fx,y(t + s) ≥ (Fx,z(t), Fz,y(s)) for all x, y, z ∈ E and t > , s > .
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