COMMON FIXED POINT THEOREM IN $b$-MENGER SPACES WITH A FULLY CONVEX STRUCTURE

Abstract

We prove, in b-Menger spaces [9] the existence of common fixed point for nonexpansive mappings in fully convex b-Menger space by using the normal structure property. We provide examples to analyze and illustrate our main results.

PROBABILISTIC $\varphi$-CONTRACTION IN $b$-MENGER SPACES WITH FULLY CONVEX STRUCTURE

“…We now recall some basic definitions and relevant lemmas in the theory of b-Menger spaces (see [11] and [12]).…”

“…(R, F, T M , 2 n−1 ) is a fully convex b-Menger space with the convex structure W (a, b; λ) = λa+(1−λ)b satisfying the condition (C 2 ) and since F is continuous then the convex structure verifies the condition (C 1 ) (see [12]). We take A = [− 1 2 , 1 2 ], we have f (∂A) ⊆Å.…”

“…They discussed some topological and geometrical properties of probabilistic b-metric spaces and they showed the fixed point and common fixed point property for a self mapping which is a nonlinear contractions in b-Menger spaces which are a particular spaces of probabilistic b-metric spaces (see [14]). Furthermore, they defined the notion of fully convex structure and established in fully convex b-Menger spaces the existence of common fixed point for nonexpansive mapping by using the normal structure property (see [12]). Also, they showed a fixed point theorem in b-Menger spaces using B-contraction with cyclical conditions (see [13]).…”