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

A. Mbarki, R. Oubrahim

2019
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International Journal of Applied Mathematics
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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. A. Mbarki, R. Oubrahim If M is a Banach space with norm . and A is a nonempty subset of M , in this context a mapping f : The nonexpansive mapping f may frequently fail to have a fixed point in a setting which permits the existence of a decreasing sequence {A i } i∈IN
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... quence {A i } i∈IN of nonempty, closed, convex and f -invariant (f (A i ) ⊂ A i , i ∈ IN)) sets having empty intersection. However, Kirk [7] observed that the presence of a geometric property called 'normal structure' (M.S. Brodskii, D.P. Milman, On the center of a convex set, Dokl. Akad. Nauk. SSSR, 59 (1948), 837-840) guarantees that the nonexpansive f such that A is nonempty, weakly compact convex subset of a Banach space M , has a fixed point in M . Since its publication in 1965, many have tried to extend it to metric spaces. But because of its strong connection to the linear convexity structure of linear spaces, it was hard to come up with a nice and flexible extension. For example, Takahashi [15] was may be the first one to give a metric analogue to Kirk's theorem. His approach was based on defining a convexity in metric spaces extremely similar to the linear convexity also known as Menger convexity [1]-[3], [8] . In 1987 Hadžić [6] offered an extension of Takahachi's structure to Menger spaces and proved fixed point theorem for nonexpansive mappings in probabilistic metric spaces with a convex structure. Following Hadžić's approach, Ješić et al. [14] have introduced, strictly convex and normal structure in Menger spaces and proved fixed point theorem for nonexpansive mappings. Recently, Mbarki et al. [9] introduced the probabilistic b-metric spaces (b-Menger spaces) as a generalization of probabilistic metric spaces (Menger spaces) and they studied some topological properties and showed the fixed point property for nonlinear contractions in these spaces. This paper is organized as follows. In Section 2, we present some basic concepts and definitions on b-Menger spaces. In Section 3, we show some geometric and topological properties in convex b-Menger spaces and we define the fully convex b-Menger spaces. We finish this section by proving the main result in this paper, i.e., the existence of common fixed point for nonexpansive mappings in fully convex b-Menger space using the normal structure property. Finally, in Section 4, we construct a significant example of fully convex b-Menger spaces from the literature and we prove a common fixed point theorem in these spaces. Our results generalize some well-known results in the literature.

doi:10.12732/ijam.v32i2.5
fatcat:7no72aislraibkk4ifwxdfotdu