Abstract
In this paper, we introduce the concepts of generalized probabilistically bounded set $\Omega^{*}$ and Menger-Hausdorff metric $\widetilde{G}^{*}$ in Menger probabilistic G-metric spaces, and prove that $(\Omega^{*},\widetilde{G}^{*},\Delta)$ is also a Menger probabilistic G-metric space. Utilizing these concepts, we establish some common fixed point theorems for three hybrid pairs of mappings satisfying the common property $(E.A)$ in Menger probabilistic G-metric spaces. Finally, an example is given to exemplify the theorems.
Highlights
1 Introduction and preliminaries As a generalization of a metric space, the concept of a probabilistic metric space has been introduced by Menger [, ]
Fixed point theory in a probabilistic metric space is an important branch of probabilistic analysis, and many results on the existence of fixed points or solutions of nonlinear equations in Menger PM-spaces have been studied by many scholars
In, Mustafa and Sims [ ] introduced the concept of a generalized metric space, and many fixed point results have been obtained by many authors
Summary
Introduction and preliminariesAs a generalization of a metric space, the concept of a probabilistic metric space has been introduced by Menger [ , ]. [ ] A Menger probabilistic G-metric space (for brevity, a PGM-space) is a triple (X, G∗, ), where X is a nonempty set, is a continuous t-norm and G∗ is a mapping from X × X × X into D (Gx∗,y,z denote the value of G∗ at the point (x, y, z)) satisfying the following conditions: (PGM- ) Gx∗,y,z(t) = for all x, y, z ∈ X and t > if and only if x = y = z; (PGM- ) Gx∗,x,y(t) ≥ Gx∗,y,z(t) for all x, y, z ∈ X with z = y and t > ; (PGM- ) Gx∗,y,z(t) = Gx∗,z,y(t) = Gy∗,x,z(t) = · · · (symmetry in all three variables); (PGM- ) Gx∗,y,z(t + s) ≥ (Gx∗,a,a(s), Ga∗,y,z(t)) for all x, y, z, a ∈ X and s, t ≥ .
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