<p>A Meir-Keeler type fixed point theorem for a family of mappings is proved in Menger probabilistic metric space (Menger PM-space). We establish that completeness the equivalent to property larger class includes continuous as well discontinuous mappings. In addition it, satisfying (ϵ - δ) non-expansive established.</p>

In the existing literature, Banach contraction theorem as well Meir-Keeler fixed point were extended to fuzzy metric spaces. However, extensions require strong additional assumptions. The purpose of this paper is determine a class spaces in which both theorems remain true without need any condition. We demonstrate wide validity new class.

In this paper we introduce a generalization of Meir-Keeler contraction forrandom mapping T : Ω×C → C, where C be a nonempty subset of a Banachspace X and (Ω,Σ) be a measurable space with being a sigma-algebra of sub-sets of. Also, we apply such type of random fixed point results to prove theexistence and unicity of a solution for an special random integral equation.

Recently, a class of mappings named as couplings was introduced in [U.P.B. Sci. Bull. Series A, 79 (2017), 1-12]. Based on this concept, we introduce symmetric Meir-Keeler and ensure the existence strong coupled fixed points. We present some concrete examples to support obtained results. Furthermore, an application our results, investigate unique solution system integral equations.

Throughout this paper, by R we denote the set of all nonnegative numbers, while N is the set of all natural numbers. Let A and B be nonempty subsets of a metric space X, d . Consider a mapping f : A ∪ B → A ∪ B, f is called a cyclic map if f A ⊆ B and f B ⊆ A. A point x in A is called a best proximity point of f in A if d x, fx d A,B is satisfied, where d A,B inf{d x, y : x ∈ A,y ∈ B}, and x ∈ ...