Abstract
The aim of this paper is to prove a common random fixed-point and some random fixed-point theorems for random weakly contractive operators in separable Banach spaces. A random Mann iterative process is introduced to approximate the fixed point. Finally, the main result is supported by an example and used to prove the existence and the uniqueness of a solution of a nonlinear stochastic integral equation system.
Highlights
PreliminariesLet (X, ‖.‖) be a separable Banach space, βX be the σ-algebra of all Borel subsets of X, and (Ω, β, μ) be a complete probability measure space with the measure μ and β be the σ-algebra of subsets of Ω
Probabilistic functional analysis is one of the essential mathematical disciplines that are applied to solving problems, characterized with uncertainties, known as probabilistic models. e random fixed-point theorems are stochastic generalizations of classical fixed-point theorems which are known as deterministic results and are required for the theory of random equations, random matrices, random partial differential equations, and various classes of random operators
E theory of random fixed point was initiated by the Prague School of Probability in the 1950s. e random fixedpoint theory finds its roots in the work of Spacek [7] and Hans [8, 9]. ey established a stochastic generalization of Banach contraction principle (BCP), and they applied their results to study the existence of a solution of random linear Fredholm integral equations
Summary
Let (X, ‖.‖) be a separable Banach space, βX be the σ-algebra of all Borel subsets of X, and (Ω, β, μ) be a complete probability measure space with the measure μ and β be the σ-algebra of subsets of Ω. We restrict our attention to the case where X is a separable Banach space In this setting, the concept of weak and strong random variables is equivalent. (i) A mapping T: Ω × C ⟶ C is said to be random operator if for each x ∈ C, the mapping T(., x): Ω ⟶ C is measurable (ii) A random operator T: Ω × C ⟶ C is continuous if the set of all ω ∈ Ω for which T(ω, .) is continuous has measure one roughout this paper, we denote RV(X) as the set of all X− valued random variables and we adopt the following definition of the random fixed point given by Joshi and Bose in [34]. Let x ∈ RV(X). x is said to be a random fixed point of T if μ{ω: T(ω, x(ω)) x(ω)} 1
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