Abstract
In this paper, we establish several random fixed point theorems for random operators satisfying some iterative condition w.r.t. a measure of noncompactness. We also discuss the case of monotone random operators in ordered Banach spaces. Our results extend several earlier works, including Itoh’s random fixed point theorem. As an application, we discuss the existence of random solutions to a class of random first-order vector-valued ordinary differential equations with lack of compactness.
Highlights
Random fixed point theorems are stochastic generalizations of deterministic fixed point theorems
We prove some new random fixed point results for monotone convex-power condensing random operators in ordered Banach spaces
A mapping ξ : Ω ⟶ X is said to be: (i) a deterministic fixed point of a random operator T : Ω × X ⟶ E if Tðω, ξðωÞÞ = ξðωÞ for every ω ∈ Ω, (ii) a random fixed point of a random operator T : Ω × X ⟶ E if it is measurable and Tðω, ξðωÞÞ = ξðωÞ for every ω ∈ Ω, we present some basic facts regarding measures of noncompactness in Banach spaces, which we will be needed in the sequel
Summary
Random fixed point theorems are stochastic generalizations of deterministic fixed point theorems. Tan and Yuan [17] established an interesting result which serves as a bridge that links the random fixed point theory with the deterministic fixed point theory. We prove some new random fixed point theorems for (countably) convex-power condensing random operators in Banach spaces. We prove some new random fixed point results for monotone (countably) convex-power condensing random operators in ordered Banach spaces. To illustrate our theoretical results, we investigate the solvability of a broad class of random first-order vector-valued ordinary differential equations. F ðt, uðt, wÞ, ð1Þ where t belongs to a bounded and closed interval in the real line R, ω belongs to a set Ω endowed with a σ-algebra, and the functions f , u have values in a Banach space E
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