On random fixed point theorems with application to boundary value problem

Abstract

We establish relation-theoretic random fixed point results for random operators acting on several classes of metric spaces: separable complete metric spaces, Hausdorff $L$-spaces, and generalized metric spaces with vector-valued distances. The key idea is to impose contraction assumptions on pairs that stand in a given binary relation, and to express measurable conditions explicitly using measurable-space structure and measurable Picard iterates. Our main contributions are threefold. First, we provide a corrected fixed-point argument that avoids circular limit reasoning. Second, we prove uniqueness using an R-directed comparison construction that requires no transitivity assumption on the relation. Third, we formulate a nonnegative measurable matrix-valued contraction principle, grounded in the induced infinity norm. We then apply these abstract results to a coupled random nonlocal boundary value problem. The application is a random integral operator on $C([0,1],\mathbb{R})\times C([0,1],\mathbb{R})$ and the boundary terms are Riemann—Stieltjes functionals of BV. We include an explicit contraction matrix and a numerical example verifying that the norm condition holds.