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Nonlinear operator equations defined on Banach spaces play an important role in the theory of differential and integral equations and have been extensively studied over the past several decades (see [1–7]). Random techniques have been crucial in diverse areas from pure mathematics to applied sciences. S̆pac̆ek [8] and Hans̆ [9] initiated the study of random operator equations. They proved random fixed point theorems of contraction type. These results were extended and applied to various problems by Hans̆ [10–12] and Bharucha-Reid [13,14]. In recent years, the theory of random nonlinear operator equations has attracted the attention of many authors (see [15–23]). In this paper, we consider some classes of random operator equations on ordered Polish spaces. We study the existence of random positive solutions to such problems by using monotone iterative techniques and the properties of cones. As applications, we use the results obtained in this paper to study the existence of random positive solutions to some stochastic integral equations. Through this paper, E is a real Banach space with norm ‖ · ‖, 0E is the zero element of E, and P is a cone in E. So, a partially ordered relation in E can be defined by

Positive solutions to a class of random operator equations and applications to stochastic integral equations