Random fixed point theorems in Banach spaces applied to a random nonlinear integral equation of the Hammerstein type

Abstract

Abstract The purpose of this paper is to define a new random operator called the generalized ϕ-weakly contraction of the rational type. This new random operator includes those studied by Khan et al. (Filomat 31(12):3611–3626, 2017) and Zhang et al. (Appl. Math. Mech. 32(6):805–810, 2011) as special cases. We prove some convergence, existence, and stability results in separable Banach spaces. Moreover, we produce some numerical examples to demonstrate the applicability of our analytical results. Furthermore, we apply our results in proving the existence of a solution of a nonlinear integral equation of the Hammerstein type.

Keywords

• Random fixed point
• Bochner integrable
• Generalized ϕ-weakly contraction of the rational type
• Random iterative process
• Almost sure T-stability
• Nonlinear integral equation of the Hammerstein type