Existence and stability behavior of random solutions of a system of nonlinear random equations

Abstract

A study of a system of nonlinear random integral equations of the Volterra type of the form x(t;ω) = h(t, x(t;ω)) + ∝ 0 t k,(t, τ, x(τ;ω); ω)ƒ(τ, x(τ,ω))dτ , is presented where t ϵ R + = { t: t ≥ 0}, ωϵΩ, Ω being the underlying set of a complete probability measure space (Ω, A, P). The random process x( t; ω) is the unknown random vector { x i ( t; ω)} i = 1 n , each component defined on R + × Ω; h(t, x(t, ω)) is the stochastic free vector { h i ( t, x i ( t; ω))} i = 1 n ; the nonlinear stochastic kernel is a matrix k( t, τ, x( τ, ω); ω) = { k ij ( t, τ, x( τ, ω); ω)}; defined, for each component, on R + × R + × R × Ω for 0 ≤ τ ≤ t < ∞ and i, j = 1, 2,…, n; and the random scalar function ƒ(t, x (t;ω)) = {ƒ i(t, x i(t; ω))} i = 1 n is defined on R + × R. The object of this paper is to study the following types of stability: 1. (i) stability in the mean; 2. (ii) asymptotic stability in the mean and; 3. (iii) stochastic asymptotic exponential stability of the above equation. In addition, sufficient hypotheses, utilizing specific Banach spaces, will be developed for the existence of a unique random solution, a second-order stochastic process, for the Volterra system described above. The present results generalize the recent results of Ahmed and Teo.