Abstract
It is well known that Stochastic equations had many useful applications in describing numerous events and problems of real world, and the nonlocal integral condition is important in physics, finance and engineering. Here we are concerned with two problems of a coupled system of random and stochastic nonlinear differential equations with two coupled systems of nonlinear nonlocal random and stochastic integral conditions. The existence of solutions will be studied. The sufficient condition for the uniqueness of the solution will be given. The continuous dependence of the unique solution on the nonlocal conditions will be proved.
Highlights
Let (Ω, z, P) be a fixed probability space, where Ω is a sample space, z is a σ-algebra and P is a probability measure.The aim of this article is to extend the results of A.M.A
The continuous dependence of a unique solution has been studied on the random initial data and the random function which ensures the stability of the solution
The coupled system (24) with nonlocal integral conditions (25) satisfies all the Assumptions 1–5 of Theorem 1. with b = 21, c = 16, there exists at least one solution of the system
Summary
Let (Ω, z, P) be a fixed probability space, where Ω is a sample space, z is a σ-algebra and P is a probability measure. Let Z (t; ω ) = Z (t), t ∈ [0, T ], ω ∈ Ω be a second order stochastic process, i.e., E( Z2 (t)) < ∞, t ∈ [0, T ]. Let C = C ([0, T ], L2 (Ω)) be the space of all second order stochastic processes which is mean square (m.s) continuous on [0, T ]. Let X = C ([0, T ], L2 (Ω)) × C ([0, T ], L2 (Ω)) be the class of all ordered pairs ( x, y), x, y ∈ C with the norm k( x, y)k X = max{ k x kC , kykC } = max{ sup k x (t)k2 , sup ky(t)k2 }. Let φi : [0, T ] → [0, T ] be continuous functions such that φi (t) ≤ t and consider the following assumptions.
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