On a Coupled System of Random and Stochastic Nonlinear Differential Equations with Coupled Nonlocal Random and Stochastic Nonlinear Integral Conditions

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.

On a Coupled System of Stochastic Ito^-Differential and the Arbitrary (Fractional) Order Differential Equations with Nonlocal Random and Stochastic Integral Conditions

“…Proof. Integrating the Equations ( 7) and ( 9) (see [12][13][14][15][16][17][18]) with substitution by (8) and using the non-local conditions (3) and ( 4) the equivalent between the problem (1)-( 3) and the integral representation (10)- (11) and the problem (1), ( 2) and (4) and the integral representation ( 12)-( 13) can be proved. Proof.…”

“…The motive of this work is to generalize the scope results of A.M.A. El-Sayed [16,17] on the stochastic fractional operators and the solution of non-local coupled systems of stochastic differential equations see [7,16].…”

Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation

“…Many authors have been interested to study fractional stochastic differential equations (see [1,2,5,8,10,11,13,15,19,21,27,28]). The existence and uniqueness of solutions to stochastic differential equations have been studied by many authors see [14,16,18,25]. In [24], the author discussed a computational method to get an approximate solution of a stochastic beam equation.…”

Existence and stability results for the integrable solution of a singular stochastic fractional-order integral equation with delay

“…These equations utilize random numbers or functions as coefficients for independent or dependent variables. Recently, El-Sayed and Fouad [15][16][17] studied a specific category of problems dealing with stochastic differential equations with nonlocal conditions. Their research shows that using Schauder's fixed point theorem, there is always at least one solution for a functional nonlocal random integral equation within the space of all squared integrable stochastic processes with a finite second moment.…”