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

Abstract

The fractional stochastic differential equations had many applications in interpreting many events and phenomena of life, and the nonlocal conditions describe numerous problems in physics and finance. Here, we are concerned with the combination between the three senses of derivatives, the stochastic Ito^-differential and the fractional and integer orders derivative for the second order stochastic process in two nonlocal problems of a coupled system of two random and stochastic differential equations with two nonlocal stochastic and random integral conditions and a coupled system of two stochastic and random integral conditions. We study the existence of mean square continuous solutions of these two nonlocal problems by using the Schauder fixed point theorem. We discuss the sufficient conditions and the continuous dependence for the unique solution.

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

“…Over the years, fractional differential equations and its applications have gotten extensive attention, it is widely used in various disciplines, interested researchers can see the work in [4,9,12,17,31,34]. 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].…”

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.…”

Stabilization for Stochastic Coupled Kuramoto Oscillators via Nonlinear Distributed Feedback Control

“…In recent years, coupled systems have been extensively studied for their applications in physics, ecology, epidemiology, etc. [1][2][3][4][5][6][7][8][9][10][11][12]. In particular, coupled Kuramoto oscillators (CKOs) have received a lot of attention, and their dynamical properties have been investigated by many scholars [13][14][15][16].…”