On a Neutral Itô and Arbitrary (Fractional) Orders Stochastic Differential Equation with Nonlocal Condition

Abstract

In this paper, we are concerned with the combinations of the stochastic Itô-differential and the arbitrary (fractional) orders derivatives in a neutral differential equation with a stochastic, nonlinear, nonlocal integral condition. The existence of solutions will be proved. The sufficient conditions for the uniqueness of the solution will be given. The continuous dependence of the unique solution will be studied.

Oscillation Test for Second-Order Differential Equations with Several Delays

“…Dynamical systems with several delays have been intensively studied in control theory; see [3,4]. To encourage special interest in the oscillatory behavior of solutions to second-order NDDEs through their applications in the natural sciences and engineering, There is a continuing need to discover new necessary conditions for the oscillation or nonoscillation of solutions varietal type equations; see, e.g., papers [5][6][7][8][9][10][11].…”

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.…”