On a fractional impulsive partial stochastic integro-differential equation with state-dependent delay and optimal controls

This paper is mainly concerned with a new class of fractional impulsive partial stochastic integro-differential equations with state-dependent delay and optimal controls in Hilbert spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence and uniqueness of mild solutions are proved by means of stochastic analysis theory, fractional calculus and the fixed point technique combined with solution operator. The existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. Finally, an example is given for demonstration.     

Solvability and Optimal Controls of a Fractional Impulsive Stochastic Partial Integro-Differential Equation with State-Dependent Delay

In this paper, a new class of fractional impulsive stochastic partial integro-differential control systems with state-dependent delay and their optimal controls in a Hilbert space is studied. We firstly prove an existence result of mild solutions for the control systems by using stochastic analysis, analytic \(\alpha \)-resolvent operator, fractional powers of closed operators and suitable fixed point theorems. Then we derive the existence conditions of optimal pairs to the fractional impulsive stochastic control systems. Finally, an example is given to illustrate the effectiveness of our main results.     

Existence and exponential stability for neutral stochastic fractional differential equations with impulses driven by Poisson jumps

The paper is mainly concerned with a class of neutral stochastic fractional integro-differential equation with Poisson jumps. First, the existence and uniqueness for mild solution of an impulsive stochastic system driven by Poisson jumps is established by using the Banach fixed point theorem and resolvent operator. The exponential stability in the pth moment for mild solution to neutral stochastic fractional integro-differential equations with Poisson jump is obtained by establishing an integral inequality.     

Existence results for a new class of fractional impulsive partial neutral stochastic integro-differential equations with infinite delay

In this paper, a new class of fractional impulsive partial neutral stochastic integro-differential equations with infinite delay is introduced.Under some dissipative conditions, we obtain the existence, uniqueness and continuous dependence of mild solutions for these equations. An application involving a fractional stochastic parabolic system with not instantaneous impulses is considered.     

Existence of Mild Solutions to Stochastic Delay Evolution Equations with a Fractional Brownian Motion and Impulses

In this article, we study the existence of mild solutions to stochastic impulsive evolution equations with time delays, driven by fractional Brownian motion with the Hurst index H > 1/2 via a new fixed point analysis approach.     

Stepanov-like pseudo almost periodic solutions for impulsive perturbed partial stochastic differential equations and its optimal control

This paper is mainly concerned with the Stepanov-like pseudo almost periodicity to a class of impulsive perturbed partial stochastic differential equations. Firstly, we prove the existence of $p$-mean piecewise Stepanov-like pseudo almost periodic mild solutions for the impulsive stochastic dynamical system in a Hilbert space under non-Lipschitz conditions. The results are obtained by using the fixed point techniques with fractional power arguments. Then the existence of optimal pairs of system governed by impulsive partial stochastic differential equations is also obtained. Finally, an example is provided to illustrate the developed theory.     

Local and global existence of mild solutions for impulsive fractional semilinear integro-differential equation

In this paper, we study the local and global existence of mild solutions for impulsive fractional semilinear integro-differential equations in an arbitrary Banach space associated with operators generating compact semigroup on the Banach space. Also, we review some applications of fractional differential equations.     

Existence of solutions for nonlinear fractional stochastic differential equations

The fractional stochastic differential equations have wide applications in various fields of science and engineering. This paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in Hilbert spaces. Using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directly from deterministic fractional equations, new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay. Further, we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions. Examples are provided to illustrate the obtained theory.     

Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition

This paper is concerned with the local and global existence of mild solution for an impulsive fractional functional integro differential equations with nonlocal condition. We establish a general framework to find the mild solutions for impulsive fractional integro-differential equations, which will provide an effective way to deal with such problems. The results are obtained by using the fixed point technique and solution operator on a complex Banach space.     

Existence Results for Impulsive Neutral Stochastic Functional Integro-Differential Equations with Infinite Delays

In this paper, we prove the existence of mild solutions for a class of impulsive neutral stochastic functional integro-differential equations with infinite delays in an abstract space by means of the Krasnoselskii-Schaefer type fixed point theorem.     

Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion

This paper deals with the existence and uniqueness of mild solutions to neutral stochastic delay functional integro-differential equations perturbed by a fractional Brownian motion B ^H , with Hurst parameter H ∈ (1/2, 1). We use the theory of resolvent operators developed by R. Grimmer to show the existence of mild solutions. An example is provided to illustrate the results of this work.     

Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays

ABSTRACT

In this paper, we investigate the existence and Hyers-Ulam stability for random impulsive stochastic functional differential equations with finite delays. Firstly, we prove the existence of mild solutions to the equations by using Krasnoselskii's fixed point. Then, we investigate the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, an example is given to illustrate our results.     

Solvability and Optimal Controls of Fractional Impulsive Stochastic Evolution Equations with Nonlocal Conditions

This paper deals with the solvability and optimal controls of a class of impulsive fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. Firstly, the existence and uniqueness of mild solutions for the considered system are investigated. Then, we derive the existence conditions of optimal pairs to the control systems. In the end, an example is presented to illustrate the effectiveness of our abstract results.     

Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays

In this paper, we establish the existence and uniqueness of mild solutions for a class of semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays in α-norm. And the main tool is the fixed point theorem due to Sadovskii. Some known results are generalized.     

Existence Results for Fractional Order Semilinear Integro-Differential Evolution Equations with Infinite Delay

This paper is concerned with existence results of mild solutions for fractional order semilinear integro-differential evolution equations (FSIDEEs) and semilinear neutral integro-differential evolution equations (FSNIDEEs in short) with infinite delay in α-norm. Our tools include the Banach contraction principle, the nonlinear alternative of Leray–Schauder type and the Krasnoselskii–Schaefer type fixed point theorem.     

Existence of Solutions for Fractional Integro-differential Equations with Impulsive and Integral Conditions

This paper presents the existence of solutions for a class of Cauchy problems with integral condition for impulsive fractional integro-differential equations. Based on definition of solution for impulsive fractional integro- differential equations, the existence theorems of solutions of fractional differ- ential equation are obtained by applying fixed point methods. Finally, three examples are given to demonstrate the feasibility of the obtained results.     

Study a class of Hilfer fractional stochastic integrodifferential equations with Poisson jumps

Abstract

In this article, we derive the sufficient conditions for the existence of mild solutions of Hilfer fractional stochastic integrodifferential equations with nonlocal conditions and Poisson jumps in Hilbert spaces. Results will be obtained in the pth mean square sense by using the fractional calculus, semigroup theory and stochastic analysis techniques. The article generalizes many of the existing results in the literature in terms of (1) Riemann–Liouville and Caputo derivatives are the special cases. (2) In the sense of pth mean square norm. (3) Stochastic integrodifferential with nonlocal conditions and Poisson jumps. A numerical example is provided to validate the obtained theoretical results.     

Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations

In this paper, we study optimal relaxed controls and relaxation of nonlinear fractional impulsive evolution equations. Firstly, existence of piecewise continuous mild solutions for the original fractional impulsive control system is presented. Secondly, fractional impulsive relaxed control system is constructed by using a regular countably additive measure and making the original control system convexified. Thirdly, optimal relaxed controls and relaxation theorems are obtained. Finally, application to initial-boundary value problem of fractional impulsive parabolic control system is considered.     

Impulsive Problems for Fractional Partial Neutral Functional Integro-Differential Inclusions with Infinite Delay and Analytic Resolvent Operators

In this paper, the existence of mild solutions for a class of impulsive fractional partial neutral functional integro-differential inclusions with infinite delay and analytic α-resolvent operators in Banach spaces is investigated. Sufficient conditions for the existence are derived with the help of the fixed-point theorem for discontinuous multi-valued operators due to Dhage and the fractional power of operators combined with approximation techniques. An example is provided to illustrate the theory.