Mean square stability for controlled hybrid neutral stochastic differential equations with infinite delay

We study the existence and asymptotic stability inpth moment of a mild solution to a class of nonlinear fractional neutral stochastic differential equations with infinite delays in Hilbert spaces. A set of novel sufficient conditions are derived with the help of semigroup theory and fixed point technique for achieving the required result. The uniqueness of the solution of the considered problem is also studied under suitable conditions. Finally, an example is given to illustrate the obtained theory.

We examine the controllability problem for a class of neutral fractional integrodifferential equations with impulses and infinite delay. More precisely, a set of sufficient conditions are derived for the exact controllability of nonlinear neutral impulsive fractional functional equation with infinite delay. Further, as a corollary, approximate controllability result is discussed by assuming compactness conditions on solution operator. The results are established by using solution operator, fractional calculations, and fixed point techniques. In particular, the controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is controllable. Finally, an example is given to illustrate the obtained theory.

SummaryIn this work, the optimal control for a class of fractional neutral stochastic differential equations with deviated arguments driven by infinite delay and Poisson jumps is studied in Hilbert space involving the Caputo fractional derivative. The sufficient conditions for the existence of mild solution results are formulated and proved by the virtue of fractional calculus, characteristic solution operator, fixed‐point theorem, and stochastic analysis techniques. Furthermore, the existence of optimal control of the proposed problem is presented by using Balder's theorem. Finally, the obtained theoretical results are applied to the fractional stochastic partial differential equations and a stochastic river pollution model.

Existence results for impulsive neutral evolution integrodifferential equations with infinite delay

In this article, we study the existence of mild solutions and the approximate controllability for a class of systems governed by neutral equations of second-order with infinite delay in infinite-dimensional Hilbert spaces. The mild solution and approximate controllability are achieved by constructing the fundamental solution for the associated linear equation and assuming that the linear system is approximately controllable. The discussion is based on the fundamental solution theory and Rothe's fixed point theorem. In addition, an example is given to illustrate our main conclusion.

Neutral stochastic differential equations driven by a fractional Brownian motion with impulsive effects and varying-time delays

We investigate in this paper the existence of mild solutions for the fractional differential equations of neutral type with nonlocal conditions and infinite delay in Hilbert spaces by employing fractional calculus and Krasnoselski-Schaefer fixed point theorem. Finally an example is provided to illustrate the application of the obtained results.

This paper is devoted to study the permanence and periodic solution of a competitive system with infinite delay, feedback control, and the Allee effect. We derive sufficient conditions for the permanence and existence of a periodic solution in a competitive system with infinite delay, feedback control, and the Allee effect by using the differential inequality theory and constructing the Lyapunov function. We provide explicit estimates of the lower and upper bounds of the population density. This study reveals that the Allee effect plays an essential role in the permanence and increases the risk of population extinction.

This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained.

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We introduced a new phase space for impulsive system with infinite delay and claim that the phase space considered by different authors are not correct.

We consider an integrodifferential reaction-diffusion system which finds application in population dynamics. The memory kernels accounting for delay effects can be of both weak and strong type. Rescaling the kernels with a time relaxation $\varepsilon>0$, we show that the original model gives raise to a one-parameter family of dynamical systems in a suitable phase-space, We prove that this family is characterized by a corresponding family of exponential attractors which is stable as the delay effects vanish, i.e., when $\varepsilon$ goes to $0$.

Existence of solutions of non-autonomous second order functional differential equations with infinite delay

Attraction, stability and robustness for stochastic functional differential equations with infinite delay

In this paper, coincidence degree theory is applied to study the existence of periodic solution of a class of impulsive functional differential equations with infinite delay. Two sufficient criterions are obtained for the existence of periodic solution, which are generalizations of some existing results.

We prove the existence and uniqueness of a periodic solution to neutral function partial differential equations with infinite delay of the form $$ \frac {\partial Fu_t}{\partial t}= A(t)Fu_t +g(t,u_t)\quad \mbox {for } t \gt 0,\quad u(t)=\phi (t)\quad \hb