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.     

Exact controllability and continuous dependence of fractional neutral integro-differential equations with state-dependent delay

In the present paper, with the help of the resolvent operator and some analytic methods, the exact controllability and continuous dependence are investigated for a fractional neutral integro-differential equations with state-dependent delay. As an application, we also give one example to demonstrate our results.     

Approximate controllability of second-order impulsive stochastic differential equations with state-dependent delay

In this paper we study a kind of second-order impulsive stochastic differential equations with state-dependent delay in a real separable Hilbert space. Some sufficient conditions for the approximate controllability of this system are formulated and proved under the assumption that the corresponding deterministic linear system is approximately controllable. The results concerning the existence and approximate controllability of mild solutions have been addressed by using strongly continuous cosine families of operators and the contraction mapping principle. At last, an example is given to illustrate the theory.     

Controllability of second-order neutral functional differential inclusions in Banach spaces

In this paper, we prove the controllability of second-order neutral functional differential inclusions in Banach spaces. The result are obtained by using the theory of strongly continuous cosine families and a fixed point theorem for condensing maps due to Martelli.     

On second order differential equations with state-dependent delay

We study existence and uniqueness of solutions for a general class of second order abstract differential equations with state-dependent delay. Some examples related to partial differential equations with state dependent delay are presented.     

Banach空间中一类分数阶微分方程边值问题

研究Banach空间中一类非线性分数阶微分方程边值问题.构建此类方程的Green函数,利用非紧测度和相关的不动点定理,得到了此类方程的mild解存在的几个充分条件,所得结果改进和推广了一些已有的结论.     

Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach spaces

In this paper, we consider the non-autonomous semilinear impulsive differential equations with state-dependent delay. The approximate controllability results of the first-order systems are obtained in a separable reflexive Banach space, which has a uniformly convex dual. In order to establish sufficient conditions of the approximate controllability of such a system, we have used the theory of linear evolution systems, properties of the resolvent operator and Schauder’s fixed point theorem. Finally, we provide two concrete examples to validate our results.     

Banach空间中含间断项的非线性积分微分方程的可解性

我们获得了 Ｂａｎａｃｈ 空间中含间断项的非线性积分微分方程初值问题的最大最小解的存在性,并推广了许多已知结果．     

Periodic solutions in periodic state-dependent delay equations and population models

Sufficient and realistic conditions are obtained for the existence of positive periodic solutions in periodic equations with state-dependent delay. The method involves the application of the coincidence degree theorem and estimations of uniform upper bounds on solutions. Applications of these results to some population models are presented. These application results indicate that seasonal effects on population models often lead to synchronous solutions. In addition, we may conclude that when both seasonality and time delay are present and deserve consideration, the seasonality is often the generating force for the often observed oscillatory behavior in population densities.

     

Existence and exact controllability of fractional evolution inclusions with damping

The aim of this paper is to deal with the existence of mild solutions and exact controllability for a class of fractional evolution inclusions with damping (FEID, for short) in Banach spaces. Firstly, we provide the representation of mild solutions for FEID by applying the method of Laplace transform and the theory of (α,κ)‐regularized families of operators. Next, we are concerned with the existence and exact controllability of FEID under some suitable sufficient conditions by using the method of measure of noncompactness and an appropraite fixed point theorem. Finally, an application to nonlinear partial differential equations with temporal fractional derivatives is presented to illustrate the effectiveness of our main results. Copyright © 2017 John Wiley & Sons, Ltd.     

On state-dependent delay partial neutral functional integro-differential equations

In this paper the existence of mild solutions for a class of abstract neutral integro-differential equations with state-dependent delay is studied.     

Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay

In this paper, we prove the existence of mild solutions for a first‐order impulsive semilinear stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay. We consider the cases in which the right hand side is convex or nonconvex valued. The results are obtained by using two different fixed point theorems for multivalued mappings. Copyright © 2015 John Wiley & Sons, Ltd.     

Decay integral solutions for neutral fractional differential equations with infinite delays

Our aim in this work is to find decay integral solutions for a class of neutral fractional differential equations in Banach spaces involving unbounded delays. By constructing a suitable measure of noncompactness on the space of solutions and establishing new estimates for fractional resolvent operators, we prove the existence of a compact set of decay integral solutions to the mentioned problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Controllability of impulsive neutral functional differential inclusions with infinite delay

The paper is concerned with the controllability of the first-order impulsive neutral functional differential inclusions with infinite delay in a Banach space. Sufficient conditions for controllability are obtained by using a fixed point theorem for condensing maps due to Martelli. As an application, we also consider an impulsive neutral partial functional differential equation.     

Approximate ontrollability of second order impulsive functional differential system with infinite delay in Banach spaces

This paper studies the approximate controllability of second order impulsive functional differential system with infinite delay in Banach spaces. Sufficient conditions are formulated and proved for the approximate controllability of such system under the assumption that the associated linear part of system is approximately controllable. The results are obtained by using strongly continuous cosine families of operators and the contraction mapping principle. An example is given to illustrate the obtained theory     

On the set of solutions of some nonconvex nonclosed hyperbolic differential inclusions

We consider a class of nonconvex and nonclosed hyperbolic differential inclusions and we prove the arcwise connectedness of the solution set.     

On controllability of fractional impulsive neutral infinite delay evolution integrodifferential systems in Banach spaces

In this work, controllability of fractional impulsive neutral evolution integrodifferential systems in a Banach space has been addressed. Sufficient conditions for the controllability are established using fractional calculus, resolvent operators and Krasnoselskii’s fixed point theorem.     

Positive periodic solutions for impulsive differential equations with infinite delay and applications to integro-differential equations

Sufficient conditions for the existence of at least one positive periodic solution are established for a family of scalar periodic differential equations with infinite delay and nonlinear impulses. Our criteria, obtained by applying a fixed-point argument to an original operator constructed here, allow to treat equations incorporating a rather general nonlinearity and impulses whose signs may vary. Applications to some classes of Volterra integro-differential equations with unbounded or periodic delay and nonlinear impulses are given, extending and improving results in the literature.