Existence results for semilinear differential equations with nonlocal and impulsive conditions

This paper is concerned with the existence for impulsive semilinear differential equations with nonlocal conditions. Using the techniques of approximate solutions and fixed point, existence results are obtained, for mild solutions, when the impulsive functions are only continuous and the nonlocal item is Lipschitz in the space of piecewise continuous functions, is not Lipschitz and not compact, is continuous in the space of Bochner integrable functions, respectively.     

Measure of noncompactness and semilinear nonlocal functional differential equations in Banach spaces

This paper is concerned with the measure of noncompactness in the spaces of continuous functions and semilinear functional differential equations with nonlocal conditions in Banach spaces.The relationship between the Hausdorff measure of noncompactness of intersections and the modulus of equicontinuity is studied for some subsets related to the semigroup of linear operators in Banach spaces.The existence of mild solutions is obtained for a class of nonlocal semilinear functional differential equations without the assumption of compactness or equicontinuity on the associated semigroups of linear operators.     

Existence and Approximate Controllability of Solutions for an Impulsive Evolution Equation with Nonlocal Conditions in Banach Space

In this article, we study the existence of mild solutions and approximate controllability for non-autonomous impulsive evolution equations with nonlocal conditions in Banach space. The existence of mild solutions and some conditions for approximate controllability of these non-autonomous impulsive evolution equations are given by using the Krasnoselskii''s fixed point theorem, the theory of evolution family and the resolvent operator. In particular,the impulsive functions are supposed to be continuous and the nonlocal item is divided into Lipschitz continuous and completely bounded. An example is given as an application of the results.     

Decay mild solutions for elastic systems with structural damping involving nonlocal conditions

This paper deals with a class of elastic systems with structural damping subject to nonlocal conditions. By using a suitable measure of noncompactness on the space of continuous functions on the half-line, we establish the existence of mild solutions with explicit decay rate of exponential type. An example is given to illustrate the abstract results.     

Existence of almost periodic solutions for strong stable impulsive differential equations

Conditions for strong stability and the existence of almostperiodic solutions of systems of impulsive differential equationswith impulsive effect at fixed moments are obtained. The investigationsare carried out by means of piecewise continuous functions whichare analogues of Lyapunov functions.     

Integral manifolds for uncertain impulsive differential–difference equations with variable impulsive perturbations

In the present paper sufficient conditions for the existence of integral manifolds of uncertain impulsive differential–difference equations with variable impulsive perturbations are obtained. The investigations are carried out by means the concepts of uniformly positive definite matrix functions, Hamilton–Jacobi–Riccati inequalities and piecewise continuous Lyapunov’s functions.     

Controllability of impulsive differential systems with nonlocal conditions

The paper is concerned with the controllability of impulsive functional differential equations with nonlocal conditions. Using the measure of noncompactness and Mönch fixed-point theorem, we establish some sufficient conditions for controllability. Firstly, we require the equicontinuity of evolution system, and next we only suppose that the evolution system is strongly continuous. Since we do not assume that the evolution system generates a compact semigroup, our theorems extend some analogous results of (impulsive) control systems.     

非局部条件下半线性微分方程的适度解

讨论了Banach空间中非局部条件下半线性微分方程的适度解的存在性,利用不动点和非紧测度的方法,给出了在不需要半群紧性条件下方程适度解的存在性,并且对f是连续紧算子和f是Lipschitz连续的情形做了统一处理,从而得到了更为广泛和一般性的结果.     

Integral manifolds of impulsive fractional functional differential systems

In this paper, a class of impulsive Caputo fractional functional differential systems with variable impulsive perturbations is investigated. Sufficient conditions for the existence of integral manifolds are obtained. The main results are proved by means of piecewise continuous Lyapunov functions and the fractional comparison principle. The demonstrated techniques can be applied in studying properties of many applied problems of diverse interest.     

Almost periodic solutions in the PC-space for uncertain impulsive dynamical systems

This paper provides sufficient conditions for the existence of almost periodic solutions for an uncertain impulsive dynamical system. The investigation is carried out by utilizing the concept of uniformly positive definite matrix functions, Hamilton-Jacobi-Riccati inequality and piecewise continuous functions of the Lyapunov functions type.     

Existence of almost periodic solutions for strongly stable nonlinear impulsive differential-difference equations

Sufficient conditions are established for the existence of almost periodic solutions for strongly stable nonlinear impulsive differential-difference equations. The investigations are carried out by means of piecewise continuous functions of Lyapunov type and by using Markoff’s sets. We provide an example to demonstrate the effectiveness of our results.     

Favardʼs theorem of piecewise continuous almost periodic functions and its application

In this paper, by constructing Bochner–Fejér polynomials for piecewise continuous almost periodic functions (PCAP, for short), the authors establish Favard?s theorem of PCAP functions, which illustrates when the primitive function of PCAP function is a PCAP function. As its application, combining coincidence degree theory, we consider the existence of PCAP solution of impulsive single population model with hereditary effects. To our best knowledge, it is the first time when coincidence degree theory is used to study the existence of PCAP solution of impulsive differential equation.     

Monotone Iterative Technique for a Class of Semilinear Evolution Equations with Nonlocal Conditions

This paper discusses the existence and uniqueness of mild solutions for a class of semilinear evolution equations with nonlocal conditions in an ordered Banach space E. Under some monotonicity conditions and noncompactness measure conditions of the nonlinearity, a new monotone iterative method on the evolution equations with nonlocal conditions has been established. Particularly, an existence result without using noncompactness measure condition is obtained in ordered and weakly sequentially complete Banach spaces, which is very convenient for application. An example to illustrate our main results is also given.     

On almost periodic processes in impulsive competitive systems with delay and impulsive perturbations

In the present paper we investigate the existence of almost periodic processes of ecological systems which are presented with the general impulsive nonautonomous Lotka–Volterra system of integro-differential equations with infinite delay. The impulses are at fixed moments of time, and by using the techniques of piecewise continuous Lyapunov’s functions, new sufficient conditions for the global exponential stability of the unique positive almost periodic solutions of these systems are given.     

Fractional Retarded Differential Equations Involving Mixed Nonlocal Plus Local Initial Conditions

Abstract

The aim of this paper is to discuss the existence of mild solutions and positive mild solutions for a general class of semilinear fractional retarded evolution equations subjected to mixed nonlocal plus local initial conditions on infinite dimensional Banach spaces. Under the situation that the nonlinear term and nonlocal function satisfy some appropriate growth conditions and a noncompactness measure condition, we obtained the existence of mild solutions and positive mild solutions by utilizing a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness. In addition, the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example about the retarded parabolic partial differential equation involving a general mixed nonlocal plus local initial conditions is also given to illustrate the feasibility of our abstract results.     

PIECEWISE CONTINUOUS SOLUTIONS OF INITIAL VALUE PROBLEMS OF SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS

Results on the existence of piecewise continuous solutions for two classes of initial value problems of impulsive singular fractional differential equations are obtained.     

Existence of Global Solutions for a Class of Abstract Second-Order Nonlocal Cauchy Problem with Impulsive Conditions in Banach Spaces

In this paper, we are concerned with a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. First, we study the existence of mild solutions for a class of second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces on an interval [0,a]. Later, we study a couple of cases where we can establish the existence of global solutions for a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. We apply our theory to study the existence of solutions for impulsive partial differential equations.     

Generalized Cauchy problems involving nonlocal and impulsive conditions

We study the abstract Cauchy problem involving a class of nonlinear differential inclusions, with impulsive and nonlocal conditions. By using MNC estimates, the existence result and continuous dependence on initial data of the solution set are proved.     

Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions

The existence, uniqueness and continuous dependence of a mild solution of an impulsive neutral functional differential evolution nonlocal Cauchy problem in general Banach spaces are studied, by using the fixed point technique and semigroup of operators.     

The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay

This paper is mainly concerned with the existence and uniqueness of mild solutions for fractional impulsive neutral functional infinite delay integrodifferential systems with nonlocal initial conditions. The results are obtained by the fixed point theorem combined with a strongly continuous operator semigroup.