Dichotomous solutions of linear impulsive differential equations

The notion of L_p(h,k) solutions of linear impulsive differential equations in Banach spaces is introduced. Sufficient conditions for existence of such solutions are derived. Possible applications to linear control systems with impulses are considered. An illustrative example is given.     

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.     

A class of impulsive nonautonomous differential equations and Ulam–Hyers–Rassias stability

In this paper, we study a model described by a class of impulsive nonautonomous differential equations. This new impulsive model is more suitable to show dynamics of evolution processes in pharmacotherapy than the classical one. We apply Krasnoselskii's fixed point theorem to obtain existence of solutions. Meanwhile, we mainly present the sufficient conditions on Ulam–Hyers–Rassias stability on both compact and unbounded intervals. Many analysis techniques are used to derive our results. Copyright © 2014 John Wiley & Sons, Ltd.     

Monotone iterative technique for impulsive delay differential equations

In this paper, by proving a new comparison result, we present a result on the existence of extremal solutions for nonlinear impulsive delay differential equations.     

Non-instantaneous impulsive fractional-order implicit differential equations with random effects

In this article, we study existence and stability of a class of non-instantaneous impulsive fractional-order implicit differential equations with random effects. First, we establish a framework to study impulsive fractional sample path associated with impulsive fractional L^p-problem, and present the relationship between them. We also derive the formula of the solution for inhomogeneous impulsive fractional L^p-problem and sample path. Second, we construct a sequence of Picard functions, which admits us to apply successive approximations method to seek the solution of impulsive fractional sample path. Further, we derive the existence of solutions to impulsive fractional L^p-problem. Third, the concepts of Ulam's type stability are introduced and sufficient conditions to guarantee Ulam–Hyers–Rassias stability are derived. Finally, an example is given to illustrate the theoretical results.     

Global stability results for impulsive differential equations

Global results concerning stability properties of impulsive differential equations are established, employing piecewise continuous Lyapunov functions which are then applied for proving stability and boundedness properties     

Periodic solutions of abstract functional differential equations with state‐dependent delay

In this paper, we are concerned with the existence of solutions of systems determined by abstract functional differential equations with infinite and state‐dependent delay. We establish the existence of mild solutions and the existence of periodic solutions. Our results are based on local Lipschitz conditions of the involved functions. We apply our results to study the existence of periodic solutions of a partial differential equation with infinite and state‐dependent delay. Copyright © 2016 John Wiley & Sons, Ltd.     

Periodic boundary value problems for second order impulsive integro-differential equations of volterra type

The existence of solutions of periodic boundary value problems for second order impulsive integro-differential equations of Volterra type is investigated. By using the method of upper and lower solutions, it is proved that the problem in whi h impulses occur at fixed times has a solution. Some impulsive integro-differential inequalities related to such problem are also established.     

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.     

On the stability of impulsive functional differential equations with infinite delays

In this paper, the stability problem of impulsive functional differential equations with infinite delays is considered. By using Lyapunov functions and the Razumikhin technique, some new theorems on the uniform stability and uniform asymptotic stability are obtained. The obtained results are milder and more general than several recent works. Two examples are given to demonstrate the advantages of the results. Copyright © 2014 John Wiley & Sons, Ltd.     

Multiplicity of positive solutions to M-point boundary value problem of second order impulsive differential equations

In this paper, by using Avery-Peterson theorem on a convex cone, we consider the m-point boundary value problems for second order impulsive differential equations with the nonlinear term depending on the first order derivative, the multiplicity result of three positive solutions are obtained.     

Note on periodic solutions of fractional differential equations

The existence and nonexistence of periodic solutions are discussed for fractional differential equations by varying the lower limits of Caputo derivatives. The developed approach is illustrated on several examples.     

Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion

Some results on the existence and uniqueness of mild solution for a system of semilinear impulsive differential equations with infinite fractional Brownian motions are proved. The approach is based on Perov's fixed point theorem and a new version of Schaefer's fixed point theorem in generalized Banach spaces. The relationship between mild and weak solutions and the exponential stability of mild solutions are investigated as well. The abstract theory is illustrated with an example.     

Almost periodic solutions for forced perturbed impulsive differential equations

Sufficirnt condition for the existence of almost periodic solutions of forced perturbed systems of impulsive differential equations with impulsive effect at fixed Moments are considered.     

The existence of mild solutions for impulsive fractional partial differential equations

This paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations. Some errors in Mophou (2010) [2] are corrected, and some previous results are generalized.     

Existence of positive solutions for singular impulsive differential equations with integral boundary conditions

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions where the nonlinearity f(t,u,v) may be singular at u = 0 and v = 0. The proof is based on the theory of Leray–Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved. Copyright © 2014 John Wiley & Sons, Ltd.     

Multiple solutions of boundary‐value problems for impulsive differential equations

In this paper, we investigate the existence of multiple solutions to a second‐order Dirichlet boundary‐value problem with impulsive effects. The proof is based on critical point theorems. Copyright © 2011 John Wiley & Sons, Ltd.     

Periodic solutions for ordinary differential equations with sublinear impulsive effects

The continuation method of topological degree is used to investigate the existence of periodic solutions for ordinary differential equations with sublinear impulsive effects. The applications of the abstract approach include the generalizations of some classical nonresonance theorem for impulsive equations, for instance, the existence theorem for asymptotically positively homogeneous differential systems and the existence theorem for second order equations with Landesman-Lazer conditions.