Exponential boundedness of solutions for impulsive delay differential equations

For an impulsive delay differential equation

exponent estimates of solutions have been obtained.     

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.     

Periodic solutions of delay impulsive differential equations

We study the following semilinear impulsive differential equation with delay:

     

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.     

EXISTENCE THEOREMS OF SOLUTIONSFOR NONLINEAR IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

In this paper, we use the Monch fixed-point theorem to prove some existence theoremsof sotuti9ns for the nonlinear impulsive Volterra integral equations in Banach spaces that improveand extend the results in [2].     

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.     

Ultimate boundedness of impulsive fractional differential equations

This article investigates nonlinear impulsive Caputo fractional differential equations. Utilizing Lyapunov functions, Laplace transforms of fractional derivatives and boundedness of Mittag-Leffler functions, several sufficient conditions are derived to ensure the global ultimate boundedness and the exponential stability of the systems. An example is given to explain the obtained results.     

Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations

In this work, the asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses is investigated. By impulsive differential inequality and Riccati transformation, sufficient conditions of asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses are obtained. An example is also inserted to illustrate the impulsive effect.     

On functional improper Volterra integral equations and impulsive differential equations in ordered Banach spaces

In this paper we derive existence and comparison results for discontinuous improper functional integral equations of Volterra type in an ordered Banach space which has a regular order cone. For this purpose we prove Dominated and Monotone Convergence Theorems for improper integrals. The obtained results are then applied to first-order impulsive differential equations. Concrete examples are also solved by using symbolic programming.     

Ultimate boundedness of impulsive fractional delay differential equations

This paper concerns with the ultimate boundedness problem for impulsive fractional delay differential equations. Based on the impulsive fractional differential inequality, the boundedness of Mittag-Leffler functions, and the successful construction of suitable Lyapunov functionals, some algebraic criteria are derived for testing the global ultimate boundedness of the equations, and the estimations of the global attractive sets are provided as well. One example is also given to show the effectiveness of the obtained theoretical results.     

Existence of solutions for impulsive partial neutral functional differential equations

This paper is concerned with partial neutral functional differential equations of first and second order with impulses. We establish some results of existence of mild solutions for these classes of equations.     

Strict stability of impulsive functional differential equations

Strict stability is the kind of stability that can give us some information about the rate of decay of the solutions. There are some results about strict stability of differential equations. In the present paper, we shall extend the strict stability to impulsive functional differential equations. By using Lyapunov functions and Razumikhin technique, we shall get some criteria for the strict stability of impulsive functional differential equations, and we can see that impulses do contribute to the system's strict stability behavior.     

A new result on impulsive differential equations involving non-absolutely convergent integrals

In this paper we obtain, as an application of a Darbo-type theorem, global solutions for differential equations with impulse effects, under the assumption that the function on the right-hand side is integrable in the Henstock sense. We thus generalize several previously given results in literature, for ordinary or impulsive equations.     

Stability of impulsive functional differential equations

In this paper the stability of impulsive functional differential equations in which the state variables on the impulses are related to the time delay is studied. By using Lyapunov functions and Razumikhin techniques, some criteria of stability, asymptotic stability and practical stability for impulsive functional differential equations in which the state variables on the impulses are related to the time delay are provided. Some examples are also presented to illustrate the efficiency of the results obtained.     

Exponential stability for impulsive delay differential equations by Razumikhin method

In this paper, we study exponential stability for impulsive delay differential equation of the form

     

Nondensely defined evolution impulsive differential inclusions with nonlocal conditions

In this paper, we shall establish sufficient conditions for the existence of integral solutions for some nondensely defined evolution impulsive differential inclusions in Banach spaces with nonlocal conditions.     

Existence and compactness of solutions to impulsive differential equations with nonlocal conditions

The new type of nonlinear integral inequalities of Volterra–Fredholm type for discontinuous functions is investigated. Then, by using these inequalities and Schaefer fixed‐point theorem, we present new existence results for impulsive semilinear differential equations with nonlocal conditions. Moreover, the compactness of solution sets can be shown in some certain conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Positive solutions of singular Caputo fractional differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions of singular super-linear (or sub-linear) integral boundary value problems for fractional differential equation involving Caputo fractional derivative. Necessary and sufficient conditions for the existence of C³[0, 1] positive solutions are given by means of the fixed point theorems on cones. Our nonlinearity f(t, x) may be singular at t = 0 and/or t = 1.