Existence of global solutions for second order impulsive abstract partial differential equations

In this paper, we study the existence of global solutions for a class of second order impulsive abstract functional differential equations. The results are obtained by using Leray-Schauder’s Alternative fixed point theorem. An application is provided to illustrate the theory.     

Boundary value problems for first order functional differential equations with impulsive integral conditions

In this paper, by using the method of lower and upper solutions coupled with the monotone iterative technique, we give conditions for existence of extreme solutions for first order functional differential equations with a new impulsive integral condition. Some comparison results are also formulated.     

Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions

This paper describes the method of quasilinearization for first-order nonlinear impulsive functional differential equations with anti-periodic boundary conditions. A monotone iterative technique coupled with lower and upper solutions is employed to obtain sequences of approximate solutions converging monotonically and quadratically to the unique solution of the problem at hand.     

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.     

Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations

This paper uses critical point theory and variational methods to investigate the subharmonic solutions with prescribed minimal period for a class of second-order impulsive differential equations. The conditions for the existence of subharmonic solutions are established. Our results rectify some known results in the literature and will allow us, in the future, to deal with the subharmonic solutions for more extensive impulsive problems.     

Nagumo type existence results of Sturm-Liouville BVP for impulsive differential equations

In this paper, the method of upper and lower solutions and the Schauder degree theory are employed in the study of Sturm-Liouville boundary value problems for second order impulsive differential equations. We obtain the existence of at least three solutions to the problem under the assumption that the nonlinear term f satisfies a Nagumo condition with respect to the first order derivative.     

Almost periodic solutions for abstract impulsive differential equations

This paper studies the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in Banach space. The investigations are carried out by means of the fractional powers of operators. We construct an example to illustrate the feasibility of our results.     

Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions

In this paper, the existence of at least three positive solutions to impulsive second order differential equations is investigated. Sufficient conditions which guarantee the existence of positive solutions are obtained, by using the Avery-Peterson theorem. An example is added to illustrate the results.     

Periodic boundary value problems of second-order impulsive differential equations

This paper discusses periodic boundary value problems of second-order impulsive differential equations. By using Schaeffer’s fixed-point theorem and monotone iterative technique, some existence results are obtained.     

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

By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for second-order nonlinear impulsive integro-differential equations of mixed type.     

A multiplicity result for periodic solutions of second order differential equations with a singularity

By the use of the Poincaré–Birkhoff fixed point theorem, we prove a multiplicity result for periodic solutions of a second order differential equation, where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity. Our main theorem is related to previous results by Rebelo (1996, 1997)  and  and Rebelo and Zanolin (1996)  and , in connection with a problem raised by del Pino et al. (1992) [1].     

Periodic solutions for some second order differential equations with singularity

In this paper, we will prove the existence of infinitely many harmonic and subharmonic solutions for the second order differential equation ẍ + g(x) = f(t, x) using the phase plane analysis methods and Poincaré–Birkhoff Theorem, where the nonlinear restoring field g exhibits superlinear conditions near the infinity and strong singularity at the origin, and f(t, x) = a(t)x ^γ + b(t, x) where 0 ≤ γ ≤ 1 and b(t, x) is bounded. This project was supported by the Program for New Century Excellent Talents of Ministry of Education of China and the National Natural Science Foundation of China (Grant No. 10671020 and 10301006).     

Almost periodic solutions for a class of non-instantaneous impulsive differential equations

Abstract

This paper is concerned with almost periodic solutions for nonlinear non-instantaneous impulsive differential equations with variable structure. With the help of the notation of non-instantaneous impulsive Cauchy matrix, mild sufficient conditions are derived to guarantee the existence, uniqueness of asymptotically stable almost periodic solutions. Both example and numerical simulation are given to illustrate our effectiveness of the above results. As one expects, the results presented here have extended and improved some previous results for instantaneous impulsive differential equations.     

On periodic solutions for even order differential equations

A class of sufficient conditions are obtained for the existence of a unique 2π$2\pi$-periodic solution of even order differential equations, employing an initial value problem.     

Existence of solutions for the integral boundary value problems of fractional order impulsive differential equations

In this paper, we study a class of integral boundary value problem for fractional order impulsive differential equations, where both the nonlinearity and the impulsive terms contain the fractional order derivatives. By using fixed‐point theorems, the existence results of solution for the boundary value problem are established. Finally, some examples are presented to illustrate the existence results. Copyright © 2015 John Wiley & Sons, Ltd.     

Oscillation of even order partial differential equations with distributed deviating arguments

Some Kamenev-type oscillation criteria are established for a class of boundary value problems associated with even-order partial differential equations with distributed deviating arguments. Our approach is to reduce the high-dimensional oscillation problem to a one-dimensional oscillation one, and the general means developed by Philos and Wong is used as the main tool. The results obtained here extend and improve some known results in the literature.     

EXISTENCE OF SOLUTIONS FOR SECONDORDER IMPULSIVE DIFFERENTIAL EQUATIONS

In this paper, we apply the coincidence degree theory to study nonlinear second orderimPulsive periodic boundary value problems (PBVP), and show some sufficient conditions for the existence of solutions.     

Partial averaging for impulsive differential equations with supremum

Partial averaging for impulsive differential equations with supremum is justified. The proposed averaging schemes allow one to simplify considerably the equations considered.