Zhiguo Luo

Periodic boundary value problem for first-order impulsive functional differential equations

This paper discusses the periodic boundary value problem for a class of first-order impulsive functional differential equations. We establish several existence results under weaker hypotheses by using the Leray-Schauder alternative, the lower and upper solution method and the monotone iterative technique. The corresponding results in the literature are improved or extended, some examples are also given to illustrate the advantage of the results.

Multi-point boundary value problems for second-order functional differential equations

This paper is concerned with the existence of extreme solutions of multi-point boundary value problem for a class of second-order functional differential equations. We introduce a new concept of lower and upper solutions. By using the method of upper and lower solutions and monotone iterative technique, we obtain the existence of extreme solutions.

Solutions to a boundary value problem of a fourth-order impulsive differential equation

This paper is concerned with the existence of solutions to a boundary value problem of a fourth-order impulsive differential equation with a control parameter λ. By employing some existing critical point theorems, we find the range of the control parameter in which the boundary value problem admits at least one solution. It is also shown that under certain conditions there exists an interval of the control parameter in which the boundary value problem possesses infinitely many solutions. The main results are also demonstrated with examples.MSC:34B15, 34B18, 34B37, 58E30.

New results on oscillation for delay differential equations with piecewise constant argument

Abstract We introduce a new technique to obtain some new oscillation criteria for the oscillating coefficients delay differential equation with piecewise constant argument of the form x′(t) + a(t)x(t) + b(t)x({t − k}) = 0, where a(t) and b(t) are right continuous functions on [−k, ∞), k is a positive integer, and [·] denotes the greatest integer function. Our results improve and generalize the known results in the literature. Some examples are also given to demonstrate the advantage of our results.

New results for oscillation of delay difference equations

In this paper, we consider the delay difference equation xn+1 − xn + pnxn−k = 0, n = 0, 1, 2, …, where pn is a sequence of nonnegative real numbers and k is a positive integer. Some new results for the oscillation of this equation are obtained. Our theorems improve all known results in the literature.

Multiple Positive Solutions of the Singular Boundary Value Problem for Second-Order Impulsive Differential Equations on the Half-Line

This paper uses a fixed point theorem in cones to investigate the multiple positive solutions of a boundary value problem for second-order impulsive singular differential equations on the half-line. The conditions for the existence of multiple positive solutions are established.

Some oscillation criteria for difference equations

Abstract Consider the difference equation x n +1 − x n + p n x n − k = 0, where { p n } is a sequence of nonnegative real numbers and k is a positive integer. It is proved that all solutions oscillate if there exists some positive integer l such that lim sup n→∞ ∑ i=0 k Pn−i+ ∏ i=0 k ∑ j=1 k Pn−i+j+ ∑ m=0 l−1 ∑ i=0 k ∏ j=0 m+1 Pn−jk−1 > 1 or lim sup n→∞ ∑ i=1 k Pn−i+ ∏ i=1 k ∑ j=1 k Pn−i+j+ ∑ m=0 l−1 ∑ i=0 k ∏ j=0 m+1 Pn−jk−1 > 1.

Oscillation for solutions of nonlinear neutral differential equations with impulses

Abstract This paper is concerned with nonlinear neutral differential equations with impulses of the form Some oscillation criteria for solutions of this equation are established. An interesting example is also given, which illustrates that impulses play an important role in giving rise to the oscillation of equations.

Stability of Bidirectional Associative Memory networks with impulses

Abstract By using Lyapunov functional and some analysis technique, sufficient conditions are obtained for the existence and asymptotic stability of a unique equilibrium of a Bidirectional Associative Memory (BAM) neutral network with Lipschitzian activation functions without assuming their boundedness, monotonicity or differentiability and subjected to impulsive state displacements at fixed instants of time. The sufficient conditions are in terms of the parameters of the network only and are easy to verify.

Existence of multiple solutions of some second order impulsive differential equations

This paper uses critical point theory and variational methods to investigate the multiple solutions of a boundary value problem for second order impulsive differential equations. The conditions for the existence of multiple solutions are established.