Periodic solutions of difference equations

We give an overview of results on the existence of periodic solutions of difference equations that have been obtained in the last two decades. The survey covers both ordinary and Volterra difference systems. Some extensions and generalizations of known result are also presented.     

Periodic solutions of autonomous systems of difference and difference-differential equations with a small delay

In the present paper the method of the equivalent differential-operator equation has been applied in the study of the existence and asymptotic representation of periodic solutions of autonomous systems of the form     

Periodic solutions for dynamic equations on time scales

Massera type criteria are established for the existence of periodic solutions of linear and nonlinear dynamic equations on time scales. Some interesting properties of the exponential function on time scales are presented. Furthermore, a sufficient condition guaranteeing the boundedness of the solutions of the equation is presented.     

Periodic solutions for fourth-order nonlinear functional difference equations

By using the critical point method, some new criteria are obtained for the existence and multiplicity of periodic solutions for fourth-order nonlinear functional difference equations. The proof is based on the linking theorem in combination with variational technique. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.     

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).     

Periodic solutions of difference equations

Consider the difference equation

(∗)     

Periodic solutions for a class of higher-dimension functional differential equations with impulses

By employing a fixed-point theorem in cones, we establish some criteria for existence of positive periodic solutions of a class of n$n$-dimension periodic functional differential equations with impulses. We also give some applications to several biomathematical models and new results are obtained.     

Periodic solutions of some second-order differential equations with desultorily sublinear nonlinearities

In this paper, we study the existence of periodic solutions of the Rayleigh equations

x^″+f(x^′)+g(x)=e(t).     

Periodic solutions of non-linear discrete Volterra equations with finite memory

In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non-linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed-point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. The effect of weak singularities is addressed in a final section. The detail that is presented, which is supplemented using appendices, reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes.     

Periodic solutions of nonlinear integral equations

The existence of a continuous periodic solution of the system

Periodic solutions for general nonlinear state-dependent delay logistic equations

In this paper, by using the continuation theorem of coincidence degree theory, we investigate the existence of periodic solutions for more general state-dependent delay logistic equations. Several sufficient conditions are given, and the obtained conditions possess important significance in both theories and applications.     

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.     

Periodic solutions of delay equations with three delays via bi-Hamiltonian systems

In this paper we give sufficient conditions for the existence of periodic solutions of delay differential equations with three delays.     

Eigenvalues of second-order difference equations with coupled boundary conditions

This paper is concerned with coupled boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues is proved, numbers of eigenvalues are calculated, and relationships between the eigenvalues of a self-adjoint second-order difference equation with three different coupled boundary conditions are established. These results extend the relevant existing results of periodic and antiperiodic boundary value problems.