On the behavior of the solutions to periodic linear delay differential and difference equations

Some new results on the behavior of the solutions to periodic linear delay differential equations as well as to periodic linear delay difference equations are given. These results are obtained by the use of two distinct roots of the corresponding (so called) characteristic equation.     

Remarks on oscillation and nonoscillation for second-order linear difference equations

In this remark, we shall show the main results of the earlier work [W.T. Li, S.S. Cheng, Remarks on two recent oscillation theorems for second-order linear difference equations, Appl. Math. Lett. 16 (2003) 161–163] are incorrect.     

Existence and multiplicity results for periodic solutions of nonlinear difference equations

We use Brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second-order and first-order difference equations. We obtain, in particular upper and lower solutions theorems, Ambrosetti–Prodi type results and sharp existence conditions for nonlinearities which are bounded from below or above.     

Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities

This paper is devoted to the study of nonlinear singular and non-singular fourth order difference equations coupled with periodic boundary value conditions. In the paper some existence and nonexistence results are given. The results are deduced from Krasnoselskii's fixed point theorems in cones. An exhaustive study of the Green's function related to a linear fourth order problem is done.     

On the oscillation and periodic character of a third order rational difference equation

We prove that every positive solution of the following difference equation:

converges to a period two solution.

     

Existence and structure results on almost periodic solutions of difference equations

We study the almost periodic solutions of Euler equations and of some more general Difference Equations. We consider two different notions of almost periodic sequences, and we establish some relations between them. We build suitable sequences spaces and we prove some properties of these spaces. We also prove properties of Nemytskii operators on these spaces. We build a variational approach to establish existence of almost periodic solutions as critical points, We obtain existence theorems fornonautonomous linear equations and for an Euler equation with a concave and coercive Lagrangian. We also use a Fixed Point approach to obtain existence results for quasi-linear Difference Equations.     

Existence of periodic and subharmonic solutions for second order functional difference equations

In this paper, by using the critical point theory, the existence of periodic and subharmonic solutions to a class of second order functional difference equations is obtained. The main approach used in our paper is variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic and subharmonic solutions of second order forward and backward difference equations.     

Nonoscillation and oscillation of solutions of a class of third order difference equations

In this paper, sufficient conditions are obtained for nonoscillation/oscillation of all solutions of a class of nonlinear third order difference equations of the form

     

Uniform asymptoic stability in linear volterra difference equations

For linear Volterra difference equations of nonconvolution type, uniform asymptotic stability of the zero solution is characterized by the summability of the resolvent matrix. Moreover, the existence of bounded solutions of nonhomogeneous linear Volterra difference equations is studied.     

Nicholson’s blowflies differential equations revisited: Main results and open problems

This review covers permanence, oscillation, local and global stability of solutions for Nicholson’s blowflies differential equation. Some generalizations, including the most recent results for equations with a distributed delay and models with periodic coefficients, are considered.