On the asymptotics of some systems of difference equations

We prove an inclusion theorem regarding a system of difference equations and apply it in getting the asymptotics of some solutions of some concrete difference equations.     

On a symmetric bilinear system of difference equations

We show in an elegant way how the main result in the recent paper Matsunaga and Suzuki (2018) follows from a known result, and discuss the system appearing therein.     

Periodicity and solutions of some rational difference equations systems

In this paper we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case. Moreover, we study the periodicity of solutions for such systems. Finally, some numerical examples are presented.     

On the asymptotics of some difference equations

We prove two inclusion theorems regarding a difference equation and apply them in getting the asymptotics of positive solutions of some concrete difference equations.     

On some class of periodic-discrete homogeneous difference equations via Fibonacci sequences

The intent of this paper is to solve the homogeneous linear difference equations with periodic coefficients. For this purpose, we turn our study in solving a class of discrete-time linear systems, in the algebra of square matrices. The tools used repose predominately on the combinatorial properties of the generalized Fibonacci sequences in the algebra of square matrices. Some explicit solutions are established and special cases are discussed. Illustrative examples are given.     

On the global periodicity of discrete dynamical systems and application to rational difference equations

A new necessary condition for global periodicity of discrete dynamical systems and of difference equations is obtained here. This condition will be applied to contribute to solving the problem of global periodicity for second order rational difference equations.     

On existence of homoclinic orbits for some types of autonomous quadratic systems of differential equations

The new existence conditions of homoclinic orbits for the system of ordinary quadratic differential equations are founded. Further, the realization of these conditions together with the Shilnikov Homoclinic Theorem guarantees the existence of a chaotic attractor at 3D autonomous quadratic system. Examples of the chaotic attractors are given.     

Exact solutions of some systems of fractional differential‐difference equations

In this paper, the ‐expansion method is proposed to establish hyperbolic and trigonometric function solutions for fractional differential‐difference equations with the modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential‐difference equation into its differential‐difference equation of integer order. We obtain the hyperbolic and periodic function solutions of the nonlinear time‐fractional Toda lattice equations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exact solutions for nonlinear fractional differential–difference equations and systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Periodically forced nonlinear systems of difference equations

The existance of nontrivial (x=0( periodic solutions of a general class of periodic nonlinear difference equations is proved using bifurcation theory methods. Specifically, the existance of a global continuum of nontrivial periodicsolutions that bifurcates from the trivial solution (x=0) is proved. Conditions are given under which the nontrivial solutions are positive. A prerrequisite Fredholm and adjoint operator theory for linear periodic systems is developed. An application to application dynamics is made.     

On the boundedness of some rational difference equations

We study the boundedness character of solutions of some third order rational difference equations and we confirm some of the conjectures posed in Camouzis et al. [“Progress report on the boundedness character of third-order rational equations”, Journal of Difference Equations and Applications 11 (2005), 1029–1035] and [“On third order rational difference equations, part 6”, Journal of Difference Equations and Applications 11 (2005), 759–777].     

Delay difference equations in Banach spaces

We derive explicit stability conditions for semilinear delay difference equations in a Banach space. It is assumed that the nonlinearities of the considered equations satisfy the local Lipschitz condition. By virtue of the new estimates for the norm of functions of quasi-Hermitian operators, explicit stability and boundedness conditions are given. Applications to infinite dimensional delay difference systems are discussed.     

Qualitative behavior of difference equation of order two

In this paper we deal with the qualitative behavior of the solutions of the following difference equation

where the initial conditions x₋₁,x₀ are arbitrary positive real numbers and a,b,c,d are positive constants. Also, we obtain the form of the solution of some special cases of this equation.     

Periodicity of some classes of holomorphic difference equations

In this paper, we investigate the periodicity character of some classes of difference equations. Among other results, we give a simpler proof of an Open Problem addressed in Csörnyei and Laczkovich, 2001, “Some periodic and non-periodic recursions”, Monath. Math., 132, 215–236. We also consider Open Problem 2.8 in Grove and Ladas, 2005, Periodicities in Nonlinear Difference Equations, (Chapman and Hall/CRC), and the case when a difference equation has periodic solutions depending on arbitrary parameters.     

The freezing method for abstract nonlinear difference equations

The freezing method for ordinary differential systems is extended to a class of semilinear difference equations in a Hilbert space, whose linear parts have slowly varying coefficients and nonlinearities satisfying local Lipschitz conditions. The main methodology is based on a combined use of recent norm estimates for operator-valued functions with the freezing method as well as the multiplicative representation of solutions. Thus, explicit stability and boundedness conditions are derived. Applications to infinite dimensional delay difference systems are discussed.