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 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 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.     

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.     

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.     

On the periodic nature of the solutions of the reciprocal difference equation with maximum

We prove that every positive solution of the third order difference equation

     

EXISTENCE OF PERIODIC SOLUTIONS OF SYSTEM OF DIFFERENCE EQUATIONS

This paper studies the nonautonomous nonlinear system of difference equationsΔx(n)=A(n)x(n)+f(n,x(n)),n∈Z,(*) where x(n)∈R~N,A(n)=(a_(ij)(n))N×N is an N×N matrix,with a-(ij)∈C(R,R) for i,j= 1,2,3,...,N,and f=(f_1,f_2,...,f_N)~T∈C(R×R~N,R~N),satisfying A(t+ω)=A(t),f(t+ω,z)=f(t,z) for any t∈R,(t,z)∈R×R~N andωis a positive integer.Sufficient conditions for the existence ofω-periodic solutions to equations (*) are obtained.     

Asymptotical stability of partial difference equations with variable coefficients

The dynamical systems considered have scalar state, are multivariate, linear, time-discrete, and time-variable and are described by an initial value problem for a class of evolutionary partial difference equations. The time set is the nonnegative part of the integer lattice in several dimensions. Parts of the asymptotical stability set in the parameter space spanned by the time-variable coefficients are explicitly found. To assess the quality of the sufficient stability criteria, a comparison with the exact stability set is made in an example.     

Oscillation for linear non-autonomous systems of difference equations with continuous arguments

In this paper, we establish sufficient conditions for the oscillation of the linear non-autonomous systems of difference equations with continuous arguments