Asymptotic behavior of solutions of functional difference equations

For linear functional difference equations, we obtain some results on the asymptotic behavior of solutions, which correspond to a Perron-type theorem for linear ordinary difference equations. We also apply our results to Volterra difference equations with infinite delay.     

Asymptotic behavior in neutral difference equations with negative coefficients

In this paper, we study the asymptotic behavior of the solutions of a neutral difference equation of the form $\Delta [x(n) + cx(\tau (n))] - p(n)x)(\sigma (n)) = 0,$ , where τ(n) is a general retarded argument, σ(n) is a general deviated argument, c ∈ ?, (?p(n)) _n≥0 is a sequence of negative real numbers such that p(n) ≥ p, p ∈ ?₊, and Δ denotes the forward difference operator Δx(n) = x(n+1)?x(n).     

Asymptotic behavior of solutions of nonlinear Volterra equations

An energy decay rate is obtained for solutions of wave type equations in a bounded region in Rⁿ whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface.     

Oscillatory behavior of the second-order nonlinear neutral difference equations

In this paper, we consider the oscillation of the second-order neutral difference equation $$\Delta ^2 \left( {x_n - px_{n - \tau } } \right) + q_n f\left( {x_{n - \sigma _n } } \right) = 0$$ as well as the oscillatory behavior of the corresponding ordinary difference equation $$\Delta ^2 z_n + q_n f\left( {R\left( {n,\lambda } \right)z_n } \right) = 0$$ .     

Asymptotic approximation of solutions of nonlinear third order difference equations

In this paper, we obtain asymptotic bounds, under appropriate conditions, of solutions of third order difference equations of the form

Asymptotic behavior of solutions of forced nonlinear neutral delay differential equations with impulses

Sufficient conditions for asymptotic behavior of the solutions of nonlinear forced neutral delay differential equations with impulses are found. The results given in [2,4,6,7] are generalized and improved.     

Asymptotic behavior results for nonlinear neutral delay difference equations

This paper is concerned with the nonlinear neutral delay difference equation

(∗)     

Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations

We consider a class of fourth-order nonlinear difference equations of the form {fx006-01} where α, β are ratios of odd positive integers and {p _n}, {q _n} are positive real sequences defined for all n ∈ ℕ(n ₀). We establish necessary and sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior under suitable combinations of convergence or divergence conditions for the sums {fx006-02}. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 8–27, January, 2008.     

Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses

This paper is concerned with a nonlinear neutral differential equations with impulses of the form

(*)     

Asymptotic properties of solutions of nonlinear difference equations

Using discrete inequalities and Schauder's fixed point theorem we study the problem of asymptotic equilibrium for difference equations.     

The asymptotic behavior of nonoscillatory solutions of nonlinear neutral type difference equations

In this paper, the authors study the asymptotic behavior of solutions of second-order neutral type difference equations of the form

Δ²(y_n+py_n−k)+f(n,y_n−ℓ)=0,n

Existence for nonoscillatory solutions of higher-order nonlinear neutral difference equations

In this paper, we consider the following forced higher-order nonlinear neutral difference equation

     

Asymptotic behavior of solutions to nonlinear Volterra integral equations

A class of operator Riccati integral equations is associated with a factorization problem in a certain Banach algebra. Recent results concerning factorization in this algebra are used to obtain existence, uniqueness, and continuous dependence results for the Riccati equations.     

Asymptotic behavior of solutions of linear homogeneous second-order difference equations

We analyze and study the asymptotic behavior (asn→∞) of the general solutionx _n of the equationx _n+2 =Ax _n +Bx _n+1 ,A≠0,n=0,1,2,..., for various possible values of coefficients and initial data. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 211–215, August, 99.     

Asymptotic behavior of solutions of difference equations in Banach spaces

In this paper we consider the first order difference equation

and give necessary and sufficient conditions so that there exist solutions which are asymptotically constant. These results generalize those given earlier by Popenda and Schmeidel. As an application we give necessary and sufficient conditions for the second order difference equation

to have asymptotically constant solutions.