Higher Order Nonlinear Neutral Difference Research Articles

On the oscillation of higher order nonlinear neutral difference equations

In this paper, we shall investigate some oscillation criteria for the solutions of mth-order nonlinear neutral difference equation where mgeq 1. The results presented here complement some of the known results reported in the literature. Examples are included to illustrate the importance of the main results.

Existence of periodic solutions for a higher-order neutral difference equation

Based on a continuation theorem of Mawhin, the existence of a periodic solution for a higher-order nonlinear neutral difference equation is studied. Our conclusion is new and interesting.

Convergence of solutions of higher order neutral difference equations with quasi-differences

Abstract A higher order nonlinear neutral difference equation with quasi-differences is studied. Sufficient conditions are established under which for every real constant there exists a solution of considered equation convergent to this constant.

Asymptotically zero solution of a class of higher nonlinear neutral difference equations with quasidifferences

A class of higher order nonlinear neutral difference equations with quasidifferences is studied. Sufficient conditions under which considered equation has a solution which converges to zero are presented.

Forced oscillation of higher-order nonlinear neutral difference equations with positive and negative coefficients

AbstractIn this paper, we study the forced oscillation of the higher-order nonlinear difference equation of the formΔm[x(n)−p(n)x(n−τ)]+q1(n)Φα(n−σ1)+q2(n)Φβ(n−σ2)=f(n),wherem≥1,τ,σ1andσ2are integers,0<α<1<βare constants,Φ∗(u)=|u|∗−1u,p(n),q1(n),q2(n)andf(n)are real sequences withp(n)>0. By taking all possible values ofτ,σ1andσ2into consideration, we establish some new oscillation criteria for the above equation in two cases: (i)q1=q1(n)≤0,q2=q2(n)>0; (ii)q1≥0,q2<0.MSC:39A10.

Oscillatory and asymptotic properties of higher order nonlinear neutral difference equations

In this paper, we study the oscillatory and asymptotic properties of solutions of higher order nonlinear neutral difference equations. Using Krasnoselskii’s fixed point theorem, Schauder’s fixed point theorem and Lebesgue’s dominated convergence theorem, we obtain some necessary and sufficient conditions for all bounded solutions to be oscillatory or tend to zero. Our results extend and improve some known results.

Oscillatory and asymptotic properties of higher order nonlinear neutral difference equations with oscillating coefficients

In this work, the oscillatory and asymptotic properties of higher order nonlinear neutral difference equations with oscillating coefficients are studied. Some new necessary and sufficient criteria, which improve several known results, are obtained.

Classification and existence of nonoscillatory solutions of higher order nonlinear neutral difference equations

In this paper, we consider the higher order nonlinear neutral delay difference equation of the form $$\Delta ^r (x_n + px_{n - \tau } ) + f(n,x_{n - \sigma _1 (n)} ,x_{n - \sigma _2 (n)} ,...,x_{n - \sigma _m (n)} ) = 0.$$ We give an integrated classification of nonoscillatory solutions of the above equation according to their asymptotic behaviours. Necessary and sufficient conditions for the existence of nonoscillatory solutions with designated asymptotic properties are also established.

Oscillatory behavior of higher order nonlinear neutral difference equation

Oscillatory behavior of higher order nonlinear neutral difference equation