Abstract
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.
Highlights
We consider the oscillation of the following mth-order forced nonlinear difference equation of the form m x(n) – p(n)x(n – τ ) + q (n) α(n – σ ) + q (n) β (n – σ ) = f (n), ( )
To the best of our knowledge, little has been known about the forced oscillation of Eq ( ) with positive and negative coefficients (q ≤, q > or q ≥, q < ) and mixed nonlinearities ( < α )
Motivated by the work in [ – ], we study the forced oscillation of Eq ( ) in this paper
Summary
We consider the oscillation of the following mth-order forced nonlinear difference equation of the form m x(n) – p(n)x(n – τ ) + q (n) α(n – σ ) + q (n) β (n – σ ) = f (n), ( ) The main contribution of this paper is that we establish some new oscillation criteria for Eq ( ) with positive and negative coefficients and mixed nonlinearities. Proof Assume to the contrary that there exists a nontrivial solution x(n) of Eq ( ) such that x(n) is nonoscillatory.
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