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Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type. (English) Zbl 1199.39022

Summary: We consider the difference equation of neutral type

Δ 3 [x(n)-p(n)x(σ(n))]+q(n)f(x(τ(n)))=0,n(n 0 ),

where p,q(n 0 ) + ; σ,τ, σ is strictly increasing and lim n σ(n)=; τ is nondecreasing and lim n τ(n)=, f, xf(x)>0. We examine the following two cases:

0<p(n)λ * <1,σ(n)=n-k,τ(n)=n-l,

and

1<λ * p(n),σ(n)=n+k,τ(n)=n+l,

where k,l are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as n with a weaker assumption on q than the usual assumption i=n 0 q(i)= that is used in the literature.

MSC:
39A22Growth, boundedness, comparison of solutions (difference equations)
39A10Additive difference equations
39A12Discrete version of topics in analysis
34K40Neutral functional-differential equations
39A21Oscillation theory (difference equations)