Abstract
In this article, a class of fourth-order difference equations with quasi-differences and deviating argument is considered. We state a new oscillation theorem for the sublinear case and we complete the existing results in the literature. Our approach is based on considering Equation (1) as a system of the four-dimensional difference system and on the cyclic permutation of the coefficients in the difference equations.
Highlights
In this article, we consider a class of fourth-order nonlinear difference equations of the form an bn cn( xn)γ β α + dnxλn+τ =, ( )where α, β, γ, λ are the ratios of odd positive integers, τ ∈ Z is a deviating argument and {an}, {bn}, {cn}, {dn} are positive real sequences defined for n ∈ N = {n, n +, . . .}, n is a positive integer, and is the forward difference operator defined by xn = xn+ – xn.By a solution of Equation ( ) we mean a real sequence {xn} satisfying Equation ( ) for n ∈ N
Where α, β, γ, λ are the ratios of odd positive integers, τ ∈ Z is a deviating argument and {an}, {bn}, {cn}, {dn} are positive real sequences defined for n ∈ N = {n, n +, . . .}, n is a positive integer, and is the forward difference operator defined by xn = xn+ – xn
Remark Corollary completes the oscillation criteria for Equation ( ) with τ = given in [ , ] where instead of the condition dn = ∞ it is assumed that both series P , Tare divergent or convergent, respectively
Summary
Such solutions are called quickly oscillatory and the following result can be seen as a necessary condition for their existence. Theorem Equation ( ) with τ even has no quickly oscillatory solutions. Lemma Equation ( ) has no solution of type (a) if any of the following conditions hold: (i) n+τ – i=n c i /γ λ n+τ – i=n c i /γ i– j=n b j /β
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