Abstract
The purpose of this paper is to present a pair of an oscillation theorem and a nonoscillation theorem for the second-order nonlinear difference equation where is continuous on ℝ and satisfies the signum condition if . The obtained results are best possible in a certain sense. Proof is given by means of the Riccati technique and phase plane analysis of a system. A discrete version of the Riemann-Weber generalization of Euler-Cauchy differential equation plays an important role in proving our results. MSC:39A12, 39A21.
Highlights
We consider the second-order nonlinear difference equation x(n) +f x(n) =, n(n + ) n ≥ n, ( . )where f (x) is a real-valued continuous function satisfying xf (x) > if x = .Here the forward difference operator is defined as x(n) = x(n + ) – x(n) and x(n) = ( x(n))
The forward difference operator is defined as x(n) = x(n + ) – x(n) and x(n) = ( x(n))
It is well known that an oscillation constant for equation ( . ) is /
Summary
Where K , K , K , K are arbitrary constants and z satisfies z – z + λ = (for the proof, see [ – ]). Other results on the oscillation constant for difference equations can be found in [ – ] and the references cited therein. It is well known that an oscillation constant for equation We note that their results are proved by using exact solutions of the Riemann-Weber version of Euler differential equation
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