Abstract
In this paper, we present some new oscillation criteria for nonlinear neutral difference equations of the form ?(b(n)?(a(n)?z(n))) + q(n)x?(?(n)) = 0 where z(n) = x(n) + p(n)x(?(n)),? > 0, b(n) > 0, a(n) > 0, q(n) ? 0 and p(n) > 1. By summation averaging technique, we establish new criteria for the oscillation of all solutions of the studied difference equation above. We present four examples to show the strength of the new obtained results.
Highlights
This paper is concerned with the oscillatory behavior of solutions of the thirdorder nonlinear neutral delay difference equation (1)∆(b(n)∆(a(n)∆z(n))) + q(n)xα(σ(n)) = 0, n ≥ n0 where z(n) = x(n)+ p(n)x(τ (n)), and n ∈ N(n0) = {n0, n0 +1, ...}, n0 is a positive integer
We present some new oscillation criteria for nonlinear neutral difference equations of the form
Most of the results obtained in [1, 2, 9, 10, 12, 14, 16, 17] for neutral type third order difference equations ensure that every solution is either oscillatory or tends to zero monotonically
Summary
This paper is concerned with the oscillatory behavior of solutions of the thirdorder nonlinear neutral delay difference equation (1). A nontrivial solution of (1) is said to be oscillatory if the terms of the sequence {x(n)} are neither eventually all positive nor eventually all negative, and nonoscillatory otherwise. Most of the results obtained in [1, 2, 9, 10, 12, 14, 16, 17] for neutral type third order difference equations ensure that every solution is either oscillatory or tends to zero monotonically. To the best of authors’ knowledge, there is no result regarding property A or the oscillation of all solutions of (1) under the conditions (H1)−(H2). Motivated by this observation, we attempt to obtain some new oscillation results for equation (1) under the condition (2) which ensures that all solutions of (1) are oscillatory. It should be noted that the research in this paper was motivated by the recent results in [3, 5] established for differential equations
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