New Oscillation Criteria for Third-Order Nonlinear Mixed Neutral Difference Equations

Abstract

Some new oscillation criteria are established for a third-order nonlinear mixed neutral difference equation. Our results improve and extend some known results in the literature. Several examples are given to illustrate the importance of the results.

Oscillation Criteria for Third Order Neutral Generalized Difference Equations with Distributed Delay

“…Let A(λ), B(λ), C(λ), and Q(λ, s) be respectively defined in (13) and (11). Then, each solution of Equation 1is either oscillatory or y(λ) → 0, as λ → ∞.…”

“…The oscillatory and asymptotic behaviors of solutions of the third order difference equations were studied by Schmeidal [9]. Behaviors of oscillation of the third order nonlinear delay difference equation by Riccati transformation technique were obtained by several authors like Aktas et al [10], Elabbasy et al [11], Saker et al [12], Selvaraj et al [13][14][15], Thandapani et al [16].…”

Oscillation Properties of Third Order Nonlinear Delay Dynamic Equations on Time Scales

“…For an excellent introduction to the calculus on time scales, see Hilger [1] , and Bohner and Peterson [2,3]. For further results concerning the oscillatory and asymptotic behavior of third order dynamic equation, we refer to the papers [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references cited therein. Since we are interested in the oscillatory behavior of solutions near infinity, we assume that 𝑠𝑢𝑝𝕋 = ∞ (unbounded above) and define the time scale interval [𝑡 0 , ∞) 𝕋 by [𝑡 0 , ∞) 𝕋 ≔ [𝑡 0 , ∞) ∩ 𝕋.…”

Qualitative Property of Third-Order Nonlinear Neutral Distributed-Delay Generalized Difference Equations

“…Notably, numerous monographs concern with issues of the existence and multiplicity of solutions using different methods, such as critical point theory, topological degree theory, fixed-point index theory, and Lie theory. In recent years, there has been a continual interest in getting sufficient conditions for oscillatory behavior of different classes of third-order difference equations with or without deviating arguments (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references cited therein). e third-order nonlinear neutral distributed-delay generalized differential equation is of the form…”