Oscillation of third-order nonlinear delay difference equations

Key Findings & Data:Here, {cn}, {dn}, {pn}, and {qn} are sequences of positive real numbers for n0 \in N, f is a continuous function such that f(u) /u\geq K > 0 for u \neq 0.

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Conclusion & Impact:Examples are given to illustrate the importance of the results.

🔬Design:Research Article

📍Scope:Comprehensive Sample

⚙️Variables:Difference equation · Delay · Third order

Abstract

Third-order nonlinear difference equations of the form D (cnD (dnD xn))+pnD xn+1+qnf(xn-s)=0, n\geq n0 are considered. Here, {cn}, {dn}, {pn}, and {qn} are sequences of positive real numbers for n0 \in N, f is a continuous function such that f(u) /u\geq K > 0 for u \neq 0. By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results.

Özet

Third-order nonlinear difference equations of the form D (cnD (dnD xn))+pnD xn+1+qnf(xn-s)=0, n\geq n0 are considered. Here, {cn}, {dn}, {pn}, and {qn} are sequences of positive real numbers for n0 \in N, f is a continuous function such that f(u) /u\geq K > 0 for u \neq 0. By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results.

Keywords

Difference equation Delay Third order Oscillation Nonoscillation Riccati transformation

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Cite This Article

AKTAS, M. F., ZAFER, A., TİRYAK, A. (2022). Oscillation of third-order nonlinear delay difference equations. Turkish Journal of Mathematics, Cilt: 36 - Sayı: 3, 422–436. https://ajindex.com/articles/608420

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References (1)

Corollary 3.2 Assume that all the assumptions of Theorem 3.5 hold, except that the condition (3.21) is replaced by lim sup m→∞Hm,m0 Kq− p2ρnd −σR d2 n+1 =∞ for every m ≥ n0, lim sup m→∞Hm,m A n=m0 ≥ n0. Then, every solution{xn} of eq. (1.1) is either oscillatory or limxn= 0. n→∞