Geodesic
Asymptotic behavior of solutions of nonlinear difference equations
Mathematica Bohemica, Tome 129 (2004) no. 4, pp. 349-359

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The nonlinear difference equation \[ x_{n+1}-x_n=a_n\varphi _n(x_{\sigma (n)})+b_n, \qquad \mathrm{(\text{E})}\] where $(a_n), (b_n)$ are real sequences, $\varphi _n\: \mathbb{R}\longrightarrow \mathbb{R}$, $(\sigma (n))$ is a sequence of integers and $\lim _{n\longrightarrow \infty }\sigma (n)=\infty $, is investigated. Sufficient conditions for the existence of solutions of this equation asymptotically equivalent to the solutions of the equation $y_{n+1}-y_n=b_n$ are given. Sufficient conditions under which for every real constant there exists a solution of equation () convergent to this constant are also obtained.
The nonlinear difference equation \[ x_{n+1}-x_n=a_n\varphi _n(x_{\sigma (n)})+b_n, \qquad \mathrm{(\text{E})}\] where $(a_n), (b_n)$ are real sequences, $\varphi _n\: \mathbb{R}\longrightarrow \mathbb{R}$, $(\sigma (n))$ is a sequence of integers and $\lim _{n\longrightarrow \infty }\sigma (n)=\infty $, is investigated. Sufficient conditions for the existence of solutions of this equation asymptotically equivalent to the solutions of the equation $y_{n+1}-y_n=b_n$ are given. Sufficient conditions under which for every real constant there exists a solution of equation () convergent to this constant are also obtained.
DOI : 10.21136/MB.2004.134043
Classification : 39A10, 39A11
Keywords: difference equation; asymptotic behavior 
Migda, Janusz. Asymptotic behavior of solutions of nonlinear difference equations. Mathematica Bohemica, Tome 129 (2004) no. 4, pp. 349-359. doi: 10.21136/MB.2004.134043
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[1] R. P. Agarwal: Difference Equations and Inequalities. Marcel Dekker, New York, 1992. | MR | Zbl

[2] M. P. Chen, J. S. Yu: Oscillations for delay difference equations with variable coefficients. Proc. of the First International Conference of Difference Equations (1995), 105–114. | MR

[3] S. S. Cheng, G. Zhang, S. T. Liu: Stability of oscillatory solutions of difference equations with delays. Taiwanese J. Math. 3 (1999), 503–515. | DOI | MR

[4] J. R. Graef, C. Qian: Asymptotic behavior of a forced difference equation. J. Math. Anal. Appl. 203 (1996), 388–400. | DOI | MR

[5] G. Ladas: Explicit conditions for the oscillation of difference equations. J. Math. Anal. Appl. 153 (1990), 276–287. | DOI | MR | Zbl

[6] M. Migda, J. Migda: Asymptotic properties of the solutions of second order difference equation. Arch. Math. (Brno) 34 (1998), 467–476. | MR

[7] E. Schmeidel: Asymptotic properties of solutions of a nonlinear difference equations. Comm. Appl. Nonlinear Anal. 4 (1997), 87–92. | MR

[8] J. Shen, J. P. Stavroulakis: Oscillation criteria for delay difference equations. Electronic J. Differ. Eq. 10 (2001), 1–15. | MR

[9] X. H. Tang, J. S. Yu: Oscillation of nonlinear delay difference equations. J. Math. Anal. Appl. 249 (2000), 476–490. | DOI | MR | Zbl

[10] X. H. Tang, Y. Liu: Oscillation for nonlinear delay difference equations. Tamkang J. Math. 32 (2001), 275–280. | MR

[11] X. H. Tang, J. S. Yu: Oscillation of delay difference equations. Comput. Math. Appl. 37 (1999), 11–20. | DOI | MR | Zbl

[12] J. Yan, C. Qian: Oscillation and comparison results for delay difference equations. J. Math. Anal. Appl. 165 (1992), 346–360. | DOI | MR

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