×

Growth estimates for solutions of nonlinear second-order difference equations. (English) Zbl 1069.39016

The author investigates the asymptotic behavior of solutions of the nonlinear second order difference equation \[ \Delta (c_{n-1}\Delta x_{n-1}) = f(n,x_n,\Delta x_{n-1}) + g(n,x_n,\Delta x_{n-1}). \tag{*} \] A typical result is the following statement.
Theorem. Suppose that
\(c_n>0\) and \(\sum_{k=1}^{\infty} 1/c_{k} =\infty\);
\(| g(n,u,v)| \leq g_n\), where \(g_n\) is a real sequence such that \(\sum_{k=1}^{\infty}g_k<\infty\);
\(| f(n,u,v)| \leq d_n| u| ^{\alpha}\), \(u,v\in \mathbb R\) for some \(\alpha\in [0,1)\), where the sequence \(d_n\) satisfies
\[ \sum_{k=1}^{\infty}| d_k| \left( \sum_{i=1}^{k-1}\frac{1}{c_k}\right)^{\alpha} <\infty. \]
Then for any solution of (*)
\[ x_n={\mathcal O}\left( \sum_{k=1}^{n-1}\frac{1}{c_k}\right)\quad \text{as\;} n\to \infty \]
and \(\lim_{n\to \infty}c_n\Delta x_n\) exists finite. The main results of the paper are illustrated by a number of examples and are proved using estimates based on Bellman-Gronwall-type inequalities.

MSC:

39A11 Stability of difference equations (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kurpinar, Indian J. Math. 37 pp 113– (1995)
[2] Elaydi, An introduction to difference equations (1996) · Zbl 0840.39002 · doi:10.1007/978-1-4757-9168-6
[3] De Blasi, Analele stintificae ale Universitatii Al. Cuza 20 pp 56– (1974)
[4] Agarwal, Difference equations and inequalities 228 (2000)
[5] DOI: 10.1016/0022-0396(84)90041-X · Zbl 0489.34035 · doi:10.1016/0022-0396(84)90041-X
[6] DOI: 10.1007/BF01297622 · Zbl 0145.06003 · doi:10.1007/BF01297622
[7] Stević, ANZIAM J. 43 pp 559– (2002)
[8] Papaschinopoulos, Math. Japonica 33 pp 457– (1988)
[9] Mitrinović, Differential and integral inequalities (1988)
[10] Stević, ANZIAM J. 46 pp 157– (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.