Thandapani, E.; Manuel, M. M. S.; Graef, J. R.; Spikes, P. W. Monotone properties of certain classes of solutions of second-order difference equations. (English) Zbl 0933.39014 Comput. Math. Appl. 36, No. 10-12, 291-297 (1998). Summary: The authors consider the difference equations \[ \Delta(a_n\Delta x_n)=q_nx_{n+1} \tag{*} \] and \[ \Delta(a_n\Delta x_n)= q_nf(x_{n+1}), \tag{**} \] where \(a_n>0\), \(q_n>0\), and \(f:\mathbb{R} \to\mathbb{R}\) is continuous with \(uf(u)>0\) for \(u\neq 0\). They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included. Cited in 18 Documents MSC: 39A11 Stability of difference equations (MSC2000) Keywords:second-order difference equations; nonlinear; monotone solutions; asymptotic behavior; nonoscillatory solutions PDFBibTeX XMLCite \textit{E. Thandapani} et al., Comput. Math. Appl. 36, No. 10--12, 291--297 (1998; Zbl 0933.39014) Full Text: DOI References: [1] Cheng, S. S.; Li, H. J.; Patula, W. T., Bounded and zero convergent solutions of second order difference equations, J. Math. Anal. Appl., 141, 463-483 (1989) · Zbl 0698.39002 [2] Thandapani, E.; Graef, J. R.; Spikes, P. W., Monotonicity and summability of solutions of a second order nonlinear difference equation, Bull. Inst. Math. Acad. Sinica, 23, 343-356 (1995) · Zbl 0845.39003 [3] Agarwal, R. P., Difference Equations and Inequalities (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0784.33008 [4] Cheng, S. S.; Patula, W. T., An existence theorem for a nonlinear difference equation, Nonlinear Anal., 20, 193-203 (1993) · Zbl 0774.39001 [5] Györi, I.; Ladas, G., Oscillation Theory of Delay Differential Equations with Applications (1991), Clarendon Press: Clarendon Press Oxford · Zbl 0780.34048 [6] Lakshmikantham, V.; Trigiante, D., Theory of Difference Equations: Numerical Methods and Applications, (Math. in Science and Engineering, Volume 181 (1988), Academic Press: Academic Press New York) · Zbl 0683.39001 [7] Griffel, D. H., Applied Functional Analysis (1981), Ellis Harwood: Ellis Harwood Chichester · Zbl 0461.46001 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.