Oscillation and nonoscillation criteria for a half-linear difference equation of the second order and extended discrete Hardy inequality

A. Kalybay; D.  Karatayeva

doi:10.37863/umzh.v74i1.2298

A. Kalybay KIMEP Univ., Almaty, Kazakhstan https://orcid.org/0000-0003-3011-1783
D. Karatayeva L. N. Gumilyov Eurasian Nat. Univ., Nur-Sultan, Kazakhstan

DOI: https://doi.org/10.37863/umzh.v74i1.2298

Keywords: Generalized zeros, oscillatory and non-oscillatory properties, conjugate, disconjugate, discrete weighted inequality.

Abstract

UDC 517.9

We establish the oscillatory properties of a half-linear difference equation of the second order by using a suitable extension of the weighted discrete Hardy inequality.

References

A. Z. Alimagambetova, R. Oinarov, Two-sided estimates for solutions of a class of second-order nonlinear difference equations (in Russian), Mat. Zh., 8, № 3(29), 12 – 21 (2008).

G. Bennett, Some elementary inequalities, Quart. J. Math. Oxford Ser. (2), 38, 401 – 425 (1987), https://doi.org/10.1093/qmath/38.4.401 DOI: https://doi.org/10.1093/qmath/38.4.401

G. Bennett, Some elementary inequalities, II, Quart. J. Math. Oxford Ser. (2), 39, 385 – 400 (1989), https://doi.org/10.1093/qmath/39.4.385 DOI: https://doi.org/10.1093/qmath/39.4.385

G. Bennett, Some elementary inequalities, III, Quart. J. Math. Oxford Ser. (2), 42, 149 – 174 (1991), https://doi.org/10.1093/qmath/42.1.149 DOI: https://doi.org/10.1093/qmath/42.1.149

M. S. Braverman, V. D. Stepanov, On the discrete Hardy’s inequality, Bull. London Math. Soc., 26, 283 – 287 (1994), https://doi.org/10.1112/blms/26.3.283 DOI: https://doi.org/10.1112/blms/26.3.283

O. Došlý, S. Fišnarová, Summation characterization of the recessive solution for half-linear difference equations, Adv. Difference Equat., 2009, Article 521058 (2009); https://doi.org/10.1155/2009/521058. DOI: https://doi.org/10.1155/2009/521058

O. Došlý, P. Řehák, Nonoscillation criteries for half-linear second-order difference equations, Comput. Math. Appl., 42, 453 – 464 (2001), https://doi.org/10.1016/S0898-1221(01)00169-9 DOI: https://doi.org/10.1016/S0898-1221(01)00169-9

H. A. El-Morshedy, Oscillation and nonoscillation criteria for half-linear second order difference equations, Dynam. Systems and Appl., 15, 429 – 450 (2006).

P. Hasil, M. Vesel´y, Oscillation constants for half-linear difference equations with coefficients having mean values, Adv. Differerence Equat., 2015, Article 210 (2015); https://doi.org/10.1186/s13662-015-0544-1. DOI: https://doi.org/10.1186/s13662-015-0544-1

J. Jiang, X. Tang, Oscillation of second order half-linear difference equations, I, Appl. Math. Sci., 8, №40, 1957 – 1968 (2014), https://doi.org/10.12988/ams.2014.1270 DOI: https://doi.org/10.12988/ams.2014.1270

J. Jiang, X. Tang, Oscillation of second order half-linear difference equations, II, Appl. Math. Lett., 24, 1495 – 1501 (2011), https://doi.org/10.1016/j.aml.2011.03.029 DOI: https://doi.org/10.1016/j.aml.2011.03.029

A. Kalybay, D. Karatayeva, R. Oinarov, A. Temirkhanova, Oscillation of a second order half-linear difference equation and the discrete Hardy inequality, Electron. J. Qual. Theory Different. Equat., № 43, 1 – 16 (2017); http://dx.doi.org/10.14232/ejqtde.2017.1.43. DOI: https://doi.org/10.14232/ejqtde.2017.1.43

A. Kufner, L. Maligranda, L.-E. Persson, The Hardy inequality. About its history and some related results, Vydavatelsk´y servis, Pilsen (2007). DOI: https://doi.org/10.2307/27642033

R. Oinarov, A. P. Stikharnyi, Boundedness and compactness criteria for a certain difference inclusion, Math. Notes, 50, Issue 5, 1130 – 1135 (1991), https://doi.org/10.1007/BF01157699 DOI: https://doi.org/10.1007/BF01157699

R. Oinarov, K. Ramazanova, A. Tiryaki, An extension of the weighted Hardy inequalities and its application to half-linear equations, Taiwanese J. Math., 19, № 6, 1693 – 1711 (2015); http://dx.doi.org/10.11650/tjm.19.2015.5764. DOI: https://doi.org/10.11650/tjm.19.2015.5764

P. Řehák, Hartman –Winter type lemma, oscillation, and conjugacy criteria for half-linear difference equations, J. Math. Anal. and Appl., 252, 813 – 827 (2000), https://doi.org/10.1006/jmaa.2000.7124 DOI: https://doi.org/10.1006/jmaa.2000.7124

P. Řehák, Half-linear discrete oscillation theory, Electron. J. Qual. Theory Different. № 24, 1 – 14 (2000). DOI: https://doi.org/10.14232/ejqtde.1999.5.24

P. Řehák, Oscillatory properties of second order half-linear difference equations, Czechoslovak Math. J., 51, № 126, 303 – 321 (2001), https://doi.org/10.1023/A:1013790713905 DOI: https://doi.org/10.1023/A:1013790713905

P. Řehák, Comparision theorems and strong oscillation in the half-linear discrete oscillation theory, Rocky Mountain J. Math., 33, № 1, 333 – 352 (2003), https://doi.org/10.1216/rmjm/1181069996 DOI: https://doi.org/10.1216/rmjm/1181069996

C. Sturm, Sur les equations differentielles lineares du second ordre, J. Math. Pures et Appl., № 1, 106 – 186 (1836).

Y. G. Sun, Oscillation and nonoscillation for half-linear second order difference equations, Indian J. Pure and Appl. Math., 35, № 2, 133 – 142 (2004).

E. Thandapani, K. Rarc, J. R. Graef, Oscillation and comparison theorems for half-linear second order difference equations, Comput. Math. Appl., 42, 953 – 910 (2001), https://doi.org/10.1016/S0898-1221(01)00211-5 DOI: https://doi.org/10.1016/S0898-1221(01)00211-5

M. Vesel´y, P. Hasil, Oscillation and nonoscillation of asymptotically almost periodic half-linear difference equations, Abstr. Appl. Anal., 2013, Article 432936 (2013); http://dx.doi.org/10.1155/2013/432936. DOI: https://doi.org/10.1155/2013/432936