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

Анотація

UDC 517.9

КРИТЕРІЇ КОЛИВАННЯ ТА НЕКОЛИВАННЯ ДЛЯ НАПІВЛІНІЙНОГО РІЗНИЦЕВОГО РІВНЯННЯ ДРУГОГО ПОРЯДКУ ТА РОЗШИРЕННЯ ДИСКРЕТНОЇ НЕРІВНОСТІ ГАРДІ

За допомогою відповідного розширення дискретної нерівності Гарді встановлено коливні властивості напівлінійного різницевого рівняння другого порядку.

Посилання

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