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Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales. (English) Zbl 1211.34115

Summary: The purpose of this paper is to establish oscillation criteria for the second order nonlinear dynamic equation
\[ (r(t)(x^\Delta(t))^\gamma)^\Delta+f(t,x(g(t)))=0, \]
on an arbitrary time scale \(\mathbb T\), where \(\gamma\) is a quotient of odd positive integers and \(r\) is a positive \(rd\)-continuous function on \(\mathbb T\). The function \(g:\mathbb T\to\mathbb T\) satisfies \(g(t)\geq t\) and \(\lim_{t\to\infty}g(t) =\infty\) and \(f\in C(\mathbb T\times \mathbb R,\mathbb R)\). We establish some new sufficient conditions under which the above equation is oscillatory by using the generalized Riccati transformation. Our results in the special cases when \(\mathbb T=\mathbb R\) and \(\mathbb T=\mathbb N\) involve and improve some oscillation results for second-order differential and difference equations; and when \(\mathbb T=h\mathbb N\), \(\mathbb T=q^{\mathbb N_0}\) and \(\mathbb T=\mathbb N^2\) our oscillation results are essentially new. Some examples illustrating the importance of our results are included.

MSC:

34N05 Dynamic equations on time scales or measure chains
34K11 Oscillation theory of functional-differential equations
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References:

[1] Agarwal, R. P.; Bohner, M.; Saker, S. H., Oscillation of second order delay dynamic equation, Canadian Appl. Math. Quart., 13, 1-17 (2005) · Zbl 1126.39003
[2] Agarwal, R. P.; O’Regan, D.; Saker, S. H., Philos-type oscillation criteria for second order half linear dynamic equations, Rocky Mountain J. Math., 37, 1085-1104 (2007) · Zbl 1139.34029
[3] Agarwal, R. P.; Shien, S. L.; Yeh, C. C., Oscillation criteria for second-order retarded differential equations, Math. Comput. Model., 26, 1-11 (1997) · Zbl 0902.34061
[4] Atkinson, F. V., On second-order nonlinear oscillations, Pacific J. Math., 5, 643-647 (1955) · Zbl 0065.32001
[5] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[6] Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales (2003), Birkhäuser: Birkhäuser Boston · Zbl 1025.34001
[7] Došlý, O., Qualitative theory of half-linear second order differential equations, Math. Boh., 10, 181-195 (2001) · Zbl 1016.34030
[8] Erbe, L.; Hassan, T. S.; Peterson, A.; Saker, S. H., Oscillation criteria for half-linear delay dynamic equations on time scales, Nonlinear Dyn. Syst. Theor., 10, 51-68 (2009) · Zbl 1173.34037
[9] Erbe, L.; Hassan, T. S.; Peterson, A.; Saker, S. H., Oscillation criteria for sublinear half-linear delay dynamic equations on time scales, Int. J. Differ. Equat., 3, 227-245 (2008)
[10] Erbe, L.; Peterson, A.; Saker, S. H., Hille-Kneser-type criteria for second-order dynamic equations on time scales, Adv. Differ. Equat., 2006, 1-18 (2006) · Zbl 1229.34136
[11] Erbe, L.; Peterson, A.; Saker, S. H., Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London Math. Soc., 76, 701-714 (2003) · Zbl 1050.34042
[12] Erbe, L.; Peterson, A.; Saker, S. H., Oscillation criteria for second-order nonlinear delay dynamic equations on time scales, J. Math. Anal. Appl., 333, 505-522 (2007) · Zbl 1125.34046
[13] Fite, W. B., Concerning the zeros of solutions of certain differential equations, Trans. Amer. Math. Soc., 19, 341-352 (1917)
[14] Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities (1988), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0634.26008
[15] Han, Z.; Shi, B.; Sun, S., Oscillation criteria for second order delay dynamic equations on time scales, Adv. Differ. Equat., 2007, 1-16 (2007)
[16] Hassan, T. S., Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl., 345, 176-185 (2008) · Zbl 1156.34022
[17] Hassan, T. S., Oscillation of third order nonlinear delay dynamic equations on time scales, Math. Comput. Model., 49, 1573-1586 (2009) · Zbl 1175.34086
[18] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math., 18, 18-56 (1990) · Zbl 0722.39001
[19] V. Kac, P. Cheung, Quantum Calculus, Universitext, 2002.; V. Kac, P. Cheung, Quantum Calculus, Universitext, 2002. · Zbl 0986.05001
[20] Kamenev, I. V., Integral criterion for oscillation of linear differential equations of second order, Math. Zemetki, 23, 249-251 (1978), (in Russian) · Zbl 0386.34032
[21] Leighton, W., On self-adjoint differential equations of second-order, J. London Math. Soc., 27, 37-47 (1952) · Zbl 0048.06503
[22] Manojlovic, J. V., Oscillation criteria for second-order half-linear differential equations, Math. Comput. Model., 30, 109-119 (1999) · Zbl 1042.34532
[23] Nehari, Z., Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc., 85, 428-445 (1957) · Zbl 0078.07602
[24] Ohriska, J., Oscillation of second order delay and ordinary differential equations, Czech. Math. J., 34, 107-112 (1984) · Zbl 0543.34054
[25] Philos, C. G., Oscillation theorems for linear differential equation of second order, Arch. Math., 53, 483-492 (1989) · Zbl 0661.34030
[26] Sahiner, Y., Oscillation of second-order delay dynamic equations on time scales, Nonlinear Anal.: Theor. Meth. Appl., 63, 1073-1080 (2005) · Zbl 1224.34294
[27] Saker, S. H., Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comput. Appl. Math., 177, 375-387 (2005) · Zbl 1082.34032
[28] Thandapani, E.; Ravi, K.; Graef, J., Oscillation and comparison theorems for half-linear second order difference equations, Comput. Math. Appl., 42, 953-960 (2001) · Zbl 0983.39006
[29] Wang, Q. R., Oscillation and asymptotics for second-order half-linear differential equations, Appl. Math. Comput., 122, 253-266 (2001) · Zbl 1030.34031
[30] Waltman, P., An oscillation criterion for a nonlinear second-order equation, J. Math. Anal. Appl., 10, 439-441 (1965) · Zbl 0131.08902
[31] Wintner, A., On the nonexistence of conjugate points, Amer. J. Math., 73, 368-380 (1951) · Zbl 0043.08703
[32] Yan, J., A note on an oscillation criterion for an equation with damping term, Proc. Amer. Math. Soc., 90, 277-280 (1984) · Zbl 0542.34028
[33] Zhang, B. G.; Zhu, S., Oscillation of second-order nonlinear delay dynamic equations on time scales, Comput. Math. Appl., 49, 599-609 (2005) · Zbl 1075.34061
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