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Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain. (English) Zbl 1056.34050

This paper is concerned with oscillation theory for selfadjoint second-order scalar dynamic equations of the form \[ (p(t)x^\Delta(t))^\Delta+q(t)x^\sigma(t)=0(\ast) \] on time scales \({\mathbb T}\). Here, a time scale is an arbitrary nonempty closed subset of the real numbers, denoted as measure chain by the authors, \(x^\Delta\) stands for the \(\Delta\)-derivative of \(x\) and \(x^\sigma\) is the composition of \(x\) with the forward jump operator \(\sigma\). Furthermore, \(p,q:{\mathbb T}\rightarrow{\mathbb R}\) are assumed to be rd-continuous.
The authors provide Kamenev-type and interval oscillation criteria for such linear dynamic equations on time scales. These criteria generalize corresponding theorems for ODEs by Ch. G. Philos [Arch. Math. 53, 482–492 (1989; Zbl 0661.34030)] or the second author [J. Math. Anal. Appl. 229, 258–270 (1999; Zbl 0924.34026)], respectively, and are new for difference equations in particular. The paper closes with four examples illustrating the obtained results, two of them for difference equations and one on a time scale with unbounded graininess.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
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[1] Agarwal, R. P.; Bohner, M., Basic calculus on time scales and some of its applications, Results Math., 35, 3-22 (1999) · Zbl 0927.39003
[2] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales, an Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[3] Chen, S.; Erbe, L. H., Riccati techniques and discrete oscillations, J. Math. Anal. Appl., 142, 468-487 (1989) · Zbl 0686.39001
[4] Chen, S.; Erbe, L. H., Oscillation and nonoscillation for systems of self-adjoint second order difference equation, SIAM J. Math. Anal., 20, 939-949 (1989) · Zbl 0687.39001
[5] Coles, W. J., An oscillation criterion for second order differential equations, Proc. Amer. Math. Soc., 19, 755-759 (1968) · Zbl 0172.11702
[6] Erbe, L. H., Oscillation criteria for second order linear equations on a time scale, Canad. Appl. Math. Quart., 9, 345-375 (2001) · Zbl 1050.39024
[7] Erbe, L. H.; Hilger, S., Sturmian theory on measure chains, Differential Equations Dynam. Systems, 1, 223-246 (1993) · Zbl 0868.39007
[8] L.H. Erbe, L. Kong, Q. Kong, A telescoping principle for oscillations of second order differential equations on time scales, Rocky Mountain J. Math., in press; L.H. Erbe, L. Kong, Q. Kong, A telescoping principle for oscillations of second order differential equations on time scales, Rocky Mountain J. Math., in press · Zbl 1156.34021
[9] Erbe, L. H.; Kong, Q.; Ruan, S., Kamenev type theorems for second order matrix differential systems, Proc. Amer. Math. Soc., 117, 957-962 (1993) · Zbl 0777.34024
[10] Erbe, L.; Peterson, A., Averaging techniques for self-adjoint matrix equations on a measure chain, J. Math. Anal. Appl., 271, 31-58 (2002) · Zbl 1014.39005
[11] Erbe, L.; Peterson, A., Oscillation criteria for second-order matrix dynamic equations on a time scale. Dynamic equations on time scales, J. Comput. Appl. Math., 141, 169-185 (2002) · Zbl 1017.34030
[12] Erbe, L.; Peterson, A.; Řehák, P., Comparison theorems for linear dynamic equations on time scales, J. Math. Anal. Appl., 275, 418-438 (2002) · Zbl 1034.34042
[13] Fite, W. B., Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc., 19, 341-352 (1918) · JFM 46.0702.02
[14] Hartman, P., On nonoscillatory linear differential equations of second order, Amer. J. Math., 74, 389-400 (1952) · Zbl 0048.06602
[15] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math., 18, 18-56 (1990) · Zbl 0722.39001
[16] Hooker, J. W.; Kwong, M. K.; Patula, W. T., Oscillatory second order linear difference equations and Riccati equations, SIAM J. Math. Anal., 18, 54-63 (1987) · Zbl 0619.39005
[17] Kamenev, I. V., An integral criterion for oscillation of nonlinear differential equations of second order, Mat. Zametki. Mat. Zametki, Math. Notes, 23, 136-138 (1978), translation in · Zbl 0408.34031
[18] Kong, Q., Interval criteria for oscillation of second order linear ordinary differential equations, J. Math. Anal. Appl., 229, 258-270 (1999) · Zbl 0924.34026
[19] Kong, Q.; Zettl, A., Interval oscillation conditions for difference equation, SIAM J. Math. Anal., 26, 1047-1060 (1995) · Zbl 0828.39002
[20] Kwong, M. K.; Zettl, A., Integral inequalities and second order linear oscillation, J. Differential Equations, 45, 16-23 (1982) · Zbl 0498.34022
[21] Li, H. J., Oscillation criteria for second order linear differential equations, J. Math. Anal. Appl., 194, 217-234 (1995) · Zbl 0836.34033
[22] Philos, Ch. G., Oscillation theorems for linear differential equations of second order, Arch. Math., 53, 482-492 (1989) · Zbl 0661.34030
[23] Yan, J., Oscillation theorems for second order differential equations with damping, Proc. Amer. Math. Soc., 98, 276-282 (1986) · Zbl 0622.34027
[24] Willett, D. W., On the oscillatory behavior of the solutions of second order linear differential equations, Ann. Polon. Math., 21, 175-194 (1969) · Zbl 0174.13701
[25] Wintner, A., A criterion of oscillatory stability, Quart. Appl. Math., 7, 115-117 (1949) · Zbl 0032.34801
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