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A nonoscillation theorem for half-linear differential equations with periodic coefficients. (English) Zbl 1217.34056

Summary: The half-linear differential equation \((\phi_p(x'))'+a(t)\phi_p(x')+b(t)\phi_p(x)=0\) is considered under the assumption that the coefficient \(a(t)\) and an indefinite integral \(B(t)\) of \(b(t)\) are periodic functions with period \(T>0\). It is proved that \(\{(p-1)\phi_{p^*}*(B(t))-a(t)\}B(t)\leq 0\) \((0\leq t\leq T)\) is sufficient for all nontrivial solutions to be nonoscillatory. Here, \(p>1\) and \(\phi_ q(y)=|y|^{q-2}y\) for \(q=p\) or \(q=p^*=p/(p-1)\). The proof is given by means of the Riccati technique. The condition is shown to be sharp. Sufficient conditions are also presented for all nontrivial solutions to be oscillatory in the linear case \(p=2\). Some examples and simulations are included to illustrate our results.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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[1] Agarwal, R. P.; Grace, S. R.; O’Regan, D., Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations (2002), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht-Boston-London · Zbl 1073.34002
[2] J. Differ. Eq., 221, 272-274 (2006) · Zbl 1096.34023
[3] Coppel, W. A., Disconjugacy. Disconjugacy, Lect. Notes in Math., vol. 220 (1971), Springer-Verlag: Springer-Verlag Berlin
[4] Došlý, O., Half-linear differential equations, (Cañada, A.; Drábek, P.; Fonda, A., Handbook of Differential Equations, Ordinary Differential Equations, vol. I (2004), Elsevier: Elsevier Amsterdam), 161-357 · Zbl 1090.34027
[5] Došlý, O.; Elbert, Á., Conjugacy of half-linear second-order differential equations, Proc. Roy. Soc. Edin. Sect. A, 130, 517-525 (2000) · Zbl 0961.34020
[6] Došlý, O.; Pátíková, Z., Hille-Wintner type comparison criteria for half-linear second order differential equations, Arch. Math. (Brno), 42, 185-194 (2006) · Zbl 1164.34386
[7] Došlý, O.; Řehák, P., Half-linear Differential Equations. Half-linear Differential Equations, North-Holland Mathematics Studies, vol. 202 (2005), Elsevier: Elsevier Amsterdam · Zbl 1090.34001
[8] Á. Elbert, A half-linear second order differential equation, in: M. Farkas (Ed.), Qualitative Theory of Differential Equations, vol. I (Szeged, 1979), Colloq. Math. Soc. János Bolyai, 30, North-Holland, Amsterdam-New York, 1981, pp. 153-180.; Á. Elbert, A half-linear second order differential equation, in: M. Farkas (Ed.), Qualitative Theory of Differential Equations, vol. I (Szeged, 1979), Colloq. Math. Soc. János Bolyai, 30, North-Holland, Amsterdam-New York, 1981, pp. 153-180.
[9] Elbert, Á.; Schneider, A., Perturbations of the half-linear Euler differential equation, Results Math., 37, 56-83 (2000) · Zbl 0958.34029
[10] Hille, E., Non-oscillation theorems, Tran. Amer. Math. Soc., 64, 234-252 (1948) · Zbl 0031.35402
[11] Jaroš, J.; Kusano, T.; Tanigawa, T., Nonoscillatory half-linear differential equations and generalized Karamata functions, Nonlinear Anal., 64, 762-787 (2006) · Zbl 1103.34017
[12] Kwong, M. K., On certain Riccati integral equations and second-order linear oscillation, J. Math. Anal. Appl., 85, 315-330 (1982) · Zbl 0504.34019
[13] Kwong, M. K.; Wong, J. S.W., Oscillation and nonoscillation of Hill’s equation with periodic damping, J. Math. Anal. Appl., 288, 15-19 (2003) · Zbl 1039.34026
[14] Kwong, M. K.; Wong, J. S.W., On the oscillation of Hill’s equations under periodic forcing, J. Math. Anal. Appl., 320, 37-55 (2006) · Zbl 1102.34021
[15] Mirzov, J. D., On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Appl., 53, 418-425 (1976) · Zbl 0327.34027
[16] Swanson, C. A., Comparison and Oscillation Theory of Linear Differential Equations (1968), Academic Press: Academic Press New York and London · Zbl 0191.09904
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