×

Note on forced oscillation of \(n\)th-order sublinear differential equations. (English) Zbl 1064.34020

The authors consider the \(n\)th-order nonlinear differential equation \[ x^{(n)}+q(t)| x| ^{\lambda}\operatorname {sgn}x=e(t),\;t\in [t_{0},\infty ), \tag{E} \] where \(q(t)\) and \(e(t)\) are continuous maps on \([t_{0},\infty ),\) and \( \lambda \in (0,1)\) (sublinear case). The main result provides a sufficient condition for the oscillatory character of (E), that is, to ensure that all its solutions have arbitrarily large zeros. The statement is very involved to be described here, and its proof is based on similar arguments to that of R. P. Agarwal and S. R. Grace [Appl. Math. Lett. 13, 53–57 (2000; Zbl 0978.39012)] and C. H. Ou and J. S. W. Wong [J. Math. Anal. Appl. 262, 722–732 (2001; Zbl 0997.34059)]. Moreover, as an application of the main theorem, the authors study the oscillatory nature of \[ x^{\prime \prime }+t^{\alpha }\sin t | x| ^{\lambda } \operatorname{sgn}x=mt^{\beta }\cos t,\;t\geq 0, \tag{E\('\)} \] where \(\alpha \geq 0,\) \(\beta >0,\) \(m\) and \(0<\lambda <1\) are real constants. It is proved that if \(\beta >\frac{\alpha +2}{1-\lambda }, \) then (E\('\)) is oscillatory. This gives an analogue example to that one of A.H. Nasr [Proc. Am. Math. Soc. 126, 123-125 (1998; Zbl 0891.34038)].

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agarwal, R. P.; Grace, S. R., Forced oscillation of \(n\) th-order nonlinear differential equations, Appl. Math. Lett., 13, 53-57 (2000) · Zbl 0958.34050
[2] El-Sayed, M. A., An oscillation criterion for a forced second order linear differential equation, Proc. Amer. Math. Soc., 118, 813-817 (1993) · Zbl 0777.34023
[3] Erdelyi, A., Asymptotic Expansions (1956), Dover: Dover New York · Zbl 0070.29002
[4] Grace, S. R.; Lalli, B. S., Asymptotic and oscillatory behavior of \(n\) th order forced functional differential equations, J. Math. Anal. Appl., 140, 10-25 (1989) · Zbl 0719.34119
[5] Hamedani, G. G., Oscillatory behavior of \(n\) th order forced functional differential equations, J. Math. Anal. Appl., 195, 123-134 (1995) · Zbl 0844.34068
[6] Kartsatos, A. G., On the maintenance of oscillation under the effect of a small forcing term, J. Differential Equations, 10, 355-363 (1971) · Zbl 0211.11902
[7] Kartsatos, A. G., Maintenance of oscillations under the effect of a periodic forcing term, Proc. Amer. Math. Soc., 33, 377-383 (1972) · Zbl 0234.34040
[8] Kartsatos, A. G., The oscillation of a forced equation implies the oscillation if the unforced equation-small forcing, J. Math. Anal. Appl., 76, 98-106 (1980) · Zbl 0443.34032
[9] Keener, M. S., Solutions of a certain linear nonhomogeneous second order differential equations, Appl. Anal., 1, 57-63 (1971) · Zbl 0215.43802
[10] Nasr, A. H., Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. Math. Soc., 126, 123-125 (1998) · Zbl 0891.34038
[11] Ou, C. H.; Wong, J. S.W., Forced oscillation of \(n\) th-order functional differential equations, J. Math. Anal. Appl., 262, 722-732 (2001) · Zbl 0997.34059
[12] Rainkin, S. M., Oscillation theorems for second order nonhomogeneous linear differential equations, J. Math. Anal. Appl., 53, 550-553 (1976) · Zbl 0328.34033
[13] Skidmore, A.; Bowers, J. J., Oscillatory behaviour of solutions of \(y'' + p(x) y = f(x)\), J. Math. Anal. Appl., 49, 317-323 (1975) · Zbl 0312.34025
[14] Skidmore, A.; Leighton, W., On the differential equation \(y'' + p(x) y = f(x)\), J. Math. Anal. Appl., 43, 45-55 (1973) · Zbl 0287.34031
[15] Sun, Y. G., A note on Nasr’s and Wong’s papers, J. Math. Anal. Appl., 286, 363-367 (2003) · Zbl 1042.34096
[16] Y.G. Sun, R.P. Agarwal, Forced oscillation of \(n\); Y.G. Sun, R.P. Agarwal, Forced oscillation of \(n\)
[17] Y.G. Sun and R.P. Agarwal, Interval oscillation criteria for higher-order forced nonlinear differential equations, Nonlinear Func. Anal. Appl., in press; Y.G. Sun and R.P. Agarwal, Interval oscillation criteria for higher-order forced nonlinear differential equations, Nonlinear Func. Anal. Appl., in press · Zbl 1075.34031
[18] Y.G. Sun, C.H. Ou, J.S.W. Wong, Interval oscillation theorems for a second order linear differential equation, Comput. Math. Appl., in press; Y.G. Sun, C.H. Ou, J.S.W. Wong, Interval oscillation theorems for a second order linear differential equation, Comput. Math. Appl., in press · Zbl 1069.34049
[19] Teufel, H., Forced second order nonlinear oscillations, J. Math. Anal. Appl., 40, 148-152 (1972) · Zbl 0211.12001
[20] Wong, J. S.W., Second order nonlinear forced oscillations, SIAM J. Math. Anal., 19, 667-675 (1988) · Zbl 0655.34023
[21] Wong, J. S.W., Oscillation criteria for a forced second order linear differential equations, J. Math. Anal. Appl., 231, 235-240 (1999) · Zbl 0922.34029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.