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

Sun, Yuan Gong; Wong, James S. W.

J. Math. Anal. Appl. 298, No. 1, 114-119 (2004).

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)].

Reviewer: Antonio Linero Bas (Murcia)

MSC:

34C10

Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

oscillatory solutions; zeros; nth-order nonlinear differential equations; forcing term; oscillatory potentials; asymptotic expansions; kernel function

Citations:

Zbl 0978.39012; Zbl 0997.34059; Zbl 0891.34038

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