×

Oscillation criteria for second order superlinear differential equations. (English) Zbl 0880.34033

Summary: A new oscillation criterion is given for general superlinear ordinary differential equations of second-order of the form \(x''(t)+ a(t)f(x(t))= 0\), where \(a(t)\in C[t_0,\infty)\), \(f(x)\in C(\mathbb{R})\) and \(xf(x)>0\), \(f'(x)\geq 0\) for \(x\neq 0\). Furthermore, \(f(x)\) also satisfies a superlinear condition, which covers the prototype nonlinear function \(f(x)=|x|^\gamma\text{sgn }x\) with \(\gamma>1\) known as the Emden-Fowler case. The coefficient \(a(t)\) is not assumed to be eventually nonnegative. The oscillation criterion involving integral averages of \(a(t)\) gives a positive answer to a question asked by J. S. W. Wong [Can. J. Math. 45, No. 5, 1094-1103 (1993; Zbl 0797.34037)].

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C29 Averaging method for ordinary differential equations

Citations:

Zbl 0797.34037
PDF BibTeX XML Cite