Oscillation criteria for second order nonlinear differential equations of Euler type (Q5927577)

The authors investigate oscillatory properties of the Euler-type nonlinear differential equation NEWLINE\[NEWLINE t^2 x'' +g(x)=0, \tag{*} NEWLINE\]NEWLINE where the nonlinearity \(g\) satisfies \(xg(x)>0\) for \(x\neq 0\). A typical result is the following theorem: Suppose that there exists a \(\lambda>1/16\) such that NEWLINE\[NEWLINE {g(x)\over x}\geq {1\over 4}+{\lambda\over(\log|x|)^2} NEWLINE\]NEWLINE for \(|x|\) sufficiently large. Then all nontrivial solutions to (*) are oscillatory. NEWLINENEWLINENEWLINEThe results here are proved using the phase plane analysis of the Liénard system associated with (*). In the last section of the paper, some oscillation results on (*) are extended to the more general equation \(x''+a(t)g(x)=0\) and the oscillation criteria by \textit{J. S. W. Wong} [Methods Appl. Anal. 3, No. 4, 476-485 (1996; Zbl 0946.34028)] are generalized.