Generalization of an oscillation criterion for sublinear differential equations
Consider the equation (*) \(x''(t)+a(t)F(| x(t)|)sgn x(t))=0\), \(t\in [0,\infty)\), where a(t)\(\in C[0,\infty)\), \(F(x)=\int^{x}_{0}f(y)dy\) with \(f(x)>0\) \((x>0)\), \(x^{-\alpha}f(x)\) is nondecreasing in (0,\(\infty)\), and \(x^{-\beta}f(x)\) is nonincreasing in (0,\(\infty)\) for constants \(\alpha\) and \(\beta\) such that \(-1<\alpha \leq \beta <0\). Suppose that \(\phi\) (t)\(\in C[0,\infty)\), \(\phi (t)>0\), \(\phi'(t)>0\), \(\phi''(t)<0\) for \(t>0\), \((\alpha - 2\beta)/(1+\beta)>(2(m+1)/(m-1)^ 2)((\beta -\alpha)/(1+\alpha))\) and \[ (\phi | \phi''| /\phi^{'2}-((\beta -\alpha)/(1+\alpha)) > 2(m+1)(\beta - \alpha)((1+\beta)/(1+\alpha)) \] with \(m>1\) a constant. Let \[ A(s)=\lim_{T\to \infty}1/T\int^{T}_{s}\int^{t}_{s}a(\tau)\theta (\tau)F(\phi (\tau)\quad)d\tau dt \] and \(A_+(s)=\max \{A(s),0\}\) \((s>0)\) be well- defined, where \(\theta (\tau)=| \beta |\), \(a(\tau)\geq 0\), \(=| \alpha |\), \(a(\tau)<0\). The authors show that in addition to the above assumptions, if either \(A(s)=+\infty\) or A(s)\(\neq -\infty\) and \[ \{\lim_{T\to \infty}\int^{T}_{1}(A^ 2_+(s)/s)ds\}\{\lim_{T\to \infty}\int^{T}_{1}(\phi'/\phi)^ 2sds\}^{-1}=+\infty, \] then all continuable solutions of equation (*) are oscillatory.
scientific article; zbMATH DE number 4057963