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Integral averages and oscillation of second order sublinear differential equations. (English) Zbl 0721.34026
Consider the second order sublinear ODE $(1)\quad x''(t)+a(t)f[x(t)]=0,$ where a is a continuous function on $[t\sb 0,\infty)$, $t\sb 0>0$, and f is a continuous function on the real line ${\bbfR}$. Suppose that f has a continuous derivative on ${\bbfR}\setminus \{0\}$ and satisfies $yf(y)>0,\quad f'(y)\ge 0,\quad \forall y\ne 0;\quad \int\sb{+0}\frac{dy}{f(y)}<\infty,\quad \int\sb{- 0}\frac{dy}{f(y)}<\infty.$ The purpose of this paper is to present two new oscillation criteria, namely. Theorem 1. Let n be an integer with $n\ge 2$ and $\phi$ be a positive and twice continuously differentiable function on $[t\sb 0,\infty)$ wth $\phi '\ge 0$ and $\phi ''\le 0$ on $[t\sb 0,\infty)$. Equation (1) is oscillatory if $$\limsup\sb{t\to \infty}\frac{1}{t\sp{n- 1}}\int\sp{t}\sb{t\sb 0}(t-s)\sp{n-1}[\phi (s)]\sp{\lambda}a(s)ds=\infty.$$ Theorem 2. Suppose that $\lambda >0$. Let n be an integer with $n\ge 2$ and $\phi$ be a positive function (twice continuously differentiable) on $[t\sb 0,\infty)$ such that $(\phi ')\sp 2\le C\phi (-\phi '')$ on $[t\sb 0,\infty)$, where C is a positive constant. Equation (1) is oscillatory if there exists a continuous function A on $[t\sb 0,\infty)$ with $\int\sp{\infty}\sb{t\sb 0}([A\sb+(T)]\sp 2/T)dT=\infty,$ where $A\sb+(T)=\max \{A(T),0\}$, $T>t\sb 0$, and such that $$\limsup\sb{t\to \infty}\frac{1}{t\sp{n-1}}\int\sp{t}\sb{T}(t-s)\sp{n-1}[\phi (s)]\sp{\lambda}a(s)ds\ge A(T),\quad \forall T\ge t\sb 0.$$

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory