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Note on a non-oscillation theorem of Atkinson. (English) Zbl 1058.34035
The authors consider the second-order nonlinear differential equation $$ y''(x)+F(x,y(x))=0,\quad x\in[0,\infty), \tag{*} $$ where $F:\Bbb{R}^+\times \Bbb{R}\to \Bbb{R}$ is continuous. The following theorem is proved: Let $X:=\{u\in C[0,\infty):\ 0\leq u(t)\leq M, t\geq 0\}$, where $M>0$ is a constant. Assume that for any $u\in X$ $$ \int_0^\infty tF(t,u(t))\,dt\leq M $$ and that there exists a continuous function $k:\Bbb{R}^+\to \Bbb{R}^+$ such that $\int_0^\infty tk(t)\,dt<1$ and $$ \vert F(t,u)-F(t,v)\vert \leq k(t)\vert u-v\vert \quad\text{for } t\geq 0. $$ Then $(*)$ has a positive monotone solution on $(0,\infty)$ such that $y(x)\to M$ as $x\to \infty$. Several examples including sublinear, superlinear and transcendental cases illustrate this result.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34D05Asymptotic stability of ODE
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