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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 \(F:\mathbb{R}^+\times \mathbb{R}\to \mathbb{R}\) 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 \(k:\mathbb{R}^+\to \mathbb{R}^+\) such that \(\int_0^\infty tk(t)\,dt<1\) and \[ | F(t,u)-F(t,v)| \leq k(t)| u-v| \quad\text{for } t\geq 0. \] 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.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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