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Uniqueness of positive solutions of nonlinear second-order equations. (English) Zbl 0857.34034
The author studies the question of uniqueness for positive solutions \(u\in C^2([-R,R])\) of the two-point boundary value problem (1) \(u''(t)+ f(|t|,u(t))=0\), \(-R<t<R\), \(u(\pm R)=0\), where \(R>0\) is fixed and the function \(f:[0,R]\times [0,\infty)\to\mathbb{R}^1\) is in \(\mathbb{C}^1 ([0,R]\times [0,\infty))\) and satisfies the following hypotheses: a) \(uf_u(t,u)> f(t,u)\) on \([0,R]\times [0,\infty)\); b) \(f_t(t,u)\leq 0\) on \([0,R]\times [0,\infty)\); c) there exists \(u>0\) such that \(f(R,u)\geq 0\).
The result of uniqueness for solutions of (1) is illustrated by examples for which existence is well known.

34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI
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