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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.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
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