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Positive solutions of two-point boundary value problems for second-order differential equations with the nonlinearity dependent on the derivative. (English) Zbl 1145.34015

Summary: The existence of positive solutions is established for two-point boundary value problems for second-order differential equations with the nonlinearity dependent on the derivative:
\[ \begin{cases} (L\varphi)(x)=h(x)f(\varphi(x),\varphi'(x)),\quad & 0\leq x\leq 1,\\ R_1(\varphi)\equiv\alpha_1\varphi(0)+\beta_1\varphi'(0)=0,\\ R_2(\varphi)\equiv\alpha_2\varphi(1)+\beta_2\varphi'(1)=0,\end{cases} \]
where \((L\varphi)(x)=-(p(x)\varphi'(x))'+q(x)\varphi(x)\). Conditions are given in terms of the relative behavior of the quotient \(\frac{f(u,v)}{|u|+|v|}\) for \(|u|+|v|\) near 0 and \(+\infty\).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

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