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Existence of optimal positive solutions for Neumann boundary value problems of second order differential equations. (Chinese. English summary) Zbl 1463.34107

Summary: By using the fixed point exponential theory of cone mapping, we show the optimal conditions for the existence of positive solutions for second-order continuous Neumann boundary value problems \[ \begin{cases} u'' (t)+a (t)u (t) = g (t)f(u (t)),\, t \in [0, T], \\ u' (0) = u' (t) = 0, \end{cases} \] with nonnegative Green’s function, where \(f \in C (\mathbb{R}^+, \mathbb{R}^+)\), \(a (\cdot) \in C ([0, T], (0, +\infty))\) satisfying the corresponding homogeneous linear problems have only trivial solutions, \(g \in C ((0, T), \mathbb{R}^+)\), and \(g (t)\) is allowed to be singular at \(t = 0\) and \(t = T\), \(\mathbb{R}^+: = [0, \infty)\).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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