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Positive solutions for an m-point boundary-value problem. (English) Zbl 1171.34014

Summary: We obtain sufficient conditions for the existence of a positive solution, and infinitely many positive solutions, of the m-point boundary-value problem
\[ x''(t) = f(t, x(t)), \quad 0 < t < 1, \]
\[ x'(0) = 0, \quad x(1)=\sum_{i=1}^{m-2}\alpha _{i}x(\eta _{i}). \]
Our main tools are the Guo-Krasnoselskii’s fixed point theorem and the monotone iterative technique. We also show that the set of positive solutions is compact.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint 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
34A45 Theoretical approximation of solutions to ordinary differential equations
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