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Multiplicity of positive solutions for singular three-point boundary value problems at resonance. (English) Zbl 1188.34032
The paper deals with the problem of existence of positive solutions for the three-point boundary value problem
\[ x''+m^{2}x=f(t,x)+e(t),\quad t\in (0,1), \]
with conditions
\[ x'(0)= 0, \qquad x(\eta)=x(1), \] where \(m \in (0,\frac{\pi}{2})\) and \(\eta \in (0,1)\) are given.
The problem is assumed to be with singular nonlinear perturbations at resonance. The proof is based on a nonlinear Leray-Schauder principle and the fixed point theorem in cones for completely continuous operators.

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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