Multiplicity of positive solutions for a singular second-order three-point boundary value problem with a parameter.(English)Zbl 1442.34046

Summary: This paper is concerned with the following second-order three-point boundary value problem $$u''(t) + \beta^2 u(t) + \lambda q (t) f (t, u (t)) = 0, t \in (0, 1), u (0) = 0, u(1) = \delta u(\eta)$$, where $$\beta \in(0, \pi / 2), \delta > 0, \eta \in(0,1)$$, and $$\lambda$$ is a positive parameter. First, Green’s function for the associated linear boundary value problem is constructed, and then some useful properties of Green’s function are obtained. Finally, existence, multiplicity, and nonexistence results for positive solutions are derived in terms of different values of $$\lambda$$ by means of the fixed point index theory.

MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:

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