Multiplicity of Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem with a Parameter

Abstract

This paper is concerned with the following second-order three-point boundary value problem $u^{″}\left(t\right)+{\beta }^{2}u\left(t\right)+\lambda q\left(t\right)f\left(t,u\left(t\right)\right)=0$, $t\in \left(0\mathrm{ },\mathrm{ }1\right)$, $u\left(0\right)=0$, $u(1)=\delta u(\eta )$, where $\beta \in (0,\pi /2)$, $\delta >0$, $\eta \in (\mathrm{0,1})$, 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.