Positive Solutions for Second-Order Three-Point Eigenvalue Problems

Abstract

With the help of the fixed point index theorem in cones, we get an existence theorem concerning the existence of positive solution for a second-order three-point eigenvalue problem $x^{\prime \prime }\left(t\right)+\lambda f\left(t,x\left(t\right)\right)=0$, $0\le t\le 1,x\left(0\right)=0,x\left(1\right)=x\left(\eta \right)$, where $\lambda$ is a parameter. An illustrative example is given to demonstrate the effectiveness of the obtained result.