Coexistence of positive solutions of nonlinear three-point boundary value and its conjugate problem

In this paper, the three-point boundary value problem−u″(t)=f(t,u(t)),0⩽t⩽1,u′(0)=0,u(1)=αu(η), and its conjugate boundary value problem−v″(s)=f(s,v(s)),0⩽s⩽1,s≠η,v′(0)=0,v+′(η)−v−′(η)=αv′(1),v(1)=0, are studied under some conditions concerning the same first eigenvalue of the both linear problems. By applying the fixed point index theory, the coexistence of single and multiple positive solutions of the above mentioned problems is verified. As an application, some examples are given to illustrate our results.