Necessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems

We study the nonlinear boundary value problem with multi-point boundary condition (|u″|p−1u″)″=f(t,u,u′,u″),t∈(0,1),u(2i)(0)=u(2i)(1)=∑j=1maiju(2i)(tj),i=0,1. Necessary and sufficient conditions are obtained for the existence of symmetric positive solutions of this problem using fixed point theorems on cones. Applications of our results to the special case where ff is a power function of uu and its derivatives are also discussed. Moreover, similar conclusions for a more general higher order boundary value problem are established. Our results extend some recent work in the literature on boundary value problems for ordinary differential equations.