Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems

Necessary and sufficient conditions are obtained for the existence of symmetric positive solutions to the boundary value problem(|u″|p−1u″)″=f(t,u,u′,u″),t∈(0,1),u(2i)(0)=u(2i)(1)=0,i=0,1. Applications of our results to the special case where f is a power function of u 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. The analysis of this paper mainly relies on fixed point theorems on cones.