Symmetric positive solutions of nonlinear boundary value problems

We study the nonlinear boundary value problemu(2m)=f(t,u,u′,…,u(2m−2)),t∈(0,1),u(2i)(0)=u(2i)(1)=0,i=0,…,m−1. The existence of symmetric positive solutions of the above problem is discussed. Sufficient conditions are obtained for the problem to have one, any finite number, and a countably infinite number of such solutions. Our results extend some recent work in the literature on boundary value problems of ordinary differential equations. We illustrate our results by two examples, none of which can be handled using the existing results.