A necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems

We study the nonlinear boundary value problem (ϕ(u′′))′′=f(t,u,u′,u′′),t∈(0,1),u(2i)(0)=u(2i)(1)=0,i=0,1, and obtain a necessary and sufficient condition for the existence of symmetric positive solutions. We also discuss the application of our result to the special case where ff is a power function of uu and its derivatives. Moreover, similar conclusions for a more general higher order boundary value problem are established. Our analysis mainly relies on the lower and upper solution method.