Optimal existence theory for single and multiple positive solutions to fourth-order periodic boundary value problems

This paper presents a new optimal existence theory for single and multiple positive solutions to a fourth-order periodic boundary value problem u(4)-βu″+αu=F(t,u(τ(t))),0⩽t⩽1u(4)-βu″+αu=F(t,u(τ(t))),0⩽t⩽1, with u(i)(0)=u(i)(1)u(i)(0)=u(i)(1), i=0,1,2,3i=0,1,2,3, where F:[0,1]×R+→R+F:[0,1]×R+→R+ and τ:[0,1]→[0,1]τ:[0,1]→[0,1] are continuous, α,β∈Rα,β∈R and satisfy α>0α>0, α<β/2+2π2, α/π4+β/π2+1>0α/π4+β/π2+1>0. Existence is established using a fixed-point theorem in cones.