The Existence of Countably Many Positive Solutions for Nonlinear nth-Order Three-Point Boundary Value Problems

We consider the existence of countably many positive solutions for nonlinear nth-order three-point boundary value problem u(n)(t)+a(t)f(u(t))=0, t∈(0,1), u(0)=αu(η), u′(0)=⋯=u(n−2)(0)=0, u(1)=βu(η), where n≥2,α≥0,β≥0,0<η<1,α+(β−α)ηn−1<1, a(t)∈Lp[0,1] for some p≥1 and has countably many singularities in [0,1/2). The associated Green's function for the nth-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearity f which yield the existence of countably many positive solutions by using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone