Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems

We study the existence of positive solutions for equations of the form −u″(t)=f(t,u(t)),a.e. t∈(0,1), when f(t,u)+ω2u≥0f(t,u)+ω2u≥0 for u≥0u≥0, for some constant ω>0ω>0, and for the perturbed equation −u″(t)+ω2u(t)=h(t,u(t)),a.e. t∈(0,1), when h(t,u)≥0h(t,u)≥0, subject to various non-local boundary conditions. In particular, we establish existence and multiplicity of positive solutions for non-perturbed boundary value problems at resonance by considering equivalent non-resonant perturbed problems with the same boundary conditions.