MULTIPLE POSITIVE SOLUTIONS OF RESONANT AND NON-RESONANT NON-LOCAL FOURTH-ORDER BOUNDARY VALUE PROBLEMS

We study the existence of positive solutions for equations of the form \begin{linenomath}$$u^{(4)}(t)-{\omega}^4 u(t)=f(t, u(t)),\;\;\text{a.e.} \;\; t \in (0,1),$$\end{linenomath} where 0 < ω < π, subject to various non-local boundary conditions defined in terms of the Riemann–Stieltjes integrals. We prove the existence and multiplicity of positive solutions for these boundary value problems in both resonant and non-resonant cases. We discuss the resonant case by making a shift and considering an equivalent non-resonant problem.