The Dirichlet BVP for the second order nonlinear ordinary differential equation at resonance (Q2890662)

The author considers the Dirichlet BVP NEWLINE\[NEWLINE\begin{aligned} &u''(t) = p(t)u(t) + f(t,u(t)) + h(t), \quad t \in [a,b],\\ &u(a) = 0, u(b) = 0, \end{aligned}NEWLINE\]NEWLINE where \(h\), \(p \in L([a,b],{\mathbb R})\) and \(f \in K([a,b]\times{\mathbb R},{\mathbb R})\). There are obtained sufficient conditions for the existence of a solution in the case when the problem NEWLINE\[NEWLINE u''(t) = p(t)u(t), \quad u(a) = 0, u(b) = 0NEWLINE\]NEWLINE has a nontrivial solution.