Further study on solvability of two-point boundary value problems with reflection of the argument

For all $k\in {\mathbf{N}}$ , we consider the following two-point boundary value problems: $$ u''(x) +k^{2}u(\pi -x)+g(x,u(\pi -x))=h(x) \text{ in } (0,\pi ), u(0)=0 =u(\pi ) $$ and $$ u''(x) +k^{2}u(x)-g(x,u(\pi -x))=-h(x) \text{ in } (0,\pi ), u(0)=0 =u( \pi ), $$ where $g:(0,\pi )\times {\mathbf{R}}\to {\mathbf{R}}$ is a Carathéodory function that grows linearly in u as $|u|\to \infty $ and $h\in L^{1}(0,\pi )$ satisfies a generalized Landesman–Lazer condition. In particular, the Leray–Schauder continuation method is used to prove the existence of solutions to these problems.