A two-point boundary value problem with reflection of the argument (Q6595747)

In the paper is studied solvability of two-point boundary value problems \N\[ u''(x)+u(\pi-x)+g(x,u(\pi-x)) =h(x),\,\,\,x\in (0,\pi),\,\,\,u(0)=u(\pi)=0 \] \Nand \N\[ u''(x)+u(\pi-x)-g(x,u(\pi-x)) =-h(x),\,\,\,x\in (0,\pi),\,\,\,u(0)=u(\pi)=0, \] \Nwhere \(h\in L^1(0,\pi)\) and \(g\) is a Carathéodory function. Assuming that there exists \(r_0>0\), \(a,b,c,d\in L^1(0,\pi)\), \(a\ge0\), \(b\ge0\), and \(a(x)\le 3\) for \(x\in(0,\pi)\) a.e. with strict inequality on a positive measurable subset of \((0,\pi)\), \N\[ c(x)\le g(x,u)\le a(x)|u|+b(x) \] \Nfor \(x\in(0,\pi)\) a.e. and \(\forall u\ge r_0\), \N\[ -a(x)|u|-b(x)\le g(x,u)\le d(x) \] \Nfor \(x\in(0,\pi)\) a.e. and \(\forall u\le -r_0\), the first problem has a solution provided that \N\[ \int_0^{\pi}g_-(x)\sin x dx < \int_0^{\pi}h(x)\sin x dx < \int_0^{\pi}g_+(x)\sin x dx, \] \Nwhere \(g_-(x)=\limsup_{u\to - \infty} g(x,u)\), \(g_+(x)=\liminf_{u\to \infty}g(x,u)\). The proof uses the Leray-Schauder continuation method. Several other existence results (where, e.g. inequalities for \(g\) are modified) are derived for both problems indicated. Results remain valid for some similar problems.