A Two-Point Boundary Value Problem with Reflection of the Argument

Abstract

We consider the following two-point boundary value problems $u^{″}\left(x\right)+u\left(\pi -x\right)+g\left(x,u\left(\pi -x\right)\right)=h\left(x\right)$ in $\left(0,\pi \right),u\left(0\right)=0=u\left(\pi \right),$ and $u^{″}\left(x\right)+u\left(\pi -x\right)-g\left(x,u\left(\pi -x\right)\right)=-h\left(x\right)$ in $\left(0,\pi \right),u\left(0\right)=0=u\left(\pi \right),$ by setting $h\in L^1\left(0,\pi \right)$ and $g:\left(0,\pi \right)\times R⟶R$ being a Caratheodory function. When $a,b\in L^1\left(0,\pi \right)$ , $a\left(x\right)\le 3$ for $x\in \left(0,\pi \right)$ a.e. with strict inequality on a positive measurable subset of $\left(0,\pi \right)$ , and $\left|g\left(x,u\right)\right|\le a\left(x\right)\left|u\right|+b\left(x\right)$ for $x\in \left(0,\pi \right)$ a.e. as well as sufficiently large $\left|u\right|$ , several existence theorems will be obtained, with or without a sign condition.