On upper and lower functions for a fourth-order ordinary differential equation. II (Q2459748)

Consider the following boundary value problem (BVP) for a fourth-order ordinary differential equation: \[ x^{(\text{iv})}=f(t,x,x',x'',x'''),\;t\in [0,T],\quad x'(0)=x'''(0)=0,\quad x(T)=x'(T)=0, \] where \(f:[0,T]\times \mathbb{R}^4\to\mathbb{R}\) is a continuous function. After introducing a quite complicated notion of upper and lower solutions for this BVP, an existence result is provided. The concepts of lower and upper solutions are then simplified for the special case when \(f\) only depends on \(t\), \(x\) and \(x'''\). [For part I, see Differ. Equ. 41, No. 8, 1126--1136 (2005); translation from Differ. Uravn. 41, No. 8, 1074--1083 (2005; Zbl 1145.34009).]