The method of lower and upper solutions for fourth order equations with the Navier condition (Q1678058)

The authors study the boundary value problem \[ \begin{aligned} Lu\equiv u^{(4)}+(k_1+k_2)u''+k_1k_2u =f(x,u(x)), \quad x\in(0,1),\\ u(0)=u''(0)=u(1)=u''(1)=0 \end{aligned} \] under the assumption that there exist lower and upper solutions \(\alpha\leq\beta\) (that is, \[ L(\alpha(x)\leq f(x,\alpha(x)),\quad\alpha(0)\leq0, \quad\alpha(1)\leq0, \quad\alpha''(0)\geq0, \quad\alpha''(1)\geq0 \] and reverse inequalities for \(\beta\)). By looking at the Green's function and using the equivalent fixed point formulation they prove that if \(k_1<0<k_2<\pi^2\) and \(f(x,\cdot)\) is increasing in \([\alpha(x),\beta(x)]\) for \(x\in[0,1]\), then the problem has a solution \(u\) such that \(\alpha(x)\leq u(x)\leq\beta(x)\) for \(x\in[0,1]\). This completes earlier results, see e. g. [\textit{R. Vrabel}, J. Math. Anal. Appl. 421, No. 2, 1455--1468 (2015; Zbl 1408.34031)] and [\textit{P. Habets} and the reviewer, Port. Math. (N.S.) 64, No. 3, 255--279 (2007; Zbl 1137.34315)].