The method of upper-lower solutions for nonlinear second order differential inclusions
The paper concerns a system which consists of the scalar differential inclusion \[ (| x'|^{p-2}x')'\in\partial g(t,x)+F(t,x,x')+\theta(x) \] on an interval \([0,b]\) and of the boundary conditions \[ x'(0)\in\xi_1(x(0)), \qquad -x'(b)\in\xi_2(x(b)). \] Here: \(1<p<\infty\); the function \(g\) is \(t\)-measurable and \(x\)-convex; the closed convex valued multifunction \(F\) is \(t\)-graph measurable and \((x,y)\)-graph closed; the possibly discontinuous function \(\theta\) satisfies a decreasing-like condition; and finally, \(\xi_1\) and \(\xi_2\) are maximal monotone graphs in \(\mathbb R^2\). There are considered also certain functions \(\psi\) and \(\varphi\), ``lower'' and ``upper'' appropriate for the system, such that \(\psi(t)\leq\varphi(t)\) on \([0,b]\). The main result establishes the existence of solutions \(x\) to the system such that \(\psi(t)\leq x(t)\leq\varphi(t)\) on \([0,b]\). The proof makes use of a fixed point theorem in ordered Banach spaces [\textit{S.~Heikkilä} and \textit{S.~Hu}, Appl. Anal. 51, 115--127 (1993; Zbl 0839.54033)]. In case the function \(\theta\) is continuous, the set of \(x\)-solutions has a smallest and a greatest element. The system incorporates Dirichlet, Neumann, and Sturm-Liouville problems, whereas the method of proof applies to periodic problems.
scientific article; zbMATH DE number 5159581
scientific article