On upper and lower solutions of the differential inclusion \(x''\in F(t,x)\) (Q1279417)

The author extends some results of the reviewer and \textit{F. Zanolin} [J. Math. Anal. Appl. 181, No. 3, 684-700 (1994; Zbl 0801.34029) and Boll. Unione Mat. Ital., VII. Ser., A 9, No. 2, 273-286 (1995; Zbl 0841.34017)] to the Dirichlet problem for differential inclusions \[ x''\in F(t,x), \quad x(0)=a,\quad x(\pi)=b. \tag{1} \] Here, \(x\) is scalar and \(F\) satisfies Davy conditions. A first theorem concerns the method of upper and lower solutions. The author proves the existence of a solution using upper and lower solutions with a finite number of angles. This result is the main tool to investigate positive solutions to problem (1) with \(a=b=0\), in case the function \(F\) has possible singularities both in the \(t\)-variable, for \(t=0\) and \(t=\pi\), and in the \(x\)-variable for \(x=0\).