About the existence of solutions of a boundary value problem for a Carathéodory differential system (Q1368797)

The authors consider boundary value problems (BVP) of the type \(x'= f(t,x)\) for a.e. \(t\in I=[a,b]\subset\mathbb{R}\), \(x\in\mathbb{R}^n\), \(x\in S\) \((S\subset AC(I,\mathbb{R}^n))\), where \(f\) is a Carathéodory function. This BVP is connected to an equivalent nonlinear Urysohn integral equation. A related integral equation with an exact number of solutions is studied in order to show the existence of solutions of the BVP through a fixed point theorem for \(w\)-maps in the Darboux sense. In applications, the authors consider the two-point BVP \[ -u''= g(t,u)+ h(t)u\quad (t\in [0,\pi],x\in\mathbb{R}), \] \(u(0)= u(\pi)\), where \(g\) is a Carathéodory function, \(h\in L^2(0,\pi)\).