On discontinuous first-order implicit boundary value problems (Q1268406)

This paper is an interesting contribution to the theory of discontinuous implicit differential equations. More precisely, the authors consider boundary value problems of the form \[ F(t,u(t),u'(t)-g(t,u(t))) = 0, \text{ a.e. in } J = [t_{0},t_{1}], \quad B(u(t_{0},u(t_{1})))= 0, \] where the functions \(g: J \times \mathbb{R} \rightarrow \mathbb{R}, F: J \times \mathbb{R}^{2} \rightarrow \mathbb{R}\) and \(B: \mathbb{R}^{2} \rightarrow \mathbb{R}\) may be discontinuous and they are not necessarily monotone in any of their arguments. It is shown the existence of extremal solutions and the monotone dependence of these solutions on the functions \(F\) and \(B.\) Moreover, the authors establish a method of successive approximation in order to obtain, in a constructive way, a maximal solution. The general results are illustrated with some examples.