Existence of solution for discontinuous third order boundary value problems (Q1807769)

Using the method of upper and lower solutions, the authors investigate the existence of solutions for two types of boundary value problems for a discontinuous nonlinear third-order differential equation, namely \[ u'''(t)=q(u''(t))f(t,u(t)), \text{for a.e. } t\in I=[a,b],\tag{1} \] where \(q:\mathbb{R}\to (0,\infty)\) is bounded and \(f:I\times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function. The existence of a solution to equation (1) with the boundary conditions (2) \(u(a)=A\), \(u(b)=B\), \(u''(a)=C\), where \(A,B,C\) are real constants, is proved by means of fixed point theory. Moreover, the existence of extremal solutions to (1), (2) in a sector is obtained via monotone iterative technique. Then, using the results obtained for (1), (2) and limit arguments, the authors demonstrate the existence of a solution to problem (1) with the boundary conditions \(u(a)=u(b)\), \(u'(a)=u'(b)\), \(u''(a)=C\). Finally, a change of variables allows the authors to consider some different boundary conditions.