A global bifurcation theorem for convex-valued differential inclusions (Q1884380)

The authors prove a global bifurcation theorem for convex-valued completely continuous maps. They prove as well an existence theorem for convex-valued differential inclusions with Sturm-Liouville boundary conditions \[ u''(t)\in\varphi(t,u(t),u'(t))\text{ for a.e., }t\in (a,b)\quad l(u)=0. \] The assumptions refer to the appropriate asymptotic behaviour of \(\varphi(t,x,y)\) for \(| x|+| y|\) close to 0 and to \(+\infty\), and they are independent from the Bernstein type conditions. In the last section, they give a set of examples of \(\varphi\) satisfying the assumptions of the theorem but not satisfying the Bernstein conditions.