Solvability of boundary value problems for the inclusion \(u_{tt}- u_{xx}\in g(t,x,u)\) via the theory of multivalued A-proper maps (Q1823444)

The work studies the existence of a coincidence point x for the inclusion \[ (1)\quad Lx\in \Gamma (x). \] Here L: D(L)\(\subseteq E\to F\) is a linear operator between the Banach spaces E and F and \(\Gamma\) : \(E\to 2^ F\) is a convex-valued function. In contrast to the well-studied case of the Fredholm operator L, the authors study the case when \(\dim Ker L=co\dim Im L=\infty.\) For the multivalued map \(\Gamma\) suitable compact conditions (called L-compacity and weak L-compacity) are introduced. An existence theorem for (1) is proved under the assumption that \(\Gamma\) is L-compact and A-proper. It turns out that any monotone demi-continuous map \(\Gamma\) : \(H\to 2^ H\) is weakly A-proper according to one of the main results. An application to the problem \[ u_{tt}-u_{xx}\in g(t,x,u) \] for a multifunction g is given.