A coincidence point theorem for sequentially continuous mappings (Q892344)

The following abstract coincidence theorem is proved: Theorem. Let \(X,Y\) be real Banach spaces, \(K\) be a weakly compact, convex subset of \(X\), and let \(F,G:K\to 2^Y\) be multifunctions such that their graphs are sequentially closed with \(F(K)\) sequentially weakly compact. If, for every \(x\in K\), the set \(\{ u\in K: G(x)\cap F(u)\neq \emptyset \}\) is nonempty and convex, then there exists \(x_0\in K\) such that \(F(x_0)\cap G(x_0)\neq \emptyset\). As applications, an existence result of classical solution to some two-point boundary problem of second order is given. Also, some semilinear elliptic problems with mixed boundary conditions are considered.