  In the present paper we establish a fixed point result of Krasnoselskii type
for the sum $A+B$, where $A$ and $B$ are continuous maps acting on locally
convex spaces. Our results extend previous ones. We apply such results to
obtain strong solutions for some quasi-linear elliptic equations with lack of
compactness. We also provide an application to the existence and regularity
theory of solutions to a nonlinear integral equation modeled in a Banach space.
In the last section we develop a sequentially weak continuity result for a
class of operators acting on vector-valued Lebesgue spaces. Such a result is
used together with a geometric condition as the main tool to provide an
existence theory for nonlinear integral equations in $L\sp p(E)$.
