Common fixed point theorems for set-valued mappings in normed spaces

Let Phi be the class of all real functions phi : [0, infinity[x[0, infinity[-> [0, infinity[ that satisfy the following condition: there exists alpha is an element of]0, 1[ such that phi((1-alpha)r, alpha r) < r, for all r > 0. In this paper, we show that if X is a nonempty compact convex subset of a real normed vector space, any two closed set-valued mappings T, S : X paired right arrows X, with nonempty and convex values, have a common fixed point whenver there exists a function phi is an element of Phi such that parallel to y - u parallel to <= phi(parallel to y - x parallel to, parallel to u - x parallel to), for all x is an element of X, y is an element of T (x), u is an element of S(x). Next, we prove that the same conclusion holds when at least one of the set-valued mappings is lower semicontinuous with nonempty closed and convex values. Our common fixed point theorems turn out to be useful for a unitary treatment of several problems from optimization and nonlinear analysis (quasi-equilibrium problems, quasi-optimization problems, constrained fixed point problems, quasi-variational inequalities).