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

Let Φ be the class of all real functions φ: [0 , ∞[× [0 , ∞[→ [0 , ∞[that satisfy the following condition: there exists α∈]0,1[such thatφ((1-α)r,αr)<r,for allr>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⇉ X, with nonempty and convex values, have a common fixed point whenver there exists a function φ∈ Φ such that ‖y-u‖≤φ(‖y-x‖,‖u-x‖),for allx∈X,y∈T(x),u∈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).