Let $$\Phi $$ Φ be the class of all real functions $$\varphi : [0, \infty [ \times [0, \infty [ \rightarrow [0, \infty [$$ φ : [ 0 , ∞ [ × [ 0 , ∞ [ → [ 0 , ∞ [ that satisfy the following condition: there exists $$\alpha \in ]0, 1[ \text { such that } \varphi ((1 - \alpha ) r, \alpha r) < r,\; \text { for all } r > 0$$ α ∈ ] 0 , 1 [ such that φ ( ( 1 - α ) r , α 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 $$T, S: X \rightrightarrows X$$ T , S : X ⇉ X , with nonempty and convex values, have a common fixed point whenver there exists a function $$\varphi \in \Phi $$ φ ∈ Φ such that $$\begin{aligned} \Vert y - u\Vert \le \varphi (\Vert y - x\Vert , \Vert u - x\Vert ), \; \text { for all } x\in X, y\in T(x), u\in S(x). \end{aligned}$$ ‖ y - u ‖ ≤ φ ( ‖ y - x ‖ , ‖ u - x ‖ ) , for all x ∈ 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).