The aim of this paper is to solve the fixed point problems: $$ v=\mathcal{O}v,\quad \hbox{with}\, \mathcal{O}v(x) \mathop{=}^{\rm def} \max (Lv(x), Bv(x) ), x \in \varepsilon, \quad (1)$$ where $$\varepsilon$$ is a finite set, L is contractive and B is a nonexpansive operator and $$...