Ky-Fan inequality, Nash equilibria in some idempotent and harmonic convex structure

B-convexity is defined as a suitable Peano-Kuratowski limit of linear convexities. An alternative idempotent convex structure called inverse B-convexity was recently proposed in the literature. This paper continues and extends some investigation started in these papers. In particular we focus on the Ky-Fan inequality and prove the existence of a Nash equilibrium for inverse B-convex games. This we do by considering a suitable “harmonic” topological structure which allows to establish a KKM theorem as well as some important related properties. Among other things a coincidence theorem is established. The paper also establishes fixed point results and Nash equilibriums properties in the case where two different convex topological structures are merged. It follows that one can consider a large class of games where the players may optimize their payoff subject to different forms of convexity. Among other things an inverse B-convex version of the Debreu-Gale-Nikaido theorem is proposed.