In this paper, we consider set optimization problems involving set relations. Using the wellknown KKM-Fan Lemma and relaxed convexity assumptions, we study existence conditions for these problems. Moreover, we introduce parametric nonlinear scalarization functions for sets and study their properties. By utilizing the concerning functions, we investigate
relationships between set optimization problems and equilibrium problems. Finally, sufficient conditions for the Hölder continuity of solution maps to such problems via equilibrium problems are established.