Quasi-equilibrium problems with lower and upper bounds in ordered topological spaces

In this paper, we consider quasi-equilibrium problem and implicit quasi-equilibrium problem with lower and upper bounds in the setting of ordered topological spaces. To prove the existence of their solutions, we establish a Fan-Browder type fixed point theorem and its equivalent maximal element theorem in the setting of ordered topological spaces. We introduce the concept of (α,β)-pseudodissipative maps. By using our maximal element theorem, we prove the existence of solutions of quasi-equilibrium problem with lower and upper bounds under (α,β)-pseudodissipative assumption. By using the selection of a multivalued map, we extend our results for implicit quasi-equilibrium problem for lower and upper bounds.