On equilibrium multivalued problems in the general model with the Boolean-valued bifunction
Vo Viet Tri · Numerical Algebra Control and Optimization · 2023
In this paper, we establish some sufficient conditions for the existence of a solution of the equilibrium problem. The problem is understood in general form: find $ \overline{x}\in E $ such that$ \begin{equation*} \Phi(\overline{x},y) \vartriangleleft (-D\backslash\{0\})^c \text{ for all }y \in E; \end{equation*} $and some of its variants, where $ \vartriangleleft $ is a Boolean-valued bifunction and $ \Phi $ is a multivalued mapping with values in vector space without topological structure. We use this result to show that $ (C,\vartriangleleft) $-saddle point exists for multi-valued mappings. The results are established by combining the concepts of cyclic quasi-monotonicity and 'algebraic' semicontinuity. Our results are interesting and refreshing because we do not need to use the convex hypothesis.