On equilibrium multivalued problems in the general model with the Boolean-valued bifunction

In this paper, we establish some sufficient conditions for the existence of a solution 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{ all }y \in E; \end{equation*} $and its variants, where Boolean-valued bifunction and \Phi multivalued mapping with values vector space without topological structure. We use result to show that (C,\vartriangleleft) $-saddle point exists multi-valued mappings. results are established by combining concepts cyclic quasi-monotonicity 'algebraic' semicontinuity. Our interesting refreshing because do not need convex hypothesis.