System Of Generalized Multi-valued Variational Inclusions And Equilibrium Existence Theorem In FC-spaces

This paper mainly further studies system of multi-valued variational inclusions,equilibrium problems with pseudomonotone multi-valued bifunctions in FC-spaces and equilibrium problem of abstract economies in FC-spaces,it also unites and generalizes some known results in recent literatures.In the first place,we first introduce and study two classes of system of generalized set-valued variational inclusions with(A,η)-accretive operators in real q-uniformly smooth Banach spaces.By using the resolvent operator technique associated with(A,η)-accretive operators,we construct iterative algorithms for solving these systems of generalized set-valued variational inclusions in real q-uniformly smooth Banach spaces respectively.We also prove the existence of solutions for these generalized set-valued variational inclusions and the convergence of iterative sequences generated by algorithms.Next,we introduce a class of(A,η)-accretive operator in real Banach spaces for overcoming the limitedness of(A,η)-accretive operator defined in q-uniformly Banach spaces[10].The resolvent operator associated with(A,η)-accretive operator is defined and its Lipschitz continuity is showed.As applications,we introduce and study a new system of generalized quasi-variational-like inclusions with(A,η)-accretive operators in Banach spaces.By using the resolvent operator technique associated with(A,η)-accretive operators,we construct an iterative algorithm for solving the system of generalized quasi-variational-like inclusions.The existence of solutions for the system of generalized quasi-variational-like inclusions and the convergence of iterative sequences generated by the algorithm are proved.Finally,we prove some properties of A-monotone operator in Banach space,give the notion of proximal mapping associated with the A-monotone operator and show its Lipschitz continuity.We also consider a new system of generalized nonlinear quasi-variational-like inclusions with A-monotone operators in Banach spaces and constructed a new iterative algorithm for solving the system of generalized nonlinear quasi-variational-like inclusions in Banach spaces.Under suitable sssumptions,we prove the convergence of the iterative sequence generated by the algorithm.Secondly,we first study some classes of generalized vector equilibrium problems with pseudomonotone multi-valued bifunctions in FC-spaces. By using an KKM theorem[48],some existence results of equilibrium points for the generalized vector equilibrium problems are established in FC-spaces. Next,we study some classes of generalized vector quasi-equilibrium problems with pseudomonotone multi-valued bifunctions in FC-spaces.By using an existence theorem of maximal elements in FC-spaces[57],some existence results of equilibrium points for generalized vector quasi-equilibrium problems are established in FC-spaces.Lastly,we first establish some fixed point theorems in noncompact FC-spaces.Next two existence theorems of maximal elements for L_F~*-correspondence and L_F~*-majorized correspondence are obtained.Finally, by applying the existence theorems of maximal elements,some new equilibrium existence theorems for one person game,qualitative games and noncompact abstract economies with L_F~*-majorized correspondences are obtained in FC-spaces.