Study On The Generalized Equilibrium Problems

Equilibrium is a central concept to study some systems governing many real-life phenomena in fields ranging from economics and networks to mechanics. Andthe study of equilibrium in real-world applications have promoted the develop-ment of fixed point theory and optimization. The abstract variational inequality isa simple and natural formulation for those systems. Ky Fan’s minimax inequalityis another statement. And the equilibrium problem formulation is a generaliza-tion for them. The equilibrium problem formulation not only proposed a uni-fied model including optimization, fixed point theorems, variational inequalitiesand complementarity problems, but also contained other important mathemati-cal models, such as best approximations, saddle point theorems, hemi-variationalinequalities, Nash equilibrium, etc. The vector equilibrium problem, containingmulti-objective optimization and vector variational inequalities, is an importantextension for the equilibrium problem. In addition, the equilibrium problem alsohas been extended more case, such as generalized equilibrium problems, system ofequilibrium problems, quasi-equilibrium problems and like-equilibrium problems.They have close ties with some non-linear analysis problems in optimization, con-trol theory, game theory, engineering and mechanics. And then they have wideapplication background.This paper studies the solution existence for vector equilibrium problems,system of equilibrium problems and generalized equilibrium problems and searchessolving methods for them. The main points of this paper are as follows:1. Based on Gerstewitz nonlinear scalarization function and the nonlin-ear scalarization function in variable control structure, the concept of nonlinearscalarization function for subsets of ordered topological vector space is presented.Some properties are investigated. The concept of upper semi-continuous-like ofset-valued mappings in topological spaces, which extended the concept of uppersemi-continuous of set-valued mappings, is proposed. The relationships betweenupper semi-continuous and upper semi-continuous-like is discussed.2. The solution existence of the vector equilibrium problem in topologicalvector spaces is studied. An Iterative algorithm for the vector equilibrium prob- lem in European spaces, which is a vector version of Alfredo’s projection iterativealgorithm for the scalar equilibrium problem, is established. By using the non-linear scalarization method, the convergence of the algorithm is proved. As anextension of Tada’s algorithm for the scalar equilibrium problem, two algorithmsfor vector equilibrium problems in Hilbert spaces are proposed. Using the non-linear scalarization method, it is proved that one of the algorithms convergesstrongly and the other converges weakly.3. As an extension for the well-known Ekland’s variational principle, the gen-eralized Ekland’s variational principle of set-valued mappings which includes thevector Ekland’s variational principle is gained. System of vector equilibrium prob-lems in metric spaces are studied. By using the generalized Ekland’s variationalprinciple of set-valued mappings, some new conclusions of solution existence forthe system of vector equilibrium problems are obtained. The conclusions are onlyrequired the functions in the system of vector equilibrium problems satisfying acertain continuity, without the domain satisfies any convexity, nor does it requirethe functions satisfies the monotonicity and generalized convexity conditions.4. A hybrid iterative scheme for finding the common elements of the fixedpoints set for an infinite family of nonexpansive mappings and the set of solutionsfor a system of mixed equilibrium problems, which is an extension for Moudafi’sviscosity iterative method and explicit viscosity iterative scheme, is introduced.As corollaries, the algorithms for solving system of equilibrium problems andequilibrium problems are obtained.5. The generalized vector equilibrium problems for set-valued mappings inmetric spaces are studied. By using the generalized Ekland’s variational princi-ple of set-valued mappings, some existence theorems are obtained. An iterativeprojection algorithm for solving the generalized vector equilibrium problem inEuropean spaces with variable control structure is presented. This algorithmis a generalization of Alfredo’s projection algorithm for the scalar equilibriumproblem.6. As an extension for equilibrium problems, vector equilibrium problemsand generalized vector equilibrium problems in metric spaces, a kind of gener- alized equilibrium problem in a G-convex space is introduced. By means of thefixed-point theorems, some existence theorems with upper semi-continuous-likeset-valued mappings are obtained. As corollaries, some new existence results forequilibrium problems, vector equilibrium problems and generalized vector quasi-equilibrium problems are obtained.