The Study Of Equilibrium Problems And Set Optimization Problems

Both the equilibrium theory and the set optimization theory are the impor-tant contents in nonlinear analysis and provide important tools for the develop-ment of many theories. The former has been widely used in mathematical physics,economics, transportation, network system, engineering, mathematics and otherdisciplines, and the latter has been widely used in economics, management sci-ence, engineering design, optimal control, diferential inclusion, etc. The generalabstract equilibrium problem contains optimization problems, Nash equilibriumproblems, complementarity problems, fxed-point problems, variational inequalityproblems, saddle point problems and some vector minimization problems as itsspecial cases, and so the equilibrium theory is closely related to fxed-point the-ory, game theory, variational inequality, optimization, diferential equation, etc.The vector optimization, whose criterion is equivalent to fnding efcient pointsof the image set, has been deeply discussed and widely used. But this criterionis not always suitable for all types of optimization problems, for example, sometypes of optimization problems for set-valued mappings. Thus the set optimizationcriterion, which is to seek the efcient sets of the value set, has been introduced.In this thesis, by using scalarization method and variational principle, theexistence of solutions of generalized vector equilibrium problems is proved, re-spectively. The iterative schemes for a system of equilibrium problems (and asystem of fxed-point problems) are constructed by using auxiliary techniques,and their strong convergence is testifed. Finally, the well-posedness for (systemsof)(parametrically) generalized vector quasi-equilibrium problems and for l-setoptimization problems is investigated. The full thesis is divided into six chapters and the frst chapter is introduction.Chapter1mainly introduces the research backgrounds and progress of equilibriumproblems and of set optimization problems. The research progress of equilibriumproblems is separated into three aspects: existence of solutions, iterative schemeand well-posedness, and that of set optimization problems includes two aspects:existence of solutions and well-posedness.Chapter2mainly provides preliminaries for Chapters3-6. First, some knownconceptions and conclusions are recalled. Second, three types of scalarizationfunctions for set-valued mappings, namely, l-, u-and p-scalarization functions, aredefned by set relations and generalized Gerstewizt functions, and their propertiesare discussed. Finally, the variational principles for generalized vector equilibriumproblems are established by applying p-scalarization functions.Chapter3mainly investigates the existence of solutions of generalized vectorequilibrium problems. First, the existence of solutions of a class of generalizedvector equilibrium problems is shown by applying the variational principles estab-lished in Chapter2; Second, the existence of solutions of two classes of generalizedvector quasi-equilibrium problems is proved by using scalarization method; Fi-nally, a result of existence of solutions of a variational inclusion problem is givenas an application. Some previous results are improved and generalized.The auxiliary technique is one of the important techniques of iteration con-struction for variational inequalities. In chapter4, we apply this technique to con-struct the iterative schemes for a system of equilibrium problems (and a system offxed-point problems). First, an iteration for a system of mixed equilibrium-likeproblems is constructed and the strong convergence of the constructed iteration istestifed in real refexive Banach spaces. Second, an iteration for the common so- lutions of a system of equilibrium problems and a system of fxed-point problemsis constructed by hybrid method, and its strong convergence is proved. At thesame time, the framework of real uniformly smooth and uniformly convex Banachspaces involving in most literatures is generalized to real uniformly smooth andstrictly convex Banach spaces with Kadec-Klee property.In Chapter5, the notions of generalized Tykhonov well-posedness for gen-eralized vector quasi-equilibrium problems, generalized Hadamard well-posednessfor parametrically generalized vector quasi-equilibrium problems and generalizedTykhonov well-posedness for systems of parametrically generalized vector quasi-equilibrium problems are introduced. Their sufcient criteria and/or their metriccharacterizations are provided by using continuity of set-valued mappings. Thenthe relation between the generalized Tykhonov well-posedness for generalized vec-tor quasi-equilibrium problems and that for constrained minimization problems isstudied by using the gap functions of generalized vector quasi-equilibrium prob-lems. Finally, the relations among these types of well-posedness defned in thischapter are discussed by fgures and counter-examples. Some conclusions on thewell-posedness for vector equilibrium problems with single-valued objective map-pings are generalized.In Chapter6, the conceptions of well-posedness for l-set optimization prob-lems are introduced. The metric characterizations and the (necessary and) suf-fcient criteria of the introduced well-posedness are discussed. The equivalentrelations between the well-posedness for l-set optimization problems and that forminimization problems are established by applying the gap functions of l-set opti-mization problems. Finally, the well-posedness for l-set convex optimization prob-lems is studied further by discussing the convexity and the lower semi-continuity of the gap functions of l-set convex optimization problems.