Minimax Theorems And Variational Inclusions

Minimax theory is an important content of nonlinear analysis. It has been ap-plied in many fields, such as games theory, mathematical economics, optimizationtheory, variational inequality, di?erential equations and fixed point theory, etc.In this thesis, we shall generalize some classical minimax theorems by using fixedpoint theorem, discuss some minimax theorems of two functions defined on FC-spaces and some generalized minimax theorems of set-valued mappings defined onG-convex spaces. Then we shall study the existence of the solution of some gener-alized variational inclusions and fixed point problems of nonexpansive mappingsand pseudocontractive mappings, which have a close relation to minimax theorem.In Chapter 2, we shall extend some classical minimax theorems to the statesof two functions defined on FC-spaces and set-valued mappings defined on G-convex spaces. On the one hand, we prove some collectively fixed point theoremsfor a family of set-valued mappings defined on the product space of noncompactFC-space by applying the existence theorem of maximal elements. The notion ofF-g-quasiconvexity (F-g-quasiconcavity) for two functions is introduced. By usingthe obtained fixed point theorem, we prove some minimax theorems involving twofunctions. On the other hand, we introduce the notions of generalized weak loosesaddle point and generalized loose saddle point for set-valued mappings, applythe fixed point theorem and the sacralization method to establish some existencetheorems in locally G-convex spaces and show some applications of the saddlepoint existence theorem.In Chapter 3, we shall extend the resolvent operators technique, apply it todiscuss the existence of the solution of two classes of generalized nonlinear vari-ational inclusions, and improve and generalize some previous results about theexistence theorems of the solution of generalized nonlinear variational inclusions. Firstly, we introduce the notion of G-η-monotone mapping, study the propertiesof this mapping and the existence of the solution of a class of generalized implicitvariational-like inclusions involving the G-η-monotone mapping, construct somealgorithms to approximate the solution of the generalized implicit variational-likeinclusion and show the strong convergence of the iterative sequences in Hilbertspace. Secondly, a new class of g-η-accretive mapping is introduced in Banachspace. Some properties of this mapping are shown. The resolvent operator asso-ciated with the m-accretive operators is extended. An iterative algorithm is con-structed to approximate the solution of a class of generalized implicit variational-like inclusion involving g-η-accretive mapping and the strong convergence of theiterative sequence is proved in q-uniformly smooth Banach space.In Chapter 4, we shall construct iterative schemes for approximating thefixed point of nonexpansive non-self mapping and pseudocontractive mappings,prove its convergence, and improve and generalize some previous results. Firstly,we construct a Mann's iterative scheme with errors to approximate the commonfixed point of two nonexpansive non-self mappings, and prove that the iterativesequences strongly converge to the common fixed point in uniformly convex Banachspace under weaker conditions. Secondly, we introduce the modified Ishikawaiterative sequence to approximate the common fixed point of a finite family ofLipschitz pseudocontractive self-mappings on a closed convex subset, and provethat the iterative sequence strongly converges to the common fixed point in Hilbertspace.