The Generalized F-projection Operator With Applications

It is well known that metric projection operators in Hilbert and Banach spaces arewidely used in different areas of mathematics such as functional analysis and numericalanalysis, theory of optimization and approximation, and also for the problems of optimalcontrol and operations research, nonlinear and stochastic programming and game theory.Metric projection operators can be defined in similar way in Hilbert and Banachspaces. Meanwhile, they differ significantly in their properties. Metric projection opera-tors in Hilbert spaces are monotone (accretive) and nonexpansive. They yield absolutelybest approximations for arbitrary elements by the points of convex closed sets. This leadsto a variety of applications of such operators to investigation of theoretical and appliedquestions of Numerical and Functional Analysis. Metric projection operators in Banachspaces are unlikely to have all the properties mentioned above and their applicationsare more limited. In order to overcome the difficulty, in 1994, Alber [2] introduced thegeneralized projections from Hilbert spaces to uniformly convex and uniformly smoothBanach spaces and studied their properties in detail. In [3], Alber presented some ap-plications of the generalized projections to approximate solving variational inequalitiesand Von-Neumann intersection problem in Banach spaces. Recently, Li [45] extendedthe definition of the generalized projection operator from uniformly convex and uni-formly smooth Banach spaces to re?exive Banach space and studied some properties ofthe generalized projection operator with applications to solving the variational inequalityin Banach spaces.On the other hand, Since generalized variational inequalities have the nonlinearterm, the above method cannot be applicable. In order to overcome the difficulty, weshall introduce a new concept of generalized f-projection operator and studied its prop- erties in detail. As applications, we shall use it to solve the generalized variationalinequalities in Banach spaces.In Chapter 1,we present the background and some known results on variationalinequalities and projection operators. In Chapter 2, we shall introduce a new concept ofgeneralized f- projection operator and studied its properties in detail such as existence,monotonicity, single-valued, continuity. We also construct the equivalence between pro-jection operator equation and generalized variational inequalities. In Chapter 3, we shallstudy the existence of solutions for generalized variational inequalities. Using the gen-eralized f- projection and KKM technique, we gain the existence results of solutions forgeneralized variational inequalities. In Chapter 4, by employing the notion of general-ized f-projection operator and the well-known KKM Theorem, we establish some exis-tence results for the generalized set-valued variational inequality problem and the gen-eralized set-valued quasivariational inequality problem in re?exive and smooth Banachspaces. In Chapter 5, we shall propose an iterative method of approximating solutionsfor a class of generalized variational inequalities and give a convergence result for theiterative method in uniformly convex and uniformly smooth Banach spaces. In Chapter6, we introduce a new notion of exceptional family of elements for J-completely con-tinuous field, and utilize this notion to study the solvability for a class of generalized f-complementarity problems in Banach spaces. Employing the Leray-Schauder alternativetheorem and the generalized f-projection operator introduced by Wu and Huang, we ob-tain some solvability results for the generalized f-complementarity problems in Banachspaces under suitable conditions. In Chapter 7, Employing the Leray-Schauder degreetheorem and the generalized f-projection operator, we present results for generalizedvariational inequalities without monotonicity or coercivity assumptions. In particular,we gain the existence result of non-zero solution for generalized variational inequalities.