Research On Ky Fan 's Best Approximation Theory In Banach Spaces

Nonlinear functional analysis is a research field of mathematics with profound the-ories and extensive applications. It constructs many general theories and methods to deal with nonlinear problems on the basis of the study of the nonlinear problems which appeared in mathematics and the natural sciences. Its rich theories and advanced meth-ods are widely used in studies of solving many kinds of nonlinear differential equations, nonlinear integral equations and some other types of equations, and handling many nonlinear problems in computational mathematics, cybernetics, optimized theory, dy-namic system, economical mathematics, etc. At present, nonlinear functional analysis mainly covers topology degree theory, critical point theory, partial order method, anal-ysis method, monotone mapping theory, and so on. In recent years nonlinear problems have received highly attention of the domestic and foreign mathematics and natural science field, so the research on nonlinear functional analysis and its applications is very important in both theory and applications.Ky Fan’s best approximation theory is important subject in the theory of nonlinear functional analysis. Owing to the importance in both theory and applications, Ky Fan’s best approximation theory has attracted many researchers’attention, and a large number of results have been obtained. In recent years, under the impetus of functional analysis and practical problems, the development of the research on Ky Fan’s best approximation theory is rapid.The present paper employs monotone characterizations of the metric projection operator and the generalized metric projection operator. Then we use the nonlinear functional analysis theory and methods, such as cone theory, fixed point theory, geome-try of Banach spaces, lattice theory and monotone iterative technique, to investigate the existence, maximal solution and minimal solution, uniqueness, and iterative approxi-mation results on Ky Fan’s best approximation theory and the variational inequality problems, including coupled best approximation theorem, coupled coincidence best approximation theorem, best proximity point theorem, general variational inequality problem, and fixed point theorem of non-self mappings. Having studied thorough-ly, some new interesting results under weaker conditions have been obtained, most of which have been published in 《Sci. China Math.》 (SCI),《Fixed Point Theory Appl.》 (SCI) and《Abstr. Appl. Anal》, ect.The dissertation is divided into six chapters. In Chapter Ⅰ, the background of non-linear functional analysis and some basic concepts and theorems have been introduced. In Chapter Ⅱ. we research the best approximation theorems for discontinuous map-pings with respect to Lyapunov functional W(x, y) and fixed point theorems of non-self mappings. In 2.2, the monotone characterizations of the generalized metric projection operator are considered and the best approximation theorems for discontinuous map-pings with respect to Lyapunov functional W(x, y) are established. In 2.3, the fixed point theorems of non-self mappings and the general best approximation problems are considered as applications of the best approximation theorems. In Chapter Ⅲ, the existence of solutions for the best approximation theorems and variational inequality problems in Banach spaces have been studied. In 3.2, the monotone characterizations of the metric projection operator and Ky Fan’s best approximation theorems under some new assumptions have been discussed. In 3.3, the existence of solutions for varia-tional inequality problems has been proved, and by using the new boundary condition, the fixed point theorems of non-self mappings are obtained. In Chapter IV, the best approximation theorems and best proximity point theorems for the generalized metric projection operator have been dealt with. In 4.2, the monotone characterizations of the generalized metric projection operator are considered, as applications, the best ap-proximation theorems are established. In 4.3, the existence results for best proximity point have been studied. In Chapter V, the variational inequality problems and best proximity point theorems are discussed. In 5.2, the monotone characterizations of the metric projection operator under new conditions have been studied, as applications, the variational inequality problems are considered. In 5.3, the best proximity point problems and fixed point theory of non-self mappings have been researched. In Chap-ter VI, the coupled best approximation and coupled coincidence best approximation problems have been focused on. In 6.2. the existence for the coupled best approxima-tion point has been discussed. In 6.3, by using the monotone characterizations of the metric projection operator and generalized metric projection operator, the existence for coupled coincidence best approximation point has been investigated.