The Fixed Point Problems In Hilbert Space

Fixed point problem is an important part of the nonlinear functional analysis theory. From Banach proposed and proved Banach contraction principle in1920’s, fixed point problem caused widespread concern. A series of improved or new con-cepts of contractive mappings and fixed point theorems about them were proposed, and some results have been applied to financial mathematics and random operator theory, and so on. Recently, it becomes a main study direction at equilibrium prob-lems and iterative approximation of fixed point problems. At the same time, many kinds of new iteration schemes also have been concerned by many scholars.The fixed point problems in Hilbert space are discussed in this paper, the paper is contained with four parts as following:Firstly, background, main contents that we will discuss and significance are introduced.Secondly, a new Ishikawa iteration scheme is proposed for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of two infinitely nonexpansive mappings in Hilbert space. And a weak convergence theorem is established for this iteration. The results extend and improve the corre-sponding ones.Thirdly, iteration algorithms are introduced for Lipschitzian asymptotically pseudo-contractive mappings and semigroups in Hilbert space. Some strong con-vergence theorems are proved. Our results improve and extend some results in the aspects of mappings and semigroups. Finally, a concept of u-distance is introduced. A fixed point theorem and its corollary are proved for multivalued map by using u-distance. At the same time, four different types of condition are gotten. Any one of these conditions are satisfied, this theorem and corollary can be obtained. Our results improve and extend some important results.