Iterative Algorithms For Fixed Points Of Several Classes Of Generalized Nonexpansive Mappings In Banach Space

In this thesis,we mainly study several kinds of generalized nonexpansive mappings and prove the existence of fixed points and convergence of iteration sequences for these nonexpansive mappings on compact convex sets.We also study the existence,unique-ness and convergence of solutions for a class of variational inequalities in Banach spaces on fixed point sets of strong T-(D_a,1⁺)mappings.This thesis consists of seven chapters.In Chapter 1,we introduce the research background and current situation of fixed point problem,and expound the main motivation of our topic selection and the main work of this thesis.In Chapter 2,we introduce some basic definitions and concepts that need to be used in this thesis.In Chapter 3,we introduce a new class of generalized nonexpansive mappings T-?D_a?,expound the difference between T-?D_a?mappings and Suzuki's generalized nonexpansive mappings,and introduce the basic properties of T-?D_a?mappings.We prove the existence and convergence of fixed points of T-?D_a?mappings on compact convex sets of Banach space by means of quadratic iteration method.In Chapter 4,we optimize the iteration algorithm for fixed point of generalized nonexpansive mappings T-?D_a?,improve the proof method in Chapter 3,and obtain more general results.In Chapter 5,we generalize the T-?D_a?mappings,introduce a new kind of generalized nonexpansive mappings T-?D_a⁺?,point out the difference between the T-?D_a⁺?mappings and the T-?D_a?mappings,and introduce the basic properties of the T-?D_a⁺?mappings in Banach space.We prove the existence and convergence of the T-?D_a⁺?mappings on compact convex sets.In Chapter 6,we study a class of variational inequalities on compact convex sets and approximate solutions on fixed point sets of strong T-(D_a,1⁺)mappings in Banach spaces.Many scholars discuss these problems in uniformly convex Banach spaces or Hilbert spaces.Similar to the definition of positive bounded linear operators in Hilbert spaces,we give the concept of positive bounded linear operators in Banach spaces.We introduce an iterative algorithm on compact convex sets and give a strong convergence theorem for solutions of variational inequalities on fixed point sets of strong T-(D_a,1⁺)mappings.Our results are obtained in general Banach spaces with less space requirements.In particular,we use a special proof method in the proof process,which can be used for reference in the future study of some problems in Banach space.In Chapter 7,we summarize the whole paper and elaborate on the follow-up work we intend to carry out.