Research On Fixed Point Properties Of Asymptotically Nonexpansive Mappings And Fixed Points For Strongly Invariant Mappings

The fixed point theory is an important part of nonlinear functional analysis theory, it has closely connected with many branches in modern mathematics. E-specially it plays important roles in solving various kinds of equations questions about the existence and uniqueness of the solutions(including linear or nonlinear, determine or undetermine the types of differential equations and integral equa-tions, and various kinds of operator equations). The main research direction of fixed point theory is to study the existence of fixed point and approximate al-gorithm of operators in metric space. Many scholars have combined fixed point problems of nonlinear operator with many specific practical problems, by using the method of fixed point theory to solve practical problems in mathematics, such as convex feasibility problem, split feasibility problem. Recently, being a general-ization of the split feasibility problem (SFP), Moudaf proposed split common fixed point problem, which promote the development of fixed point theory. In this thesis, we want to study the split common fixed point problem of asymp-totically nonexpansive mappings and firmly nonexpansive mappings in Hilbert spaces, and introduce the concept of multi-valued asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces. We also obtain the existence theorem, demiclosed principle and convergence theorem of multi-valued asymp-totically nonexpansive mappings. The content of this thesis is organized by four sections.In the first section, we make a brief overview of the research background and status of the fixed point theory.In the second section, an iterative algorithm is introduced to solve the split common fixed point problem for asymptotically nonexpansive mappings in Hilbert spaces, and it possesses strong convergence for split common fixed point problem of asymptotically nonexpansive mappings although the mappings do not have semi-compactness.In the third section, we prove the existence of fixed points and demiclosed principle for multi-valued asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces with monotone modulus of uniform convexity. We also obtain a A-convergence theorem for multi-valued asymptotically nonexpansive mapping.In the last section, we construct an iterative scheme which converges strong-ly to a solution of a split equality fixed point problems of firmly nonexpansive mappings in Hilbert spaces.