Fixed Point Properties For Generalized Nonexpansive Type Maps In Geodesic Metric Spaces

The fixed point theory is an important part of the nonlinear functional analysis that is developing rapidly.Especially,it plays a key role in solving the existence problem of different kinds of equations.The fixed point theory dated from the early twentieth century,when Brouwer and Banach proposed “Brouwer Fixed Point Theorem” and“Banach Contraction Mapping Principle”respectively.After that,researchers all over the world devoted themselves to the study of the fixed point theory,which makes it become a significant branch of mathematics.Traditionally,the fixed point theory largely used the Banach space theory and the topological degree theory to study the fixed point properties.In recent decades,the study has been extended to a variety of metric spaces,such as generalized metric spaces,probabilistic metric spaces etc.Geodesic metric space is a spatial framework combined of differential geometric,Banach space properties and metric space properties,which mainly includes CAT(0)spaces(the letter C,A,T stand for Cartan,Alexandrov and Toponogov),W-hyperbolic spaces and Busemann spaces.However,compared with the rich content of the fixed point theory in Banach spaces,the study on the fixed point properties in geodesic metric spaces is still in its infancy and problems waiting for the in-depth discussion abundantly exist.The fixed point theory in geodesic metric space has important applications both in solving variational inequalities and computer graph theory,so it is of great theoretical and actual value to study the fixed point properties of nonlinear operators in geodetic metric space.The dissertation focuses on the study of the fixed point problems of several kinds of generalized nonexpansive type mappings in geodesic metric spaces.The main content is as follows.Firstly,to study the fixed point properties for mean nonexpansive mappings in CAT(0)spaces.Existence theorem,convergence theorems and the demiclosedness principle of fixed points are proved for mean nonexpansive single-valued mappings defined on a bounded closed convex subset in CAT(0)spaces.Furthermore,the criterion of stationary points is discussed for mean nonexpansive set-valued mappings defined on a bounded closed convex subset in CAT(0)spaces.Secondly,to study the fixed point properties for C-type set-valued mappings in geodesic metric spaces.Existence theorem of common fixed points is proved for a pair of commuting single-valued and set-valued mappings satisfying the condition(C),which are defined on a bounded closed convex subset in CAT(0)spaces.Then,two convergence theorems are discussed for set-valued mappings satisfying the condition(C)in CAT(0)spaces and W-hyperbolic spaces.Thirdly,to study the common fixed point properties for pairwise mappings in CAT(0)spaces.Two new types of generalized nonexpansive pairwise mappings are defined in metric spaces,which are called pairwise mappings satisfying the condition(PC?)and(PE?)respectively.Examples are shown to indicate that they are more general than the nonexpansive mappings.The equivalent condition of the existence of common fixed points is given for pairwise mappings satisfying condition(PC?).Besides,the demiclosedness principle is studied for pairwise mappings satisfying condition(PE?).Furthermore,by use of the S-iteration,the asymptotic behavior is studied for pairwise mappings satisfying condition(PC?)in CAT(0)spaces.Finally,to study the fixed point properties for L-type mappings in CAT(0)spaces.The relationship between L-type mappings and other generalized nonexpansive mappings is discussed in CAT(0)spaces.Existence theorem of fixed points is proved for L-type single-valued mappings.Furthermore,existence theorems of common fixed points are also discussed.Then,approximation theorems of a three-step iteration scheme are shown for mappings satisfying condition(L).