Study On Fixed Point Theorems And Properties Of Stationary Points Of Nonexpansive Mappings In Banach Spaces

Since the fixed point theory solves a series of application problems such as the existence theorem of implicit function and the existence and uniqueness of the solution of the initial value problem of differential equation,which prompted the mathematicians to carry out in-depth and extensive research.Especially in recent decades,with the continuous development of computers,domestic and foreign scholars introduced various iterative methods to approximate nonlinear mappings and application to solve some practical problems.Consequently,the research for the fixed point problem of expansive type mappings and iterative algorithms is very necessary.In this thesis,we mainly study the fixed point theorems and properties of stability point of nonexpansive mappings in Banach spaces from the following aspects.The whole thesis involves four parts as follows:Firstly,the background of fixed point theory and iterative schemes and its development in Banach spaces are discussed,which provides the correct direction for the research work in this thesis.Secondly,the Nadler theorem and the Lim theorem are generalized to the case of mean nonexpansive set-valued mappings.Meanwhile,by using Banach geometric properties,asymptotic stability point sequences,asymptotic centers,Asymptotic Radius and other properties,the sufficient and necessary conditions for mean nonexpansive set-valued mappings are given.Hence,the results related to nonexpansive set-valued mappings are successfully extended to the case of mean non-expansive set-valued mappings.Thirdly,a new iteration scheme is introduced.By using I condition,semi-compact mappings and Opial properties,the strong convergence and the weak convergence problems of(?,?)-generalized hybrid mappings in above iteration scheme are discussed.Meanwhile,an example of(?,?)-generalized hybridmapping which satisfies the conditions of the convergence theorem but is not a nonexpansive mapping is shown.Lastly,by improving the iterative schemes which have existed,two new iterative schemes are introduced.In these two iterative schemes,the strong convergence and weak convergence theorems of generalized nonexpansive mappings which satisfy C_?condition in uniformly convex Banach spaces are studied.In addition,an example of generalized nonexpansive mapping which satisfies the conditions of above theorem is given,and the convergence rates and stability of two iterations are compared with the existing iterative scheme by using the mapping.