Research On The Properties Of CAT(0) Space And The Zero Point Approximation Of Monotone Operators

The fixed point theory mainly studies the existence and approximation algorithm of fixed points of operators in metric space.It influenced the whole field of mathematics for more than half a century in the last century,and Banach fixed point theory and its application became the focus of research.In recent years,the study of fixed point theory in metric space has been extended to CAT(0)space.Therefore,this paper mainly studies the fixed point problems of nonexpansive mappings in CAT(0)space,which is divided into the following three parts:Firstly,this paper expounds the background and research status of the fixed point theory in CAT(0)space,including its origin and properties,and clarifies the research direction of this paper.Secondly,according to the properties that the subdifferential of proper,lower semicontinuous and convex function in CAT(0)space is monotone operator and the resolvents of the monotone operator are nonexpansive mappings,on the basis of Manntype and Halpern-type iterative algorithms,the Mann-Halpern-type and Halpern-Manntype approximation algorithms for a zero point of the monotone operator are established,and it is proved that the two iterative sequences have strong convergence and ?-convergence to the zero point of the monotone operator respectively.Besides,in CAT(0)space,variational inequality problem,nonexpansive mapping and Mann and Ishikawa iterative algorithm are combined,then the inexact approximation algorithm is established,and it is proved that the sequences of the two algorithms are ?-convergent to the fixed point of the nonexpansive mapping.Finally,the more general metric space-b-metric-like space is discussed,and several mappings are defined,and the existence and uniqueness of fixed points are also proved.In addition,in order to show the rationality of the defined mapping,the calculation process of the related parameters of the mapping is also given.