Research On Some Fixed Point Properties Of B-metric Spaces

Fixed point theory has a long history and is full of vitality.Because of its high unity and wide application,it has been favored by many scholars at home and abroad,and has become one of the most active research fields in mathematical analysis.The study of fixed point problem has been from the extension of spaces to the change of mapping types and iteration methods to find the fixed points,which makes the content of the fixed point theory more and more rich.This thesis is devoted to to study the fixed point problem in b-metric spaces and rectangular b-metric spaces.Firstly,the research results of domestic and foreign scholars in the field of fixed point theory in recent years and the development status of fixed point theory are introduced,which provide the background knowledge and the research direction for the research work in this thesis.Secondly,the convex b-metric space with the linear characteristic is defined by introducing the convex structure in a b-metric space,and the Mann iteration scheme is given in the above space.Furthermore,by using the sequence generated by Mann iteration,it is proved that the existence and the uniqueness of the fixed points for some contractive type mappings in the complete convex b-metric space under certain conditions.Then,the concept of weak stability in convex b-metric spaces is defined.Moreover,it is proved that Mann iterative sequences are weakly T-stable.What is more,Agrawal iteration scheme is generalized to the convex b-metric spaces.The existence and the strong convergence theorems of fixed points for non-expansive set-valued mappings by virtue of Agrawal iterative sequence are given.The stability of Agrawal iterative sequence with respect to the mapping T is discussed.Besides,the above results are applied to the existence and uniqueness of the solutions for Fredholm linear integral equations.Last but not least,some of the above results are extended to rectangular b-metric spaces,and the fixed point theorems of rectangular convex metric spaces are given by using Mann iterative algorithm.It is proved that the fixed point problems of some contractive mappings are well-posedness under certain conditions.Furthermore,the results are applied to dynamic programming problems.