Researches On Fixed Point Theorems And The Stability Of Operators Iterations In Generalized Metric Spaces

The fixed point theory and nonlinear operators theory is an important part of non-linear functional analysis, which is applied to many fields such as differential equations, integral equations, control theory, optimization theory, operator spectral theory, mathe-matical programming, equilibrium problems of economic and traffic, modern mechan-ics and nonlinear evolution equations and so on. Especially, the application of fixed point theory in modern mechanics and nonlinear evolution equations is more extensive than in any other areas. And the research of fixed point theory has become a hot issue in recent years.Recently, the study of fixed point theory of mappings satisfying certain contractive conditions has been researched extensively by many mathematicians since fixed point theory plays a major role in mathematics and applied sciences. In2006, Mustafa and Sims introduced a new generalized metric space, called G-metric space, as the exten-sion of metric space. People have obtained a lot of new important results of fixed point theorems and common fixed point theorems satisfying the different contractive condi-tions in G-metric spaces. On the other hand, some authors get a lot of initial results when discussing the problems of coupled fixed point theorems and coupled common fixed point theorems in G-metric spaces. Inspired by these new results, we did some deep researching on the problems of fixed point and coupled fixed point on the basis of the existed conclusions. The article has been divided into six parts as follow:The first part is the preface, which introduces the background of researching and the current status of the fixed point in metric space and G-metric space.The second part studies the common fixed point theorems using the condition of R-weakly commuting of type (Ag) and incompatible mappings in G-metric spaces while we delete the completeness and the continuity of mappings which are required for proving some common fixed point theorems in G-metric spaces by other authors. Therefore, the new obtained results developed and improved the fixed point theorems in a large extent.The third part introduced some power type contractive mappings includes the sec-ond power type contractive mappings, the third power type contractive mappings and the forth power type contractive mappings in generalized metric spaces, and we proved some fixed point theorems and common fixed point theorems which are proved by some real examples. To be mentioned, these types of power contractive mappings are firstly presented and researched under the framework of G-metric spaces, so the obtained results are neoteric.The forth part introduced a new concept of weakly commuting mappings called φ-weakly commuting mappings, by which we give some new common fixed point the-orems of Altman integral mappings in G-metric spaces, and we also give some exam-ples to support our results. Specially, the concept of φ-weakly commuting mappings is first proposed in G-metric spaces while it is the first time that the Altman integral mappings in G-metric space was put forward. Therefore, the results obtained are the further development and improvement of fixed point theory in G-metric spaces.In the fifth part, we talked about the coupled common fixed point of commuting mappings of another type of contraction condition, taking advantage of which we prove a new coupled common fixed point theorem and we also give an example to support our theorem in G-metric space. The obtained results improve and extend the relative results of which Shatanawi put forward in2011.In the sixth part, we mainly prove some new theorems of the stability of operators iterations, which enrich and develop the theory of fixed point in G-metric space.In short, this article obtained some new fixed point theorems, common fixed point theorems and convergence theorems of iteration by introducing new classes of map-pings, constructing new iterative schemes, using new iterative techniques. And all the types of contractive mappings and the new relative concepts are the first proposed and discussed in G-metric spaces. These results improved, extended, complemented and completed related results, furthermore, enriched and developed the fixed point theory and the theory of nonlinear operators iterations, containing superior theoretical signifi-cance and strong application value.