The Research Of The Common Fixed Point For Some Maps In The G-metric Space

In recent years, the research about the common fixed point theory and application is more and more widely attentioned, and achieved many important research results. It is applied to many important fields such as differential equations, integral equations, variational inequalitoes, optimization theorey, applied mechanics and operations research and and so on. So the research of the common fixed point make it more theoretical significance and application value in the nonlinear analysis.In2006, Mustafa and Sims introduced a new notion of generalized metric spaces, which are called G-metric spaces, and deeply discussed the topological properties in this space. Then after, many authors obtained many fixed point and common fixed point theorems in G-metric spaces, and give some application in integral equation. Later, many authors have also been researched coupled coincidence and coupled fixed point theorems in partially ordered G-metric spaces, and achieved many significant research results. Inspired by these results, this article studied three parts.The first part is the preface, which introduces research background and current status of the fixed point in G-metric space and the partially ordered G-metric space.The second part.(1) By using the concept of weakly commuting mappings and weakly compatible mappings to discuss the existence and uniqueness problems for six self-mappings in G-metric space. Some new common fixed point theorems are obtained. An example is provided to support our results. The result only need one is continuous, a pair is weakly commuting, two pairs are weakly compatible for six self-mappings. The result extend and improve the corresponding results of Mustafa,Obiedat and Awawdeh(2008),Mustafa and Sims (2009),Abbas and Rhoades(2009), Mustafa, Shatanawi and Bataineh (2009), Abbas, Nazir and Saadati(2011), Abbas、Nazir and Vetro (2011).(2) By using the Ag type R-weak commutativity and noncompatible conditions of self-mapping pairs, we establishes some new common fixed point theorem with φ-contractive condition in the framework of G-metric spaces. Our results not demanding for completeness of spaces and continuity of mappings. The results obtained in this paper largely improves and extends some relative results.(3) We use the concept of weakly commuting mappings and weakly compatible mappings to discuss a new common fixed point problem for the six self-mappings with third power type in complete G-metric spaces.The third part discussed the coupled common fixed points problem for a pair of commuting mappings in partially ordered G-metric space. An example is given to support the usability of our results. The result generalize and improve the results of Nashine and Shatanawi (2011) from partially ordered metric spaceds to a partially ordered G-metric space.In short, this article obtained some new fixed point theorems and common fixed point theorems by constructing a new class of mappings, introducing a new definition and weakening the condition of the space and the contrative mappings in the literature. These results improved, extended and generationed the fixed point theorey results in the metric place, broaderd the range of applications, and make it have more theoretical significance and application value.