Common Fixed Points For Two Mappings Satisfying Implicit Conditions On 2-metric Spaces

Banach fixed point theorem is the most basic theories in the banach fixed point theory.And it has a wide range of applications in mathematics and other fields.Many scholars have extend and improved the Banach fixed point theorem,Especially they have given some impotant conclusions about fixed points,common fixed points and coincident points on 2-metric spaces.Many typical theorems has been formed through banach fixed point theory devel-oped.for example,banach fixed point theorem,kannan type fixed point theorem,chatterjea type fixed point theorem,integral type fixed point theorem,geraghty type fixed point theorem and some variant or generalized results.In this paper,firstly,we introduce and combine the class of geraghty type functions with the well-know kannan and charrerjea type contraction condition,to discuss and ob-tian the exist theorems of common fixed points for two mappings that satisfy Geraghty-Kannan type or Geraghty-Chatterjea type contraction condition.the above two results are extended results of the Geraghty-Banach fixed point theorem.the next,we introduce some real functions and establish integral type implicit contraction condition to give the unique common fixed points theorem for two mappings with the above contraction conditionThe obtained results of this paper primely generalize and improve some known results,and pointed out how to extend the results on real metric spaces to 2-metric spaces.