On The Fixed Point Theorems Of Caristi's Type About ? Function

The fixed point theorem is an important part in the area of nonlinear functional analysis,a lot of problems in math,physics and programme can be transformated into fixed point problems,which is used widely.In recent years,there are more and more researches about the studying of fixed point theorem,including the famous Caristi's fixed point theorem and Ekeland's variational principle which is equivalent to Caristi's fixed point theorem.In this thesis,firstly,we briefly discuss the relationship between generalized distance and other functions,we also summarize the developments and present situations of Caristi's fixed point theorem of ? function's type and Ekeland's variational principle which is equivalent to Caristi's fixed point theorem of ? function's type.Secondly,we use some tools such as partially ordered set and minimal element to achieve the Caristi's fixed point theorem of ? function's type based on other scholars' methods and results.Next we prove that the combination of y and d is also a metric function.At last,because of the equivalence of Caristi's fixed point theorem and Ekeland's variational principle,we generalize the Ekeland's variational principle according to the extended Caristi's fixed point theorem of ? function's type under the weakened conditions and strengthen conclusions of the Ekeland's variational principle.