On The Fixed Point Theorems Of Caristi’s Type

The research of fixed point theory in nonlinear functional analysis has caused a lot of people’s interest. And people have made great achievements, including the Caristi’s fixed point theorem and its related Ekeland’s variational principle. These two theorems provide the effective tools to solve the nonlinear problems, and they are also the important achievements in nonlinear functional analysis.In this paper, we further study the fixed point theory, on the basis of previous results with weakened and improved conditions of Caristi fixed point theorem and Ekeland variational principle. Through some nonlinear functional analysis methods, we get some promotions of the fixed point theorems of Caristi’s type and variational principle of Ekeland’s type.Firstly, the developments and present situations of the fixed point theorems of Caristi’s type and variational principle of Ekeland’s type are summarized.Secondly, we give the fixed point theorems of Caristi’s type which extend the previous results under weakened and improved conditions of its related theorems, according to the literatures due to Amini-Harandi, Khamsi and Suzuki.Finally, we generalize the variational principle of Ekeland’s type, according to the extended fixed point theorem of Caristi’s type, under the weakened conditions and strengthen conclusions of the Ekeland’s variational principle.