Fixed Point Theorems For Some Contractive Mappings Of Integral Type With W-Distance And Their Applications In Functional Equations

In 2012,Branciari proved the fixed point theorem for the contractive mapping of integral type,which is one of the most important achievements of the fixed point theory.In recent years,scholars have extended it to G-metric spaces,2-metric spaces and other fields.In 1996,Kada et al.introduced the concept of w-distance,which has attracted extensive attention of scholars.This proposal drives a brand new direction for the study of fixed point theorems.This paper studies the contractive mappings of integral type and proves several fixed point theorems of nonlinear single-valued contractive mappings and multi-valued contractive mappings of integral type by combining the ideas of w-distance.This paper consists of four parts.The first part is composed of the research status and fundamental knowledge.The foreign and domestic research status introduces the development status of w-distance and contractive mappings of integral type and the works of other authors related to this article and obtained some important conclusions.The basic knowledge mainly introduces the concepts,symbols and lemmas contained in this paper.The second part is the main part of this paper,ten theorems are demonstrated and some proof processes are provided.The proof processes of similar theorems are omitted.Firstly,on the basis of Branciari's contraction mapping inequality of integral type,five forms of contractive mappings are obtained and the fixed point theorems of five different nonlinear contractive mappings of integral type are proved by changing the upper limit function in the integral at the right of the inequality and changing the coefficient of the integral from a constant to a function.Secondly,inspired by Kutbi and Sintunavarat's ideas,the contractive mapping is extended to the integral type and the terms in the maximum at the right of the contractive inequality are added.Five fixed point theorems of multi-valued contractive mappings are obtained in complete metric spaces and ?-complete metric spaces and the existence of fixed points are proved.The third part is composed of the examples and applications.The third chapter constructs five illustrative examples.Example 2.1 illustrates that Theorem 2.1 generalizes indeed Lakzian et al.'s and Dutta and Choudhury's theorems.Examples 3.2-3.4 illustrates that Theorems 2.7-2.9 is different from Lakzian et al.'s and Dutta and Choudhury's theorems,respectively.Example 3.5 illustrates that Theorem 2.10 is a proper extension of Kutbi and Sintunavarat's theorem.In the fourth chapter,several applications of fixed point theorems ofnonlinear integral type in functional equations are given.And this article explores the applications of functional equations arising in dynamic programming with the help of fixed point theorems.It also illuminates the existence and uniqueness of solution to the kind of equation.The fourth part mainly includes the references involved in this paper,papers published during postgraduate study and acknowledgements.