Fixed Point Theorems For Some Multi-valued Contractive Mappings Of Integral Type And Single-valued Contractive Mappings Of Integral Type With W-distance And Their Applications In Nonlinear Integral Equations

The fixed point theory,which plays a prominent role in the field of mathematics,is also of great significance in the study of modern mathematical theories.Banach contraction principle is one of the most basic and essential results in this theory.In this paper,the main idea is to combine the ideas of Branciari,Kada and Nadler,by changing the conditions satisfied by the integral type contractive mappings with w-distance and trying to extend the integral coefficient to the binary function and expand the term in the maximum value at the right end of the compression inequality,then obtain new fixed point theorems.This paper consists of three parts.The first part encompasses preparatory knowledge and introduction which mainly shows the related achievements of the multi-valued contractive mappings of integral type and the single-valued contractive mappings of integral type with w-distance at home and abroad,the preparatory part mainly introduces some basic concepts,symbol description and lemma that will be used in this paper.The second part is the core of this paper.All fixed point theorems and their partial proof processes to be proved in this paper are given.Some theorems are very similar,so they are omitted.Theorems 2.1-2.4 introduce the concept of w-distance on the basis of contractive mappings of integral type.Four different fixed point theorems of single-valued contractive mappings of integral type are obtained by changing the conditions satisfied by the functions in contractive mappings of integral type with w-distance.Theorems 2.5-2.8 mainly combine Nadler's idea to extend the integral coefficient to the binary function and expand the term in the maximum value of the right end of the compression inequality,then obtain four fixed point theorems of set-valued contractive mappings of integral type.The third part consists of examples and applications.Example 3.1 shows that Theorem2.1 extends indeed Theorems 1.1 and 1.3 and differs from Theorem 1.2.Examples 3.2-3.4show that Theorems 2.2-2.4 are different from Theorem 1.3.Application part focuses on the applications of the fixed point theorems for single-valued contractive mappings of integral type in nonlinear integral equations,which solves the problem of the existence and uniqueness of the solution of nonlinear integral equations.