Fixed Point Theorems And Common Fixed Point Theorems For Integral Type Contractive Mappings In Metric Spaces And Symmetric Spaces And Applications

The Banach contraction principle has significant research value in fixed point theory,which has been extended and applied to different directions by many scholars,such as references [1-42].Among them,the w-distance idea proposed by Kada et al.in 1996 and the symmetric metric space given by Hicks and Rhoades in 1999 have aroused the interest of lots of scholars.The basic content of this paper is mainly inspired by Kada,Hicks,Rhoades and Branciari,and construct several new fixed point theorems and common fixed point theorems for integral contractive mappings by changing the form of contraction and the term in the maximum value at the right end of contraction inequality.This paper consists of five modules.The first module includes introduction and preliminary.The introduction introduces the development of symmetric space,w-distance and integral types contraction mappings,and some important results obtained by other scholars;the preparatory knowledge recommends the symbols,lemmas,axioms and definitions needed in this paper.The second and third modules are the center of this paper.The contents of the seven new theorems and their proof process are given.In the second module,inspired by the results of Kada,Branciari and others,this part obtain four fixed point theorems for contractive mappings of integral types under w-distance in complete metric spaces by changing the contractive form in fixed point theorem or the maximum value of the right end of contraction inequalities,and prove the existence and uniqueness of fixed points.The third module is based on the results of Hicks,Rhoades and Branciari,by changing the contraction form to get three common fixed point theorems of integral contraction mappings in symmetric spaces,and prove the existence and uniqueness of common fixed points.The fourth module constructs five examples.Example 4.1 illustrates that theorem 2.1 is different from Liu et al’s theorems.Example 4.2 shows that theorem 2.4 really extends Branciari’s theorem and is different from Liu et al’s theorems.Examples 4.3-4.5 show that theorems 3.1-3.3 are different from those of Aamri,Moutauakil and Aliouche.In the fifth module,the applications of the fixed point theorem and common fixed point theorem of some integral type contractive mappings in metric spaces and symmetric spaces are given,and the existence and uniqueness of bounded solutions of integral equations and functional equations are explored.