Common Fixed Point Theorems For Nonlinear Contractive Mappings Of Integral In 2-metric Spaces And Applications To Functional Equations

Fixed point theory is one of the most essential topics of nonlinear analysis.As one of the most fundamental and famous results of fixed point theory,the Banach contraction principle offers effective methods of solving the existence,uniqueness,iterative approximation and error estimation of solutions in many fields.Branciari,Nadler,Gahler and Iseki have explored the Banach contraction principle from different aspects,and obtained useful results.With the help of Branciari,Nadler and Iseki’s ideas,the main purpose of this paper is to explore the common fixed point theorems and the common stable point theorems of nonlinear integral contractive mappings in 2-metric spaces and compact 2-metric spaces,respectively.In Chapter 1,the research background is introduced from three aspects: 2-metric spaces,multi-valued contractive mappings and integral contractive mappings,and the various symbols,definitions and lemmas involved in this paper are given.In Chapter 2,we first give two common fixed point theorems for nonlinear single-valued contractive mappings of integral in 2-metric spaces and the existence and uniqueness of the common fixed points.Next adding the compactness condition to 2-metric spaces and changing some conditions of mappings,we obtain the common fixed point theorems for nonlinear single-valued contractive mappings of integral in compact 2-metric spaces and the existence and uniqueness of the common fixed points.In Chapter 3,on the basis of Chapter 2,two of four single-valued contractive mappings are transformed into multi-valued contractive mappings,and their common stable points for nonlinear multi-valued contractive mappings of integral in 2-metric spaces and compact 2-metric spaces and the existence and uniqueness of common stable points are proved.In Chapter 4,three examples are constructed.The spaces,mappings and functions of some theorems in Chapters 2 and 3 are given concretely,which can show the significance and value of the common fixed point theorems and the common stable point theorems in this paper.It is also shown that the common fixed point theorems and the common stable point theorems are the true generalizations of the partial fixed point theorems,common fixed point theorems and common stable point theorems in Chapter 1.The Chapter 5gives the applications of common fixed point theorems of nonlinear single-valued contractive mappings of integral in functional equations.