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

Fixed point theory and applications is one of the most basic topics in nonlinear analysis,among which one of the most basic and important fixed point theorem is Banach contraction principle.It is known to us all that Banach contraction principle has many generalizations and applications.The essential content of the paper is mainly combined Gahler,Iseki,Branciari with Nadler’s ideas,by changing some conditions of mappings and adding the terms in Maximum at the right end of the inequality and some new fixed point theorems are constructed.The paper is made up of four parts.The first part includes introduction and preliminaries and consists of two chapters.The Chapter 1 is introduction.The introduction chiefly formulated the development of 2-space metric,single-valued contractive mappings and multi-valued contractive mappings of nonlinear integral type,and some important results of the researchers.The Chapter 2 consists of preliminaries.The preliminaries mainly introduce the fundamental knowledge,many symbols,lemmas and definitions involved in the paper.The second part is the core of this paper,formulating sixteen theorems and their proof processes,which includes two chapters.In Chapter 3,with the help of Gahler,Iseki and Branciari’s results,changing or adding the Maximum at the right end of the inequality and changing the conditions of φ and ψ in the contractive mappings,eight fixed point theorems for signle-valued contractive mappings of nonlinear integral type in 2-metric spaces are obtained,and the existence and uniqueness of the fixed point are proved.The partial proof processes of the eight fixed point theorems are given,and the similar proof processes are omitted.In Chapter 4,under the fundamental ideas of Branciari and Nadler’s results,changing or adding the Maximum at the right end of the inequality and changing the conditions of α,β and η in the contractive mappings in complete metric spaces,four fixed point theorems of multi-valued contractive mappings of nonlinear integral type are proposed.What’ s more,the existence of the fixed point is showed.The third part contains the examples and applications,which is made up of two chapters.In Chapter 5,three examples are presented.Example 5.1 illustrates that the Theorems 4.1 and 4.2 extend indeed some important theorems of Nadler,Mizoguchi and Takahashi,Feng and Liu,and Ciric.Examples 5.2 and 5.3 illustrate that the Theorems 4.3 and 4.4 generalize substantially Klim’s and Wardowski’s theorems,respectively.The Chapter 6 includes the application of fixed point theorems of single-valued contractive mappings of partial nonlinear integral type in 2-metric spaces in a functional equation.It solves the existence and uniqueness of the functional equation.The fourth part is composed by the references in this paper,papers published during postgraduate study and acknowledgements.