Common Fixed Point Theorems Of Integral Type Contractive Mappings In Metric Spaces And G-metric Spaces And Their Applications

Fixed point theory and applications is an important part of nonlinear analysis,and is also one of the most active topics in nonlinear analysis.Banach contraction principle as the most basic result of fixed point theory is widely generalized and applied in many domains.The paper is mainly used the thought of which Branciari,Shoaib,Wardowski and Nadler explored and generalized Banach contraction principle,giving several fixed point and common fixed point theorems of integral type and F-contractive mappings in metric spaces and G-metric spaces.The paper is composed of six chapters,which could be separated into four parts.The first part contains introduction and preliminaries.The introduction mainly recommends the works and conclusions of single-valued contractive mappings of integral type in G-metric spaces,single-valued F-contractive mappings and multi-valued contractive mappings in metric spaces which the domestic and foreign researchers carried out.The preliminaries chiefly formulates the definitions,symbols,lemmas and fundamental knowledge employed in the paper.The second part consists of Chapter 3 and Chapter 4,which is the core of this paper.In Chapter 3,on the basis of Branciari,Aydi and Wardowski’s results and with the help of the innovative ideas of Shoaib,adding functions at the both ends of the contractive inequality and expanding the terms of the function coefficient and the upper limit function in the integral at the right end of the contractive inequality simultaneously,four common fixed point and fixed point theorems of single-valued integral type and single-valued F-contractive mappings in G-metric are established.Chapter 4 is inspired by Branciari,Nadler and Liu’s results,four fixed point theorems for multi-valued contractive mappings of integral type in metric spaces are obtained by modifying the function coefficient ? from one-variable functions to two-variable functions and changing the conditions of the functions,and the existence of fixed points are certified.The third part includes the examples and applications.In Chapter 5,three examples are formulated to present how the theorems obtained in the paper extend or differ from the theorems received by predecessors.Example 5.1 shows that Theorems 3.1 and 3.2 generalize substantially Phaneendra and Aydi’s theorems and are different from Shoaib’s theorems.Example 5.2 illustrates that Theorem 3.7 is different from the theorems of Phaneendra.Example 5.3 explains that Theorems 4.1-4.4 extend indeed the results of Nadler,Mizoguchi and Takahaski.The Chapter 6 is applications of the common fixed point theorems for single-valued contractive mappings of integral type in systems of functional equations and fixed point theorems for single-valued F-contractive mappings in an integral equation,respectively.The fourth part contains the references used in the paper,papers published during postgraduate study and acknowledgements.