Research On Some Fixed Point Problems In Nonlinear Analysis

It is well known that the Banach's contraction mapping principle is an important theorem in nonlinear analysis.Meanwhile,fixed point theorems are widely used in all aspects of mathematics.In this thesis,some fixed point problems in nonlinear analysis are studied.The thesis is divided into four chapters.In Chapter 1,we mainly describe the historical background of fixed pointtheory in metric spaces,b-metric spaces and fuzzy metric spaces.In Chapter 2,we construct a new class of contractive mappings in fuzzy metric spaces,and prove the existence of fixed points for these new contractive mappings in such spaces.Furthermore,the uniqueness of the fixed points is discussed.At the end of this chapter,an appropriate example is given to illustrate the main result.In Chapter 3,by introducing different new contractive mappings in metric spaces and imposing proper constraints,we prove that the fixed points for these mappings are existent and unique.In addition,by using these new theorems,the existence and uniqueness of solutions for a class of integral equations are studied.In Chapter 4,we obtain new fixed point theorems by using different contractive conditions,and prove that under the hypotheses of the theorems,the mapping T has a unique fixed point.At the end of this chapter,some examples are given to support the main results.The summary and prospect are given in the last part of this thesis.