Iterative Approximation And Stability Analysis Of Fixed Points For Contractive Type Mappings

The fixed point theory of nonlinear operators is one of the important topics in non-linear analysis,and is an important part of functional analysis theory.It has wide appli-cations in sloving of differential and integral equations,the analysis of some optimization algrothims,variation theory and so on.In this paper,we mainly study the existence of fixed points for some contractive operators and an application of iterative algorithm to the integral equation.The text is divided into four chapters:Chapter 1 introduces the background and the development of the fixed point the-ory,briefly describes the main work of this paper,and gives some basic definitions and conclusions that will be needed for this article.In Chapter 2,we study the existence of fixed points for a class of generalized con-tractive operators and Banach operator pairs in generalized convex metric spaces.Under some conditions,we give the uniqueness theorem of fixed points.An example is also given to verify the sufficiency of the conditions.Moreover,the application of the conclusion in solving the second kind of Volterra integral equation is also considered.In Chapter 3,we discuss the formula of error estimations and approximation theorems of fixed points for strongly demicontractive(SDC)operators in Hilbert spaces.Under a new lemma and the Lipschitz condition,the formula of error estimations and convergence theorems of Ishikawa iteration algorithms for strongly demicontractive(SDC)operators are given(without the asymptotically regular condition).At the same time,an example is also given to compare with the Mann iterative algorithm,it show that our algorithm converges faster than the Mann iterative algorithm.In Chapter 4,on the basis of Chapter 3,we analyze T-stability of the Ishikawa iter-ative algorithms for strongly demicontractive operators.We first extend the T-stability of the Mann iterative algorithm to the Ishikawa iterative algorithms.Secondly,the weakω2-stability theorems and almost stability theorems of the Ishikawa iterative algorithm are given by the convergence theorem in Chapter 3.Finally,we give a strongly demicon-tractive(SDC)operators to show that the Ishikawa iterative algorithm is weak ω2-stable but not T-stable.In Chapter 5,we summarizes the whole text and puts forward the prospect,assump-tion and thinking.