Iterative Approximation Of Fixed Points Of Asymptotically Pseudocontractive Nonself-mappings

From Banach fixed point theorem was proved in 1922 by Banach, the problem of approximation of fixed points of nonlinear mappings and the studying about solution of nonlinear equation become more and more extensive. In a very long time, people have been studying to approximate to fixed points of asymptotically pseudocontractive mappings by using different iterative sequences (Mann and Ishikawa iterative sequence, modified Mann and Ishikawa iterative sequence, etc.) in different spaces. The results have been more mature. However, all the results they discussed require that the mapping T is a self-mapping in the subset of Banach space E.In this paper, we introduce firstly the concept of asymptotically pseudocontractive nonself-mappings and discuss the problem that several typical iteration sequences approximate to fixed points of one, two and a finite number of asymptotically pseudocontractive nonself-mappings and we present the main contents of this article.In the first chapter, we present the introduction of several classes of nonlinear operator and several typical iteration sequences. Furthermore, we present some results about the problem of fixed points of asymptotically pseudocontractive mappings.In the second chapter, we first introduce the definition of asymptotically pseudocontractive nonself-mappings, let K be a retract of real Banach space E with retraction P :E→K. A mapping T :K→E is called asymptotically pseudocontractive if there exists a sequence {k n} ? [1,∞) with k n→1 as n→∞and for every x ,y∈K, there exists j ( x ? y )∈J ( x ? y) such that .Furthermore, we put forward modified Ishikawa iterative sequence with errors for asymptotically pseudocontractive mappings i.e. where {αn },{βn }, {λn },{δn} ? [0,1], sequences {u n },{v n } are bounded in E. In a realBanach space, we prove that the iterative sequence above converges strongly to fixed points of asymptotically pseudocontractive nonself-mappings under some restrictive conditions on the parameters. The result mainly breakthrough the restriction that all the results about asymptotically pseudocontractive mappings require that the mappings is a self-mappings.In the third chapter, we continue to discuss the problem of iterative approximation of common fixed points for two uniformly Li-Lipschitzian asymptotically pseudocontractive nonself-mappings, i=1,2.In the end, we consider to extend the result of chapter two to a finite number of asymptotically pseudocontractive mappings in the fifth chapter. We introduce N steps iterative sequence for finite asymptotically pseudocontractive nonself-mappings. Under some restrictive conditions on the parameters, strong convergence theorem for common fixed points of a finite number of uniformly L-Lipschitzian asymptotically pseudocontractive nonself-mappings is proved.