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Applied Mathematics and Mechanics

, Volume 24, Issue 12, pp 1421–1430 | Cite as

Interative approximation of fixed points for almost asymptotically nonexpansive type mappings in Banach spaces

  • Zeng Lu-chuan
Article

Abstract

A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced, which includes a number of known classes of nonlinear Lipschitzian mappings and non-Lipschitzian mappings in Banach spaces as special cases; for example, the known classes of nonexpansive mappings, asymptotically nonexpansive mappings and asymptotically nonexpansive type mappings. The convergence problem of modified Ishikawa iterative sequences with errors for approximating fixed points of almost asymptotically nonexpansive type mappings is considered. Not only S. S. Chang's inequality but also H. K. Xu's one for the norms of Banach spaces are applied to make the error estimate between the exact fixed point and the approximate one. Moreover, Zhang Shi-sheng's method (Applied Mathematics and Mechanics (English Edition), 2001,22(1):25–34) for making the convergence analysis of modified Ishikawa iterative sequences with errors is extended to the case of almost asymptotically nonexpansive type mappings. The new convergence criteria of modified Ishikawa iterative sequences with errors for finding fixed points of almost asymptotically nonexpansive type mappings in uniformly convex Banach spaces are presented. Also, the new convergence criteria of modified Mann iterative sequences with errors for this class of mappings are immediately obtained from these criteria. The above results unify, improve and generalize Zhang Shi-sheng's ones on approximating fixed points of asymptotically nonexpansive type mappings by the modified Ishikawa and Mann iterative sequences with errors.

Key words

almost asymptotically nonexpansive type mapping fixed point modified Ishikawa iterative sequence with error modified Mann iterative sequence with error 

Chinese Library Classification number

O177.91 

2000 Mathematics Subject Classification

47H09 47H10 47H17 

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Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 2003

Authors and Affiliations

  • Zeng Lu-chuan
    • 1
  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiP.R. China

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