Direct Estimate of Accumulated Errors for a General Iteration Method

Abstract

We study error analysis for an SP iteration mehod with mixed errors defined by Chugh and Kumar in [Convergence of SP iterative scheme with mixed errors for accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach spaces, Int. J. Comput. Math. 90 (2013), pp. 1865-1880]; in particular, controllability and accumulation of errors for such an iteration method are investigated.

Keywords

• SP iteration method,SP iteration method with mixed errors

References

1. W. Phuengrattana and S. Suantai, \textit{On the rate of convergence of Mann, Ishikawa, Noor and SP iterations for continuous functions on an arbitrary interval}, J. Comput. Appl. Math., 2011 (\textbf{235}): 3006-3014.
2. E. Picard, \textit{M\'{e}moire sur la th\'{e}orie des \'{e}quations aux d\'{e}riv\'{e}es partielles et la m\'{e}thode des approximations successives}, J. Math. Pure. Appl., 1890 (\textbf{6}): 145-210
3. W. R. Mann, \textit{Mean value methods in iteration}, Proc. Amer. Math. Soc., 1953 (\textbf{4}):506-510.
4. S. Ishikawa, \textit{Fixed points by a new iteration method}, Proc. Amer. Math. Soc., 1974 (\textbf{44}): 147-150.
5. M. A. Noor, \textit{New approximation schemes for general variational inequalities}, J. Math. Anal. Appl., 2000 (\textbf{251}): 217-229.
6. R. Chugh and V. Kumar, \textit{Convergence of SP iterative scheme with mixed errors for accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach spaces}, Int. J. Comput. Math., 2013 (\textbf{90}): 1865-1880.
7. Y. Xu and Z. Liu, \textit{On estimation and control of errors of the Mann iteration process}, J. Math. Anal. Appl., 2003 (\textbf{286}):804-806.
8. Y. Xu, Z. Liu and S. M. Kang, \textit{Accumulation and control of random errors in the Ishikawa iterative process in arbitrary Banach space}, Comput. Math. Appl., 2011 (\textbf{61}): 2217-2220.

9. S. M. \c{S}oltuz and T. Grosan, \textit{Data dependence for Ishikawa iteration when dealing with contractive-like operators}, Fixed Point Theory A, 2008 (\textbf{2008}): 1-7.