DOI QR코드

DOI QR Code

CONVERGENCE OF ISHIKAWA ITERATION WITH ERROR TERMS ON AN ARBITRARY INTERVAL

Yuan, Qing;Cho, Sun-Young;Qin, Xiaolong

  • Received : 2009.07.22
  • Published : 2011.04.30

Abstract

In this paper, a continuous real function on the real line is considered. The necessary and sufficient conditions for the convergence of the Ishikawa iteration with error terms for the functional are obtained.

Keywords

fixed point;real function;Mann iteration;Ishikawa iteration

References

  1. D. Borwein and J. Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl. 157 (1991), no. 1, 112-126. https://doi.org/10.1016/0022-247X(91)90139-Q
  2. R. L. Franks and R. P. Marzec, A theorem on mean value iterations, Proc. Amer. Math. Soc. 30 (1971), 324-326. https://doi.org/10.1090/S0002-9939-1971-0280656-9
  3. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
  4. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  5. Y. Qing and L. Qihou, The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval, J. Math. Anal. Appl. 323 (2006), no. 2, 1383-1386. https://doi.org/10.1016/j.jmaa.2005.11.058
  6. B. E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc. 196 (1974), 161-176. https://doi.org/10.1090/S0002-9947-1974-0348565-1

Cited by

  1. Convergence theorems for continuous functions on an arbitrary interval vol.62, pp.2, 2013, https://doi.org/10.1007/s12215-013-0121-y
  2. Approximating fixed points for continuous functions on an arbitrary interval vol.2013, pp.1, 2013, https://doi.org/10.1186/1029-242X-2013-214