Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON THE CONVERGENCE OF ISHIKAWA ITERATION WITH ERRORS FOR REAL CONTINUOUS FUNCTIONS

  • Kittithat Boonpot (Department of Mathematics Faculty of Science Khon Kaen University) ;
  • Satit Saejung (Department of Mathematics Faculty of Science Khon Kaen University, Research Center for Environmental and Hazardous Substance Management (EHSM) Khon Kaen University)
  • Received : 2022.11.28
  • Accepted : 2023.04.06
  • Published : 2024.04.30
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Abstract

We point out an error appeared in the paper of Yuan et al. [3] and present a correction of their result under a more general assumption. Moreover, we discuss the validity of the conditions imposed on the sequences of error terms.

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Acknowledgement

The authors would like to thank the referee for his/her comments and suggestions. S. Saejung was supported by National Research Council of Thailand and Khon Kaen University under grant N42A650290; and the Research and Graduate Studies at Khon Kaen University.

References

  1. D. Borwein and J. M. 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. P. Cholamjiak, On Ishikawa-type iteration with errors for a continuous real function on an arbitrary interval, Appl. Math. Sci. (Ruse) 7 (2013), no. 37-40, 1901-1907. https://doi.org/10.12988/ams.2013.13172 
  3. Q. Yuan, S. Y. Cho, and X. L. Qin, Convergence of Ishikawa iteration with error terms on an arbitrary interval, Commun. Korean Math. Soc. 26 (2011), no. 2, 229-235. https://doi.org/10.4134/CKMS.2011.26.2.229 
  4. Q. Yuan and Q. H. Liu, 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