Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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NEW ITERATIVE PROCESS FOR THE EQUATION INVOLVING STRONGLY ACCRETIVE OPERATORS IN BANACH SPACES

  • Zeng, Ling-Yan (SCHOOL OF MATHEMATICS AND INFORMATION CHINA WEST NORMAL UNIVERSITY) ;
  • Li, Jun (SCHOOL OF MATHEMATICS AND INFORMATION CHINA WEST NORMAL UNIVERSITY) ;
  • Kim, Jong-Kyu (DEPARTMENT OF MATHEMATICS EDUCATION KYUNGNAM UNIVERSITY)
  • Published : 2007.11.30
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Abstract

In this paper, under suitable conditions, we show that the new class of iterative process with errors introduced by Li et al converges strongly to the unique solution of the equation involving strongly accretive operators in real Banach spaces. Furthermore, we prove that it is equivalent to the classical Ishikawa iterative sequence with errors.

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References

  1. R. P. Agarwal, N. J. Huang, and Y. J. Cho, Stability of iterative processes with errors for nonlinear equations of $\phi$-strongly accretive type operators, Numer. Funct. Anal. Optim. 22 (2001), no. 5-6, 471-485 https://doi.org/10.1081/NFA-100105303
  2. S. S. Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal. 30 (1997), no. 7, 4197-4208 https://doi.org/10.1016/S0362-546X(97)00388-X
  3. S. S. Chang, On Chidume's open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces, J. Math. Anal. Appl. 216 (1997), no. 1, 94-111 https://doi.org/10.1006/jmaa.1997.5661
  4. C. E. Chidume, An iterative process for nonlinear Lipschitzian strongly accretive map- pings in $L_p$ spaces, J. Math. Anal. Appl. 151 (1990), no. 2, 453-461 https://doi.org/10.1016/0022-247X(90)90160-H
  5. Y. J. Cho, H. Zhou, and J. K. Kim, Iterative approximations of zeroes for accretive operators in Banach spaces, Commun. Korean Math. Soc. 21 (2006), no. 2, 237-251 https://doi.org/10.4134/CKMS.2006.21.2.237
  6. N. J. Huang, Y. J. Cho, B. S. Lee, and J. S. Jung, Convergence of iterative processes with errors for set-valued pseudocontractive and accretive type mappings in Banach spaces, Comput. Math. Appl. 40 (2000), no. 10-11, 1127-1139 https://doi.org/10.1016/S0898-1221(00)00227-3
  7. 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
  8. J. K. Kim, Convergence of Ishikawa iterative sequences for accretive Lipschitzian map- pings in Banach spaces, Taiwanese J. Math. 10 (2006), no. 2, 553-561 https://doi.org/10.11650/twjm/1500403843
  9. J. K. Kim, S. M. Jang, and Z. Liu, Convergence theorems and stability problems of Ishikawa iterative sequences for nonlinear operator equations of the accretive and strongly accretive operators, Comm. Appl. Nonlinear Anal. 10 (2003), no. 3, 85-98
  10. J. K. Kim, K. S. Kim, and Y. M. Nam, Convergence and stability of iterative processes for a pair of simultaneously asymptotically quasi-nonexpansive type mappings in convex metric spaces, J. Comput. Anal. Appl. 9 (2007), no. 2, 159-172
  11. J. K. Kim and Y. M. Nam, Modified Ishikawa iterative sequences with errors for asymp- totically set-valued pseudocontractive mappings in Banach spaces, Bull. Korean Math. Soc. 43 (2006), no. 4, 847-860 https://doi.org/10.4134/BKMS.2006.43.4.847
  12. J. Li, Inequalities on sequences with applications, Acta Math. Sinica (Chin. Ser.) 47 (2004), no. 2, 273-278
  13. J. Li, N. J. Huang, H. J. Hwang, and Y. J. Cho, Stability of iterative procedures with er- rors for approximating common fixed points of quasi-contractive mappings, Appl. Anal. 84 (2005), no. 3, 253-267 https://doi.org/10.1080/00036810412331284374
  14. L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accre- tive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), no. 1, 114-125 https://doi.org/10.1006/jmaa.1995.1289
  15. 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
  16. R. H. Martin, A global existence theorem for autonomous differential equations in a Banach space, Proc. Amer. Math. Soc. 26 (1970), 307-314 https://doi.org/10.1090/S0002-9939-1970-0264195-6
  17. Y. G. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998), no. 1, 91-101 https://doi.org/10.1006/jmaa.1998.5987