Research Article
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On the rate of convergence of different implicit iterations in convex metric spaces

Year 2018, Volume: 6 Issue: 1, 110 - 116, 15.04.2018

İsa Yıldırım

Abstract

In this paper, we introduce a new three-step implicit iteration process and prove that it is faster than the other implicit iteration processes. We prove some convergence theorem for generalized contraction mappings in convex metric space. We also support our results by a numerical example.

Keywords

Convergence rate implicit iterative processes

References

  • [1] M. Abbas, P. Vetro, S. H. Khan, On fixed points of Berinde’s contractive mappings in cone metric spaces, Carpath. J. Math., 26(2) (2010) 121-133.
  • [2] Banach S., Sur les Operations dans les Ensembles Abstraits et leur Applications aux Equations Integrales, Fund. Math. 3 (1922), 133–181.
  • [3] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasicontractive operators, Fixed Point Theory and Applications, 2004(2004), 97-105.
  • [4] V. Berinde, On the convergence of the Ishikawa iteration in the class of quasi contractive operators. Acta Math. Univ. Comen. 73, 119-126 (2004).
  • [5] S.S. Chang, L. Yang, X.R. Wang, Stronger convergence theorems for an infinite family of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comp. 217 (2010) 277–282.
  • [6] R. Chugh, P. Malik and V. Kumar, On analytical and numerical study of implicit fixed point iterations, Cogent Mathematics (2015), 2: 1021623.
  • [7] Lj. B. C´ iric´, Rafiq, A., Cakic´, N., & Ume, J. S. (2009). Implicit Mann fixed point iterations for pseudo-contractive mappings. Applied Mathematics Letters, 22, 581–584.
  • [8] Lj. B. C´ iric´, Rafiq, A., Radenovic´, S., Rajovic´,M., & Ume, J. S. (2008). OnMann implicit iterations for strongly accretive and strongly pseudo-contractive mappings. Applied Mathematics and Computation, 198, 128–137.
  • [9] Lj. B.Ciric, Ume, J. S. M., & Khan, S. (2003). On the convergence of the Ishikawa iterates to a common fixed point of two mappings. Archivum Mathematicum (Brno) Tomus, 39, 123–127.
  • [10] C.O. Imoru, M.O. Olantiwo, On the stability of Picard and Mann iteration processes. Carpath. J. Math. 19, 155-160 (2003).
  • [11] A.R. Khan, M.A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications, Comput. Math. Appl. 59 (2010) 2990-2995.
  • [12] S.H. Khan, I. Yildirim, M. Ozdemir. Convergence of an implicit algorithm for two families of nonexpansive mappings, Comput. Math. Appl. 59 (2010) 3084-3091.
  • [13] S. H. Khan, Common fixed points of quasi-contractive type operators by a generalized iterative process, IAENG Int. J. Appl. Math., 41(3) (2011), 260-264.
  • [14] J. K. Kim, K. S. Kim, S. M. Kim, Convergence theorems of implicit iteration process for for finite family of asymptotically quasi-nonexpansive mappings in convex metric space, Nonlinear Analysis and Convex Analysis, 1484 (2006) 40-51.
  • [15] U. Kohlenbach, (2004). Some logical metatherems with applications in functional analysis. Transactions of the American Mathematical Society, 357, 89–128.
  • [16] Q.Y. Liu, Z.B. Liu, N.J. Huang, Approximating the common fixed points of two sequences of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comp. 216 (2010) 883–889.
  • [17] M.O. Osilike, A. Udomene, Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings. Indian J. Pure Appl. Math. 30, 1229-1234 (1999).
  • [18] S.M. S¸ oltuz, Grosan, T., Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory Appl. 2008, Article ID 242916 (2008). doi:10.1155/2008/242916
  • [19] W. Takahashi, A convexity in metric space and nonexpansive mappings, Kodai Math. Sem. Rep. 22 (1970) 142-149.
  • [20] I. Yildirim, M. Abbas, Convergence rate of implicit iteration process and a data dependence result, European J. Pure and Appl. Math. 11 (1) (2018), 189-201.
  • [21] I. Yildirim, S. H. Khan, Convergence theorems for common fixed points of asymptotically quasi-nonexpansive mappings in convex metric spaces, Applied Mathematics and Computation, Volume 218, Issue 9 (2012), 4860-4866.
  • [22] T. Zamfirescu, Fix point theorems in metric spaces. Arch. Math. 23, 292-298 (1972).

Year 2018, Volume: 6 Issue: 1, 110 - 116, 15.04.2018

İsa Yıldırım

Abstract

References

  • [1] M. Abbas, P. Vetro, S. H. Khan, On fixed points of Berinde’s contractive mappings in cone metric spaces, Carpath. J. Math., 26(2) (2010) 121-133.
  • [2] Banach S., Sur les Operations dans les Ensembles Abstraits et leur Applications aux Equations Integrales, Fund. Math. 3 (1922), 133–181.
  • [3] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasicontractive operators, Fixed Point Theory and Applications, 2004(2004), 97-105.
  • [4] V. Berinde, On the convergence of the Ishikawa iteration in the class of quasi contractive operators. Acta Math. Univ. Comen. 73, 119-126 (2004).
  • [5] S.S. Chang, L. Yang, X.R. Wang, Stronger convergence theorems for an infinite family of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comp. 217 (2010) 277–282.
  • [6] R. Chugh, P. Malik and V. Kumar, On analytical and numerical study of implicit fixed point iterations, Cogent Mathematics (2015), 2: 1021623.
  • [7] Lj. B. C´ iric´, Rafiq, A., Cakic´, N., & Ume, J. S. (2009). Implicit Mann fixed point iterations for pseudo-contractive mappings. Applied Mathematics Letters, 22, 581–584.
  • [8] Lj. B. C´ iric´, Rafiq, A., Radenovic´, S., Rajovic´,M., & Ume, J. S. (2008). OnMann implicit iterations for strongly accretive and strongly pseudo-contractive mappings. Applied Mathematics and Computation, 198, 128–137.
  • [9] Lj. B.Ciric, Ume, J. S. M., & Khan, S. (2003). On the convergence of the Ishikawa iterates to a common fixed point of two mappings. Archivum Mathematicum (Brno) Tomus, 39, 123–127.
  • [10] C.O. Imoru, M.O. Olantiwo, On the stability of Picard and Mann iteration processes. Carpath. J. Math. 19, 155-160 (2003).
  • [11] A.R. Khan, M.A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications, Comput. Math. Appl. 59 (2010) 2990-2995.
  • [12] S.H. Khan, I. Yildirim, M. Ozdemir. Convergence of an implicit algorithm for two families of nonexpansive mappings, Comput. Math. Appl. 59 (2010) 3084-3091.
  • [13] S. H. Khan, Common fixed points of quasi-contractive type operators by a generalized iterative process, IAENG Int. J. Appl. Math., 41(3) (2011), 260-264.
  • [14] J. K. Kim, K. S. Kim, S. M. Kim, Convergence theorems of implicit iteration process for for finite family of asymptotically quasi-nonexpansive mappings in convex metric space, Nonlinear Analysis and Convex Analysis, 1484 (2006) 40-51.
  • [15] U. Kohlenbach, (2004). Some logical metatherems with applications in functional analysis. Transactions of the American Mathematical Society, 357, 89–128.
  • [16] Q.Y. Liu, Z.B. Liu, N.J. Huang, Approximating the common fixed points of two sequences of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comp. 216 (2010) 883–889.
  • [17] M.O. Osilike, A. Udomene, Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings. Indian J. Pure Appl. Math. 30, 1229-1234 (1999).
  • [18] S.M. S¸ oltuz, Grosan, T., Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory Appl. 2008, Article ID 242916 (2008). doi:10.1155/2008/242916
  • [19] W. Takahashi, A convexity in metric space and nonexpansive mappings, Kodai Math. Sem. Rep. 22 (1970) 142-149.
  • [20] I. Yildirim, M. Abbas, Convergence rate of implicit iteration process and a data dependence result, European J. Pure and Appl. Math. 11 (1) (2018), 189-201.
  • [21] I. Yildirim, S. H. Khan, Convergence theorems for common fixed points of asymptotically quasi-nonexpansive mappings in convex metric spaces, Applied Mathematics and Computation, Volume 218, Issue 9 (2012), 4860-4866.
  • [22] T. Zamfirescu, Fix point theorems in metric spaces. Arch. Math. 23, 292-298 (1972).
There are 22 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Research Article
Authors

İsa Yıldırım

Submission Date July 11, 2017
Publication Date April 15, 2018
Published in Issue Year 2018 Volume: 6 Issue: 1

Cite

APA Yıldırım, İ. (2018). On the rate of convergence of different implicit iterations in convex metric spaces. Konuralp Journal of Mathematics, 6(1), 110-116.
AMA Yıldırım İ. On the rate of convergence of different implicit iterations in convex metric spaces. Konuralp J. Math. April 2018;6(1):110-116.
Chicago Yıldırım, İsa. “On the Rate of Convergence of Different Implicit Iterations in Convex Metric Spaces”. Konuralp Journal of Mathematics 6, no. 1 (April 2018): 110-16.
EndNote Yıldırım İ (April 1, 2018) On the rate of convergence of different implicit iterations in convex metric spaces. Konuralp Journal of Mathematics 6 1 110–116.
IEEE İ. Yıldırım, “On the rate of convergence of different implicit iterations in convex metric spaces”, Konuralp J. Math., vol. 6, no. 1, pp. 110–116, 2018.
ISNAD Yıldırım, İsa. “On the Rate of Convergence of Different Implicit Iterations in Convex Metric Spaces”. Konuralp Journal of Mathematics 6/1 (April2018), 110-116.
JAMA Yıldırım İ. On the rate of convergence of different implicit iterations in convex metric spaces. Konuralp J. Math. 2018;6:110–116.
MLA Yıldırım, İsa. “On the Rate of Convergence of Different Implicit Iterations in Convex Metric Spaces”. Konuralp Journal of Mathematics, vol. 6, no. 1, 2018, pp. 110-6.
Vancouver Yıldırım İ. On the rate of convergence of different implicit iterations in convex metric spaces. Konuralp J. Math. 2018;6(1):110-6.
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