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Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces

Yıl 2017, Cilt: 46 Sayı: 3, 373 - 388, 01.06.2017

Öz

In this paper, we simplify the iterative scheme introduced by Fukhar-ud-din and Berinde [Iterative Methods for the Class of Quasi-Contractive Type Operators and Comparison of their Rate of Convergence in Convex Metric Spaces, Filomat 30 (1), 223230, 2016] and study convergence and data dependency of the new proposed scheme of a quasi-contractive operator on a hyperbolic space. It is shown that our results provide better convergence rate.

Kaynakça

  • Abbas, M., Khan, S. H. Some $\Delta-$convergence theorems in $CAT(0)$ spaces, Hacet. J. Math. Stat. 40 (4), 563569, 2011.
  • Agarwal, R. P., O' Regan, D. and Sahu, D. R. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (1), 6179, 2007.
  • Berinde, V. Iterative approximation of fixed points (Springer-Verlag, Berlin, 2007).
  • Bridson, M. and Haeiger, A. Metric spaces of non-positive curvature (Springer-Verlag, Berlin, 1999).
  • Chugh, R. and Kumar, V. Data dependence of Noor and SP iterative schemes when dealing with quasi-contractive operators, Int. J. Comput. Appl. 40 (15), 4146, 2011.
  • Dhompongsa, S. and Panyanak, B. On $\Delta-$convergence theorems in $CAT(0)$ spaces, Comput. Math. Appl. 56 (10), 25722579, 2008.
  • Fukhar-ud-din H. One step iterative scheme for a pair of nonexpansive mappings in a convex metric space, Hacet. J. Math. Stat. 44 (5), 10231031, 2015.
  • Fukhar-ud-din, H. and Berinde, V. Iterative Methods for the Class of Quasi-Contractive Type Operators and Comparsion of their Rate of Convergence in Convex Metric Spaces, Filomat 30 (1), 223230, 2016.
  • Gürsoy, F., Karakaya, V. and Rhoades, B. Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 2013 (1), 112, 2013.
  • Ishikawa, S. Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1), 147 150, 1974.
  • Karakaya, V., Gürsoy, F. and Ertürk, M. Some convergence and data dependence results for various fixed point iterative methods, Kuwait J. Sci. 43 (1), 112128, 2016.
  • Khan A. R. On modified Noor iterations for asymptotically nonexpansive mappings, Bull. Belg. Math. Soc. Simon Stevin, 17 (1), 127140, 2010.
  • Khan, A. R., Fukhar-ud-din, H. and Khan, M. A. A. An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 2012 (1), 112, 2012.
  • Khan A. R., Gürsoy, F. and Karakaya, V. Jungck-Khan iterative scheme and higher con- vergence rate, Int. J. Comput. Math. 93 (12), 20922105, 2016.
  • Khan, A. R., Gürsoy, F. and Kumar, V. Stability and data dependence results for Jungck- Khan iterative scheme, Turkish J. Math. 40 (3), 631640, 2016.
  • Khan, A. R., Khamsi, M. A. and Fukhar-ud-din, H. Strong convergence of a general iteration scheme in $CAT(0)-$spaces, Nonlinear Anal. 74 (3), 783791, 2011.
  • Khan, A. R., Kumar, V. and Hussain, N. Analytical and numerical treatment of Jungck-type iterative schemes, Appl. Math. Comput. 231, 521535, 2014.
  • Kirk, W. A. Krasnoselskii's iteration process in hyperbolic space, Numer. Funct. Anal. Optim. 4 (4) 371381, 19811982.
  • Knopp, K. Theory and Applications of Infinite Series (Berlin, 1931).
  • Kohlenbach, U. Some logical metatheorems with applications in functional analysis, Trans. Am. Math. Soc. 357 (1), 89128, 2005.
  • Mann, W. R. Mean value methods in iterations, Proc. Amer. Math. Soc. 4 (3), 506-510, 1953.
  • Phuengrattana, W. and Suantai, S. Comparison of the rate of convergence of various iter- ative methods for the class of weak contractions in Banach spaces, Thai J. Math. 11 (1), 217226, 2013.
  • Singh, S. L., Bhatnagar, C. and Mishra, S. N. Stability of Jungck-type iterative procedures, Int. J. Math. Math. Sci. 2005 (19), 30353043, 2005.
  • Soltuz, S. M. and Grosan, T. Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl. 2008, 17, 2008.
  • Takahashi, W. A convexity in metric spaces and nonexpansive mappings I, Kodai Math. Sem. Rep. 22 (2), 142149, 1970.
  • Talman, L. A. Fixed points for condensing multifunctions in metric spaces with convex structure, Kodai Math. Sem. Rep. 29 (1-2), 6270, 1977.
  • Weng, X. Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc. 113 (3), 727731, 1991.
  • Xu, B. and Noor, M. A. Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2), 444-453, 2002.
Yıl 2017, Cilt: 46 Sayı: 3, 373 - 388, 01.06.2017

Öz

Kaynakça

  • Abbas, M., Khan, S. H. Some $\Delta-$convergence theorems in $CAT(0)$ spaces, Hacet. J. Math. Stat. 40 (4), 563569, 2011.
  • Agarwal, R. P., O' Regan, D. and Sahu, D. R. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (1), 6179, 2007.
  • Berinde, V. Iterative approximation of fixed points (Springer-Verlag, Berlin, 2007).
  • Bridson, M. and Haeiger, A. Metric spaces of non-positive curvature (Springer-Verlag, Berlin, 1999).
  • Chugh, R. and Kumar, V. Data dependence of Noor and SP iterative schemes when dealing with quasi-contractive operators, Int. J. Comput. Appl. 40 (15), 4146, 2011.
  • Dhompongsa, S. and Panyanak, B. On $\Delta-$convergence theorems in $CAT(0)$ spaces, Comput. Math. Appl. 56 (10), 25722579, 2008.
  • Fukhar-ud-din H. One step iterative scheme for a pair of nonexpansive mappings in a convex metric space, Hacet. J. Math. Stat. 44 (5), 10231031, 2015.
  • Fukhar-ud-din, H. and Berinde, V. Iterative Methods for the Class of Quasi-Contractive Type Operators and Comparsion of their Rate of Convergence in Convex Metric Spaces, Filomat 30 (1), 223230, 2016.
  • Gürsoy, F., Karakaya, V. and Rhoades, B. Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 2013 (1), 112, 2013.
  • Ishikawa, S. Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1), 147 150, 1974.
  • Karakaya, V., Gürsoy, F. and Ertürk, M. Some convergence and data dependence results for various fixed point iterative methods, Kuwait J. Sci. 43 (1), 112128, 2016.
  • Khan A. R. On modified Noor iterations for asymptotically nonexpansive mappings, Bull. Belg. Math. Soc. Simon Stevin, 17 (1), 127140, 2010.
  • Khan, A. R., Fukhar-ud-din, H. and Khan, M. A. A. An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 2012 (1), 112, 2012.
  • Khan A. R., Gürsoy, F. and Karakaya, V. Jungck-Khan iterative scheme and higher con- vergence rate, Int. J. Comput. Math. 93 (12), 20922105, 2016.
  • Khan, A. R., Gürsoy, F. and Kumar, V. Stability and data dependence results for Jungck- Khan iterative scheme, Turkish J. Math. 40 (3), 631640, 2016.
  • Khan, A. R., Khamsi, M. A. and Fukhar-ud-din, H. Strong convergence of a general iteration scheme in $CAT(0)-$spaces, Nonlinear Anal. 74 (3), 783791, 2011.
  • Khan, A. R., Kumar, V. and Hussain, N. Analytical and numerical treatment of Jungck-type iterative schemes, Appl. Math. Comput. 231, 521535, 2014.
  • Kirk, W. A. Krasnoselskii's iteration process in hyperbolic space, Numer. Funct. Anal. Optim. 4 (4) 371381, 19811982.
  • Knopp, K. Theory and Applications of Infinite Series (Berlin, 1931).
  • Kohlenbach, U. Some logical metatheorems with applications in functional analysis, Trans. Am. Math. Soc. 357 (1), 89128, 2005.
  • Mann, W. R. Mean value methods in iterations, Proc. Amer. Math. Soc. 4 (3), 506-510, 1953.
  • Phuengrattana, W. and Suantai, S. Comparison of the rate of convergence of various iter- ative methods for the class of weak contractions in Banach spaces, Thai J. Math. 11 (1), 217226, 2013.
  • Singh, S. L., Bhatnagar, C. and Mishra, S. N. Stability of Jungck-type iterative procedures, Int. J. Math. Math. Sci. 2005 (19), 30353043, 2005.
  • Soltuz, S. M. and Grosan, T. Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl. 2008, 17, 2008.
  • Takahashi, W. A convexity in metric spaces and nonexpansive mappings I, Kodai Math. Sem. Rep. 22 (2), 142149, 1970.
  • Talman, L. A. Fixed points for condensing multifunctions in metric spaces with convex structure, Kodai Math. Sem. Rep. 29 (1-2), 6270, 1977.
  • Weng, X. Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc. 113 (3), 727731, 1991.
  • Xu, B. and Noor, M. A. Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2), 444-453, 2002.
Toplam 28 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Faik Gursoy

Abdul Rahim Khan

Hafiz Fukhar-ud-din

Yayımlanma Tarihi 1 Haziran 2017
Yayımlandığı Sayı Yıl 2017 Cilt: 46 Sayı: 3

Kaynak Göster

APA Gursoy, F., Khan, A. R., & Fukhar-ud-din, H. (2017). Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces. Hacettepe Journal of Mathematics and Statistics, 46(3), 373-388.
AMA Gursoy F, Khan AR, Fukhar-ud-din H. Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces. Hacettepe Journal of Mathematics and Statistics. Haziran 2017;46(3):373-388.
Chicago Gursoy, Faik, Abdul Rahim Khan, ve Hafiz Fukhar-ud-din. “Convergence and Data Dependence Results for Quasi-Contractive Type Operators in Hyperbolic Spaces”. Hacettepe Journal of Mathematics and Statistics 46, sy. 3 (Haziran 2017): 373-88.
EndNote Gursoy F, Khan AR, Fukhar-ud-din H (01 Haziran 2017) Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces. Hacettepe Journal of Mathematics and Statistics 46 3 373–388.
IEEE F. Gursoy, A. R. Khan, ve H. Fukhar-ud-din, “Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces”, Hacettepe Journal of Mathematics and Statistics, c. 46, sy. 3, ss. 373–388, 2017.
ISNAD Gursoy, Faik vd. “Convergence and Data Dependence Results for Quasi-Contractive Type Operators in Hyperbolic Spaces”. Hacettepe Journal of Mathematics and Statistics 46/3 (Haziran 2017), 373-388.
JAMA Gursoy F, Khan AR, Fukhar-ud-din H. Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces. Hacettepe Journal of Mathematics and Statistics. 2017;46:373–388.
MLA Gursoy, Faik vd. “Convergence and Data Dependence Results for Quasi-Contractive Type Operators in Hyperbolic Spaces”. Hacettepe Journal of Mathematics and Statistics, c. 46, sy. 3, 2017, ss. 373-88.
Vancouver Gursoy F, Khan AR, Fukhar-ud-din H. Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces. Hacettepe Journal of Mathematics and Statistics. 2017;46(3):373-88.