Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces
Yıl 2017,
Cilt: 46 Sayı: 3, 373 - 388, 01.06.2017
Faik Gursoy
,
Abdul Rahim Khan
,
Hafiz Fukhar-ud-din
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
Faik Gursoy
,
Abdul Rahim Khan
,
Hafiz Fukhar-ud-din
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.