ON THE NEW MULTI-STEP ITERATION PROCESS FOR MULTI-VALUED MAPPINGS IN A COMPLETE GEODESIC SPACE
Year 2015,
Volume: 64 Issue: 2, 77 - 87, 01.08.2015
Aynur Şahin
Metin Başarır
Abstract
The purpose of this paper is to prove the strong and 4-convergencetheorems of the new multi-step iteration process for multi-valued quasi-nonexpansivemappings in a complete geodesic space. Our results extend and improve someresults in the literature
References
- Ba¸sarır, M. and ¸Sahin, A., On the strong and 4-convergence of new multi-step and S-iteration processes in a CAT(0) space, J. Inequal. Appl. 2013, Article ID 482, 13 pages.
- Bridson, M. and Hae*iger, A., Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999.
- Espinola, R. and Fernandez-Leon, A., CAT( )-spaces, weak convergence and …xed points, J. Math. Anal. Appl. 353 (2009), 410-427.
- Gürsoy, F., Karakaya, V. and Rhoades, B. E., Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 2013, Article ID 76, 12 pages.
- He, J. S., Fang, D. H., López, G. and Li, C., Mann’s algorithm for nonexpansive mappings in CAT( ) spaces, Nonlinear Anal. 75 (2012), 445-452.
- Khan, S. H., Fukhar-ud-din, H. and Kalsoom, A., Common …xed points of two multivalued nonexpansive maps by a one-step implicit algorithm in hyperbolic spaces, Matematiµcki Vesnik 66(4) (2014), 397-409.
- Kimura, Y. and Nakagawa, K., Another type of Mann iterative scheme for two mappings in a complete geodesic space, J. Inequal. Appl. 2014, Article ID 72, 9 pages.
- Kimura, Y., Saejung, S. and Yotkaew, P., The Mann algorithm in a complete geodesic space with curvature bounded above, Fixed Point Theory Appl. 2013, Article ID 336, 13 pages.
- Kimura, Y. and Satô, K., Halpern iteration for strongly quasinonexpansive mappings on a geodesic space with curvature bounded above by one, Fixed Point Theory Appl. 2013, Article ID 7, 14 pages.
- Kirk, W. A. and Panyanak, B., A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), 3689-3696.
- Panyanak, B., On the Ishikawa iteration processes for multivalued mappings in some CAT( ) spaces, Fixed Point Theory Appl. 2014, Article ID 1, 9 pages.
- Phuengrattana, W. and Suantai, S., On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235 (2011), 3006-3014.
- Puttasontiphot, T., Mann and Ishikawa iteration schemes for multi-valued mappings in CAT(0) spaces, Appl. Math. Sci. 4 (2010), 3005-3018.
- Salahifard, H., Vaezpour, S. M. and Dhompongsa, S., Fixed point theorems for some general- ized nonexpansive mappings in CAT(0) spaces, J. Nonlinear Anal. Optim. 4 (2013), 241-248. [15] Shabanian, S. and Vaezpour, S. M., A minimax inequality and its applications to …xed point theorems in CAT(0) spaces, Fixed Point Theory Appl. 2011, Article ID 61, 9 pages.
- Shahzad, N. and Zegeye, H., On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal. 71 (2009), 838-844.
- Song, Y. and Cho, Y. J., Some notes on Ishikawa iteration for multi-valued mappings, Bull. Korean Math. Soc. 48(3) (2011), 575-584.
- Thianwan, S., Common …xed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space, J. Comput. Appl. Math. 224 (2009), 688-695.
- Current address : Aynur SAH·IN, Department of Mathematics, Faculty of Sciences and Arts, Sakarya University, Sakarya, 54050, Turkey.
Year 2015,
Volume: 64 Issue: 2, 77 - 87, 01.08.2015
Aynur Şahin
Metin Başarır
References
- Ba¸sarır, M. and ¸Sahin, A., On the strong and 4-convergence of new multi-step and S-iteration processes in a CAT(0) space, J. Inequal. Appl. 2013, Article ID 482, 13 pages.
- Bridson, M. and Hae*iger, A., Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999.
- Espinola, R. and Fernandez-Leon, A., CAT( )-spaces, weak convergence and …xed points, J. Math. Anal. Appl. 353 (2009), 410-427.
- Gürsoy, F., Karakaya, V. and Rhoades, B. E., Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 2013, Article ID 76, 12 pages.
- He, J. S., Fang, D. H., López, G. and Li, C., Mann’s algorithm for nonexpansive mappings in CAT( ) spaces, Nonlinear Anal. 75 (2012), 445-452.
- Khan, S. H., Fukhar-ud-din, H. and Kalsoom, A., Common …xed points of two multivalued nonexpansive maps by a one-step implicit algorithm in hyperbolic spaces, Matematiµcki Vesnik 66(4) (2014), 397-409.
- Kimura, Y. and Nakagawa, K., Another type of Mann iterative scheme for two mappings in a complete geodesic space, J. Inequal. Appl. 2014, Article ID 72, 9 pages.
- Kimura, Y., Saejung, S. and Yotkaew, P., The Mann algorithm in a complete geodesic space with curvature bounded above, Fixed Point Theory Appl. 2013, Article ID 336, 13 pages.
- Kimura, Y. and Satô, K., Halpern iteration for strongly quasinonexpansive mappings on a geodesic space with curvature bounded above by one, Fixed Point Theory Appl. 2013, Article ID 7, 14 pages.
- Kirk, W. A. and Panyanak, B., A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), 3689-3696.
- Panyanak, B., On the Ishikawa iteration processes for multivalued mappings in some CAT( ) spaces, Fixed Point Theory Appl. 2014, Article ID 1, 9 pages.
- Phuengrattana, W. and Suantai, S., On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235 (2011), 3006-3014.
- Puttasontiphot, T., Mann and Ishikawa iteration schemes for multi-valued mappings in CAT(0) spaces, Appl. Math. Sci. 4 (2010), 3005-3018.
- Salahifard, H., Vaezpour, S. M. and Dhompongsa, S., Fixed point theorems for some general- ized nonexpansive mappings in CAT(0) spaces, J. Nonlinear Anal. Optim. 4 (2013), 241-248. [15] Shabanian, S. and Vaezpour, S. M., A minimax inequality and its applications to …xed point theorems in CAT(0) spaces, Fixed Point Theory Appl. 2011, Article ID 61, 9 pages.
- Shahzad, N. and Zegeye, H., On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal. 71 (2009), 838-844.
- Song, Y. and Cho, Y. J., Some notes on Ishikawa iteration for multi-valued mappings, Bull. Korean Math. Soc. 48(3) (2011), 575-584.
- Thianwan, S., Common …xed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space, J. Comput. Appl. Math. 224 (2009), 688-695.
- Current address : Aynur SAH·IN, Department of Mathematics, Faculty of Sciences and Arts, Sakarya University, Sakarya, 54050, Turkey.