Browse > Article
http://dx.doi.org/10.4134/CKMS.c170031

SOME RESULTS OF THE NEW ITERATIVE SCHEME IN HYPERBOLIC SPACE  

Basarir, Metin (Department of Mathematics Sakarya University)
Sahin, Aynur (Department of Mathematics Sakarya University)
Publication Information
Communications of the Korean Mathematical Society / v.32, no.4, 2017 , pp. 1009-1024 More about this Journal
Abstract
In this paper, we consider the new faster iterative scheme due to Sintunavarat and Pitea ([32]) for further investigation and we prove its strong and ${\Delta}$-convergence theorems, data dependence and stability results in hyperbolic space. Our results extend, improve and generalize several recent results in CAT(0) space and uniformly convex Banach space.
Keywords
hyperbolic space; fixed point; iterative scheme; strong convergence; ${\Delta}$-convergence; stability; data dependence;
Citations & Related Records
연도 인용수 순위
  • Reference
1 F. E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660-665.   DOI
2 T. Cardinali and P. Rubbioni, A generalization of the Caristi fixed point theorem in metric spaces, Fixed Point Theory 11 (2010), no. 1, 3-10.
3 S. K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci. 25 (1972), 727-730.
4 H. Fukhar-ud-din and M. A. Khamsi, Approximating common fixed points in hyperbolic spaces, Fixed Point Theory Appl. 2014 (2014), Article ID 113, 15 pages.   DOI
5 K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, In: Topological methods in nonlinear functional analysis (Toronto, Ont., 1982), 115-123, Contemp. Math., 21, Amer. Math. Soc., Providence, RI, 1983.
6 K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York, 1984.
7 F. Gursoy, V. Karakaya, and B. E. Rhoades, Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 2013 (2013), Article ID 76, 12 pages.   DOI
8 A. M. Harder and T. L. Hicks, Stability results for fixed point iteration procedures, Math. Japon. 33 (1988), no. 5, 693-706.
9 C. O. Imoru and M. O. Olantinwo, On the stability of Picard and Mann iteration processes, Carpathian J. Math. 19 (2003), no. 2, 155-160.
10 S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150.   DOI
11 R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 10 (1968), 71-76.
12 S. H. Khan and M. Abbas, Strong and $\Delta$-convergence of some iterative schemes in CAT(0) spaces, Comput. Math. Appl. 61 (2011), no. 1, 109-116.   DOI
13 T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179-182.   DOI
14 A. R. Khan, H. Fukhar-ud-din, and M. A. A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 2012 (2012), Article ID 54, 12 pages.   DOI
15 A. R. Khan, F. Gursoy, and V. Kumar, Stability and data dependence results for the Jungck-Khan iterative scheme, Turkish J. Math. 40 (2016), no. 3, 631-640.   DOI
16 U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 (2005), no. 1, 89-128.   DOI
17 T. Kuczumow, An almost convergence and its applications, Ann. Univ. Mariae Curie- Sklodowska, Sect. A 32 (1978), 79-88.
18 L. Leustean, Nonexpansive iterations in uniformly convex W-hyperbolic spaces, In: Nonlinear analysis and optimization I. Nonlinear analysis, 193-210, Contemp. Math., 513, Israel Math. Conf. Proc., Amer. Math. Soc., Providence, RI, 2010.
19 W. R. Mann, Mean value methods in iterations, Proc. Amer. Math. Soc. 4 (1953), 506-510.   DOI
20 M. O. Osilike, Some stability results for fixed point iteration procedures, J. Nigerian Math. Soc. 14/15 (1995/96), 17-29.
21 S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), no. 6, 537-558.   DOI
22 B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl. Math. 21 (1990), no. 1, 1-9.
23 A. Sahin and M. Basarir, Convergence and data dependence results of an iteration process in a hyperbolic space, Filomat 30 (2016), no. 3, 569-582.   DOI
24 H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), no. 2, 375-380.   DOI
25 W. Takahashi, A convexity in metric spaces and nonexpansive mappings, Kodai Math. Semin. Rep. 22 (1970), 142-149.   DOI
26 T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal. 8 (1996), no. 1, 197-203.   DOI
27 W. Sintunavarat and A. Pitea, On a new iteration scheme for numerical reckoning fixed points of Berinde mappings with convergence analysis, J. Nonlinear Sci. Appl. 9 (2016), no. 5, 2553-2562.   DOI
28 S. M. Soltuz and T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive-like operators, Fixed Point Theory Appl. 2008 (2008), Article ID 242916, 7 pages.
29 I. Timis, On the weak stability of Picard iteration for some contractive type mappings, An. Univ. Craiova Ser. Mat. Inform. 37 (2010) no. 2, 106-114.
30 T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math. (Basel) 23 (1972), 292-298.   DOI
31 M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999.
32 R. P. Agarwal, D. O'Regan, and D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (2007), no. 1, 61-79.
33 H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer-Verlag, New York, 2011.
34 V. Berinde, On the stability of some fixed point procedures, Bul. Stiint. Univ. Baia Mare Ser. B Fasc. Mat.-Inform. 18 (2002), no. 1, 7-14.
35 V. Berinde, A convergence theorem for some mean value fixed point iterations procedures, Demonstratio Math. 38 (2005), no. 1, 177-184.
36 V. Berinde, Iterative Approximation of Fixed Points, Springer, 2007.