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http://dx.doi.org/10.4134/CKMS.c180093

SOME CONVERGENCE RESULTS FOR GENERALIZED NONEXPANSIVE MAPPINGS IN CAT(0) SPACES  

Garodia, Chanchal (Department of Mathematics Jamia Millia Islamia)
Uddin, Izhar (Department of Mathematics Jamia Millia Islamia)
Publication Information
Communications of the Korean Mathematical Society / v.34, no.1, 2019 , pp. 253-265 More about this Journal
Abstract
The aim of this paper is to study convergence behaviour of Thakur iteration scheme in CAT(0) spaces for generalized nonexpansive mappings. In process, several relevant results of the existing literature are generalized and improved.
Keywords
CAT(0) space; fixed point; ${\Delta}$-convergence and nonexpansive mapping;
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1 T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), no. 2, 1088-1095.   DOI
2 B. S. Thakur, D. Thakur, and M. Postolache, A new iteration scheme for approximating fixed points of nonexpansive mappings, Filomat 30 (2016), no. 10, 2711-2720.   DOI
3 B. S. Thakur, D. Thakur, and M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, Appl. Math. Comput. 275 (2016), 147-155.   DOI
4 M. Abbas, S. H. Khan, and M. Postolache, Existence and approximation results for SKC mappings in CAT(0) spaces, J. Inequal. Appl. 2014 (2014), 212, 10 pp.   DOI
5 K. S. Brown, Buildings, Springer-Verlag, New York, 1989.
6 M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik 66 (2014), no. 2, 223-234.
7 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.
8 M. Basarur and A. Sahin, On the strong and ${\Delta}$-convergence of S-iteration process for generalized nonexpansive mappings on CAT(0) space, Thai J. Math. 12 (2014), no. 3, 549-559.
9 M. Basarur and A. Sahin, A new-three step iteration for generalized nonexpansive mappings in a CAT(0) space, AIP Conf. Proc. 1611, 310, 2014.
10 M. R. Bridson and A. Hafliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999.
11 D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001.
12 S. Dhompongsa, W. Inthakon, and A. Kaewkhao, Edelstein's method and fixed point theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 350 (2009), no. 1, 12-17.   DOI
13 S. Dhompongsa, W. A. Kirk, and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal. 65 (2006), no. 4, 762-772.   DOI
14 S. Dhompongsa and B. Panyanak, On ${\Delta}$-convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56 (2008), no. 10, 2572-2579.   DOI
15 M. Eslamian and A. Abkar, One-step iterative process for a finite family of multivalued mappings, Math. Comput. Modelling 54 (2011), no. 1-2, 105-111.   DOI
16 B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957-961.   DOI
17 J. Garcia-Falset, E. Llorens-Fuster, and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375 (2011), no. 1, 185-195.   DOI
18 K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, 83, Marcel Dekker, Inc., New York, 1984.
19 M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, translated from the French by Sean Michael Bates, Progress in Mathematics, 152, Birkhauser Boston, Inc., Boston, MA, 1999.
20 S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150.   DOI
21 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
22 S. H. Khan, D. Agbebaku, and M. Abbas, Three step iteration process for two multivalued nonexpansive maps in hyperbolic spaces, J. Math. Ext. 10 (2016), no. 4, 87-109.
23 S. H. Khan and H. Fukhar-ud-din, Convergence theorems for two finite families of some generalized nonexpansive mappings in hyperbolic spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 2, 734-743.   DOI
24 W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), no. 12, 3689-3696.   DOI
25 W. Laowang and B. Panyanak, Approximating fixed points of nonexpansive nonself mappings in CAT(0) spaces, Fixed Point Theory Appl. 2010 (2010), Art. ID 367274, 11 pp.
26 Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.   DOI
27 T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179-182 (1977).   DOI
28 W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.   DOI
29 B. Nanjaras, B. Panyanak, and W. Phuengrattana, Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT(0) spaces, Nonlinear Anal. Hybrid Syst. 4 (2010), no. 1, 25-31.   DOI
30 M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), no. 1, 217-229.   DOI
31 A. Razani and H. Salahifard, Invariant approximation for CAT(0) spaces, Nonlinear Anal. 72 (2010), no. 5, 2421-2425.   DOI
32 Ritika and S. H. Khan, Convergence of Picard-Mann hybrid iterative process for generalized nonexpansive mappings in CAT(0) spaces, Filomat 31 (2017), no. 11, 3531-3538.   DOI
33 Ritika and S. H. Khan, Convergence of RK-iterative process for generalized nonexpansive mappings in CAT(0) spaces, Asian-Eur. J. Math., 2018.
34 H. F. Senter and W. G. Dotson, Jr., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-380.   DOI