Existence and convergence results for a class of nonexpansive type mappings in hyperbolic spaces

Rajendra Pant, Rameshwa Pandey

Abstract

We consider a wider class of nonexpansive type mappings and present some fixed point results for this class of mappingss in hyperbolic spaces. Indeed, first we obtain some existence results for this class of mappings. Next, we present some convergence results for an iteration algorithm for the same class of mappings. Some illustrative non-trivial examples have also been discussed.


Keywords

Reich-Suzuki type nonexpansive mapping; hyperbolic metric space; iteration process; nonexpansive mapping

Subject classification

47H10; 54H25.

Full Text:

PDF

References

M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik 66, no. 2 (2014), 223-234.

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, no. 1 (2007), 61-79.

A. Amini-Harandi, M. Fakhar and H. R. Hajisharifi, Weak fixed point property for nonexpansive mappings with respect to orbits in Banach spaces, J. Fixed Point Theory Appl. 18, no. 3 (2016), 601-607. https://doi.org/10.1007/s11784-016-0310-3

K. Aoyama and F. Kohsaka, Fixed point theorem for α-nonexpansive mappings in Banach spaces, Nonlinear Anal. 74, no. 13 (2011), 4387-4391. https://doi.org/10.1016/j.na.2011.03.057

B. A. Bin Dehaish and M. A. Khamsi, Browder and Göhde fixed point theorem for monotone nonexpansive mappings, Fixed Point Theory Appl. 2016:20 (2016). https://doi.org/10.1186/s13663-016-0505-8

H. Busemann, Spaces with non-positive curvature, Acta Math. 80 (1948), 259-310. https://doi.org/10.1007/BF02393651

T. Butsan, S. Dhompongsa and W. Takahashi, A fixed point theorem for pointwise eventually nonexpansive mappings in nearly uniformly convex Banach spaces, Nonlinear Anal. 74, no. 5 (2011), 1694-1701. https://doi.org/10.1016/j.na.2010.10.041

J. García-Falset, E. Llorens-Fuster and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375, no. 1 (2011), 185-195. https://doi.org/10.1016/j.jmaa.2010.08.069

H. Fukhar-ud-din and M. A. Khamsi, Approximating common fixed points in hyperbolic spaces, Fixed Point Theory Appl. 2014:113 (2014). https://doi.org/10.1186/1687-1812-2014-113

K. Goebel and M. Japón-Pineda, A new type of nonexpansiveness, Proceedings of 8-th international conference on fixed point theory and applications, Chiang Mai, 2007.

K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174. https://doi.org/10.1090/S0002-9939-1972-0298500-3

K. Goebel, T. Sekowski and A. Stachura, Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball, Nonlinear Anal. 4, no. 5 (1980), 1011-1021. https://doi.org/10.1016/0362-546X(80)90012-7

K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982), Contemp. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1983, pp. 115-123. https://doi.org/10.1090/conm/021/729507

K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. https://doi.org/10.1112/blms/17.3.293

M. Gregus, Jr., A fixed point theorem in Banach space, Boll. Un. Mat. Ital. A (5) 17, no. 1 (1980), 193-198.

M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, english ed., Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates. https://doi.org/10.1007/978-0-8176-4583-0

B. Gunduz and S. Akbulut, Strong convergence of an explicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces, Miskolc Math. Notes 14 (2013), no. 3, 905-913. https://doi.org/10.18514/mmn.2013.641

S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59, no. 1 (1976), 65-71. https://doi.org/10.1090/S0002-9939-1976-0412909-X

M. A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106, no. 3 (1989), 723-726. https://doi.org/10.1090/S0002-9939-1989-0972234-4

M. A. Khamsi and A. R. Khan, Inequalities in metric spaces with applications, Nonlinear Anal. 74 (2011), no. 12, 4036-4045. https://doi.org/10.1016/j.na.2011.03.034

S. H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl. 2013:69 (2013), 10. https://doi.org/10.1186/1687-1812-2013-69

S. H. Khan, D. Agbebaku and M. Abbas, Three step iteration process for two multivalued nonexpansive maps in hyperbolic spaces, J. Math. Ext. 10, no. 4 (2016), 87-109.

S. H. Khan and M. Abbas, Common fixed point results for a Banach operator pair in CAT(0) spaces with applications, Commun. Fac. Sci. Univ. Ank. S'{e}r. A1 Math. Stat. 66 (2017), no. 2, 195-204. https://doi.org/10.1501/commua1_0000000811

S. H. Khan, M. Abbas and T. Nazir, Existence and approximation results for skc mappings in busemann spaces, Waves Wavelets Fractals Adv. Anal. 3 (2017), 48-60. https://doi.org/10.1515/wwfaa-2017-0005

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, no. 2 (2017), 734-743. https://doi.org/10.22436/jnsa.010.02.34

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. 2012:54 (2012), 12. https://doi.org/10.1186/1687-1812-2012-54

W. A. Kirk, Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math. 17 (1974), 339-346. https://doi.org/10.1007/BF02757136

W. A. Kirk, Fixed point theory for nonexpansive mappings, Fixed point theory (Sherbrooke, Que., 1980), Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 484-505. https://doi.org/10.1007/bfb0092201

W. A. Kirk, Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory Appl. 2004:4 (2004), 309-316. https://doi.org/10.1155/S1687182004406081

W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68, no. 12 (2008), 3689-3696. https://doi.org/10.1016/j.na.2007.04.011

U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357, no. 1 (2005), 89-128. https://doi.org/10.1090/S0002-9947-04-03515-9

L. Leustean, Nonexpansive iterations in uniformly convex W-hyperbolic spaces, Nonlinear analysis and optimization I. Nonlinear analysis, Contemp. Math., vol. 513, Amer. Math. Soc., Providence, RI, 2010, pp. 193-210. https://doi.org/10.1090/conm/513/10084

L. Leustean, A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl. 325, no. 1 (2007), 386-399. https://doi.org/10.1016/j.jmaa.2006.01.081

T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179-182. https://doi.org/10.1090/S0002-9939-1976-0423139-X

E. Llorens-Fuster, Orbitally nonexpansive mappings, Bull. Austral. Math. Soc. 93, no. 3 (2016), 497-503. https://doi.org/10.1017/S0004972715001318

W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3

K. Menger, Untersuchungen über allgemeine metrik, Math. Ann. 100, no. 1 (1928), 75-163. https://doi.org/10.1007/BF01448840

S. A. Naimpally, K. L. Singh and J. H. M. Whitfield, Fixed points in convex metric spaces, Math. Japon. 29, no. 4 (1984), 585-597.

A. Nicolae, Generalized asymptotic pointwise contractions and nonexpansive mappings involving orbits, Fixed Point Theory Appl. (2010), Art. ID 458265, 19. https://doi.org/10.1155/2010/458265

M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251, no. 1 (2000), 217-229. https://doi.org/10.1006/jmaa.2000.7042

R. Pant and R. Shukla, Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim. 38, no. 2 (2017), 248-266. https://doi.org/10.1080/01630563.2016.1276075

S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), no. 6, 537-558. https://doi.org/10.1016/0362-546x(90)90058-o

Ritika and S. H. Khan, Convergence of Picard-Mann hybrid iterative process for generalized nonexpansive mappings in CAT(0) spaces, Filomat 31, no. 11 (2017), 3531-3538. https://doi.org/10.2298/FIL1711531R

Ritika and S. H. Khan, Convergence of RK-iterative process for generalized nonexpansive mappings in CAT(0) spaces, Asian-European Journal of Mathematics, to appear. https://doi.org/10.1142/s1793557119500773

H. F. Senter and W. G. Dotson, Jr., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-380. https://doi.org/10.1090/S0002-9939-1974-0346608-8

T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340, no. 2 (2008), 1088-1095. https://doi.org/10.1016/j.jmaa.2007.09.023

W. Takahashi, A convexity in metric space and nonexpansive mappings. I, Kodai Math. Sem. Rep. 22 (1970), 142-149. https://doi.org/10.2996/kmj/1138846111

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. https://doi.org/10.1016/j.amc.2015.11.065

Abstract Views

919
Metrics Loading ...

Metrics powered by PLOS ALM




Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt