Strong and \(\Delta\)-Convergence for Asymptotically \(k\)-Strictly Pseudo-Contractive Mappings in CAT(0) Spaces

Authors

  • Nuttapol Pakkaranang Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140
  • Poom Kumam Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140

DOI:

https://doi.org/10.26713/cma.v7i3.418

Keywords:

Fixed point, Asymptotically \(k\)-strictly pseudo-contracttive mappings, Convergence theorems, CAT(0) spaces

Abstract

In this paper, we study and prove fixed point and convergence theorems for asymptotically \(k\)-strictly pseudo-contracttive mappings in CAT(0) spaces. Our result extend and improve many results in the literature.

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References

M.R. Bridon and A. Haefliger, Metric Spaces of Non-Positive Curvature, Vol. 319, Grundlehren der Mathematischen Wissenchaften, Springer, Berlin, Germany (1999).

F.E. Browder and W.V. Pertryshyn, Construction of fixed points of nonlinear mapping in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197–228.

F. Bruhat and J. Tits, Groupes ré ductifs sur un corps local. I. Données radicielles valuées, Publ. Math. IHES. 41 (1972), 5–251.

Y.J. Cho, H.Y. Zho and G. Guo, Weak and strong convergence theorems for three step iteration with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47 (2004), 707–717.

S. Dhompongsa, W.A. Kirk and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal. 8 (2007), 35–45.

S. Dhompongsa,W.A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., Theory Methods Appl. 65 (2006), 762–772.

S. Dhompongsa and B. Panyanak, On (Delta)-convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56 (2008), 2572–2579.

T.L. Hicks and J.R. Kubicek, On the Mann iterative process in Hilbert space, J. Math. Anal. Appl. 59 (1977), 498–504.

S.H. Khan and M. Abbas, Strong and (Delta)-convergence of some iterative schemes in CAT(0) spaces, Comput. Math. Appl. 61 (2011), 109–116.

W.A. Kirk, Geodesic geometry and fixed point theory, in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colecc, Abierta, 64, 195–225, Universidad de Sevilla Secretariado de Publicaciones, Sevilla (2003).

W.A. Kirk, Geodesic geometry and fixed point theory II, in International Conference on Fixed Point Theo. Appl. 113–142, Yokohama Publ., Yokohama (2004).

W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Analysis: Theory, Methods and Applications, 68 (12) (2008), 3689–3696.

P. Kumam, G.S. Saluja and H.K. Nashine, Convergence of modified S-iteration process for two asymptotically nonexpansive mappings in the intermediate sense in CAT(0) spaces, J. Inequalities Appl. (2014), 2014:368.

T.C. Lim, Remarks on some fixed point theorems, Proceeding of the American Mathematical Society 60 (1976), 179–182.

W.R. Mann, Mean value mathods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510.

G. Marino and H.K. Xu, Weak and strong convergence theorem for strict preudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), 336–346.

B. Nanjaras and B. Panyanak, Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces, Fixed Point Theory and Applications 2010, Article ID 268780 (2010), 14 pages.

M.O. Osilike and A. Udomene, Demiclosedness principle and convergence result for strictly pseudocontractive mappings of Browder-Petryshyn type, J. Math. Anal. Appl. 256 (2002), 77–78.

L. Qihou, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mapping, Nonlinear Anal. 26 (11) (1996), 1835–1842.

A. Sahin and M. Basarir, Some iterative algorithms for k-strickly pseudo-contractive mappings in a CAT(0) space, WSEAS Transactions on Mathematics 13 (2014), 577–585.

A. Sahin and M. Basarir, On the strong and (Delta)-convergence of SP-iteration on a CAT(0) space, J. Inequal. Appl. 2013, Article ID 311 (2013), doi:10.1186/1029-242X-2013-311.

P. Saipara, P. Chaipunya, Y.J. Cho and P. Kumam, On strong and ¢-convergence of modified S-iteration for uniformly continuous total asymptotically nonexpansive mappings in CAT((kappa)) spaces, The Journal of Nonlinear Science and Applications, 8 (2015), 965–975.

H.K. Xu and M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002), 444–453.

Y. Yao and M. Postolache, Iterative methods for pseudomonotone variational inequalities and fixed point problems, J. Optim. Theory Appl. 155 (1) (2012), 273–287.

CMA Vol 7 issue 3 (2016)

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Published

14-11-2016
CITATION

How to Cite

Pakkaranang, N., & Kumam, P. (2016). Strong and \(\Delta\)-Convergence for Asymptotically \(k\)-Strictly Pseudo-Contractive Mappings in CAT(0) Spaces. Communications in Mathematics and Applications, 7(3), 189–197. https://doi.org/10.26713/cma.v7i3.418

Issue

Vol. 7 No. 3 (2016)

Section

Research Article

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