Existence and convergence results for monotone nonexpansive type mappings in‎ ‎partially ordered hyperbolic metric spaces

Document Type: Research Paper

Authors

1 Department of Mathematics‎, ‎Visvesvaraya National Institute of Technology‎, ‎Nagpur 440010‎, ‎India.

2 Faculty of Mathematics‎, ‎University of Belgrade‎, ‎Studentski trg 16/IV‎, ‎11000 Beograd‎, ‎Serbia.

3 Department of Mathematics‎, ‎Texas A & M University‎, ‎Kingsville‎, ‎78363-8202‎, ‎Texas‎, ‎USA.

Abstract

‎We present some existence and convergence results for a general class of nonexpansive mappings in partially ordered hyperbolic metric spaces‎. ‎We also give some examples to show the generality of the mappings considered herein.

Keywords

Main Subjects


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.

B.A. Bin Dehaish and M.A. Khamsi, Browder and Gohde fixed point theorem for monotone nonexpansive mappings, Fixed Point Theory Appl. 2016 (2016), no. 20, 9 pages.

F.E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1272--1276.

F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041--1044.

H. Busemann, Spaces with non-positive curvature, Acta Math. 80 (1948) 259--310.

R. Espnola and P. Lorenzo, Metric fixed point theory on hyperconvex spaces: recent progress, Arab. J. Math. 1 (2012), no. 4, 439--463.

K. Goebel and W.A. Kirk, Iteration processes for nonexpansive mappings, in: Topological Methods in Nonlinear Functional Analysis (Toronto, Ont., 1982), pp. 115--123.

Contemp. Math. 21, Amer. Math. Soc. Providence, RI, 1983.

K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math. 28, Cambridge Univ. Press, Cambridge, 1990.

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics 83, Marcel Dekker, New York, 1984.

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 (1980), no. 5, 1011--1021.

D. Gohde, Zum prinzip der kontraktiven abbildung, Math. Nachr. 30 (1965) 251--258.

M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates, Birkhäuser Boston, Boston, MA, 2007.

S. Itoh, Some fixed-point theorems in metric spaces, Fund. Math. 102 (1979), no. 2, 109--117.

M.A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106 (1989), no. 3, 723--726.

M.A. Khamsi and A.R. Khan, Inequalities in metric spaces with applications, Nonlinear Anal. 74 (2011), no. 12, 4036--4045.

A.R. Khan, H. Fukhar-ud-din and M.A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 2012 (2012), no. 54, 12 pages.

W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly. 72 (1965) 1004--1006.

W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), no. 12, 3689--3696.

U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 (2005), no. 1, 89--128.

L. Leuştean, Nonexpansive iterations in uniformly convex W-hyperbolic spaces, in: Non-linear Analysis and Optimization I. Nonlinear Nnalysis, pp. 193--210, Contemp. Math. 513, Amer. Math. Soc. Providence, RI, 2010.

T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179--182 (1977).

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

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.

J.J. Nieto and R. Rodríguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), no. 3, 223--239.

W. Phuengrattana, Approximating fixed points of Suzuki-generalized nonexpansive mappings, Nonlinear Anal. Hybrid Syst. 5 (2011), no. 3, 583--590.

A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435--1443.

A. Razani and H. Salahifard, Approximating fixed points of generalized nonexpansive mappings, Bull. Iranian Math. Soc. 37 (2011), no. 1, 235--246.

S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), no. 6, 537--558.

H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974) 375--380.

Y. Song, P. Kumam and Y.J. Cho, Fixed point theorems and iterative approximations for monotone nonexpansive mappings in ordered Banach spaces, Fixed Point Theory Appl. 2016 (2016), no. 73, 11 pages.

T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), no. 2, 1088--1095.

W. Takahashi, A convexity in metric space and nonexpansive mappings. I, Kodai Math. Sem. Rep. 22 (1970) 142--149.


Volume 43, Issue 7
November and December 2017
Pages 2547-2565
  • Receive Date: 10 December 2016
  • Revise Date: 16 December 2017
  • Accept Date: 23 December 2017