The Finite Family \(L\)-Lipschitzian Suzuki-Generalized Nonexpansive Mappings
Keywords:Fixed point set, \(L\)-Lipschitzian Suzuki-generalized nonexpansive mappings, Iteration and hyperbolic spaces
AbstractIn this paper, we propose and analyze a \(L\)-Lipschitzian Suzuki-generalized nonexpansive mapping on a nonempty subset of a hyperbolic space and prove \(\Delta\)-convergence theorems and convergence theorems for a \(L\)-Lipschitzian Suzuki-generalized nonexpansive mapping in a hyperbolic space.
M. Abbas, Z. Kadelburg and D.R. Sahu, Fixed point theorems for Lipschitzian type mappings in CAT(0) spaces, Math. Comput. Modeling 55 (2012), 1418 – 1427, DOI: 10.1016/j.mcm.2011.10.019.
M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik 66 (2014), 223 – 234, http://hdl.handle.net/2263/43663.
R.P. Agarwal, D. O'Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Convex Anal. 8(1) (2007), 61 – 79, http://www.ybook.co.jp/online2/opjnca/vol8/p61.html.
M.R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, 319 (2013), Springer Science and Business Media, https://www.springer.com/us/book/9783540643241.
S.S. Chang, G. Wang, L. Wang, Y.K. Tang and Z.L. Ma, (Delta)-convergence theorems for multivalued nonexpansive mappings in hyperbolic spaces, Appl. Math. Comp. 249 (2014), 535 – 540, DOI: doi.org/10.1186/1687-1812-2014-33.
K. Goebel and W.A. Kirk, Iteration Processes for Nonexpansive Mappings, in Topological Methods in Nonlinear Functional Analysis, S.P. Singh, S. Thomeier and B. Watson (eds.), (Toronto, 1982), 115 – 123, Contemp. Math. 21, Amer. Math. Soc. (1983).
K. Goebel and S. Reich, Uniform Convexity Hyperbolic Geometry and Nonexpansive Mappings, Dekker (1984).
M. Imdad and S. Dashputre, Fixed point approximation of Picard normal S-iteration process for generalized nonexpansive mappings in hyperbolic spaces, Math. Sci. 10 (2016), 131 – 138, DOI: 10.1007/s40096-016-0187-8.
S. Ishikawa, Fixed points by new iteration method, Proc. Am. Math. Soc. 149 (1974), 147 – 150, DOI: 10.1090/S0002-9939-1974-0336469-5.
N. Kadioglu and I. Yildirim, Approximating fixed points of nonexpansive mappings by faster iteration process, J. Adv. Math. Stud. 8(2) (2015), 257 – 264, https://arxiv.org/abs/1402.6530.
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. 54 (2012), DOI: 10.1186/1687-1812-2012-54.
W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., Theory Methods Appl. 68(12) (2008), 3689 – 3696, DOI: 10.1016/j.na.2007.04.011.
U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Am. Math. Soc. 357(1) (2005), 89 – 128, DOI: 10.7146/brics.v10i21.21791.
M.A. Krasnosel'ski, Two remarks on the method of successive approximations, Usp. Mat. Nauk. 10 (1955), 123 – 127, http://mi.mathnet.ru/eng/umn7954.
L. Leustean, Nonexpansive iteration in uniformly convex Whyperbolic spaces, in Nonlinear Analysis and Optimization I, Nonlinear Analysis, A. Leizarowitz, B.S. Mordukhovich, I. Shafrir and A. Zaslavski (eds.), Contemporary Mathematics, Vol. 513, pp. 193 – 210, Ramat Gan American Mathematical Society, Bar Ilan University, Providence (2010).
W.R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953), 506 – 510, DOI: 10.1090/S0002-9939-1953-0054846-3.
M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217 – 229, DOI: 10.1006/jmaa.2000.7042.
M.O. Osilike and S.C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modeling 32 (2000), 1181 – 1191, DOI: 10.1016/S0895-7177(00)00199-0.
S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), 537 – 558, DOI: 10.1016/0362-546X(90)90058-O.
D.R. Sahu, Application of the S-iteration process to constrained minimization problem and split feasibility problems, Fixed Point Theory 12 (2011), 187 – 204.
G.S. Saluja, Strong and (Delta)-convergence of modified two-STEP iteration for nearly asymptotically nonexpansive mapping in hyperbolic spaces, International Journal of Analysis and Applications 8(1) (2015), 39 – 52.
H. Schaefer, íœber die methode sukzessiver approximationen, ber. Dtsch. Math. 59 (1957), 131 – 140.
C. Suanoom and C. Klin-eam, Fixed point theorems for generalized nonexpansive mappings in hyperbolic spaces, Journal of Fixed Point Theory and Applications (2017), 2511 – 2528, DOI: 10.1007/s11784-017-0432-2.
C. Suanoom and C. Klin-eam, Remark on fundamentally nonexpansive mappings in hyperbolic spaces, Bull. Austral J. Nonlinear Sci. Appl. 9 (2016), 1952 – 1956, DOI: 10.22436/jnsa.009.05.01.
T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), 1088 – 1095, DOI: 10.1016/j.jmaa.2007.09.023.
W.A. Takahashi, A convexity in metric spaces and nonexpansive mappings I, Kodai Math. Sem. Rep. 22 (1970), 142 – 149, DOI: 10.2996/kmj/1138846111.
How to Cite
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.