A New Iterative Algorithm for Total Asymptotically Non-Expansive Mapping in CAT(0) Spaces
Samir Dashputre *
Department of Mathematics, Shahid Durwasha Nishad Govt. College, Arjunda, Dist.- Balod (C.G.), 491225, India.
Rakesh Tiwari
Department of Mathematics, Govt. V.Y.T. PG Autonomous College, Durg (C.G.), 491001, India.
Jaynendra Shrivas
Department of Mathematics, Govt. V.Y.T. PG Autonomous College, Durg (C.G.), 491001, India.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we provide certain fixed point results for a total asymptotically non-expansive mapping, as well as a new iterative algorithm for approximating the fixed point of this class of mappings in the setting of CAT(0) spaces. Furthermore, we establish strong and \(\Delta\)-converges theorem for total asymptotically non-expansive mapping in CAT(0) space. Our result, generalizes, improve, extend and unify the results of Thakur et al. [1], Izhar et al. [2] and many more in this direction.
Keywords: CAT (0) space, total asymptotically non-expansive mappings, strong and \(\Delta\)-convergence theorems.
How to Cite
Downloads
References
Thakur BS, Thakur D, Postolache M. Modi ed Picard-Mann hybrid iteration process for total asymptotically non-expansive mappings. Fixed Point Theory and Applications. 2015;2015:1-1.
Uddin I, Khatoon S, Mlaiki N, Abdeljawad T. A modi ed iteration for total asymptotically non-expansive mappings in hadamard spaces. AIMS Mathematics. 2021;6(5):4758{4770.
Mann WR. Mean value methods in iteration. Proceedings of the American Mathematical Society. 1953;4(3):506{510.
Ishikawa S. Fixed points by a new iteration method. Proceedings of the American Mathematical Society. 1974;44(1):147-50.
Noor MA. New approximation schemes for general variational inequalities. Journal of Mathematical Analysis and applications. 2000;251(1):217{229.
Agarwal RP, O Regan D, Sahu D. Iterative construction of xed points of nearly asymptotically non- expansive mappings. Journal of Nonlinear and convex Analysis. 2007;8(1):61.
Kumam W, Pakkaranang N, Kumam P. Modi ed viscosity type iteration for total asymptotically non- expansive mappings in CAT (0) spaces and its application to optimization problems. J. Nonlinear Sci. Appl. 2018;11(2):288-302.
Dhompongsa S, Panyanak B. On-convergence theorems in CAT (0) spaces. Computers Mathematics with Applications. 2008;56(10):2572-9.
Goebel K, Kirk WA. A xed point theorem for asymptotically non-expansive mappings. Proceedings of the American Mathematical Society. 1972;35(1):171-4.
Bruhat F, Tits J. Groupes r´eductifs sur un corps local. Publications Math´ematiques de l'Institut des Hautes Etudes Scienti ques. 1972;41:5{251.
Brown KS, Brown KS. Buildings. Springer; 1989.
Bridson MR, Hae iger A. Metric Spaces of Non-Positive Curvature. Springer Science and Business Media. 2013;319.
Basarir M, Sahin A. On the strong and -convergence theorems for total asymptotically non-expansive mappings on CAT (0) space. Carpathian Mathematical Publications. 2013 Dec 30;5(2):170-9.
Fukhar-ud-din H, Saleh K. One-step iterations for a nite family of generalized non-expansive mappings in CAT (0) spaces. Bulletin of the Malaysian Mathematical Sciences Society. 2018;41(2):597-608.
Khatoon S, Uddin I, Pakkaranang N, Wairojjana N. Common xed points of modi ed Picard-s iteration process involving two g-non-expansive mapping in CAT(0) space with directed graph. Indian J. Pure Appl. Math. 2019;18(1):1-13.
Pakkaranang N, Kumam P. Strong and -convergence for asymptotically K-strictly pseudo-contractive mappings in CAT(0) spaces. Communications in Mathematics and Applications. 2016;7(3):189{197.
Bruck R, Kuczumow T, Reich S. Convergence of iterates of asymptotically non-expansive mappings in Banach spaces with the uniform Opial property. InColloquium Mathematicae. 1993;65(2):169-179.
Gunduz B, Akbulut S. On the convergence of an iteration process for totally asymptotically I-non-expansive mappings. Journal of Nonlinear Analysis and Optimization: Theory & Applications. 2016;7(1):17-30.
Alber Y, Chidume CE, Zegeye H. Approximating xed points of total asymptotically non-expansive mappings. Fixed point theory and applications. 2006;2006:1-20.
Chidume CE, Ofoedu EU. Approximation of common xed points for nite families of total asymptotically non-expansive mappings. Journal of Mathematical Analysis and Applications. 2007;333(1):128-41.
Chidume CE, Ali B. Approximation of common xed points for nite families of nonself asymptotically non- expansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications. 2007;326(2):960- 73.
Chang SS, Wang L, Lee HJ, Chan CK, Yang L. Demiclosed principle and -convergence theorems for total asymptotically non-expansive mappings in CAT (0) spaces. Applied Mathematics and Computation. 2012;219(5):2611-7.
Plubtieng S, Ungchittrakool K. Strong convergence of modi ed ishikawa iteration for two asymptotically non-expansive mappings and semigroups. Nonlinear Analysis: Theory, Methods & Applications. 2007;43(1):2306{2315.
Kim GE, Kim TH. Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces. Computers & Mathematics with Applications. 2001;42(12):1565-70.
Dhompongsa S, Kirk WA, Sims B. Fixed points of uniformly Lipschitzian mappings. Nonlinear analysis: theory, methods & applications. 2006;65(4):762-72.
Dhompongsa S, Kirk WA, Panyanak B. non-expansive set-valued mappings in metric and Banach spaces. Journal of nonlinear and convex analysis. 2007;8(1):35-45.
Karapnar E, Salahifard H, Vaezpour SM. Demiclosedness principle for total asymptotically non-expansive mappings in spaces. Journal of Applied Mathematics. 2014;22(3):353{355.
Schu J. Weak and strong convergence to xed points of asymptotically non-expansive mappings. Bulletin of the Australian Mathematical Society. 1991;43(1):153-9.
Qihou L. Iterative sequences for asymptotically quasi-non-expansive mappings. Journal of Mathematical Analysis and applications. 2001;259(1):1-7.
Senter HF, Dotson W. Approximating xed points of non-expansive mappings. Proceedings of the American Mathematical Society. 1974;44(2):375-80.
Shahzad N, Al-Dubiban R. A Approximating common xed points of non-expansive mappings in Banach spaces. Science Asia. 2006;41:289{294.
Kim GE. Strong convergence to a xed point of a total asymptotically non-expansive mapping. Fixed Point Theory and Applications. 2013;2013(1):1-0.