Convergence theorems of new three-step iterations scheme for $I$-asymptotically nonexpansive mappings

Document Type : Research Paper

Author

Department of Mathematics, Art and Science Faculty, Ad\i yaman University, 02040, Ad\i yaman, Turkey

Abstract

Abstract. The purpose of this paper is to establish weak and strong convergence theorems of new three-step iterations for I-asymptotically nonexpansive mappings in Banach space.Also we introduce and study convergence theorems of the three-step iterative sequence for three I-asymptotically nonexpansive mappings in an uniformly convex Banach space. The results obtained in this paper extend and improve the recent ones announced by Chen and Guo [1], S. Temir [14], Yao and Noor[16] and many others.

Keywords

[1] W. Chen and W Guo, Convergence theorems for two finite fimilies of asymptotically nonexpansive mappings,
Math. Comp. Model. 54 (2011), 1311–1319.
[2] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math.
Soc. 35 (1972), 171–174.
[3] R. Glowinski, P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM,
Philadelphia, 1989.
[4] J. Gornicki, Weak convergence theorems for asymptotically nonexpansive mappings in uniformly Banach spaces,
Comment. Math. Univ. Carolin. 301 (1989), 249–252.
[5] I. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150.
[6] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510.
[7] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Analy. Appl. 251 (2000),
217–229.
[8] Z. Opial, Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73
(1967), 591–597.[9] B.E. Rhoades and S. Temir, Convergence theorems for I-nonexpansive mapping, Int. J. Math. Math. Sci. 2006
(2006), Article ID 63435, 1–4.
[10] J. Schu, Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math.
Soc. 43 (1991), 153–159.
[10] H.F. Senter, W.G. Dotson, Aproximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44
(1974), 375–380.
[11] S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings,
J. Math. Analy. Appl. 311 (2005), no. 2, 506–517.
[12] K.K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process,
J. Math. Anal. Appl. 178 (1993), 301–308.
[13] S. Temir and O. Gul, Convergence theorem for I-asymptotically quasi-nonexpansive mapping in Hilbert space, J.
Math. Anal. Appl. 329 (2007), 759–765.
[14] S. Temir, On the convergence theorems of implicit iteration process for a finite family of I-asymptotically nonexpansive mappings, J. Comp. Appl. Math. 225 (2009), 398–405.
[15] B.L. Xu and M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J.
Math. Anal. Appl. 267 (2002), 444–453.
[16] Y.Yao and M.A. Noor, Convergence of three-step iterations for asymptotically nonexpansive mappings, Appl.
Math. Comput. 187 (2007), 883–892.
[17] S.Yao and L.Wang, Strong convergence theorems for nonself I-asymptotically quasi-nonexpansive mappings, Appl.
Math. Sci. 2 (2008), 919–928.
Volume 13, Issue 2
July 2022
Pages 1849-1863
  • Receive Date: 05 December 2020
  • Revise Date: 20 December 2020
  • Accept Date: 12 April 2021