Convergence theorems for a general class of nonexpansive mappings in Banach spaces

Document Type : Research Paper


Faculty of Arts and Sciences, Adiyaman University, 02040, Adiyaman, Turkey


In this paper, we introduce a new iteration process for the approximation of fixed points. We show that our iteration process is faster than the existing iteration processes like the M-iteration process and the K-iteration process for contraction mappings. Also, we prove that the new iteration process is stable. Finally, we study the convergence of a new iterative scheme to a fixed point for the $(\alpha,\beta)$ -Reich-Suzuki nonexpansive type mappings in Banach space.


[1] M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik 66 (2014), no. 2, 223–234.
[2] 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.
[3] A. Amini-Harandi, M. Fakhar and H.R. Hajisharifi, Approximate fixed points of α-non-expansive mappings, J. Math. Analy. Appl. 467 (2018), no. 2, 1168–1173.
[4] K. Aoyama and F. Kohsaka, Fixed point theorem for α-nonexpansive mappings in Banach spaces, Nonlinear Anal. 74 (2011), no. 13, 4378–4391.
[5] V. Berinde, On the stability of some fixed point procedures, Bul. Stiint. Univ. Baia Mare, Ser. B, Mat. Inf. 18 (2002), no. 1, 7—14.
[6] J.A. Clarkson, Uniformly Convex Spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414.
[7] M. Edelstein, Fixed point theorems in uniformly convex Banach spaces, Proc. Amer. Math. Soc. 44 (1974), no. 2, 369–374.
[8] A. Ekinci and S. Temir, Convergence theorems for Suzuki generalized nonexpansive mapping in Banach spaces, Tamkang J. Math. 54 (2023), no. 1, 57–67.
[9] A. M. Harder and T. L. Hicks, Stability Results for Fixed Point Iteration Procedures, Math. Japon. 33 (1988), no. 5, 693–706.
[10] N. Hussain, K. Ullah and M. Arshad, Fixed point approximation of Suzuki generalized non-expansive mappings via new faster iteration process, J. Nonlinear Convex Analy. 19 (2018), 1383–1393.
[11] J. Garcia-Falset, E. Llorens-Fuster and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Analy. Appl. 375 (2011), no. 1, 185–195.
[12] M. Gregus Jr., A fixed point theorem in Banach space, Boll. Un. Mat. Ital. A 17 (1980), no. 1, 193–198.
[13] F. Gursoy and V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, eprint arXiv 1403.2546 (2014), 1–16.
[14] I. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150.
[15] R. Kannan, Fixed point theorems in reflexive Banach spaces, Proc. Amer. Math. Soc. 38 (1973), 111–118.
[16] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510.
[17] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Analy. Appl. 251 (2000),217–229.
[18] Z. Opial, Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc.   (1967), 591–597.
[19] R. Pandey, R. Pant, W. Rakocevic and R. Shukla, Approximating fixed points of a general class of nonexpansive mappings in Banach spaces with applications, Results Math. 74 (2019), no. 7, 24 pages.
[20] R. Pant and R. Shukla, Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces, Numer. Funct. Analy. Optim. 38 (2017), no. 2, 248–266.
[21] E. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures Appl. 6 (1890), 149–210.
[22] S. Reich, Kannan’s fixed point theorem, Boll. Un. Mat. Ital. 4 (1971), no. 4, 1–11. 23] J. Schu, Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), 153–159.
[24] H.F. Senter and W.G. Dotson Jr., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375–380.
[25] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Analy. Appl. 340 (2008), no. 2, 1088–1095.
[26] S. Temir, Weak and strong convergence theorems for three Suzuki’s generalized nonexpansive mappings, Pub. Inst. Math. 110 (2021), no. 124, 121–129.
[27] B.S. Thakur, D. Thakur and M. Postolache, A new iteration scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings, Appl. Math. Comput. 275 (2016), 147–155.
[28] K. Ullah and M. Arshad, Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat 32 (2018), no. 1, 187–196.
[29] K. Ullah, J. Ahmad and M. de la Sen, On generalized nonexpansive maps in Banach spaces, Computation 8, 61 (2020).
Volume 14, Issue 6
June 2023
Pages 371-386
  • Receive Date: 01 May 2021
  • Revise Date: 03 August 2021
  • Accept Date: 21 August 2021
  • First Publish Date: 18 March 2023