Convergence theorems and stability results of pointwise asymptotically nonexpansive mapping in Banach space


Qiansheng Feng - Department of Mathematics, Tianjin University, Tianjin, 300354, China
Nan Jiang - Department of Mathematics, Tianjin University, Tianjin, 300354, China
Yongfu Su - Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China


The purpose of this paper is to approximate the fixed point of pointwise asymptotically nonexpansive mapping using the generalized Mann and generalized Ishikawa iterative scheme. And under the condition that the pointwise asymptotically nonexpansive mapping is compact, the stability results of the two iterative schemes are studied. The main results of this paper modify and improve many important recent results in the literature.



[1] I. D. Arandjelović, Note on asymptotic contractions, Appl. Anal. Discrete Math., 1 (2007), 211–216.
[2] J. Balooee, Weak and strong convergence theorems of modified Ishikawa iteration for an infinitely countable family of pointwise asymptotically nonexpansive mappings in Hilbert spaces, Arab J. Math. Sci., 17 (2011), 153–169.
[3] A. O. Bosede, B. E. Rhoades, Stability of Picard and Mann iteration for a general class of functions, J. Adv. Math. Stud., 3 (2010), 23–25.
[4] H. Dehghan, Demiclosed principle and convergence of a hybrid algorithm for multivalued *-nonexpansive mappings, Fixed Point Theory, 14 (2013), 107–115.
[5] K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171–174.
[6] A. M. Harder, T. L. Hicks, Stability results for fixed point iteration procedures, Math. Japon., 33 (1988), 693–706.
[7] G. Khalilzadeh, R. Sarikhani, Fixed Point for pointwise asymptotically nonexpansive mapping in Banach Space which has Frechet Differential Norm, Int. J. Math. Anal., 7 (2013), 425–432.
[8] W. A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl., 277 (2003), 645–650.
[9] W. A. Kirk, Asymptotic pointwise contractions, Plenary Lecture, the 8th International Conference on Fixed Point Theory and Its Applications, Chiang Mai University, Thailand, (2007), 16–22.
[10] W. A. Kirk, H.-K. Xu, Asymptotic pointwise contractions, Nonlinear Anal., 69 (2008), 4706–4712.
[11] W. M. Kozlowski, Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 377 (2011), 43–52.
[12] Z. Ma, L. Wang, Demiclosed principle and convergence theorems for asymptotically strictly pseudononspreading mappings and mixed equilibrium problems, Fixed Point Theory Appl., 2014 (2014), 20 pages.
[13] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597.
[14] S. Rezapour, R. H. Haghi, B. E. Rhoades, Some results about T-stability and almost T-stability, Fixed Point Theory, 12 (2011), 179–186.
[15] B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl. Math., 21 (1990), 1–9.
[16] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153–159.
[17] K.-K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301–308.
[18] R.Wangkeeree, H. Dehghan, Strong and-convergence of Moudafi’s iterative scheme in CAT(0) spaces, J. Nonlinear Conv. Anal., 16 (2015), 299-309.
[19] H.-K. Xu, Asymptotic and weakly asymptotic contractions, Indian J. Pure Appl. Math., 36 (2005), 145–150.
[20] Q. Yuan, B. E. Rhoades, T-Stability of Picard Iteration in Metric Spaces, Fixed Point Theory and Appl., 2008 (2008), 4 pages.
[21] T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math., 23 (1972), 292–298.