A new two step iterative scheme for a finite family of nonself I-asymptotically nonexpansive mappings in Banach space

Let E be a real uniformly convex Banach space, K be a nonempty closed convex subset of E and let Ti : K → E be N Ii-asymptotically nonexpansive nonself mappings and Ii be N asymptotically nonexpansive nonself mappings. It is proved that a new two step iterative algorithm converges weakly to a q ∈ F in a real uniformly convex Banach space such that its dual has the Kadec-Klee property and strongly under condition (B) in a real uniformly convex Banach space. It presents some new results in this paper.

Keywords:

Nonself I-asymptotically nonexpansive mapping, Fr´echet differentiable norm, Opial property, Kadec-Klee property B condition,

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K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174.
S. Temir, On the convergence theorems of implicit iteration process for a finite family of I-asymptotically nonexpansive mappings, J. Comput. Appl. Math. 225 (2009) 398–405.
S. Temir, O. Gul, Convergence theorem for I-asymptotically quasi-nonexpansive mapping in Hilbert space, J. Math. Anal. Appl. 329 (2007) 759–765.
L. Yang, X. Xie, Weak and strong convergence theorems for a finite family of I-asymptotically nonexpansive mappings, Appl. Math. Comput. 216 (2010) 1057–1064.
K.K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993.
S. Akbulut, S. H. Khan, M. Ozdemir, An iteration process for common fixed points of two nonself asymptotically nonexpansive mappings, An. S¸ t. Univ. Ovidius Constanta, Vol. 20(1), 2012, 15-30.
H.F. Senter, W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44(1974), 375-380.
J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43(1991), 153-159.
M.O. Osilike, A. Udomene, Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder–Petryshyn type, J. Math. Anal. Appl. 256 (2001) 431–445.
R.P. Agarwal, Donal O’Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J.Nonliear Convex. Anal.8(1)(2007), 61–79.
C.E. Chidume, E.U. Ofoedu, H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 280 (2003), 364-374.
J.G. Falset, W. Kaczor, T. Kuczumow, S. Reich, Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 43 (2001) 377–401.
S. Thianwan, Common fixed point of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space, J. Comput. Appl. Math. 224 (2009) 688-695.
S. Thianwan, New Iterations with Errors for Approximating Common Fixed Points for two Generalized Asymptotically Quasi-Nonexpansive Nonself-Mappings, Mathematical Notes, 2011, Vol. 89, No. 3, pp. 397-407.
M.O. Osilike, A. Udomene, Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder–Petryshyn type, J. Math. Anal. Appl. 256 (2001) 431-445.
W. Kaczor, Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups, J. Math. Anal. Appl. 272 (2002) 565–574.
B. Gunduz and S. Akbulut, Convergence theorems of a new three-step iteration for nonself asymptotically nonexpansive mappings, Thai J. Math. 13 (2015), no. 2, 465-480.
B. Gunduz and S. Akbulut, On weak and strong convergence theorems for a finite family of nonself I-asymptotically nonexpansive mappings, Math. Morav. 19 (2015), no. 2, 49-64.
B. Gunduz, S.H. Khan, and S. Akbulut, On convergence of an implicit iterative algorithm for non self asymptotically nonexpansive mappings, Hacet. J. Math. Stat. 43 (2014), no. 3, 399-411.
B. Gunduz and S. Akbulut, A one-step implicit iterative process for a finite family of I-nonexpansive mappings in Kohlenbach hyperbolic spaces, Math Sci, 2016, 10, 55-61.
B. Gunduz and S. Akbulut, On the convergence of an iteration process for totally asymptotically I-nonexpansive mappings, J. Nonlinear Anal. Optim., Vol.7, No.1, (2016), 17-30.
I. Yildirim and F. Gu, A new iterative process for approximating common fixed points of nonself I-asymptotically quasinonexpansive mappings, Appl. Math. J. Chinese Univ. Ser. B 27 (2012), no. 4, 489–502.
S. H. Khan and H. Fukhar-ud-din,Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal.TMA, 61(8) 2005, 1295-1301.
B. Gunduz, A new multistep iteration for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces, J. Nonlinear Sci. Appl. 9 (2016), 1365-1372.
B. Gunduz, Convergence of a new multistep iteration in convex cone metric spaces, Commun. Korean Math. Soc. 32(1) (2017), 39-46.
A. Sahin and M. Basarir, On the strong convergence of a modified S-iteration process for asymptotically quasi-nonexpansive mappings in a CAT(0) space, Fixed Point Theory Appl. 2013, Article ID 12, 2013.
A. Sahin and M. Basarir, Convergence and data dependence results of an iteration process in a hyperbolic space, Filomat, 30:3(2016), 569–582.