Convergence theorems of modified Ishikawa iterations in Banach spaces

Year 2019, Volume: 2 Issue: 3, 125 - 135, 01.10.2019

Shengquan Weng

Abstract

In this paper, we introduce the modified iterations of Ishikawa type for nonexpansive mappings (nonexpansive semigroups) to have the strong convergence in a uniformly convex Banach space. We study the approximation of common fixed point of nonexpansive mappings and nonexpansive semigroups in Banach space by using a new iterative scheme.

Keywords

strong convergence , modified Ishikawa iteration , uniformly convex Banach space , nonexpansive mapping , nonexpansive semigroup

References

• [1] F.E. Browder, Fixed point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad . (1965)53, 1272-1276.
• [2] W.R. Mann, Mean value methods on iteration, Proc. Amer. Math. Soc . (1953)4, 506-510.
• [3] S.Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. (1974)44, 147-150.
• [4] Y.F.Su, X.L.Qin, Strong convergence of modified Ishikawa iterations for nonlinear mappings, Proceedings MathematicalSciences.(2007)1,117, 97-107.
• [5] B.Javad, Weak and strong convergence theorems of modified Ishikawa iteration for an infinitely countable family of pointwiseasymptotically nonexpansive mappings in Hilbert spaces, Arab Journal of Mathematical Sciences. (2011) 17, 153-169.
• [6] K.Phayap,et.al, Strong Convergence theorems of modified Ishikawa iterative method for an infinite family of strict pseudocontractionsin Banach spaces, International Journal of Mathematics and Mathematical Sciences. (2011) Article ID 549364,18.
• [7] W.Kriengsak, K.Poom, Convergence theorems of modified Ishikawa iterative scheme for two nonexpansive semigroups, FixedPoint Theory and Applications. (2010) Article ID 914702, 12.
• [8] S.S.Chang,et.al, The equivalence between the convergence of modified Picard, modified mann, and modified Ishikawa iterations,Mathematical and Computer Modelling. (2003)37, 985-991.
• [9] J.Z.Chen, D.P.Wu, Convergence theorems of modified Mann iterations, Fixed Point Theory and Applications. (2013)2013,1, 282.
• [10] M.A.Noor, New approximation schemes for general variational inequalities, J.Math.Anal.Appl. (2000)251, 217-229.
• [11] W.Phuengrattana, S.Suantai, On the rate of convergence of Mann, Ishikawa,Noor and SP-iterations for continuous functionson arbitrary interval, Journalof Computational and Applied Mathematics. (2011)235, 3006-3014.
• [12] T.Suzuki, On strong convergence to a common fixed point of nonexpansive semigroup in Hilbert spaces, Proc. Amer. Math.Soc.(2003)131,7, 371-379.
• [13] H.K.Xu, A strong convergence theorem for contraction semigroups in Banach spaces, Bull Austral Math Soc,(2005)72,371-379.
• [14] N.Shioji,W.Takahashi, Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces, NonlinearAnalysis. (1998)34, 87-99.
• [15] J.B.Baillon, Un theoreme de type ergodic pour les contrations nonlineares dans unespace de Hilbert,C.r.heba.Seanc.Acad.Sci.Paris. (1975)280, 1511-1514.
• [16] K. Nammanee, M.A. Noor, S. Suantai, Convergence criteria of modified Noor iterations with errors for asymptoticallynonexpansive mappings, J. Math. Anal. Appl. (2006)314, 320-334.
• [17] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc.(1991)43, 153-159.
• [18] H.F. Senter, W.G. Dotson,Jr, Approximating fixed points of non-expansive mappings, Proc. Amer. Math. Soc. (1974)44,375-380.
• [19] R.Chen, Y.Song, Convergence to common fixed point of nonexpansive semigroups, J. Math. Anal. Appl. (2007)200,566-575.