Abstract
In this paper, we prove that the sequence {x n } generated by modified Krasnoselskii–Mann iterative algorithm introduced by Yao et al. [J Appl Math Comput 29:383–389, 2009] converges strongly to a fixed point of a nonexpansive mapping T in a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. Furthermore, we present an example that illustrates our result in the setting of a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. The results of this paper extend and improve several results presented in the literature in the recent past.
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Aoyama K., Kimura Y., Takahashi W., Toyoda M.: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 2350–2360 (2006)
Banach S.: Sur les operations dans les ensemble abstraits et leur applications aux equations integrales. Fund. Math. 3, 133–181 (1922)
Berinde, V.: Iterative approximation of fixed points. Lecture Notes in Mathematics 1912, Springer, Berlin 2007.
Browder F.E.: Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull. Am. Math. Soc. 73, 875–882 (1967)
Browder. F.E.: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In: Proceedings of Symposia in Pure Mathematics, vol. XVIII, No. Part 2, 1976
Browder F.E.: The solvability of nonlinear functional equations. Duke Math. J. 30, 557–566 (1963)
Browder F.E.: Nonlinear elliptic boundary value problems. Bull. Am. Math. Soc. 69, 862–874 (1963)
Browder F.E.: Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces. Bull. Am. Math. Soc. 73, 470–475 (1967)
Browder F.E.: Nonlinear monotone and accretive operators in Banach spaces. Proc. Natl. Acad. Sci. USA 61, 388–393 (1968)
Browder F.E.: Nonexpansive nonlinear mappings in a Banach space. Proc. Nat. Acad. Sci. USA. 54, 1041–1044 (1965)
Bruck, R.E.: Asymptotic behaviour of nonexpansive mappings. In: Sine, R.C. (ed.) Contemporary Mathematics, vol. 18. Fixed Points and Nonexpansive Mappings, AMS, Providence (1980)
Bynum W.L.: Normal structure coefficients for Banach spaces. Pac. J. Math. 86, 427–436 (1980)
Byrne C.: Unified treatment of some algorithms in signal processing and image construction. Inverse Problems 20, 103–120 (2004)
Caccioppoli R.: Un teorema generale sull’esistenza di elementi uniti in una trasformazione funzionale. Rend. Lincei 11, 794–799 (1930)
Caristi, J.: The fixed point theory for mappings satisfying inwardness conditions. Ph.D. Thesis, The University of Iowa, Iowa City (1975)
Chidume, C.E.: Geometric properties of Banach spaces and nonlinear iterations. Springer Verlag Series: Lecture Notes in Mathematics, vol. 1965 (2009), XVII, 326p, ISBN 978-1-84882-189-7
Cioranescu I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht (1990)
Deimling K.: Nonlinear Functional Analysis. Springer, Berlin (1985)
Genel A., Lindenstrass J.: An example concerning fixed points. Isr. J. Math. 22, 81–86 (1975)
Goebel, K., Kirk, W.A.: Topics in metric fixed point theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)
Ishikawa S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44((1), 147–150 (1974)
Kato T.: Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19, 508–520 (1967)
Krasnoselskii M.A.: Two remarks on the method of successive approximations (Russian). UspehiMat. Nauk 10, 123–127 (1955)
Lim T.C.: Characterization of normal structure. Proc. Am. Math. Soc. 43, 313–319 (1974)
Lim T.C., Xu H.K.: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. TMA 22, 1345–1355 (1994)
Mann W.R.: Mean value method in iteration. Proc. Am. Math. Soc 4, 506–510 (1953)
Martin R.H.Jr.: A global existence theorem for authonomous differential equations in Banach spaces. Proc. Am. Math. Soc. 26, 307–314 (1970)
Martin, Jr., R.H.: Nonlinear Operators and Differential Equations in Banach Spaces. Interscience, New York (1976)
Ray W.O.: An elementary proof of surjectivity for a class of accretive operators. Proc. Am. Math. Soc. 75, 255–258 (1979)
Picard E.: Memoire sur la theorie des equations aux derives partielles et la methode des approximations successives. J. Math. Pures et Appl. 6, 145–210 (1890)
Podilchuk C.I., Mammone R.J.: Image recovery by convex projections using a least-squares constraint. J. Opt. Soc. Am. A 7, 517–521 (1990)
Shioji S., Takahashim W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 125, 3641–3645 (1997)
Tan K.K., Xu H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)
Xu H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66(2), 240–256 (2002)
Xu H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)
Yang Q., Zhao J.: Generalized KM theorems and their applications. Inverse Problem 21, 1971–1979 (2006)
Yao Y., Zhou H., Liou Y.C.: Strong convergence of a modified Krasnoselskii–Mann iterative algorithm for nonexpansive mappings. J. Appl. Math. Comput. 29, 383–389 (2009)
Youla D.: On deterministic convergence of iterations of related projection mappings. J. Vis. Commun. Image Represent 1, 12–20 (1990)
Zalinescu C.: On uniformly convex functions. J. Math. Anal. Appl. 95, 344–374 (1983)
Zalinescu C.: Analysis in General Vector Spaces. World Scientific, River Edge (2002)
Zhang S.: On the convergence problems of Ishikawa and Mann iteration processes with error for φ-pseudo contractive type mappings. Appl. Math. Mech. 21, 1–10 (2000)
Zhao J., Yang Q.: A note on the Krasnoselskii–Mann theorem and its generalizations. Inverse Problem 23, 1011–1016 (2007)
Zegeye H., Shahzad N., Alghamdi M.A.: Convergence of Ishikawa’s iteration method for pseudocontractive mappings. Nonlinear Anal. 74, 7304–7311 (2011)
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Shehu, Y. Modified Krasnoselskii–Mann iterative algorithm for nonexpansive mappings in Banach spaces. Arab. J. Math. 2, 209–219 (2013). https://doi.org/10.1007/s40065-013-0066-1
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DOI: https://doi.org/10.1007/s40065-013-0066-1