A modified Mann iterative scheme based on the generalized explicit methods for quasi-nonexpansive mappings in Banach spaces

Year 2019, Volume: 3 Issue: 2, 90 - 101, 30.06.2019

Thierno Sow

https://doi.org/10.31197/atnaa.557149

Abstract

In this paper,  we  introduce and study a new iterative algorithm which is a combination of a modified Mann iterative scheme and a generalized    explicit methods (GEM) for finding  a common  fixed points of an infinite family of  quasi-nonexpansive mappings in Banach spaces. Under suitable conditions, some strong convergence theorems for finding  a common  fixed points of an infinite family of quasi-nonexpansive mappings are obtained without imposing any compactness assumption. Presented results improve and generalize many known results in the current literature.

Keywords

Common fixed points , Quasi-nonexpansive mappings , Midpoint rule

References

• K. Aoyama, I. H. Koji, W. Takahashi, Weak convergence of an iterative sequencefor accretive operators in Banach spaces, Fixed Point Theory Appl. (2006), Art. ID35390, 13 pp.
• F. E. Browder, Convergenge theorem for sequence of nonlinear operator in Banachspaces, Math.Z. 100 (1967). 201-225. Vol. EVIII, part 2, 1976.
• I. Cioranescu, Geometry of Banach space, duality mapping and nonlinear problems,Kluwer, Dordrecht,1990.
• C. E. Chidume, Geometric Properties of Banach spaces and Nonlinear Iterations,Springer Verlag Series: Lecture Notes in Mathematics, Vol. 1965,(2009), ISBN 978-1-84882-189-7.
• K. Goebel, and W.A. Kirk, Topics in metric fixed poit theory, Cambridge Studies,in Advanced Mathemathics,. vo, 28, University Cambridge Press, Cambridge 1990.
• E. Hairer, S.P. Nrsett, G. Wanner, Solving Ordinary Differential Equations I: NonstiffProblems, 2nd edn. Springer Series in Computational Mathematics. Springer, Berlin(1993).
• J.D. Hoffman, Numerical Methods for Engineers and Scientists, 2nd ed.; MarcelDekker, Inc.: New York, NY, USA, 2001.
• Ke, Y. Ma, C., The generalized viscosity implicit rules of nonexpansive mappings inHilbert spaces. Fixed Point Theory Appl. 2015, 2015, 190.
• T.C. Lim, H.K. Xu, Fixed point theorems for assymptoticaly nonexpansive mapping,Nonliear Anal 22(1994), no. 11, 1345-1355.
• W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953)506-510.P. E. Mainge, Strong convergence of projected subgradient methods for nonsmoothand nonstrictly convex minimization, Set-Valued Analysis, 16, 899-912 (2008).
• G. Marino, B. Scardamaglia, R. Zaccone, A general viscosity explicit midpoint rulefor quasi-nonexpansive mappings. J.Nonlinear Convex Anal. 2017, 18, 137-148.
• A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math.Anal. Appl. 241, 46-55 (2000).
• W. Nilsrakoo, S. Saejung, Weak and strong convergence theorems for countable Lipschitzian mappings and its applications,, Nonlinear Anal. 69 (2008) 2695-2708.
• Z. Opial, Weak convergence of sequence of succecive approximation of nonexpansivemapping, Bull. Am. Math. Soc. 73 (1967), 591 597.[16] J. W. Peng, J.C. Yao, Strong convergence theorems of an iterative scheme basedon extragradient method for mixed equilibruim problem and fixed point problems,Math. Com. Model. 49 (2009) 1816-1828.
• S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces,Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274276, 1979.
• C. Schneider, Analysis of the linearly implicit mid-point rule for differential-algebraicequations. Electron. Trans. Numer. Anal. 1, 1-10 (1993).
• T.M.M. Sow, N. Djitt, and C.E. Chidume, A path convergence theorem and construc-tion of fixed points for nonexpansive mappings in certain Banach spaces, CarpathianJ.Math.,32(2016),No.2,217-226,2016.
• K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinitenonexpansive mappings and applications, Taiwanese J. Math. 5 (2001).
• H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991),no. 12, 1127-1138.
• H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66(2002), no. 2, 240 - 256.
• H.K. Xu, Alghamdi, M.A. Shahzad, N., The viscosity technique for the implicit mid-point rule of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl.2015, 41 (2015).
• Y. Yao, H. Zhou, Y. C. Liou, Strong convergence of modified Krasnoselskii-Manniterative algorithm for nonexpansive mappings, J. Math. Anal. Appl. Comput. 29(2009) 383-389.
• C. Zalinescu, On uniformly convex functions, J. Math. Anal. Appl. 95 (1983), 344-374.