Strong convergence of a modified viscosity iteration for common zeros of a finite family of accretive mappings

Volume 10, Issue 9, pp 4751--4759 http://dx.doi.org/10.22436/jnsa.010.09.18

Publication Date: September 12, 2017 Submission Date: June 08, 2017

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Authors

Yuanheng Wang - Department of Mathematics, Zhejiang Normal University, 321004 Zhejiang, China. Yan Li - Department of Basic Science, Nanyang Polytechnic Institute, 473000 Henan, China. Chanjuan Pan - Department of Mathematics, Zhejiang Normal University, 321004 Zhejiang, China.

Abstract

A new modified iterative scheme \(\{x_n\}\) is given for the viscosity approximating a common zero of a finite family of accretive mappings \(\{A_{i}\}\) in reflexive Banach spaces with a weakly continuous duality mapping \(J\) in the present paper. Under certain conditions, we prove the strong convergence of the sequence \(\{x_n\}\). The results here extend and improve the corresponding recent results of some other authors.

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ISRP Style

Yuanheng Wang, Yan Li, Chanjuan Pan, Strong convergence of a modified viscosity iteration for common zeros of a finite family of accretive mappings, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4751--4759

AMA Style

Wang Yuanheng, Li Yan, Pan Chanjuan, Strong convergence of a modified viscosity iteration for common zeros of a finite family of accretive mappings. J. Nonlinear Sci. Appl. (2017); 10(9):4751--4759

Chicago/Turabian Style

Wang, Yuanheng, Li, Yan, Pan, Chanjuan. "Strong convergence of a modified viscosity iteration for common zeros of a finite family of accretive mappings." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4751--4759

Keywords

Viscosity approximation method
accretive mapping
common zero
strong convergence
reflexive Banach space.

MSC

47H10
47H09
47J25

References

[1] L. C. Ceng, A. R. Khan, Q. H. Ansari, J. C. Yao, Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach spaces, Nonlinear Anal., 70 (2009), 1830–1840.

[2] R. D. Chen, Y. J. Liu, X. L. Shen, Iterative approximation of a zero of accretive operator in Banach space, Nonlinear Anal., 71 (2009), 346–350.

[3] R. D. Chen, Z. C. Zhu, Viscosity approximation fixed point for nonexpansive and m-accretive operators, Fixed Point Theory Appl., 2006 (2006), 10 pages.

[4] R. D. Chen, Z. C. Zhu , Viscosity approximation fixed point for accretive operator in Banach space, Nonlinear Anal., 69 (2008), 1356–1363.
- View Article
- Google Scholar

[5] Y. J. Cho, X. L. Qin, Viscosity approximation methods for a family of m-accretive mappings in reflexive Banach spaces, Positivity, 12 (2008), 483–494.

[6] L. G. Hu, L. W. Liu, A new iterative algorithm for common solutions of a finite family of accretive operators, Nonlinear Anal., 70 (2009), 2344–2351.

[7] . S. Jung , Convergence of composite iterative methods for finding zeros of accretive operators, Nonlinear Anal.,, 71 (2009), 1736–1746.

[8] T. H. Kim, H. K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal., 61 (2005), 51–60.
- View Article
- Google Scholar

[9] T. C. Lim, H. K. Xu, Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal., 22 (1994), 1345–1355.
- Google Scholar

[10] A. Moudafi , Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55.
- View Article
- Google Scholar

[11] Y. H. Wang, S. Chebbi, H. K. Xu , Weak convergence of an iterative format for finding common zeros of two accretive operators in Banach spaces, J. Nonlinear Convex Anal., 16 (2015), 1937–1947.

[12] Y. H. Wang, Y. H. Xia, Strong convergence for asymptotically pseudocontractions with the demiclosedness principle in Banach spaces, Fixed Point Theory Appl., 2012 (2012), 8 pages.

[13] H. K. Xu , Iterative algorithms for nonlinear operator, J. London Math. Soc., 66 (2002), 240–256.
- View Article
- Google Scholar

[14] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279–291.
- View Article
- Google Scholar

[15] H. K. Xu , Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl., 314 (2006), 631–643.

[16] Y. H. Yao, N. Shahzad , Strong convergence of a proximal point algorithm with general errors, Optim. Lett., 6 (2012), 621–628.

[17] Y. H. Yao, N. Shahzad, Y. C. Liou , Modified semi-implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015), 15 pages.

[18] H. Zegeye, N.Shahzad , Strong convergence theorems for a common zero of a finite family of m-accretive mapping, Nonlinear Anal., 66 (2007), 1161–1169.

[19] Q. Yuan, S. Lv , Strong convergence of a parallel iterative algorithm in a reflexive banach space, Fixed Point Theory Appl., 2014 (2014), 1–9.