# STRONG CONVERGENCE OF GENERAL ITERATIVE ALGORITHMS FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES

• Jung, Jong Soo (Department of Mathematics, Dong-A University)
• Received : 2016.05.13
• Published : 2017.05.01
• 59 14

#### Abstract

In this paper, we introduce two general iterative algorithms (one implicit algorithm and other explicit algorithm) for nonexpansive mappings in a reflexive Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm. Strong convergence theorems for the sequences generated by the proposed algorithms are established.

#### Keywords

nonexpansive mapping;general iterative algorithms;strong positive linear operator;strongly pseudocontractive mapping;fixed points;uniformly $G{\hat{a}}teaux$ differentiable norm

#### Acknowledgement

Supported by : Dong-A University

#### References

1. R. P. Agarwal, D. O'Regan, and D. R. Sagu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, 2009.
2. V. Barbu and Th. Precupanu, Convexity and Optimization in Banach spaces, Editura Academiei R. S. R. Bucharest, 1978.
3. F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. Natl. Acad. Sci. U.S.A. 532 (1965), 1272-1276.
4. G. Cai and C. S. Hu, Strong convergence theorems of a general iterative process for a finite family of ${\lambda}_i$ pseudocontraction in q-uniforly smooth Banach spaces, Comput. Math. Appl. 59 (2010), no. 1, 149-160. https://doi.org/10.1016/j.camwa.2009.07.068
5. M. M. Day, Normed Linear Spaces, 3rd ed. Springer-Verlag, Berlin-New York, 1973.
6. K. Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974), 365-374. https://doi.org/10.1007/BF01171148
7. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, Inc. New York and Basel, 1984.
8. G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006), no. 1, 43-52. https://doi.org/10.1016/j.jmaa.2005.05.028
9. A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000), no. 1, 46-55. https://doi.org/10.1006/jmaa.1999.6615
10. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), no. 1, 287-292. https://doi.org/10.1016/0022-247X(80)90323-6
11. T. Suzuki, A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 135 (2007), no. 1, 99-106.
12. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, 2000.
13. R. Wangkeeree, N. Petrot, and R. Wangkeeree, The general iterative methods for nonexpansive mappings in Banach spaces, J. Global Optim. 51 (2011), no. 1, 27-46. https://doi.org/10.1007/s10898-010-9617-6
14. H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002), no. 1, 240-256. https://doi.org/10.1112/S0024610702003332
15. H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003), no. 3, 659-678. https://doi.org/10.1023/A:1023073621589
16. H. K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006), no. 2, 631-643. https://doi.org/10.1016/j.jmaa.2005.04.082