Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T02:42:23.690Z Has data issue: false hasContentIssue false
  • English
  • Français

STRONG CONVERGENCE OF SOME ALGORITHMS FOR λ-STRICT PSEUDO-CONTRACTIONS IN HILBERT SPACE

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  02 February 2012

YONGHONG YAO ,
YEONG-CHENG LIOU  and
GIUSEPPE MARINO
YONGHONG YAO
Affiliation:
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China (email: [email protected])
YEONG-CHENG LIOU
Affiliation:
Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan (email: [email protected])
GIUSEPPE MARINO*
Affiliation:
Dipartimento di Matematica, Università della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Two algorithms have been constructed for finding the minimum-norm fixed point of a λ-strict pseudo-contraction T in Hilbert space. It is shown that the proposed algorithms strongly converge to the minimum-norm fixed point of T.

Keywords

λ-strictly pseudo-contractive mappingfixed pointalgorithmHilbert space

MSC classification

Secondary: 47H05: Monotone operators and generalizations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 2 , April 2012 , pp. 232 - 240
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The first author was supported in part by the Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105. The second author was supported in part by NSC 100-2221-E-230-012.

References

[1]Bauschke, H., ‘The approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces’, J. Math. Anal. Appl. 202 (1996), 150–159.CrossRefGoogle Scholar
[2]Browder, F. E. and Petryshyn, W. V., ‘Construction of fixed points of nonlinear mappings in Hilbert spaces’, J. Math. Anal. Appl. 20 (1967), 197–228.CrossRefGoogle Scholar
[3]Ceng, L. C., Cubiotti, P. and Yao, J. C., ‘Strong convergence theorems for finitely many nonexpansive mappings and applications’, Nonlinear Anal. 67 (2007), 1464–1473.CrossRefGoogle Scholar
[4]Chancelier, J. P., ‘Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces’, J. Math. Anal. Appl. 353 (2009), 141–153.CrossRefGoogle Scholar
[5]Chang, S. S., ‘Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces’, J. Math. Anal. Appl. 323 (2006), 1402–1416.CrossRefGoogle Scholar
[6]Cho, Y. J. and Qin, X., ‘Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces’, J. Comput. Appl. Math. 228(1) (2009), 458–465.CrossRefGoogle Scholar
[7]Cui, Y. L. and Liu, X., ‘Notes on Browder’s and Halpern’s methods for nonexpansive maps’, Fixed Point Theory 10(1) (2009), 89–98.Google Scholar
[8]Jung, J. S., ‘Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces’, J. Math. Anal. Appl. 302 (2005), 509–520.CrossRefGoogle Scholar
[9]Kim, T. H. and Xu, H. K., ‘Strong convergence of modified Mann iterations’, Nonlinear Anal. 61 (2005), 51–60.CrossRefGoogle Scholar
[10]Lewicki, G. and Marino, G., ‘On some algorithms in Banach spaces finding fixed points of nonlinear mappings’, Nonlinear Anal. 71 (2009), 3964–3972.CrossRefGoogle Scholar
[11]Liu, X. and Cui, Y., ‘Common minimal-norm fixed point of a finite family of nonexpansive mappings’, Nonlinear Anal. 73 (2010), 76–83.CrossRefGoogle Scholar
[12]Lopez, G., Martin, V. and Xu, H. K., ‘Perturbation techniques for nonexpansive mappings with applications’, Nonlinear Anal. Real World Appl. 10 (2009), 2369–2383.CrossRefGoogle Scholar
[13]Mainge, P. E., ‘Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces’, J. Math. Anal. Appl. 325 (2007), 469–479.CrossRefGoogle Scholar
[14]Marino, G. and Xu, H. K., ‘Convergence of generalized proximal point algorithms’, Commun. Pure Appl. Anal. 3 (2004), 791–808.CrossRefGoogle Scholar
[15]Marino, G. and Xu, H. K., ‘Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces’, J. Math. Anal. Appl. 329 (2007), 336–349.CrossRefGoogle Scholar
[16]Moudafi, A., ‘Viscosity approximation methods for fixed-point problems’, J. Math. Anal. Appl. 241 (2000), 46–55.CrossRefGoogle Scholar
[17]Petrusel, A. and Yao, J. C., ‘Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings’, Nonlinear Anal. 69 (2008), 1100–1111.CrossRefGoogle Scholar
[18]Plubtieng, S. and Wangkeeree, R., ‘Strong convergence of modified Mann iterations for a countable family of nonexpansive mappings’, Nonlinear Anal. 70 (2009), 3110–3118.CrossRefGoogle Scholar
[19]Reich, S., ‘Weak convergence theorems for nonexpansive mappings in Banach spaces’, J. Math. Anal. Appl. 67 (1979), 274–276.CrossRefGoogle Scholar
[20]Saeidi, S., ‘Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings’, Nonlinear Anal. 70(12) (2009), 4195–4208.CrossRefGoogle Scholar
[21]Scherzer, O., ‘Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems’, J. Math. Anal. Appl. 194 (1991), 911–933.CrossRefGoogle Scholar
[22]Shang, M., Su, Y. and Qin, X., ‘Three-step iterations for nonexpansive mappings and inverse-strongly monotone mappings’, J. Syst. Sci. Complex. 22(2) (2009), 333–344.CrossRefGoogle Scholar
[23]Shioji, N. and Takahashi, W., ‘Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces’, Proc. Amer. Math. Soc. 125 (1997), 3641–3645.CrossRefGoogle Scholar
[24]Solodov, M. V. and Svaiter, B. F., ‘Forcing strong convergence of proximal point iterations in a Hilbert space’, Math. Program. Ser. A 87 (2000), 189–202.CrossRefGoogle Scholar
[25]Suzuki, T., ‘Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces’, Proc. Amer. Math. Soc. 135 (2007), 99–106.CrossRefGoogle Scholar
[26]Wu, D., Chang, S. S. and Yuan, G. X., ‘Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space’, Nonlinear Anal. 63 (2005), 987–999.CrossRefGoogle Scholar
[27]Xu, H. K., ‘Iterative algorithms for nonlinear operators’, J. London Math. Soc. 66 (2002), 240–256.CrossRefGoogle Scholar
[28]Xu, H. K., ‘Another control condition in an iterative method for nonexpansive mappings’, Bull. Aust. Math. Soc. 65 (2002), 109–113.CrossRefGoogle Scholar
[29]Xu, H. K., ‘Iterative methods for constrained Tikhonov regularization’, Comm. Appl. Nonlinear Anal. 10(4) (2003), 49–58.Google Scholar
[30]Xu, H. K., ‘Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces’, Inverse Problems 26 (2010), 105018 (17pp).CrossRefGoogle Scholar
[31]Yao, Y., Chen, R. and Xu, H. K., ‘Schemes for finding minimum-norm solutions of variational inequalities’, Nonlinear Anal. 72 (2010), 3447–3456.CrossRefGoogle Scholar
[32]Yao, Y. and Liou, Y. C., ‘An implicit extragradient method for hierarchical variational inequalities’, Fixed Point Theory Appl. 2011 (2011), 697248 (11pp).Google Scholar
[33]Yao, Y., Liou, Y. C. and Chen, R., ‘A general iterative method for an infinite family of nonexpansive mappings’, Nonlinear Anal. 69 (2008), 1644–1654.CrossRefGoogle Scholar
[34]Yao, Y. and Xu, H. K., ‘Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications’, Optimization 60(6) (2011), 645–658.CrossRefGoogle Scholar
[35]Zegeye, H. and Shahzad, N., ‘Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings’, Appl. Math. Comput. 191 (2007), 155–163.Google Scholar
[36]Zeng, L. C., Wong, N. C. and Yao, J. C., ‘Strong convergence theorems for strictly pseudo-contractive mappings of Browder-Petryshyn type’, Taiwanese J. Math. 10 (2006), 837–849.CrossRefGoogle Scholar
[37]Zeng, L. C. and Yao, J. C., ‘Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings’, Nonlinear Anal. 64 (2006), 2507–2515.CrossRefGoogle Scholar