## Taiwanese Journal of Mathematics

### IMPLICIT AND EXPLICIT ALGORITHMS FOR MINIMUM-NORM FIXED POINTS OF PSEUDOCONTRACTIONS IN HILBERT SPACES

#### Abstract

We introduce implicit and explicit iterative algorithms for the construction of fixed points of pseudocontractions $T$ in Hilbert spaces. We prove that the proposed iterative algorithms converge strongly to the minimum-norm fixed point of $T$. Moreover we show that some of the existing iterative algorithms for nonexpansive mappings fail to converge when applied to pseudocontractions.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 4 (2012), 1489-1506.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406745

Digital Object Identifier
doi:10.11650/twjm/1500406745

Mathematical Reviews number (MathSciNet)
MR2951149

Zentralblatt MATH identifier
1293.47068

#### Citation

Yao, Yonghong; Colao, Vittorio; Marino, Giuseppe; Xu, Hong-Kun. IMPLICIT AND EXPLICIT ALGORITHMS FOR MINIMUM-NORM FIXED POINTS OF PSEUDOCONTRACTIONS IN HILBERT SPACES. Taiwanese J. Math. 16 (2012), no. 4, 1489--1506. doi:10.11650/twjm/1500406745. https://projecteuclid.org/euclid.twjm/1500406745

#### References

• W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506- 510.
• S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274-276.
• A. Genel and J. Lindenstrauss, An example concerning fixed points, Isr. J. Math., 22 (1975), 81-86.
• F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl., 20 (1967), 197-228.
• G. Marino and H. K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329(1) (2007), 336-346.
• C. E. Chidume and S. A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudo-contractions, Proc. Amer. Math. Soc., 129 (2001), 2359-2363.
• S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150.
• L. C. Ceng, A. Petrusel and J. C. Yao, Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings, Applied Math. Computation, 209 (2009), 162-176.
• C. E. Chidume, M. Abbas and B. Ali, Convergence of the Mann iteration algorithm for a class of pseudocontractive mappings, Applied Math. Computation, 194 (2007), 1-6.
• L. Ciric, A. Rafiq, N. Cakic and J. S. Ume, Implicit Mann fixed point iterations for pseudo-contractive mappings, Applied Math. Let., 22 (2009), 581-584.
• K. Q. Lan and J. H. Wu, Convergence of approximants for demicontinuous pseudo-contractive maps in Hilbert spaces, Nonlinear Anal., 49 (2002), 737-746.
• C. Moore and B. V. C. Nnoli, Strong convergence of averaged approximants for Lipschitz pseudocontractive maps, J. Math. Anal. Appl., 260 (2001), 269-278.
• X. Qin, Y. J. Cho, S. M. Kang and H. Zhou, Convergence theorems of common fixed points for a family of Lipschitz quasi-pseudocontractions, Nonlinear Anal., 71 (2009), 685-690.
• Y. Yao, Y. C. Liou and G. Marino, A hybrid algorithm for pseudo-contractive mappings, Nonlinear Anal., 71 (2009), 997-5002.
• Q. Zhang and C. Cheng, Strong convergence theorem for a family of Lipschitz pseudocontractive mappings in a Hilbert space, Math. Computer Modelling, 48 (2008), 480-485.
• H. Zhou, Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces, Nonlinear Anal., 70 (2009), 4039-4046.
• S. Reich and A. J. Zaslavski, Convergence of Krasnoselskii-Mann iterations of nonexpansive operators, Math. Computer Modelling, 32 (2000), 1423-1431.
• T. Suzuki, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc., 135 (2007), 99-106
• Y. Yao, Y. C. Liou and G. Marino, Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2009 (2009), Article ID 279058, 7 pages, doi:10.1155/2009/279058.
• X. Lu, H. K. Xu and X. Yin, Hybrid methods for a class of monotone variational inequalities, Nonlinear Anal., 71 (2009), 1032-1041.
• H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279-291.
• Y. Yao and J. C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Applied Math. Computation, 186 (2007), 1551-1558.
• K. Geobel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge University Press, 1990.
• K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.
• K. Deimling, Zero of accretive operators, Manuscripta Math., 13 (1974), 365-374.
• H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optimiz., 22(5-6) (2001), 767-773.
• R. H. Martin, Differential equations on closed subsets of Banach space, Trans. Amer. Math. Soc., 179 (1973), 399-414.