Topological Methods in Nonlinear Analysis

CQ method for approximating fixed points of nonexpansive semigroups and strictly pseudo-contractive mappings

Hossein Piri and Samira Rahrovi

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Abstract

We use the CQ method for approximating a common fixed point of a left amenable semigroup of nonexpansive mappings, an infinite family of strictly pseudo-contraction mappings and the set of solutions of variational inequalities for monotone, Lipschitz-continuous mappings in a real Hilbert space. Our results are a generalization of a result announced by Nadezhkina and Takahashi [N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim. 16 (2006), 1230-1241] and some other recent results.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 50, Number 2 (2017), 513-530.

Dates
First available in Project Euclid: 27 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1506477637

Digital Object Identifier
doi:10.12775/TMNA.2017.014

Mathematical Reviews number (MathSciNet)
MR3747026

Zentralblatt MATH identifier
06836831

Citation

Piri, Hossein; Rahrovi, Samira. CQ method for approximating fixed points of nonexpansive semigroups and strictly pseudo-contractive mappings. Topol. Methods Nonlinear Anal. 50 (2017), no. 2, 513--530. doi:10.12775/TMNA.2017.014. https://projecteuclid.org/euclid.tmna/1506477637


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References

  • R.P. Agarwal, D. O'Regan and D.R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Topological Fixed Point Theory and Its Applications, vol. 6, Springer, New York, 2009.
  • F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228.
  • S.-S. Chang, H.W.J. Lee and C.K. Chan, A new method for solving equilibrium problem, fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal. 70 (2009), 3307–3319.
  • H. Che, M. Li and X. Pan, Convergence theorems for equilibrium problems and fixed-point problems of an infinite family of strictly pseudocontractive mapping in Hilbert spaces, J. Appl. Math. 2012 (2012), Article ID 416476, 23 pages.
  • B.S. He, Z.H. Yang and X.M. Yuan, An approximate proximal- extragradient type method for monotone variational inequalities, J. Math. Anal. Appl. 300 (2004), 362–374.
  • N. Hirano, K. Kido and W. Takahashi, Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces, Nonlinear Anal. 12 (1988), 1269–1281.
  • A.N. Iusem, An iterative algorithm for the variational inequality problem, Comput. Appl. Math. 13 (1994), 103–114.
  • O. Kada and W. Takahashi, Strong convergence and nonlinear ergodic theorems for commutative semigroups of nonexpansive mappings, Nonlinear Anal. 28 (1997) 495–511.
  • G.M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonomika i Matematicheskie Metody 12 (1976), 747–756.
  • P. Katchang and P. Kumam, A composite explicit iterative process with a viscosity method for Lipschitzian semigroup in smooth Banach space, Bull. Iranian Math. Soc. 37 (2011), 143–159.
  • A.T. Lau, H. Miyake and W. Takahashi, Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces, Nonlinear Anal. 67 (2007), 1211–1225.
  • A.T. Lau, N. Shioji and W. Takahashi, Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces, J. Funct. Anal. 161 (1999), 62–75.
  • J.L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493–517.
  • N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim. 16 (2006), 1230–1241.
  • H. Piri, Hybrid pseudo-viscosity approximation schemes for systems of equilibrium problems and fixed point problems of infinite family and semigroup of non-expansive mappings, Nonlinear Anal. 74 (2011), 6788–6804.
  • H. Piri, Strong convergence of the CQ method for fixed points of semigroups of nonexpansive mappings, J. Nonlinear Funct. Anal. 2015 (2015), Article ID 18.
  • H. Piri, Approximating fixed points of semigroups of nonexpansive mappings and solving systems of variational inequalities, Math. Reports. 16 (2014), 295–317.
  • H. Piri and A.H. Badali, Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities, Fixed Point Theory Appl. 2011 (2011), doi:10.1186/1687-1812-2011-55.
  • R.T. Rockafellar, On the maximality of sums of nonlinear monotone operator, Trans. Amer. Math. Soc. 149 (1970), 75–88.
  • N. Shioji and W. Takahashi, Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces, J. Approx. Theory 97 (1999), 53–64.
  • K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math. 5 (2001), 387–404.
  • W. Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981), 253–256.
  • C.M. Yanes and H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), 2400–2411.
  • Y. Yao, Y.C. Liou and J.C. Yao, Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, Fixed Point Theory and Applications 2007 (2007), Article ID 64363, 12 pages, doi:10.1155/2007/64363.
  • Y. Yao and M.A. Noor, On viscosity iterative methods for variational inequalities, J. Math. Anal. Appl. 325 (2007), 776–787.
  • Y. Yao and M.A. Noor, On modified hybrid steepest-descent methods for general variational inequalities, J. Math. Anal. Appl. 334 (2007), 1276–1289.
  • Y. Yao and M.A. Noor, On modified hybrid steepest-descent method for variational inequalities, Carpathian J. Math. 24 (2008), 139–148.
  • Y. Yao and J.C. Yao, On modified iterative method for nonexpansive map- pings and monotone mappings, Appl. Math. Comput. 186 (2007), 1551–1558.