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Applicable Analysis
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Volume 95, 2016 - Issue 5
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Articles

Hybrid viscosity methods for equilibrium problems, variational inequalities, and fixed point problems

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Pages 1088-1117 | Received 02 Mar 2015, Accepted 12 May 2015, Published online: 05 Jun 2015

References

  • Ceng LC, Al-Homidan S, Ansari QH, et al. An iterative scheme for equilibrium problems and fixed points problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 2009;223:967–974.
  • Ceng L-C, Ansari QH, Schaible S. Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems. J. Global Optim. 2012;53:69–96.
  • Ceng L-C, Ansari QH, Wong MM, et al. Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems. Fixed Point Theory. 2012;13:403–422.
  • Ceng L-C, Ansari QH, Yao J-C. Viscosity approximation methods for generalized equilibrium problems and fixed point problems. J. Global Optim. 2009;43:487–502.
  • Ceng L-C, Ansari QH, Yao J-C. Hybrid pseudoviscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Nonlinear Anal.: Hybrid Syst. 2010;4:743–754.
  • Ceng L-C, Ansari QH, Yao J-C. Hybrid proximal-type and hybrid shrinking projection algorithms for equilibrium problems, maximal monotone operators and relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 2010;31:763–797.
  • Ceng LC, Guu SM, Yao JC. Finding common solutions of a variational inequality, a general system of variational inequalities, and a fixed-point problem via a hybrid extragradient method. Fixed Point Theory Appl. 2011;2011:22. Article ID 26159.
  • Ceng LC, Hadjisavvas N, Wong NC. Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Global Optim. 2010;46:635–646.
  • Ceng LC, Yao JC. A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem. Nonlinear Anal. 2010;72:1922–1937.
  • Ceng LC, Yao JC. A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Computat. Appl. Math. 2008;214:186–201.
  • Cianciaruso F, Marino G, Muglia L, et al. A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory Appl. 2010;2010:19. Article ID 383740.
  • Colao V, Marino G, Xu HK. An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 2008;344:340–352.
  • Latif A, Ceng L-C, Ansari QH. Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solutions of equilibrium problem and fixed point problems. Fixed Point Theory Appl. 2012;2012:186.
  • Marino G, Muglia L, Yao Y. Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points. Nonlinear Anal. 2012;75:1787–1798.
  • Rattanaseeha K. The general iterative methods for equilibrium problems and fixed point problems of countable family of nonexpansive mappings in Hilbert spaces. J. Inequal. Appl. 2013;2013:153.
  • Wang XJ, Ceng LC, Hu HY, et al. General iterative algorithms for mixed equilibrium problems, variational inequalities and fixed point problems. Fixed Point Theory Appl. 2014;2014:80.
  • Yao Y, Liou YC, Yao JC. Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl. 2007;2007:12. Article ID 64363.
  • Zeng L-C, Ansari QH, Schaible S, et al. Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces. Fixed Point Theory. 2011;12:293–308.
  • Zeng L-C, Ansari QH, Wong NC, et al. An extragradient-like approximation method for variational inequalities and fixed point problems. Fixed Point Theory Appl. 2011;2011:22.
  • Blum E, Oettli W. From optimization and variational inequalities to equilibrium problems. Math. Student. 1994;63:123–145.
  • Chadli O, Schaible S, Yao JC. Regularized equilibrium problems with application to noncoercive hemivariational inequalities. J. Optim. Theory Appl. 2004;121:571–596.
  • Chadli O, Wong NC, Yao JC. Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 2003;117:245–266.
  • Chadli O, Ansari QH, Yao J-C. Mixed equilibrium problems and anti-periodic solutions for nonlinear evolution equations. J. Optim. Theory Appl. 2015. doi:10.1007/s10957-015-0707-y.
  • Ansari QH, Wong NC, Yao J-C. The existence of nonlinear inequalities. Appl. Math. Lett. 1999;12:89–92.
  • Takahashi S, Takahashi W. Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008;69:1025–1033.
  • Ansari QH, Amini-Harandi A, Farajzadeh AP. Existence of equilibria in complete metric spaces. Taiwanese J. Math. 2012;16:777–785.
  • Ansari QH, Lalitha CS, Mehta M. Generalized convexity, nonsmooth variational inequalities and nonsmooth optimization. Boca Raton (FL): CRC Press, Taylor & Francis Group; 2014.
  • Verma RU. On a new system of nonlinear variational inequalities and associated iterative algorithms. Math. Sci. Res. Hot-Line. 1999;3:65–68.
  • Marino G, Xu HK. Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Math. Appl. 2007;329:336–346.
  • Ceng LC, Wang CY, Yao JC. Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 2008;67:375–390.
  • Moudafi A. Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 2000;241:46–55.
  • Xu HK. Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004;298:279–291.
  • Marino G, Xu HK. A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2006;318:43–52.
  • Takahashi W, Toyoda M. Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003;118:417–428.
  • Shimoji K, Takahashi W. Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J. Math. 2001;5:387–404.
  • Goebel K, Kirk WA. Topics on metric fixed-point theory. Cambridge: Cambridge University Press; 1990.
  • Xu HK. Iterative algorithms for nonlinear operators. J. London Math. Soc. 2002;66:240–256.
  • Moudafi A, Mainge P-E. Towards viscosity approximations of hierarchical fixed point problems. Fixed Point Theory Appl. 2006;2006:1–10. Article ID 95453.
  • Moudafi A, Mainge P-E. Strong convergence of an iterative method for hierarchical fixed point problems. Pacific J. Optim. 2007;3:529–538.
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