A viscosity splitting algorithm for solving inclusion and equilibrium problems

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• 作者：Buthinah A Bin Dehaish (1)
  Abdul Latif (2)
  Huda O Bakodah (1)
  Xiaolong Qin (3)
  
  1. Department of Mathematics ; Faculty of Science for Girls ; King Abdulaziz University ; P.O. Box 80203 ; Jeddah ; 21589 ; Saudi Arabia
  2. Department of Mathematics ; King Abdulaziz University ; P.O. Box 80203 ; Jeddah ; 21589 ; Saudi Arabia
  3. Department of Mathematics ; Faculty of Science ; King Abdulaziz University ; Jeddah ; Saudi Arabia
  
• 关键词：equilibrium problem ; variational inequality ; splitting algorithm ; nonexpansive mapping ; fixed point
• 刊名：Journal of Inequalities and Applications
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：2015
• 期：1
• 全文大小：1,170 KB
• 参考文献：1. Lions, PL, Mercier, B: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964-979 (1979) CrossRef
  2. Fattorini, HO: Infinite-Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge (1999) CrossRef
  3. Iiduka, H: Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 236, 1733-1742 (2012) CrossRef
  4. Byrne, C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103-120 (2008) CrossRef
  5. Martinet, B: R茅gularisation d鈥檌n茅quations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154-159 (1970)
  6. Rockafellar, RT: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 877-898 (1976) CrossRef
  7. Spingarn, JE: Applications of the method of partial inverses to convex programming decomposition. Math. Program. 32, 199-223 (1985) CrossRef
  8. Eckstein, J, Bertsekas, DP: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293-318 (1992) CrossRef
  9. Lions, PL, Mercier, B: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964-979 (1979) CrossRef
  10. Passty, GB: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383-390 (1979) CrossRef
  11. Takahashi, S, Takahashi, W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506-515 (2007) CrossRef
  12. Takahashi, S, Takahashi, W, Toyoda, M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 147, 27-41 (2010) CrossRef
  13. Kammura, S, Takahashi, W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226-240 (2000) CrossRef
  14. Iiduka, H, Takahashi, W: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341-350 (2005) CrossRef
  15. Qin, X, Cho, SY, Wang, L: A regularization method for treating zero points of the sum of two monotone operators. Fixed Point Theory Appl. 2014, Article ID 75 (2014) CrossRef
  16. Colao, V, Acedo, GG, Marino, G: An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. Nonlinear Anal. 71, 2708-2715 (2009) CrossRef
  17. Wang, ZM, Zhang, X: Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems. J. Nonlinear Funct. Anal. 2014, 15 (2014) CrossRef
  18. Magin茅, PM: A hybrid extragradient-viscosity method for monotone operator and fixed point problems. SIAM J.聽Optim. 47, 1499-1515 (2008)
  19. Moudafi, A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46-55 (2000) CrossRef
  20. Mainge, PE: A viscosity method with no spectral radius requirements for the split common fixed point problem. Eur. J. Oper. Res. 235, 17-27 (2014) CrossRef
  21. Cho, SY, Kang, SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 32, 1607-1618 (2012) CrossRef
  22. Cho, SY, Kang, SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 24, 224-228 (2011) CrossRef
  23. Zegeye, H, Shahzad, N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2, 374-397 (2012)
  24. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123-145 (1994)
  25. Fan, K: A minimax inequality and applications. In: Shisha, O (ed.) Inequalities III, pp.聽103-113. Academic Press, New York (1972)
  26. Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54, 1041-1044 (1965) CrossRef
  27. Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75-88 (1970) CrossRef
  28. Suzuki, T: Strong convergence of Krasnoselskii and Mann鈥檚 type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305, 227-239 (2005) CrossRef
  29. Liu, LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J.聽Math. Anal. Appl. 194, 114-125 (1995) CrossRef
  30. Rockafellar, RT: Convex Analysis. Princeton University Press, Princeton (1970)
  
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X

文摘

In this paper, we present a viscosity splitting algorithm with computational errors for solving common solutions of inclusion and equilibrium problems. Strong convergence theorems are established in the framework of real Hilbert spaces. Applications are also provided to support the main results.