Strong convergence of a splitting algorithm for treating monotone operators

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• 作者：Sun Young Cho (12)
  Xiaolong Qin (13) (14)
  Lin Wang (15)
  
  12. Department of Mathematics ; Gyengsang National University ; Jinju ; 660-701 ; Korea
  13. Department of Mathematics ; Hangzhou Normal University ; Hangzhou ; 310036 ; China
  14. Department of Mathematics ; Faculty of Science ; King Abdulaziz University ; Jeddah ; Saudi Arabia
  15. College of Statistics and Mathematics ; Yunnan University of Finance and Economics ; Kunming ; 650221 ; China
  
• 关键词：maximal monotone operator ; fixed point ; nonexpansive mapping ; proximal point algorithm ; zero point
• 刊名：Fixed Point Theory and Applications
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：2014
• 期：1
• 全文大小：208 KB
• 参考文献：1. Browder, FE (1976) Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math 18: pp. 78-81
  2. Browder, FE, Petryshyn, WV (1967) Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl 20: pp. 197-228 CrossRef
  3. Cho, SY, Kang, SM (2011) Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci 32: pp. 1607-1618 CrossRef
  4. Cho, SY (2010) Approximation of solutions of a generalized variational inequality problem based on iterative methods. Commun. Korean Math. Soc 25: pp. 207-214 CrossRef
  5. Cho, SY, Qin, X, Kang, SM (2013) Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim 57: pp. 1429-1446 CrossRef
  6. Cho, SY, Kang, SM (2011) Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett 24: pp. 224-228 CrossRef
  7. Luo, H, Wang, Y (2012) Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci 2: pp. 1660-1670
  8. Cho, SY, Li, W, Kang, SM (2013) Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl.
  9. Zegeye, H, Shahzad, N (2012) Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2: pp. 374-397
  10. Rodjanadid, B, Sompong, S (2013) A new iterative method for solving a system of generalized equilibrium problems, generalized mixed equilibrium problems and common fixed point problems in Hilbert spaces. Adv. Fixed Point Theory 3: pp. 675-705
  11. Qin, X, Cho, SY, Kang, SM (2011) An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim 49: pp. 679-693 CrossRef
  12. Qin, X, Cho, SY, Kang, SM (2009) Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math 233: pp. 231-240 CrossRef
  13. Qin, X, Cho, SY, Kang, SM (2010) Iterative algorithms for variational inequality and equilibrium problems with applications. J.聽Glob. Optim 48: pp. 423-445 CrossRef
  14. Browder, FE (1966) Existence and approximation of solutions of nonlinear variational inequalities. Proc. Natl. Acad. Sci. USA 56: pp. 1080-1086 CrossRef
  15. Rockafellar, RT (1976) Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res 1: pp. 97-116 CrossRef
  16. Rockafellar, RT (1976) Monotone operators and proximal point algorithm. SIAM J. Control Optim 14: pp. 877-898 CrossRef
  17. Eckstein, J (1998) Approximate iterations in Bregman-function-based proximal algorithms. Math. Program 83: pp. 113-123
  18. Khan, MA, Yannelis, NC (1991) Equilibrium Theory in Infinite Dimensional Spaces. Springer, New York CrossRef
  19. Combettes, PL The convex feasibility problem in image recovery. In: Hawkes, P eds. (1996) Advanced in Imaging and Electron Physcis. Academic Press, New York, pp. 155-270
  20. Dautray, R, Lions, JL (1990) 1. Mathematical Analysis and Numerical Methods for Science and Technology. Springer, New York
  21. Fattorini, HO (1999) Infinite-Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge CrossRef
  22. G眉ler, O (1991) On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim 29: pp. 403-409 CrossRef
  23. Qin, X, Kang, SM, Cho, YJ (2010) Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim 46: pp. 75-87 CrossRef
  24. Song, J, Chen, M (2013) A modified Mann iteration for zero points of accretive operators. Fixed Point Theory Appl.
  25. Qing, Y, Cho, SY (2014) Proximal point algorithms for zero points of nonlinear operators. Fixed Point Theory Appl.
  26. Qing, Y, Cho, SY (2013) A regularization algorithm for zero points of accretive operators. Fixed Point Theory Appl.
  27. Cho, SY, Kang, SM (2012) Zero point theorems for m-accretive operators in a Banach space. Fixed Point Theory 13: pp. 49-58
  28. Wu, C, Lv, S (2013) Bregman projection methods for zeros of monotone operators. J. Fixed Point Theory.
  29. Zhang, M (2012) Iterative algorithms for common elements in fixed point sets and zero point sets with applications. Fixed Point Theory Appl.
  30. Qin, X, Su, Y (2007) Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl 329: pp. 415-424 CrossRef
  31. Kim, JK, Anh, PN, Nam, YM (2012) Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems. J. Korean Math. Soc 49: pp. 187-200 CrossRef
  32. Qin, X, Cho, SY, Wang, L (2013) Iterative algorithms with errors for zero points of m-accretive operators. Fixed Point Theory Appl.
  33. Solodov, MV, Svaiter, BF (2000) Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program 87: pp. 189-202
  34. Zhang, M (2014) Strong convergence of a viscosity iterative algorithm in Hilbert spaces. J. Nonlinear Funct. Anal.
  35. Qin, X, Shang, M, Su, Y (2008) Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model 48: pp. 1033-1046 CrossRef
  36. Cho, SY (2013) Strong convergence of an iterative algorithm for sums of two monotone operators. J. Fixed Point Theory.
  37. Barbu, V (1976) Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff, Groningen CrossRef
  38. Suzuki, T (2005) Strong convergence of Krasnoselskii and Mann鈥檚 type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl 305: pp. 227-239 CrossRef
  39. Xue, Z, Zhou, H, Cho, YJ (2000) Iterative solutions of nonlinear equations for m-accretive operators in Banach spaces. J.聽Nonlinear Convex Anal 1: pp. 313-320
  40. Zhou, H (2008) Convergence theorems of fixed points for 魏-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal 69: pp. 456-462 CrossRef
  41. Lions, PL, Mercier, B (1979) Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal 16: pp. 964-979 CrossRef
  42. Blum, E, Oettli, W (1994) From optimization and variational inequalities to equilibrium problems. Math. Stud 63: pp. 123-145
  43. Fan, K A minimax inequality and applications. In: Shisha, O eds. (1972) Inequality. Academic Press, New York, pp. 103-113
  44. Takahashi, S, Takahashi, W, Toyoda, M (2010) Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl 147: pp. 27-41 CrossRef
  
• 刊物主题：Analysis; Mathematics, general; Applications of Mathematics; Differential Geometry; Topology; Mathematical and Computational Biology;
• 出版者：Springer International Publishing
• ISSN：1687-1812

文摘

In this paper, we investigate a splitting algorithm for treating monotone operators. Strong convergence theorems are established in the framework of Hilbert spaces.