Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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CONVERGENCE ANALYSIS OF PARALLEL S-ITERATION PROCESS FOR A SYSTEM OF VARIATIONAL INEQUALITIES USING ALTERING POINTS

  • JUNG, CHAHN YONG (Department of Business Administration, Gyeongsang National University) ;
  • KUMAR, SATYENDRA (Department of Mathematics, Institute of Science, Banaras Hindu University) ;
  • KANG, SHIN MIN (Department of Mathematics and RINS, Gyeongsang National University)
  • Received : 2018.02.18
  • Accepted : 2018.03.27
  • Published : 2018.09.30
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Abstract

In this paper we have considered a system of mixed generalized variational inequality problems defined on two different domains in a Hilbert space. It has been shown that the solution of a system of mixed generalized variational inequality problems is equivalent to altering point formulation of some mappings. A new parallel S-iteration type process has been considered which converges strongly to the solution of a system of mixed generalized variational inequality problems.

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References

  1. R.P. Agarwal, D. O'Regan, and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (2007), 61-79.
  2. R.P. Agarwal, D. O'Regan, and D.R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, Series: Topological Fixed Point Theory and Its Applications, 6, Springer, New York, 2009.
  3. E. Allevi, A. Gnudi, and I.V. Konnov, Generalized vector variational inequalities over product sets, Nonlinear Anal. 47 (2001), 573-582. https://doi.org/10.1016/S0362-546X(01)00202-4
  4. Q.H. Ansari, S. Schaible, and J.C. Yao, System of vector equilibrium problems and its applications, J. Optim. Theory Appl. 107 (2000), 547-557. https://doi.org/10.1023/A:1026495115191
  5. Q.H. Ansari and J.C. Yao, A fixed point theorem and its applications to a system of variational inequalities, Bull. Australian Math. Soc. 59 (1999), 433-442. https://doi.org/10.1017/S0004972700033116
  6. M. Bianchi, Pseudo P-monotone operators and variational inequalities, Tech. Rep. 6, Istituto di econometria e Matematica per le Decisioni Economiche, Universita Cattolica del Sacro Cuore, Milan, Italy, 1993.
  7. S.S. Chang, H.W.J. Lee, and C.K. Chan, Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. Math. Lett. 20 (2007), 329-334. https://doi.org/10.1016/j.aml.2006.04.017
  8. Y.J. Cho, Y.P. Fang, N.J. Huang, and H.J. Hwang, Algorithms for systems of nonlinear variational inequalities, J. Korean Math. Soc. 41 (2004), 489-499. https://doi.org/10.4134/JKMS.2004.41.3.489
  9. G. Cohen and F. Chaplais, Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms, J. Optim. Theory Appl. 59 (1988), 369-390. https://doi.org/10.1007/BF00940305
  10. X.P. Ding, Generalized quasi-variational-like inclusions with nonconvex functionals, Appl. Math. Comput. 122 (2001), 267-282.
  11. X.P. Ding and C.L. Luo, Perturbed proximal point algorithms for general quasi-variationallike inclusions, J. Comput. Appl. Math. 113 (2000), 153-165. https://doi.org/10.1016/S0377-0427(99)00250-2
  12. Y.P. Fang, N.J. Huang, Y.J. Cao, and S.M. Kang, Stable iterative algorithms for a class of general nonlinear variational inequalities, Adv. Nonlinear Var. Inequal. 5 (2002), 1-9.
  13. K. Guo, Y. Jiang, and S.Q. Feng, A parallel resolvent method for solving a system of nonlinear mixed variational inequalities, J. Inequal. Appl. 2013 (2013), Paper No. 509, 9 pages. https://doi.org/10.1186/1029-242X-2013-9
  14. Z. He and F. Gu, Generalized system for relaxed cocoercive mixed variational inequalities in Hilbert spaces, Appl. Math. Comput. 214 (2009), 26-30.
  15. Z.Y. Huang and M.A. Noor, An explicit projection method for a system of nonlinear variational inequalities with different ( ${\gamma}$, r)-cocoercive mappings, Appl. Math. Comput. 190 (2007), 356-361.
  16. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
  17. G. Kassay and J. Kolumban, System of multi-valued variational inequalities, Publ. Math. Debrecen 54 (1999), 267-279.
  18. J.K. Kim and D.S. Kim, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Convex Anal. 11 (2004), 235-243.
  19. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic press, New York, 1980.
  20. V. Kumar, A. Latif, A. Rafiq, and N. Hussain, S-iteration process for quasi-contractive mappings, J. Ineqal. Appl. 2013 (2013), Paper No. 206, 15 pages. https://doi.org/10.1186/1029-242X-2013-15
  21. W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  22. P. Narin, A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems, Appl. Math. Lett. 23 (2010), 440-445. https://doi.org/10.1016/j.aml.2009.12.001
  23. M.A. Noor, Nonconvex functions and variational inequalities, J. Optim. Theory Appl. 87 (1995), 615-630. https://doi.org/10.1007/BF02192137
  24. J.S. Pang, Asymmetric variational inequality problems over product sets: applications and iterative methods, Math. Program. 31 (1985), 206-219. https://doi.org/10.1007/BF02591749
  25. R. Pant and R. Shukla, Approximating fixed points of generalized ${\alpha}$-nonexpansive mappings in Banach spaces, Numerical Funct. Anal. Optim. 38 (2017), 248-266. https://doi.org/10.1080/01630563.2016.1276075
  26. E. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures Appl. 6 (1890), 145-210.
  27. D.R. Sahu, Applications of the S-iteration process to constrained minimization problems and split feasibility problems, Fixed Point Theory 12 (2011), 187-204.
  28. D.R. Sahu, Altering points and applications, Nonlinear Stud. 21 (2014), 349-365.
  29. D.R. Sahu, S.M. Kang, and A. Kumar, Convergence analysis of parallel S-iteration process for system of generalized variational inequalities, J. Function Spaces 2017 (2017), Article ID 5847096, 10 pages.
  30. D.R. Sahu and A. Petrusel, Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces, Nonlinear Anal. 74 (2011), 6012-6023. https://doi.org/10.1016/j.na.2011.05.078
  31. D.R. Sahu, V. Sagar, and K.K. Singh, Convergence of generalized Newton method using S-operator in Hilbert spaces, In: Algebra and Analysis: Theory and Applications, Narosa Publishing House Pvt. Ltd., New Delhi, India, 2015.
  32. D.R. Sahu, K.K. Singh, and V.K. Singh, S-iteration process of Newton-like and applications, In: Algebra and Analysis: Theory and Applications, Narosa Publishing House Pvt. Ltd., New Delhi, India, 2015.
  33. D.R. Sahu, J.C. Yao, V.K. Singh, and S. Kumar, Semilocal convergence analysis of S-iteration process of Newton-Kantorovich like in Banach spaces, J. Optim. Theory Appl. 172 (2017), 102-127. https://doi.org/10.1007/s10957-016-1031-x
  34. G. Stampacchia, Formes bilineaires coercivities sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416.
  35. R. Suparatulatorn, W. Cholamjiak, and S. Suantai, A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs, Numer. Algor. 77 (2018), 479-490. https://doi.org/10.1007/s11075-017-0324-y
  36. R.U. Verma, On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot-Line 3 (1999), 65-68.
  37. R.U. Verma, Projection methods, algorithms, and a new system of nonlinear variational inequalities, Comput. Math. Appl. 41 (2001), 1025-1031. https://doi.org/10.1016/S0898-1221(00)00336-9
  38. R.U. Verma, Iterative algorithms and a new system of nonlinear quasi-variational inequalities, Adv. Nonlinear Var. Ineqal. 4 (2001), 117-124.
  39. R.U. Verma, General convergence analysis for two-step projection methods and applications to variational problems, Appl. Math. Lett. 18 (2005), 1286-1292. https://doi.org/10.1016/j.aml.2005.02.026
  40. R.U. Verma, Generalized system for relaxed cocoercive variational inequalities and projection methods, J. Optim. Theory Appl. 121 (2004), 203-210. https://doi.org/10.1023/B:JOTA.0000026271.19947.05
  41. B. Wan and X. Zhan, A proximal point algorithm for a system of generalized mixed variational inequalities, J. Syst. Sci. Complex 25 (2012), 964-972. https://doi.org/10.1007/s11424-012-0157-7