Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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SYSTEM OF GENERALIZED NONLINEAR REGULARIZED NONCONVEX VARIATIONAL INEQUALITIES

  • Received : 2016.03.10
  • Accepted : 2016.05.25
  • Published : 2016.06.30
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Abstract

In this work, we suggest a new system of generalized nonlinear regularized nonconvex variational inequalities in a real Hilbert space and establish an equivalence relation between this system and fixed point problems. By using the equivalence relation we suggest a new perturbed projection iterative algorithms with mixed errors for finding a solution set of system of generalized nonlinear regularized nonconvex variational inequalities.

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References

  1. M. K. Ahmad and Salahuddin, Perturbed three step approximation process with errors for a general implicit nonlinear variational inequalities, Int. J. Math. Math. Sci., Article ID 43818, (2006), 1-14.
  2. M. K. Ahmad and Salahuddin, A stable perturbed algorithms for a new class of generalized nonlinear implicit quasi variational inclusions in Banach spaces, Advances in Pure Mathematics 2 (2) (2012), 139-148. https://doi.org/10.4236/apm.2012.23021
  3. S. S. Chang, H. W. J. Lee and C. K. Chan, Generalized system for relaxed cocoercive variational inequalities in Hilbert space, Applied Math. Lett. 20 (3) (2007), 329-334. https://doi.org/10.1016/j.aml.2006.04.017
  4. F. H. Clarke, Optimization and nonsmooth analysis, Wiley Int. Science, New York, 1983.
  5. F. H. Clarke, Y. S. Ledyaw, R. J. Stern and P. R. Wolenski, Nonsmooth analysis and control theory, Springer-Verlag, New York 1998.
  6. X. P. Ding, Existence and algorithm of solutions for a system of generalized mixed implicit equilibrium problem in Banach spaces, Appl. Math. Mech. 31 (9) (2010), 1049-1062. https://doi.org/10.1007/s10483-010-1341-z
  7. M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Review 39 (4) (1997), 669-713. https://doi.org/10.1137/S0036144595285963
  8. N. J. Huang and Y. P. Fang, Generalized m-accretive mappings in Banach spaces, J. Sichuan University 38 (4) (2001), 591-592.
  9. I. Inchan and N. Petrot, System of general variational inequalities involving different nonlinear operators related to fixed point problems and its applications, Fixed point Theory, Vol. 2011, Article ID 689478, pages 17, doi 10.1155/2011/689478.
  10. J. K. Kim, Salahuddin and A. Farajzadeh, A new system of extended nonlinear regularized nonconvex set valued variational inequalities, Commun. Appl. Nonlinear Anal. 21 (3) (2014), 21-28.
  11. M. M. Jin, Iterative algorithms for a new system of nonlinear variational inclusions with (A,${\eta}$)-accretive mappings in Banach spaces, Comput. Math. Appl. 54 (2007), 579-588. https://doi.org/10.1016/j.camwa.2006.12.030
  12. B. S. Lee and Salahuddin, A general system of regularized nonconvex variational inequalities, Appl. Comput. Math. 3 (4) (2014), 1-7.
  13. B. S. Lee, M. F. Khan and Salahuddin, Generalized Nonlinear Quasi-Variational Inclusions in Banach Spaces, Comput. Maths. Appl 56 (5) (2008), 1414-1422. https://doi.org/10.1016/j.camwa.2007.11.053
  14. H. Y. Lan, N. J. Huang and Y. J. Cho, New iterative approximation for a system of generalized nonlinear variational inclusions with set valued mappings in Banach spaces, Math. Inequal. Appl. 1 (2006), 175-187.
  15. S.B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-488. https://doi.org/10.2140/pjm.1969.30.475
  16. G. Stampacchia, An Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416.
  17. R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000), 5231-5249. https://doi.org/10.1090/S0002-9947-00-02550-2
  18. R. U. Verma, Projection methods, and a new system of cocoercive variational inequality problems, Int. J. Diff. Equa. Appl. 6 (3) (2002), 359-367.
  19. R. U. Verma, Generalized system for relaxed cocoercive variational inequalities and projection methods, J. Optim. Theo. Appl. 121 (2004), 203-210. https://doi.org/10.1023/B:JOTA.0000026271.19947.05
  20. D. J. Wen, Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators, Nonlinear Anal. 73 (2010), 2292-2297. https://doi.org/10.1016/j.na.2010.06.010
  21. X. Weng, Fixed point iteration for local strictly pseudo contractive mapping, Proceedings of American Mathematical Society 113 (3) (1991), 727-737. https://doi.org/10.1090/S0002-9939-1991-1086345-8