East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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FUZZY GENERAL NONLINEAR ORDERED RANDOM VARIATIONAL INEQUALITIES IN ORDERED BANACH SPACES

  • Received : 2016.01.29
  • Accepted : 2016.06.15
  • Published : 2016.09.30
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Abstract

The main object of this work to introduced and studied a new class of fuzzy general nonlinear ordered random variational inequalities in ordered Banach spaces. By using the random B-restricted accretive mapping with measurable mappings ${\alpha},{\alpha}^{\prime}:{\Omega}{\rightarrow}(0,1)$, an existence of random solutions for this class of fuzzy general nonlinear ordered random variational inequality (equation) with fuzzy mappings is established, a random approximation algorithm is suggested for fuzzy mappings, and the relation between the first value $x_0(t)$ and the random solutions of fuzzy general nonlinear ordered random variational inequality is discussed.

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References

  1. M. K. Ahmad and Salahuddin, On generalized multivalued random variational like in-clusions, Appl. Math. 2(2011), no. 8, 1011-1018. https://doi.org/10.4236/am.2011.28140
  2. M. K. Ahmad and Salahuddin, Collectively random fixed point theorem and applications, Pan. Amer. Math. J. 20(2010), no. 3, 69-84.
  3. M. K. Ahmad and Salahuddin, A fuzzy extension of generalized implicit vector varia-tional like inequalities, Positivity 11 (2007), no. 3, 477-484. https://doi.org/10.1007/s11117-007-2042-5
  4. R. Ahmad and F. F. Bazan, An iterative algorithm for random generalized nonlin-ear mixed variational inclusions for random fuzzy mappings, Appl. Math. Comput. 167(2005), 1400-1411.
  5. R. P. Agarwal, M. F. Khan, D. O' Regan and Salahuddin, On generalized multivalued nonlinear variational like inclusions with fuzzy mappings, Adv. Nonlinear Var. Inequal. 8(2005), 41-55.
  6. H. Amann, On the number of solutions of nonlinear equations in ordered Banach space, J. Funct. Anal. 11 (1972), 346-384. https://doi.org/10.1016/0022-1236(72)90074-2
  7. G. A. Anastassiou, M. K. Ahmad and Salahuddin, Fuzzified random generalized nonlin-ear variational inequalities, J. Concreate Appl. Math. 10(2012), no. 3-4, 186-206.
  8. S. S. Chang, Fixed point theory with applications, Chongqing Publishing House, Chongqing, 1984.
  9. S. S. Chang and Y. G. Zhu, On Variational Inequalities for fuzzy mappings, Fuzzy Sets and Systems 32 (1989), no. 3, 359-367.
  10. S. S. Chang and N. J. Huang, Generalized random multivalued quasi complementarity problems, Indian J. Math. 33 (1993), 305-320.
  11. S. S. Chang and Salahuddin, Existence of vector quasi variational like in-equalities for fuzzy mappings, Fuzzy Sets and Systems 233 (2013), 89-95, http://dx.doi.org/10.1016/j.fss2013.04.006.
  12. S. S. Chang, Salahuddin and X. R. Wang, Generalized vector variational like inequalities in fuzzy environment, Fuzzy Sets and Systems 265 (2015), 110-120. https://doi.org/10.1016/j.fss.2014.04.004
  13. H. H. Schaefer, Banach lattices and positive operators, Springer, 1974.
  14. Y. J. Cho, N. J. Huang and S. M. Kang, Random generalized set valued strongly non-linear implicit quasi variational inequalities, J. Inequal. Appl. 5(2000), 515-531.
  15. X. P. Ding and J. Y. Park, A new class of generalized nonlinear implicit quasi variational inclusions with fuzzy mappings, J. Comput. Appl. Math. 138(2002), 243-257. https://doi.org/10.1016/S0377-0427(01)00379-X
  16. Y. H. Du, Fixed points of incresing operators in ordered Banach spaces and applications, Appl. Anal. 38(1990), 1-20. https://doi.org/10.1080/00036819008839957
  17. D. J. Ge, Fixed points of mixed monotone operators with applications, Appl. Anal. 31(1988), 215-224. https://doi.org/10.1080/00036818808839825
  18. D. J. Ge and V. Lakshmikantham, Couple fixed points of nonlinear operators with applications, Nonlinear Anal. TMA, 11(1987), 623-632. https://doi.org/10.1016/0362-546X(87)90077-0
  19. A. Hassouni and A. Moudafi, A perturbed algorithms for variational inequalities, J. Math. Anal. Appl. 185(2001), 706-712.
  20. N. J. Huang, Random generalized nonlinear variational inclusions for random fuzzy mappings, Fuzzy Sets ans Systems 105(1999), 437-444. https://doi.org/10.1016/S0165-0114(97)00222-4
  21. N. J. Huang and Y. P. Fang, Fixed point theorems and a new system of multivalued generalized order complementarity problems, Positivity 7 (2003), 257-265. https://doi.org/10.1023/A:1026222030596
  22. H. G. Li, Approximation solution for general nonlinear ordered variational inequalities and ordered equations in ordered Banach space, Nonlinear Anal. Forum 13 (2008), no. 2, 205-214.
  23. H. G. Li, Approximation solution for a new class of general nonlinear ordered varia-tional inequalities and ordered equations in ordered Banach space, Nonlinear Anal. Forum 14(2009), 1-9.
  24. H. G. Li, Nonlinear inclusions problems for ordered RME set valued mappings in ordered Hilbert space, Nonlinear Funct. Anal. Appl. 16(2011), no. 1, 1-8.
  25. B. S. Lee, M. F. Khan and Salahuddin, Fuzzy generalized nonlinear mixed random variational like inclusions, Pacific J. Optim. 6(2010), no. 3, 579-590.
  26. B. S. Lee, G. M. Lee and D. S. Kim, Generalized vector valued variational inequalities and fuzzy extension, Preprint.
  27. B. S. Lee, S. H. Kim and Salahuddin, Fuzzy variational inclusions with (H,${\phi},{\psi}$)-${\eta}$-monotone mapping in Banach spaces, J. Adv. Res. Appl. Math. 4(2012), no. 1, 10-22. https://doi.org/10.5373/jaram.870.040511
  28. Salahuddin, Random hybrid proximal point algorithm for fuzzy nonlinear set valued inclusions, J. Concrete Appl. Math. 12(2014), no. 1-2, 47-62.
  29. Salahuddin and R. U. Verma, A common fixed point theorem for fuzzy mappings, Trans. Math. Prog. Appl. 1 (2013), no. 1, 59-68.
  30. Salahuddin and M. K. Ahmad, Random variational like inequalities, Adv. Nonlinear Var. Inequal. 11(2008), no. 2, 15-24.
  31. Salahuddin, M. K. Ahmad and R. U. Verma, Existence theorem for fuzzy mixed vector F-variational inequalities, Adv. Nonlinear Var. Inequal. 16(2013), no. 1, 53-59.
  32. 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
  33. C. Zhang and Z. S. Bi, Random generalized nonlinear variational inclusions for fuzzy random mappings, J. Schuan Univ. 6(2007), 499-502.
  34. L. A. Zadeh, Fuzzy Sets, Information and Control 8 (1965), 238-353.

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