Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ITERATIVE ALGORITHMS FOR A FUZZY SYSTEM OF RANDOM NONLINEAR EQUATIONS IN HILBERT SPACES

  • Received : 2016.04.19
  • Published : 2017.04.30
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Abstract

In this research work, by using the random resolvent operator techniques associated with random ($A_t$, ${\eta}_t$, $m_t$)-monotone operators, is to established an existence and convergence theorems for a class of fuzzy system of random nonlinear equations with fuzzy mappings in Hilbert spaces. Our results improve and generalized the corresponding results of the recent works.

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References

  1. R. Ahmad, M. F. Khan, and Salahuddin, Generalized nonlinear random variational inclusions, Adv. Nonlinear Var. Inequal. 3 (2000), no. 2, 125-135.
  2. R. P. Agarwal, M. F. Khan, D. O. Regan, and Salahuddin, On generalized multivalued nonlinear variational inclusions with fuzzy mappings, Adv. Nonlinear Var. Inequal. 8 (2005), no. 1, 41-55.
  3. R. P. Agarwal, M. F. Khan, D. O. Regan, and Salahuddin, Completely generalized nonlinear variational inclusions with fuzzy setvalued mappings, J. Concr. Appl. Math. 4 (2006), no. 2, 215-228.
  4. G. A. Anastassiou, M. K. Ahmad, and Salahuddin, Fuzziffied random generalized non-linear variational like inequalities, J. Concr. Appl. Math. 10 (2012), no. 3-4, 186-206.
  5. A. T. Bharucha-Reid, Random Integral Equations, Academic Press, New York, 1972.
  6. S. S. Chang, Fixed Point Theory with Applications, Chongging Publishing House, Chongging, 1984.
  7. S. S. Chang, Variational Inequality and Complementarity Problems: Theory with Applica-tions, Shanghai Scientific and Tech. Literature Publ. House, Shanghai, 1991.
  8. S. S. Chang and N. J. Huang, On the problems for a class of random variational in-equalities and quasi variational inequalities, J. Math. Res. Exposition 9(1989), 385-393.
  9. S. S. Chang and Y. G. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems 32 (1989), no. 3, 359-367. https://doi.org/10.1016/0165-0114(89)90268-6
  10. Y. J. Cho, M. F. Khan, and Salahuddin, Notes on random fixed point theorems, J. Korean Soc. Math. Educ. Ser B: Pure Appl. Math. 13 (2006), no. 3, 227-236.
  11. H. K. Dai, Generalized mixed variational like inequality for random fuzzy mappings, J. Comput. Appl. Math. 224 (2009), no. 1, 20-28. https://doi.org/10.1016/j.cam.2008.03.049
  12. X. P. Ding, Algorithm of solutions for mixed implicit variational inequalities with fuzzy mappings, Comput. Math. Appl. 38 (1999), no. 5-6, 231-241. https://doi.org/10.1016/S0898-1221(99)00229-1
  13. X. P. Ding, M. K. Ahmad, and Salahuddin, Fuzzy generalized vector variational in-equalities and complementarity problems, Nonlinear Funct. Anal. Appl. 13 (2008), no. 2, 253-263.
  14. Y. P. Fang and N. J. Huang, Iterative algorithm for a system of variational inclusions involving H-accretive operators in Banach spaces, Acta Math. Hungar. 108 (2005), no. 3, 183-195. https://doi.org/10.1007/s10474-005-0219-6
  15. P. R. Halemous, Measure Theory, Springer Verlag, New York, 1974.
  16. S. Heilpern, Fuzzy mappings and fixed point theorems, J. Math. Anal. Appl. 83 (1981), no. 2, 566-569. https://doi.org/10.1016/0022-247X(81)90141-4
  17. G. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72. https://doi.org/10.4064/fm-87-1-53-72
  18. N. J. Huang, Random generalized nonlinear variational inclusions for random fuzzy mappings, Fuzzy Sets and Systems 105 (1999), no. 3, 437-444. https://doi.org/10.1016/S0165-0114(97)00222-4
  19. N. J. Huang and Y. P. Fang, Fixed point theorems and a new system of multivalued generalized order complementarity problems, Positivity 7 (2003), no. 3, 257-265. https://doi.org/10.1023/A:1026222030596
  20. M. F. Khan and Salahuddin, Completely generalized nonlinear random variational in-clusions, South East Asian Bull. Math. 30 (2006), no. 2, 261-276.
  21. J. K. Kim, H. Y. Lan, and Y. J. Cho, Solution sensitivity of generalized nonlinear parametric ($A,\;{\eta},\;m$)-proximal operator system of equations in Hilbert spaces, J. Inequal. Appl. 2014 (2014), 362, 17 pp. https://doi.org/10.1186/1029-242X-2014-362
  22. J. K. Kim and K. S. Kim, A new system of generalized nonlinear mixed quasivariational inequalities and iterative algorithms in Hilbert spaces, J. Korean Math. Soc. 44 (2007), no. 4, 823-834. https://doi.org/10.4134/JKMS.2007.44.4.823
  23. S. H. Kim, B. S. Lee, and Salahuddin, Fuzzy variational inclusions with ($H,\;{\phi},\;{\psi}$) - ${\eta}$ - monotone mappings in Banach spaces, J. Adv. Res. Appl. Math. 4 (2012), no. 1, 10-23. https://doi.org/10.5373/jaram.870.040511
  24. H. Y. Lan, A class of nonlinear ($A,\;{\eta}$)-monotone operator inclusion problems with re-laxed cocoercive mappings, Adv. Nonlinear Var. Inequal. 9 (2006), no. 2, 1-11.
  25. B. S. Lee, M. F. Khan, and Salahuddin, Fuzzy nonlinear mixed random variational like inclusions, Pac. J. Optim. 6 (2010), no. 3, 573-590.
  26. B. S. Lee, M. F. Khan, and Salahuddin, Fuzzy nonlinear set valued variational inclusions, Comput. Math. Appl. 60 (2010), no. 6, 1768-1775. https://doi.org/10.1016/j.camwa.2010.07.007
  27. T. C. Lim, On fixed point stability for set valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985), no. 2, 436-441. https://doi.org/10.1016/0022-247X(85)90306-3
  28. S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475
  29. J. W. Peng, D. L. Zhu, and X. P. Zheng, Existence of solutions and convergence of a multistep iterative algorithm for a system of variational inclusions, Fixed Point Theory Appl. 2007 (2007), Art ID 93678, pages 20.
  30. Salahuddin, Random hybrid proximal point algorithm for fuzzy nonlinear set valued inclusions, J. Concr. Appl. Math. 12 (2014), no. 1-2, 47-62.
  31. Salahuddin, Fuzzy regularized nonlinear nonconvex random variational inequalities, Non-linear Anal. Forum 20 (2015), 1-14.
  32. Salahuddin, Geneal random variational inequalities and applications, Trans. Math. Prog. Appl. 4 (2016), no. 1, 25-33.
  33. Salahuddin and R. U. Verma, Existence of random variational inequalities, PanAmer. Math. J. 26 (2016), no. 2, 95-103.
  34. X. Z. Tan, Random quasi-variational inequality, Math. Nachr. 125 (1986), 319-328. https://doi.org/10.1002/mana.19861250124
  35. N. O. Uea and P. Kuman, A generalized nonlinear random equations with random fuzzy mappings in uniformly smooth Banach spaces, J. Inequal. Appl. 2010 (2010), Art ID 728452, pages 15.
  36. R. U. Verma and Salahuddin, System of random nonlinear variational inclusions with fuzzy mappings in q-uniformly smooth Banach spaces, Nonlinear Funct. Anal. Appl. 19 (2014), no. 3, 455-478.
  37. X. Z. Yuan, Noncompact random generalized games and random quasi variational in-equalities, J. Math. Stochastic Anal. 7 (1994), no. 4, 467-486. https://doi.org/10.1155/S1048953394000377
  38. L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
  39. C. Zhang and Z. S. Bi, Random generalized nonlinear variational inclusions for random fuzzy mappings, Sichuan Daxue Xuebao 44 (2007), no. 3, 499-502.