GENERAL NONLINEAR RANDOM SET-VALUED VARIATIONAL INCLUSION PROBLEMS WITH RANDOM FUZZY MAPPINGS IN BANACH SPACES

Title & Authors
GENERAL NONLINEAR RANDOM SET-VALUED VARIATIONAL INCLUSION PROBLEMS WITH RANDOM FUZZY MAPPINGS IN BANACH SPACES

Abstract
This paper is dedicated to study a new class of general nonlinear random A-maximal $\small{m}$-relaxed $\small{{\eta}}$-accretive (so called (A, $\small{{\eta}}$)-accretive [49]) equations with random relaxed cocoercive mappings and random fuzzy mappings in $\small{q}$-uniformly smooth Banach spaces. By utilizing the resolvent operator technique for A-maximal $\small{m}$-relaxed $\small{{\eta}}$-accretive mappings due to Lan et al. and Chang's lemma [13], some new iterative algorithms with mixed errors for finding the approximate solutions of the aforesaid class of nonlinear random equations are constructed. The convergence analysis of the proposed iterative algorithms under some suitable conditions are also studied.
Keywords
variational inclusions;A-maximal m-relaxed $\small{{\eta}}$-accretive mapping;random iterative algorithm;random relaxed cocoercive mapping;resolvent operator technique;random fuzzy mapping;q-uniformly smooth Banach space;
Language
English
Cited by
References
1.
R. P. Agarwal, Y. J. Cho, and N. J. Huang, Generalized nonlinear variational inclusions involving maximal ${\eta}$-monotone mappings, Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. Vol. 1, 2, 59-73, Kluwer Acad. Publ., Dordrecht, 2003.

2.
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), no. 1, 41-55.

3.
R. Ahmad, Q. H. Ansari, and S. S. Irfan, Generalized variational inclusions and generalized resolvent equations in Banach spaces, Comput. Math. Appl. 49 (2005), no. 11-12, 1825-1835.

4.
R. Ahmad and F. F. Bazan, An iterative algorithm for random generalized nonlinear mixed variational inclusions for random fuzzy mappings, Appl. Math. Comput. 167 (2005), no. 2, 1400-1411.

5.
M. Alimohammady, J. Balooee, Y. J. Cho, and M. Roohi, A new system of nonlinear fuzzy variational inclusions involving (A, ${\eta}$)-accretive mappings in uniformly smooth Banach spaces, J. Inequal. Appl. 2009 (2009), Article ID 806727, 33 pp.; doi:10.1155/2010/806727.

6.
M. Alimohammady, J. Balooee, Y. J. Cho, and M. Roohi, Iterative algorithms for a new class of extended general nonconvex set-valued variational inequalities, Nonlinear Anal. 73 (2010), no. 12, 3907-3923.

7.
M. Alimohammady, J. Balooee, Y. J. Cho, and M. Roohi, Generalized nonlinear random equations with random fuzzy and relaxed co-coercive mappings in Banach spaces, Adv. Nonlinear Var. Inequal. 13 (2010), no. 2, 37-58.

8.
M. Alimohammady, J. Balooee, Y. J. Cho, and M. Roohi, New perturbed finite step iterative algorithms for a system of extended generalized nonlinear mixed-quasi variational inclusions, Comput. Math. Appl. 60 (2010), no. 11, 2953-2970.

9.
J. P. Aubin, Mathematical Methods of Game and Economics Theory, North-Holland, Amsterdam, 1979.

10.
J. Balooee, Y. J. Cho, and M. K. Kang, The Wiener-Hopf equation technique for solving general nonlinear regularized nonconvex variational inequalities, Fixed Point Theory Appl. 2011 (2011). 2011:86, 34 pp.; doi:10.1186/1687-1812-2011-86.

11.
J. Balooee, Y. J. Cho, and M. K. Kang, Projection methods and a new system of extended general regularized nonconvex set-valued variational inequalities, J. Appl. Math. 2012 (2012), Article ID 690648, 18 pp.; doi:10.1155/2012/690648.

12.
A. Bensoussan, M. Goursat, and J. L. Lions, Controle impulsionnel et inequations quasi-variationnelles stationnaires, C. R. Acad. Sci. Paris Ser. A-B 276 (1973), 1279-1284.

13.
S. S. Chang, Fixed Point Theory with Applications, Chongqing Publishing House, Chongqing, 1984.

14.
S. S. Chang, Variational Inequality and Complementarity Problem Theory with Applications, Shanghai Scientific and Tech. Literature Publishing House, Shanghai, 1991.

15.
M. S. Chang and H. Y. Chen, A fuzzy user-optimal route choice problem using a link-based fuzzy variational inequality formulation, Fuzzy Sets and Systems 114 (2000), no. 2, 339-345.

16.
S. S. Chang and N. J. Huang, Generalized complementarity problems for fuzzy mappings, Fuzzy Sets and Systems 55 (1993), no. 2, 227-234.

17.
S. S. Chang and N. J. Huang, Generalized strongly nonlinear quasi-complementarity problems in Hilbert spaces, J. Math. Anal. Appl. 158 (1991), no. 1, 194-202.

18.
S. S. Chang and N. J. Huang, Generalized multivalued implicit complementarity problems in Hilbert spaces, Math. Japon. 36 (1991), no. 6, 1093-1100.

19.
S. S. Chang and N. J. Huang, Generalized random multivalued quasi-complementarity problems, Indian J. Math. 35 (1993), no. 3, 305-320.

20.
S. S. Chang and N. J. Huang, Random generalized set-valued quasi-complementarity problems, Acta Math. Appl. Sinica 16 (1993), 396-405.

21.
S. S. Chang and Y. G. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems 32 (1989), no. 3, 359-367.

22.
S. S. Chang and Y. G. Zhu, On the problems for a class of random variational inequalities and quasi-variational inequalities, J. Math. Res. Exposition 9 (1989), 385-393.

23.
Y. J. Cho, N. J. Huang, and S. M. Kang, Random generalized set-valued strongly nonlinear implicit quasi-varitional inequalities, J. Inequal. Appl. 5 (2000), no. 5, 515-531.

24.
Y. J. Cho and H. Y. Lan, A new Class of generalized nonlinear multi-valued quasi-variational-like-inclusions with H-monotone mappings, Math. Inequal. Appl. 10 (2007), no. 2, 389-401.

25.
Y. J. Cho and H. Y. Lan, Generalized nonlinear random (A, ${\eta}$)-accretive equations with random relaxed cocoercive mappings in Banach spaces, Comput. Math. Appl. 55 (2008), no. 9, 2173-2182.

26.
Y. J. Cho and X. Qin, Systems of generalized nonlinear variational inequalities and its projection methods, Nonlinear Anal. 69 (2008), no. 12, 4443-4451.

27.
X. P. Ding, Algorithm of solutions for mixed implicit quasi-variational inequalities with fuzzy mappings, Comput. Math. Appl. 38 (1999), no. 5-6, 231-249.

28.
X. P. Ding, Generalized quasi-variational-like inclusions with nonconvex functionals, Appl. Math. Comput. 122 (2001), no. 3, 267-282.

29.
X. P. Ding and J. Y. Park, A new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mapping, J. Comput. Appl. Math. 138 (2002), no. 2, 243-257.

30.
D. Dubois and H. Prade, Fuzzy Sets Systems, Theory and Applications, Academic Press, London, 1980.

31.
A. Ganguly and K. Wadhawa, On random variational inequalities, J. Math. Anal. Appl. 206 (1997), no. 1, 315-321.

32.
A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185 (1994), no. 3, 706-712.

33.
C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72.

34.
N. J. Huang, A new method for a class of nonlinear variational inequalities with fuzzy mappings, Appl. Math. Lett. 10 (1997), no. 6, 129-133.

35.
N. J. Huang, Generalized nonlinear variational inclusions with noncompact valued mappings, Appl. Math. Lett. 9 (1996), no. 3, 25-29.

36.
N. J. Huang, Random generalized nonlinear variational inclusions for random fuzzy mappings, Fuzzy Sets and Systems 105 (1999), no. 3, 437-444.

37.
N. J. Huang and Y. J. Cho, Random completely generalized set-valued implicit quasi-variational inequalities, Positivity 3 (1999), no. 3, 201-213.

38.
N. J. Huang and Y. P. Fang, Generalized m-accretive mappings in Banach spaces, J. Sichuan Univ. 38 (2001), 591-592.

39.
N. J. Huang and Y. P. Fang, A new class of general variational inclusions involving maximal ${\eta}$-monotone mappings, Pub. Math. Debrecen 62 (2003), no. 1-2, 83-98.

40.
N. J. Huang and H. Y. Lan, A couple of nonlinear equations with fuzzy mappings in fuzzy normed spaces, Fuzzy Sets and Systems 152 (2005), no. 2, 209-222.

41.
N. J. Huang, X. Long, and Y. J. Cho, Random completely generalized nonlinear variational inclusions with non-compact valued random mappings, Bull. Korean Math. Soc. 34 (1997), no. 4, 603-615.

42.
J. U. Jeong, Generalized set-valued variational inclusions and resolvent equations in Banach spaces, Comput. Math. Appl. 47 (2004), no. 8-9, 1241-1247.

43.
M. M. Jin and Q. K. Liu, Nonlinear quasi-variational inclusions involving generalized m-accretive mappings, Nonlinear Funct. Anal. Appl. 9 (2004), no. 3, 485-494.

44.
M. F. Khan, Salahuddin, and R. U. Verma, Generalized random variational-like inequalities with randomly pseudo-monotone multivalued mappings, Panamer. Math. J. 16 (2006), no. 3, 33-46.

45.
H. Y. Lan, Approximation solvability of nonlinear random (A, ${\eta}$)-resolvent operator equations with random relaxed cocoercive operators, Comput. Math. Appl. 57 (2009), no. 4, 624-632.

46.
H. Y. Lan, On multivalued nonlinear variational inclusion problems with (A, ${\eta}$)-accretive mappings in Banach spaces, J. Inequal. Appl. 2006 (2006), Art. ID 59836, 12 pp.

47.
H. Y. Lan, Projection iterative approximations for a new class of general random implicit quasi-variational inequalities, J. Inequal. Appl. 2006 (2006), Article ID 81261, 17 pp.; doi:10.1155/JIA/2006/81261.

48.
H. Y. Lan, Y. J. Cho, and R. U. Verma, On solution sensitivity of generalized relaxed cocoercive implicit quasivariational inclusions with A-monotone mappings, J. Comput. Anal. Appl. 8 (2006), no. 1, 75-87.

49.
H. Y. Lan, Y. J. Cho, and R. U. Verma, Nonlinear relaxed cocoercive variational inclusions involving (A, ${\eta}$)-accretive mappings in Banach spaces, Comput. Math. Appl. 51 (2006), no. 9-10, 1529-1538.

50.
H. Y. Lan, Y. J. Cho, and W. Xie, General nonlinear random equations with random multivalued operator in Banach spaces, J. Inequal. Appl. 2009 (2009), Article ID 865093, 17 pp.; doi:10.1155/2009/865093.

51.
H. Y. Lan, J. I. Kang, and Y. J. Cho, Nonlinear (A, ${\eta}$)-monotone operator inclusion systems involving non-monotone set-valued mappings, Taiwanese J. Math. 11 (2007), no. 3, 683-701.

52.
H. Y. Lan, J. H. Kim, and Y. J. Cho, On a new system of nonlinear A-monotone multivalued variational inclusions, J. Math. Anal. Appl. 327 (2007), no. 1, 481-493.

53.
H. Y. Lan and R. U. Verma, Iterative algorithms for nonlinear fuzzy variational inclusion systems with (A, ${\eta}$)-accretive mappings in Banach spaces, Adv. Nonlinear Var. Inequal. 11 (2008), no. 1, 15-30.

54.
M. A. Noor, Two-step approximation schemes for multivalued quasi variational inclusions, Nonlinear Funct. Anal. Appl. 7 (2002), no. 1, 1-14.

55.
M. A. Noor, Variational inequalities for fuzzy mappings. I, Fuzzy Sets and Systems 55 (1993), no. 3, 309-312.

56.
J. Y. Park and J. U. Jeong, A perturbed algorithm of variational inclusions for fuzzy mappings, Fuzzy Sets and Systems 115 (2000), no. 3, 419-424.

57.
J. Y. Park and J. U. Jeong, Iterative algorithm for finding approximate solutions to completely generalized strongly quasivariational inequalities for fuzzy mappings, Fuzzy Sets and Systems 115 (2000), no. 3, 413-418.

58.
N. Onjai-Uea and P. Kumam, A generalized nonlinear random equations with random fuzzy mappings in uniformly smooth Banach spaces, J. Inequal. Appl. 2010 (2010), Art. ID 728452, 15 pp.; doi:10.1155/2010/728452.

59.
R. U. Verma, A-monotonicity and applications to nonlinear inclusion problems, J. Appl. Math. Stoch. Anal. 17 (2004), no. 2, 193-195.

60.
R. U. Verma, Approximation solvability of a class of nonlinear set-valued variational inclusions involving (A, ${\eta}$)-monotone mappings, J. Math. Anal. Appl. 337 (2008), no. 2, 969-975.

61.
R. U. Verma, Sensitivity analysis for generalized strongly monotone variational inclusions based on the (A, ${\eta}$)-resolvent operator technique, Appl. Math. Lett. 19 (2006), no. 12, 1409-1413.

62.
R. U. Verma, The over-relaxed A-proximal point algorithm and applications to nonlinear variational inclusions in Banach spaces, Fixed Point Theory 10 (2009), no. 1, 185-195.

63.
H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127-1138.

64.
Y. Yao, Y. J. Cho, and Y. Liou, Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems, European J. Oper. Res. 212 (2011), no. 2, 242-250.

65.
Y. Yao, Y. J. Cho, and Y. Liou, Iterative algorithms for variational inclusions, mixed equilibrium problems and fixed point problems approach to optimization problems, Cent. Eur. J. Math. 9 (2011), no. 3, 640-656.

66.
L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965), 338-358.

67.
H. I. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishing Group, Boston, MA, 1988.