An iterative algorithm for random generalized nonlinear mixed variational inclusions for random fuzzy mappings. (English) Zbl 1081.65060
The aim of this paper is to introduce and study the random generalized nonlinear mixed variational inclusions for random fuzzy mappings. An iterative algorithm for finding the approximate solutions of this class of variational inclusions is defined.
Main result: By using the definition of multivalued relaxed Lipschitz and relaxed monotone operator, the authors prove that the approximate solutions obtained by the iterative algorithm converge to the exact solution of the random generalized nonlinear mixed variational inclusions for random fuzzy mappings.
Main result: By using the definition of multivalued relaxed Lipschitz and relaxed monotone operator, the authors prove that the approximate solutions obtained by the iterative algorithm converge to the exact solution of the random generalized nonlinear mixed variational inclusions for random fuzzy mappings.
Reviewer: Jan Lovíšek (Bratislava)
MSC:
| 65K10 | Numerical optimization and variational techniques |
| 49J40 | Variational inequalities |
| 49J53 | Set-valued and variational analysis |
| 26E50 | Fuzzy real analysis |
Keywords:
Variational inclusions; Algorithm; Random fuzzy mappings; Relaxed Lipschitz operator; Relaxed monotone operators; iterative algorithmReferences:
| [1] | Ahmad, R.; Ansari, Q. H., An iterative algorithm for generalized nonlinear variational inclusions, Appl. Math. Lett., 13, 23-26 (2000) · Zbl 0954.49006 |
| [2] | Chang, S. S.; Zhu, Y., On variational inequalities for fuzzy mappings, Fuzzy Set. Syst., 32, 359-367 (1989) · Zbl 0677.47037 |
| [3] | Chang, S. S., Variational Inequality and Complementarity Problem Theory with Applications (1991), Shanghai Scientific and Tech. Literature Publishing House: Shanghai Scientific and Tech. Literature Publishing House Shanghai |
| [4] | Chang, S. S.; Huang, N. J., Generalized random multivalued quasi-complementarity problems, Ind. J. Math., 35, 305-320 (1993) · Zbl 0811.60049 |
| [5] | Chang, S. S.; Huang, N. J., Random generalized set-valued quasi-complementarity problems, Acta Math. Appl. Sinica, 16, 396-405 (1993) · Zbl 0793.47063 |
| [6] | Chang, S. S.; Zhu, Y. G., On the problems for a class of random variational inequalities and quasi-variational inequalities, J. Math. Res. Exposition, 9, 385-393 (1989) · Zbl 1008.49506 |
| [7] | Chang, S. S.; Huang, N. J., Generalized strongly nonlinear quasi-complementarity problems in Hilbert spaces, J. Math. Anal. Appl., 158, 194-202 (1991) · Zbl 0739.90067 |
| [8] | Chang, S. S.; Huang, N. J., Generalized multivalued implicit complementarity problems in Hilbert spaces, Math. Jpn., 36, 1093-1100 (1991) · Zbl 0748.49006 |
| [9] | Chang, S. S., Fixed Point Theory with Applications (1984), Chongqing Publishing House: Chongqing Publishing House Chongqing |
| [10] | Hassouni, A.; Moudafi, A., A perturbed algorithm for variational inclusions, J. Math. Anal. Appl., 185, 706-712 (1994) · Zbl 0809.49008 |
| [11] | Husain, T.; Tarafdar, E.; Yuan, X. Z., Some results on random generalized games and random quasi-variational inequalities, Far East. J. Math. Sci., 2, 35-55 (1994) · Zbl 0952.91500 |
| [12] | Heilpern, S., Fuzzy mappings and fixed point theorems, J. Math. Anal. Appl., 83, 566-569 (1981) · Zbl 0486.54006 |
| [13] | Huang, N. J., Random general set-valued strongly nonlinear quasi-variational inequalities, J. Sichuan Univ., 31, 420-425 (1994) · Zbl 0836.47046 |
| [14] | Huang, N. J., Random generalized set-valued implicit variational inequalities, J. Liaoning Normal Univ., 18, 89-93 (1995) |
| [15] | Huang, N. J., Random generalized nonlinear variational inclusions for random fuzzy mappings, Fuzzy Set. Syst., 105, 437-444 (1999) · Zbl 0961.49004 |
| [16] | Huang, N. J.; Hu, X.-Q., Generalized multi-valued nonlinear quasi-complementarity problems in Hilbert spaces, J. Sichuan Univ., 31, 306-310 (1994) · Zbl 0808.47030 |
| [17] | Himmelberg, C. J., Measurable relations, Fund. Math., 87, 53-72 (1975) · Zbl 0296.28003 |
| [18] | S.S. Irfan, Existence results and solution methods for certain variational inequalities and complementarity problems, Ph.D. Thesis, Aligarh Muslim University, Aligarh, 2002.; S.S. Irfan, Existence results and solution methods for certain variational inequalities and complementarity problems, Ph.D. Thesis, Aligarh Muslim University, Aligarh, 2002. |
| [19] | Isac, G., A special variational inequality and the implicit complementarity problem, J. Fac. Sci. Univ. Tokyo, 37, 109-127 (1990) · Zbl 0702.49008 |
| [20] | Lassonde, M., On the use of KKM multifunction in fixed point theory and related topics, J. Math. Anal. Appl., 97, 151-201 (1983) · Zbl 0527.47037 |
| [21] | Mosco, U., Implicit variational problems and quasi-variational inequalities, (Lecture Notes in Mathematics, vol. 543 (1976), Springer Verlag: Springer Verlag Berlin) · Zbl 0346.49003 |
| [22] | Shih, M. H.; Tan, K. K., Generalized quasi-variational inequalities in locally convex spaces, J. Math. Anal. Appl., 108, 333-343 (1985) · Zbl 0656.49003 |
| [23] | Siddiqi, A. H.; Ansari, Q. H., Strongly nonlinear quasi-variational inequalities, J. Math. Anal. Appl., 149, 444-450 (1990) · Zbl 0712.49009 |
| [24] | Siddiqi, A. H.; Ansari, Q. H., General strongly nonlinear variational inequalities, J. Math. Anal. Appl., 166, 386-392 (1992) · Zbl 0770.49006 |
| [25] | Takahashi, W., Nonlinear variational inequalities and fixed point theorems, J. Math. Soc. Jpn., 28, 168-181 (1976) · Zbl 0314.47032 |
| [26] | Tan, K. K.; Tarafdar, E.; Yuan, X. Z., Random variational inequalities and applications to random minimization and nonlinear boundary problems, Panamer. Math. J., 4, 55-71 (1994) · Zbl 0847.60047 |
| [27] | Tan, N. X., Random quasi-variational inequality, Math. Nachr., 125, 319-328 (1986) · Zbl 0618.49003 |
| [28] | Yuan, X. Z., Non-compact random generalized games and random quasi-variational inequalities, J. Appl. Stochastic Anal., 7, 467-486 (1994) · Zbl 0821.47049 |
| [29] | Yen, C. L., A minimax inequality and its application to variational inequality, Pacific J. Math., 97, 142-150 (1981) |
| [30] | Zadeh, L. A., Fuzzy Sets, Inform. Contr., 8, 338-353 (1965) · Zbl 0139.24606 |
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