Mixed Vector Equilibrium Problems with Fuzzy Mappings
In this paper, we study fuzzy generalized mixed vector equilibrium problem and fuzzy mixed
vector equilibrium problem. We prove existence results for fuzzy generalized mixed vector equilibrium
problem and fuzzy mixed vector equilibrium problem by using some basic tools as KKM
theory and Maximal element lemma. We provide sufficient conditions that ensure the existence of
the solution of these problems. The results presented in this paper generalize, improve and unify
the previously known results in this area. An example is given.
[1] G.A. Anastassiou, Salahuddin, Weakly set valued generalized vector variational inequalities,
J. Comput. Anal. Appl. 15(4) (2013) 622-632.
[2] Q.H. Ansari, W. Oettli, D. Schlager, A generalization of vectorial equilibria, Math. Methods
Oper. Res. 46 (1997) 147–152.
[3] Q.H. Ansari, I.V. Konnov, J.C. Yao, Existence of a solution and variational principles for
vector equilibrium problems, J. Optim. Theory Appl. 110 (2001) 481–492.
[4] Q.H. Ansari, I.V. Konnov, J.C. Yao, Characterizations of solutions for vector equilibrium
problems, J. Optim. Theory Appl. 113 (2002) 435–447.
[5] Q.H. Ansari, F. Flores-Bazan, Recession methods for generalized vector equilibrium problems,
J. Math. Anal. Appl. 321 (2006), 132–146.
[6] Q.H. Ansari, Vector equilibrium problems and vector variational inequalities. In: F. Giannessi
(Ed.), Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers,
Dordrecht (2000), pp. 1–15.
[7] J.P. Aubin, Applied Functional Analysis, John Wiley & Sons, New York, 2000.
[8] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems,
Math. Stud. 63 (1994) 123–145.
[9] S.S. Chang, Y.G. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets Syst. 32
(1989) 359–367.
[10] S.S. Chang, N.J. Huang, Generalized complementarity problem for fuzzy mappings, Fuzzy
Sets Syst. 55 (1993), 227–234.
[11] S.S. Chang, K.K. Tan, Equilibria and maximal elements of abstract fuzzy economics and
qualitative fuzzy games, Fuzzy Sets Syst. 125 (2002) 389–399.
[12] S.S. Chang, Salahuddin, Existence theorems for vector quasi variational-like inequalities for
fuzzy mappings, Fuzzy Sets Syst. 233 (2013) 89–95.
[13] Y. Chiang, O. Chadli, J.C. Yao, Generalized vector equilibrium problems with trifunctions, J.
Glob. Optim. 30 (2004) 135–154.
[14] X.P. Ding, Quasi-equilibrium problems in noncompact generalized convex spaces, Appl.
Math. Mech. 21(6) (2000) 637—644.
[15] X.P. Ding, J.Y. Park, A new class of generalized nonlinear implicit quasi-variational inclusions
with fuzzy mapping, J. Comput. Appl. Math. 138 (2002) 243–257.
[16] K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142 (1961) 303–310.
[17] K. Fan, A minimax inequality and its applications. In: O. Shisha (Ed.), Inequalities, Vol. (3),
Academic Press, New York, 1972, pp. 103–113.
[18] F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Mathematical theories,
Kluwer, Dordrecht, 2000.
[19] X. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory
Appl. 108 (2001) 139–154.
[20] X. Gong, Strong vector equilibrium problems, J. Global Optim. 36 (2006) 339–349.
[21] A. Gopfert, H. Riahi, C. Tammer, C. Zalinescu, Variational Methods in Partially Ordered
Spaces, Springer, New York, 2003.
[22] S. Heilpern, Fuzzy mappings and fixed point theorems, J. Math. Anal. Appl. 83 (1981) 566–569.
[23] N.J. Huang, H.Y. Lan, A couple of nonlinear equations with fuzzy mappings in fuzzy normed
spaces, Fuzzy Sets Syst. 152 (2005), 209–222.
[24] M.F. Khan, S. Husain, Salahuddin, A fuzzy extension of generalized multi-valued h-mixed
vector variational-like inequalities on locally convex Hausdorff topological vector spaces,
Bull. Cal. Math. Soc. 100(I) (2008) 27–36.
[25] W.K. Kim, K.H. Lee, Generalized fuzzy games and fuzzy equilibria, Fuzzy Sets Syst. 122
(2001), 293–301.
[26] H.Y. Lan, R.U. Verma, Iterative algorithms for nonlinear fuzzy variational inclusions with
(A;h)-accretive mappings in Banach spaces, Adv. Nonlinear Var. Inequal. 11(1) (2008) 15–30.
[27] G.M. Lee, D.S. Kim, B.S. Lee, Vector variational inequality for fuzzy mappings, Nonlinear
Anal. Forum 4 (1999) 119–129.
[28] B.S. Lee, M.F. Khan, Salahuddin, Fuzzy nonlinear set-valued variational inclusions, Comput.
Math. Appl. 60(6) (2010) 1768–1775.
[29] J. Li, N. J. Huang, J. K. Kim, On implicit vector equilibrium problems, J. Math. Anal. Appl.
283 (2003), 501–502.
[30] A. Moudafi, Mixed equilibrium problems: sensitivity analysis and algorithmic aspect, Comput.
Math. Appl. 44 (2002) 1099–1108.
[31] M. Rahaman, R. Ahmad, Fuzzy vector equilibrium problems, Iranian J. Fuzzy Syst. 12(1)
(2015) 115–122.
[32] E. Shivanan, E. Khorram, Optimization of linear objective function subject to fuzzy relation
inequalities constraints with max-product composition, Iranian J. Fuzzy Syst. 7(5) (2010),51–71.
[33] C.H. Su, V.M. Sehgal, Some fixed point theorems for condensing multi-functions in locally
convex spaces, Proc. Natl. Acad. Sci. USA 50 (1975) 150–154.
[34] E. Tarafdar, Fixed point theorems in H-spaces and equilibrium points of abstract economies,
J. Austral. Math. Soc. Ser. A 53 (1992), 252–260.
[35] G. Xiao, Z. Fan, R. Qi, Existence results for generalized nonlinear vector variational-like
inequalities with set valued mapping, Appl. Math. Lett. 23 (2010) 44–47.
[36] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.
[37] H.J. Zimmermann, Fuzzy set Theory and Its Applications, Kluwer Academic Publishers,
Dordrecht, 1988.