Mixed Vector Equilibrium Problems with Fuzzy Mappings

Mohd. Akram

Keywords: Vector equilibrium problem, KKM theory, Maximal element lemma, Fuzzy setting, Existence, AMS Subject Classification: 47J22; 47J25; 47S40

Issue I, Volume I, Pages 107-120

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.