GENERALIZED VARIATIONAL-LIKE INEQUALITIES WITH COMPOSITELY MONOTONE MULTIFUNCTIONS

- Journal title : Journal of the Korean Mathematical Society
- Volume 45, Issue 3, 2008, pp.841-858
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2008.45.3.841

Title & Authors

GENERALIZED VARIATIONAL-LIKE INEQUALITIES WITH COMPOSITELY MONOTONE MULTIFUNCTIONS

Ceng, Lu-Chuan; Lee, Gue-Myung; Yao, Jen-Chih;

Ceng, Lu-Chuan; Lee, Gue-Myung; Yao, Jen-Chih;

Abstract

In this paper, we introduce two classes of generalized variational-like inequalities with compositely monotone multifunctions in Banach spaces. Using the KKM-Fan lemma and the Nadler's result, we prove the existence of solutions for generalized variational-like inequalities with compositely relaxed monotone multifunctions in reflexive Banach spaces. On the other hand we also derive the solvability of generalized variational-like inequalities with compositely relaxed semimonotone multi functions in arbitrary Banach spaces by virtue of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper extend and improve some earlier and recent results in the literature.

Keywords

generalized variational-like inequalities;compositely (semi) monotone multifunctions;KKM mappings;Hausdorff metric-hemicontinuity;coercivity;

Language

English

Cited by

References

1.

O. Chadli, X. Q. Yang, and J. C. Yao, On generalized vector pre-variational and prequasivariational inequalities, J. Math. Anal. Appl. 295 (2004), no. 2, 392-403

2.

S. S. Chang, B. S. Lee, and Y. Q. Chen, Variational inequalities for monotone operators in nonreflexive Banach spaces, Appl. Math. Lett. 8 (1995), no. 6, 29-34

3.

Y. Q. Chen, On the semi-monotone operator theory and applications, J. Math. Anal. Appl. 231 (1999), no. 1, 177-192

4.

R. W. Cottle and J. C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl. 75 (1992), no. 2, 281-295

5.

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), no. 4, 519-537

6.

Y. P. Fang and N. J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl. 118 (2003), no. 2, 327-338

7.

F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht, 2000

8.

D. Goeleven and D. Motreanu, Eigenvalue and dynamic problems for variational and hemivariational inequalities, Comm. Appl. Nonlinear Anal. 3 (1996), no. 4, 1-21

9.

N. Hadjisavvas and S. Schaible, Quasimonotone variational inequalities in Banach spaces, J. Optim. Theory Appl. 90 (1996), no. 1, 95-111

10.

P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math. 115 (1966), 271-310

11.

I. V. Konnov and J. C. Yao, On the generalized vector variational inequality problem, J. Math. Anal. Appl. 206 (1997), no. 1, 42-58

13.

A. H. Siddiqi, Q. H. Ansari, and K. R. Kazmi, On nonlinear variational inequalities, Indian J. Pure Appl. Math. 25 (1994), no. 9, 969-973

14.

R. U. Verma, Nonlinear variational inequalities on convex subsets of Banach spaces, Appl. Math. Lett. 10 (1997), no. 4, 25-27

15.

R. U. Verma, On monotone nonlinear variational inequality problems, Comment. Math. Univ. Carolin. 39 (1998), no. 1, 91-98

16.

X. Q. Yang and G. Y. Chen, A class of nonconvex functions and pre-variational inequalities, J. Math. Anal. Appl. 169 (1992), no. 2, 359-373

17.

J. C. Yao, Existence of generalized variational inequalities, Oper. Res. Lett. 15 (1994), no. 1, 35-40

18.

G. X. Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Monographs and Textbooks in Pure and Applied Mathematics, 218. Marcel Dekker, Inc., New York, 1999

19.

L. C. Zeng, S. M. Guu, and J. C. Yao, Iterative algorithm for completely generalized set-valued strongly nonlinear mixed variational-like inequalities, Comput. Math. Appl. 50 (2005), no. 5-6, 935-945