We introduce the concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity. We consider a class of generalized vector variationa-llike inequality problem involving generalized relaxed semimonotone mapping. By using Kakutani-Fan-Glicksberg’s fixed-point theorem, we prove the solvability for this class of vector variational-like inequality with relaxed monotonicity assumptions. The results presented in this paper generalize some known results for vector variational inequality in recent years. 1. Introduction Vector variational inequalities were initially introduced and considered by Giannessi [1] in a finite-dimensional Euclidean space in 1980. Due to its wide application the theory of the vector variational inequality is generalized in different directions and many existence results and algorithms for vector variational inequality problems have been established under various conditions; see for examples [2–7] and references therein. The concept monotonicity and the compactness operators are very useful in nonlinear functional analysis and its applications. In 1968, Browder [8] first combined the compactness and accretion of operators and posed the concept of a semiaccretive operator. Motivated by this idea, Chen [9] studied the concept of a semimonotone operator, which combines the compactness and monotonicity of an operator and posed it to the study of variational inequalities. Recently in 2003, Fang and Huang [10] introduced relaxed - -semimonotone mapping, a generalized concept of semimonotonicity, and they established several existence results for the variational-like inequality problem. In this paper, we pose two new concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity as well as two classes of generalized vector variational-like inequalities with generalized relaxed monotone mappings and generalized relaxed semimonotone mappings. We investigate the solvability of vector variational-like inequalities with generalized relaxed semimonotone mappings by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper generalize the results of Chen [9], Fang and Huang [10], Usman and Khan [11], and Zheng [7]. 2. Preliminaries Throughout the paper unless otherwise specified, let and be two real Banach spaces, be a nonempty closed and convex subset of . is said to be a closed convex and pointed cone with its apex at the origin, if the following conditions hold: (i) , for？all , (ii) ,(iii) . The partial order in , induced by the pointed cone , is defined by declaring
References
[1]
F. Giannessi, “Theorems of alternative, quadratic programs and complementarity problems,” in Variational Inequalities and Complementarity Problems, R. W. Cottle, F. Giannessi, and J. L. Lions, Eds., pp. 151–186, John Wiley & Sons, New York, NY, USA, 1980.
[2]
G. Y. Chen, “Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia theorem,” Journal of Optimization Theory and Applications, vol. 74, no. 3, pp. 445–456, 1992.
[3]
G. Y. Chen and X. Q. Yang, “The vector complementary problem and its equivalences with the weak minimal element in ordered spaces,” Journal of Mathematical Analysis and Applications, vol. 153, no. 1, pp. 136–158, 1990.
[4]
F. Giannessi, Vector Variational Inequalities and Vector Equilibrium, Kluwer Academic Press, 1999.
[5]
K. L. Lin, D. P. Yang, and J. C. Yao, “Generalized vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 92, no. 1, pp. 117–126, 1997.
[6]
X. Q. Yang, “Vector variational inequality and its duality,” Nonlinear Analysis, vol. 21, no. 11, pp. 869–877, 1993.
[7]
F. Zheng, “Vector variational inequalities with semi-monotone operators,” Journal of Global Optimization, vol. 32, no. 4, pp. 633–642, 2005.
[8]
F. E. Browder, “Semi-contract and semi-accretive nonlinear mappings in Banach space,” Bulletin of the American Mathematical Society, vol. 74, pp. 660–665, 1968.
[9]
Y. Q. Chen, “On the semi-monotone operator theory and applications,” Journal of Mathematical Analysis and Applications, vol. 231, no. 1, pp. 177–192, 1999.
[10]
Y. P. Fang and N. J. Huang, “Variational-like inequalities with generalized monotone mappings in Banach spaces,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 327–338, 2003.
[11]
F. Usman and S. A. Khan, “A generalized mixed vector variational-like inequality problem,” Nonlinear Analysis, vol. 71, no. 11, pp. 5354–5362, 2009.
[12]
D. Goeleven and D. Motreanu, “Eigenvalue and dynamic problems for variational and hemivariational inequalities,” Communications on Applied Nonlinear Analysis, vol. 3, no. 4, pp. 1–21, 1996.
[13]
R. U. Verma, “On monotone nonlinear variational inequality problems,” Commentationes Mathematicae Universitatis Carolinae, vol. 39, no. 1, pp. 91–98, 1998.
[14]
R. U. Verma, “Nonlinear variational inequalities on convex subsets of banach spaces,” Applied Mathematics Letters, vol. 10, no. 4, pp. 25–27, 1997.
[15]
K. Fan, “A generalization of Tychonoff's fixed point theorem,” Mathematische Annalen, vol. 142, no. 3, pp. 305–310, 1961.
[16]
E. Zeidle, Nonlinear Functional Analysis and Its Applications, vol. 4, Springer, Berlin, Germany, 1993.
