Abstract
In this paper, a degree theory for set-valued vector variational inequalities is built in reflexive Banach spaces. By using the method of degree theory, some existence results of solutions for set-valued vector variational inequalities are established under suitable conditions. Furthermore, some equivalent characterizations for the nonemptiness and boundedness of solution sets to single-valued vector variational inequalities are obtained under pseudomonotonicity assumption. To the best of our knowledge, there are still no papers dealing with the degree theory for vector variational inequalities.
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References
Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Kien, B.T., Wong, M.-M., Wong, N.C., Yao, J.C.: Degree theory for generalized variational inequalities and applications. Eur. J. Oper. Res. 193, 12–22 (2009)
Qiao, F.S., He, Y.R.: Strict feasibility of pseudomonotone set-valued variational inequalities. Optimization 60, 303–310 (2011)
Wang, Z.B., Huang, N.J.: Degree theory for a generalized set-valued variational inequality with an application in Banach spaces. J. Glob. Optim. 49, 343–357 (2011)
Robinson, S.M.: Localized normal maps and the stability of variational conditions. Set-Valued Anal. 12, 259–274 (2004)
Gowda, M.S.: Inverse and implicit function theorems for H-differentiable and semismooth functions. Optim. Methods Softw. 19, 443–461 (2004)
Giannessi, F.: Theorems of alternative, quadratic program and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.C. (eds.) Variational Inequality and Complementarity Problems. Wiley, New York (1980)
Giannessi, F.: Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrechet, Holland (2000)
Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-Valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (2005)
Chen, G.Y.: Existence of solutions for a vector variational inequality: an extension of the Hartman–Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992)
Chen, G.Y., Li, S.J.: Existence of solutions for a generalized quasi-vector variational inequality. J. Optim. Theory Appl. 90, 321–334 (1996)
Daniilidis, A., Hadjisavvas, N.: Existence theorems for vector variational inequalities. Bull. Aust. Math. Soc. 54, 473–481 (1996)
Lee, G.M., Kim, D.S., Lee, B.S., Yen, N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. Theory Methods Appl. 34, 745–765 (1998)
Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206, 42–58 (1997)
Fang, Y.P., Huang, N.J.: Feasibility and solvability of vector variational inequalities with moving cones in Banach spaces. Nonlinear Anal. Theory Methods Appl. 70, 2024–2034 (2009)
Hu, S.C., Parageorgiou, N.S.: Generalization of Browders degree theory. Trans. Am. Math. Soc. 347, 233–259 (1995)
Wang, Z.B., Huang, N.J.: The degree theory for set-valued compact perturbation of monotone type mappings with an application. Appl. Anal. 92, 616–635 (2013)
Cioranescu, I.: Geometry of Banach Spaces Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers, Dordrecht (1990)
Huang, X.X., Fang, Y.P., Yang, X.Q.: Characterizing the nonemptiness and compactness of the solution set of a vector variational inequality by scalarization. J. Optim. Theory Appl. 162, 548–558 (2014)
Browder, F.E.: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. 9, 1–39 (1983)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/B Nonlinear Monotone Operators. Springer, Berlin (1990)
Kien, B.T., Wong, M.M., Wong, N.C.: On the degree theory for general mappings of monotone type. J. Math. Anal. Appl. 340, 707–720 (2008)
Cellina, A.: Approximations of set-valued functions and fixed point theorems. Ann. Mater. Pura. Appl. 82, 17–24 (1969)
Isac, G., Kalashnikov, V.V.: Exceptional family of elements, Leray–Schauder alternative, pseudomonotone operators and complementarity. J. Optim. Theory Appl. 109, 69–83 (2001)
Li, J., Whitaker, J.: Exceptional family of elements and solvability of variational inequalities for mappings defined only on closed convex cones in Banach spaces. J. Math. Anal. Appl. 310, 254–261 (2005)
Kien, B.T., Yao, J.-C., Yen, N.D.: On the solution existence of pseudomonotone variational inequalities. J. Glob. Optim. 41, 135–145 (2008)
Crouzeix, J.P.: Pseudomonotone variational inequality problems: existence of solutions. Math. Program. 78, 305–314 (1997)
Aubin, J.P.: Optima and Equilibria. Springer, Berlin (1993)
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11061006, 11226224 and 61363036), the Program for Excellent Talents in Guangxi Higher Education Institutions, the Guangxi Natural Science Foundation (2012GXNSFBA053008), the Initial Scientific Research Foundation for Ph.D. of Guangxi Normal University, and the Innovation Project of Guangxi Graduate Education (YCSZ2014088).
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Communicated by Jen-Chih Yao.
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Zhong, Ry., Dou, Z. & Fan, Jh. Degree Theory and Solution Existence of Set-Valued Vector Variational Inequalities in Reflexive Banach Spaces. J Optim Theory Appl 167, 527–549 (2015). https://doi.org/10.1007/s10957-015-0731-y
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DOI: https://doi.org/10.1007/s10957-015-0731-y
Keywords
- Set-valued vector variational inequality
- Topological degree
- Existence of solutions
- Nonempty and boundedness
- \(C\)-pseudomonotone mapping