Abstract
In this paper, we consider vector variational inequalities with set-valued mappings over countable product sets in a real Banach space setting. By employing concepts of relative pseudomonotonicity, we establish several existence results for generalized vector variational inequalities and for systems of generalized vector variational inequalities. These results strengthen previous existence results which were based on the usual monotonicity type assumptions
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References
Aliprantis, C.D., Brown, D.J. and Burkinshaw, O. (1989), Existence and Optimality of Competitive Equlibria, Springer, Berlin.
Allevi, E., Gnudi, A. and Konnov, I.V. (2001), Generalized vector variational inequalities over product sets, Nonlinear Analysis 573-582.
Ansari, Q.H. and Yao, J.-C. (1999), A fixed point theorem and its applications to a system of variational inequalities, Bull. Austral. Math. Soc. 433-442.
Ansari, Q.H. and Yao, J.-C. (2000), Systems of generalized variational inequalities and their applications, Appl. Anal., 203-217.
Bianchi, M. (1993), Pseudo P-monotone operators and variational inequalities, Research Report No.6, Istituto di Econometria Matematica per le Decisioni Economiche, Università Cattolica del Sacro Cuore, Milan.
Diestel, J. and Uhl, J.J. (1970), Vector Measures, American Mathematical Society, Providence.
Florenzano, M. (1983), On the existence of equilbria in economies with an infinite dimensional commodities space, Journal of Mathematical Economics 207-219.
Hadjisavvas, N. and Schaible, S. (1998), Quasimonotonicity and pseudomonotonicity in variational inequalities and equilibrium problems. in: eds., Crouzeix, J.-P., Martinez-Legaz, J.E. and Volle, M. Generalized Convexity, Generalized Monotonicity, Kluwer Academic Publishers, Dordrecht — Boston — London, pp. 257-275.
Konnov, I.V. (1995), Combined relaxation methods for solving vector equilibrium problems, Russ. Math. (Iz. VUZ), no. 12, 51-59.
Konnov, I.V. and Yao, J.C. (1997), On the generalized vector variational inequality problem, J. Math. Anal. and Appl. 42-58.
Konnov, I.V. (1999), Combined relaxation method for decomposable variational inequalities, Optimiz. Methods and Software 711-728.
Konnov, I.V. (2001), Relatively monotone variational inequalities over product sets, Operations Research Letters 28, 21-26.
Kreps, D.M. (1979), Arbitrage and equilbrium in economies with infinitely many commodities, Journal of Mathematical Economics, 15-35.
Fan, Ky (1961), A generalization of Tychonoff ‘s fixed-point theorem, Math. Annalen, 305-310.
Oettli, W. and Schläger, D. (1998), Generalized vectorial equilibria and generalized monotonicity. In: eds., Functional Analysis with Current Applications in Science, Technology and Industry, Brokate, M. and Siddiqi, A.H., Pitman Research Notes in Mathematical Series, No. 377, Addison Wesley Longman Ltd., Essex, 145-154.
Rosen, J.B. (1965), Existence and uniqueness of equilibrium points for concave n-person games, Econometrica, 520-534.
Yuan, G.X.Z. (1998), The Study of Minimax Inequalities and Applications to Economies and Variational Inequalities, Memoires of the AMS, 625.
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Allevi, E., Gnudi, A. & Konnov, I. Generalized Vector Variational Inequalities over Countable Product of Sets. J Glob Optim 30, 155–167 (2004). https://doi.org/10.1007/s10898-004-8272-1
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DOI: https://doi.org/10.1007/s10898-004-8272-1