Abstract
The purpose of this paper is to study the existence of zero points for set-valued pseudomonotone operators in a Banach space by using a new condition which was recently proposed by the authors (Matsushita and Takahashi, Set-Valued Analysis 15:251–264, 2007).
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Alber, Ya.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A.G.(eds) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50. Marcel Dekker, New York (1996)
Alber, Ya.I., Ryazantseva, I.P.: Nonlinear Ill-posed Problems of Monotone Type. Springer, Dordrecht (2006)
Asplund, E.: Averaged norms. Israel J. Math. 5, 227–233 (1967)
Aubin, J.-P.: Optima and Equilibria. Springer, Berlin Heidelberg New York (1993)
Browder, F.E.: Nonlinear elliptic boundary problems. Bull. Am. Math. Soc. 69, 299–301 (1963)
Cioranescu, I.: Geometry of Banach spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers Group, Dordrecht (1990)
Ceng, L.C., Yao, J.C.: Approximate proximal algorithms for generalized variational inequalities with pseudomonotone multifunctions. J. Comp. Appl. Math. in press, http://dx.doi.org/10.1016/j.cam.2007.01.034
Cottle, R.W., Yao, J.C.: Pseudo-monotone complementarity problem in Hilbert space. J. Optim. Theory Appl. 75, 281–295 (1992)
Crouzeix, J.-P.: Pseudomonotone variational inequality problems: existence of solutions. Math. Prog. 78, 305–314 (1997)
Daniilidis, A., Hadjisavvas, N.: Coercivity conditions and variational inequalities. Math. Prog. 86, 433– (1999)
Facchinei, F., Pang, J.-S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer-Verlag, New York (2003)
Fan, K.: Generalization of Tychonoff’s fixed point theorem. Mathematische Annalen 142, 305–310 (1961)
Gwinner, J.: On fixed points and variational inequalities—a circular tour. Nonlinear Analysis 5, 565–583 (1981)
Herings, P.J.J., Koshevoy, G.A., Talman, A.J.J., Yang, Z.: General existence theorem of zero points. J. Optim. Theory Appl. 120, 375–394 (2004)
Hirano, N., Takahashi, W.: Existence theorems on unbounded sets in Banach spaces. Proc. Amer. Math. Soc. 80, 647–650 (1980)
Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18, 445–454 (1976)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York-London (1980)
Kneser, H.: Sur un théoréme fondamental de la théorie des jeux, Comptes Rendus Mathématique. vol. 234, pp. 2418–2420. Académie des Sciences, Paris (1952)
Matsushita, S., Takahashi, W.: On the existence of zeros of monotone operators in reflexive Banach spaces. J. Math. Anal. Appl. 323, 1354–1364 (2006)
Matsushita, S., Takahashi, W.: Existence theorems for set-valued operators in Banach spaces. Set-Valued Analysis 15, 251–264 (2007)
Minty, G.: On a “monotonicity” method for the solution of nonlinear equations in Banach spaces. Proc. Nat. Acad. Sci. USA. vol. 50, 1038–1041 (1963)
Schaible, S., Yao, J.C., Zeng, L.C.: A proximal method for pseudomonotone type variational-like inequalities. Taiwanese J. Math. 10, 497–513 (2006)
Shih, M.-H., Tan, K.K.: Browder-Hartman-Stampacchia variational inequalities for multi-valued monotone operator. J. Math. Anal. Appl. 134, 431–440 (1988)
Takahashi, W.: Convex Analysis and Approximation Fixed Points. Yokohama Publishers, Yokohama (2000) (Japanese)
Takahashi, W.: Nonlinear Functional Analysis. Fixed Points Theory and its Applications. Yokohama Publishers, Yokohama (2000)
Yao, J.C.: Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 83, 391–403 (1994)
Yao, J.C.: Variational inequalities with generalized monotone operators. Math. Operat. Res. 19, 691–705 (1994)
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Matsushita, Sy., Takahashi, W. Existence of zero points for pseudomonotone operators in Banach spaces. J Glob Optim 42, 549–558 (2008). https://doi.org/10.1007/s10898-008-9277-y
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DOI: https://doi.org/10.1007/s10898-008-9277-y