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Generalized variational inequalities and fixed point theorems. (English) Zbl 0912.49006
In this paper, the authors deduce, from a well-known fixed point theorem of Idzik, new variational inequalities for generalized upper hemicontinuous multimaps. Consequently, all of the results of W. K. Kim and K.-K. Tan [Bull. Aust. Math. Soc. 46, No. 1, 139-148 (1992; Zbl 0747.47037)] and X. Ding [J. Suchuan Norm. Univ. 17, No. 6, 10-16 (1994; MR 96b:47064)] and some others are substantially extended and improved. Moreover, they establish new fixed point theorems on generalized upper hemicontinuous multimaps including a large number of historically well-known extensions of the Brouwer or Kakutani theorems. The fixed point theorems established by the authors are useful tools in nonlinear analysis; they have many interesting applications.

MSC:
49J40 Variational inequalities
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
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[1] Idzik, A., Almost fixed point theorems, (), 779-784 · Zbl 0691.47046
[2] Kim, W.K.; Tan, K.-K., A variational inequality in non-compact sets and its applications, Bull. austral. math. soc., 42, 139-148, (1992) · Zbl 0747.47037
[3] Ding, X.-P., A class of generalized variational inequalities and its applications, J. of sichuan norm. univ. (nat. sci), 17, 10-16, (1994)
[4] Hadžić, O., Fixed point theory in topological vector spaces, (1984), University of Novi Sad Novi Sad · Zbl 0576.47030
[5] Weber, H., Compact convex sets in non-locally convex linear spaces, Schauder-Tychonoff fixed point theorem, (), 37-40 · Zbl 0760.47030
[6] Lassonde, M., Réduction du cas multivoque au cas univoque dans LES problèmes de coïncidence, (), 293-302 · Zbl 0819.47074
[7] Ben-El-Mechaiekh, H.; Deguire, P.; Granas, A., Points fixes et coïncidences pour LES fonctions multivoque-II (applications de type ϕ et \(ϕ\^{}\{∗\}\)), C. R. acad. sci. Paris, 295, 381-384, (1982) · Zbl 0525.47043
[8] Ben-El-Mechaiekh, H., The coincidence problem for compositions of set-valued maps, Bull. austral. math. soc., 41, 421-434, (1990) · Zbl 0688.54030
[9] Ben-El-Mechaiekh, H., Fixed points for compact set-valued maps, Q & A in general topology, 10, 153-156, (1992) · Zbl 0803.54038
[10] Zhang, C.-J., Generalized variational inequalities and generalized quasi-variational inequalities, Appl. math. & mech., 14, 333-344, (1993) · Zbl 0780.49010
[11] Chang, S.-S.; Zhang, C.-J., On a class of generalized variational inequalities and quasi-variational inequalities, J. math. anal. appl., 179, 250-259, (1993) · Zbl 0803.49011
[12] Berge, C., Espaces topologique, (1959), Dunod Paris
[13] Shih, M.-H.; Tan, K.-K., Generalized bi-quasi-variational inequalities, J. math. anal. appl., 143, 66-85, (1989) · Zbl 0688.49008
[14] Ding, X.-P.; Tan, K.-K., Generalized variational inequalities and generalized quasi-variational inequalities, J. math. anal. appl., 148, 497-508, (1990) · Zbl 0714.49013
[15] Kum, S., A generalization of generalized quasi-variational inequalities, J. math. anal. appl., 182, 158-164, (1994) · Zbl 0804.49012
[16] Browder, F.E., A new generalization of the Schauder fixed point theorem, Math. ann., 174, 285-290, (1967) · Zbl 0176.45203
[17] Shih, M.-H.; Tan, K.-K., Minimax inequalities and applications, Contemp. math. amer. math. soc., 54, 45-63, (1986)
[18] Kneser, H., Sur un theoreme fondamental de la theorie des jeux, C. R. acad. sci. Paris, 234, 2418-2420, (1952) · Zbl 0046.12201
[19] Sion, M., On general minimax theorems, Pacific J. math., 8, 171-176, (1958) · Zbl 0081.11502
[20] Browder, F.E., The fixed point theory of multi-valued mappings in topological vector spaces, Math. ann., 177, 283-301, (1968) · Zbl 0176.45204
[21] Park, S. and Chen, M.-P., Generalized variational inequalities of the Hartman-Stampacchia-Browder type. J. Inequalities Appl. (To appear.) · Zbl 0890.49004
[22] Park, S., Fixed point theory of multifunctions in topological vector spaces, J. Korean math. soc., 29, 191-208, (1992) · Zbl 0758.47048
[23] Park, S., Fixed point theory of multifunctions in topological vector spaces—II, J. Korean math. soc., 30, 413-431, (1993) · Zbl 0797.47029
[24] Park, S., Remarks on fixed points of generalized upper hemicontinuous maps, (), 15-24
[25] Park, S.; Bae, J.S., On zeros and fixed points of multifunctions with non-compact convex domains, Comment. math. univ. carolinae, 34, 257-264, (1993) · Zbl 0834.47050
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