Fixed point and coincidence theorems of set-valued mappings in topological vector spaces with some applications. (English) Zbl 0989.47053
From the introduction: In most of the generalizations of Kakutani-Fan-Glicksberg type fixed-point theorems, the underlying domains are paracompact and the mappings are upper semicontinuous and self-mapping. In this paper, using the existence theorem for maximizable quasi-concave functions on convex spaces of S. Park and J. S. Bae [J. Korean Math. Soc. 28, No. 2, 285-292 (1991; Zbl 0756.47050)] (which is a generalization of Fan’s existence theorem for maximizable quasi-concave functions on convex spaces) we first prove some coincidence theorems for upper hemicontinuous non-self-mappings in topological vector spaces with sufficiently many continuous linear functionals or in locally convex topological vector spaces. These results improve and unify many results in the literature [Ky Fan, Math. Ann. 266, 519-537 (1984; Zbl 0515.47029); S. Park, Contemp. Math. 72, 183-191 (1988; Zbl 0672.47046); H. M. Ko and K.-K. Tan, Tamkang J. Math. 17, No. 2, 37-45 (1986; Zbl 0613.47051)] and references therein. Next, as applications of coincidence theorems, several matching theorems for closed coverings of convex sets are derived in locally convex topological vector spaces or topological vector spaces with sufficiently many continuous linear functionals which, in turn, imply L. S. Shapley’s theorem [in ‘Mathematical programming’, Proc. advanced Seminar, Univ. Wisconsin, Madision 1972, 261-290 (1973; Zbl 0267.90100)].
MSC:
| 47H10 | Fixed-point theorems |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54C60 | Set-valued maps in general topology |
| 47H04 | Set-valued operators |
| 47J05 | Equations involving nonlinear operators (general) |
Keywords:
Kakutani-Fan-Glicksberg type fixed-point theorems; maximizable quasi-concave functions on convex spaces; coincidence theorems; hemicontinuous non-self-mappings; matching theorems for closed coverings of convex setsCitations:
Zbl 0756.47050; Zbl 0515.47029; Zbl 0672.47045; Zbl 0613.47051; Zbl 0267.90100; Zbl 0672.47046References:
| [1] | Park, S.; Bae, J. S., Existence of maximizable quasiconcave functions on convex spaces, J. Korean Math. Soc., 28, 285-292 (1991) · Zbl 0756.47050 |
| [2] | Fan, K., Some properties of convex sets related to fixed point theorems, Math. Ann., 266, 519-537 (1984) · Zbl 0515.47029 |
| [3] | Park, S., Fixed point theorems on compact convex sets in topological vector spaces, Contemporary Mathematics, 72, 183-191 (1988) · Zbl 0672.47046 |
| [4] | Ko, H. M.; Tan, K. K., A coincidence theorem with applications to minimax inequalities and fixed point theorems, Tamkang J. Math., 17, 37-45 (1986) · Zbl 0613.47051 |
| [5] | Shapley, L. S., On balanced games without side payments, (Hu, T. C.; Robinson, S. M., Mathematical Programming (1973), Academic Press: Academic Press New York), 260-290 · Zbl 0267.90100 |
| [6] | Halpern, B., Fixed-point theorems for outward maps, (Doctoral Thesis (1965), U.C.L.A) · Zbl 0191.14701 |
| [7] | Fan, K., Extensions of two fixed point theorems of F. E. Browder, Math. Z., 112, 234-240 (1969) · Zbl 0185.39503 |
| [8] | Tiel, J. V., Convex Analysis: An Introductory Text (1984), Wiley · Zbl 0565.49001 |
| [9] | Halpern, B.; Bergman, G., A fixed point theorem for inward and outward maps, Trans. Amer. Math. Soc., 130, 353-358 (1968) · Zbl 0153.45602 |
| [10] | Browder, F. E., A new generalization of the Schauder fixed point theorem, Math. Ann., 174, 285-290 (1967) · Zbl 0176.45203 |
| [11] | Rockafellar, P. T., Convex Analysis (1970), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0193.18401 |
| [12] | Fan, K., A minimax inequality and its applications, (Shisha, O., Inequality (1972), Academic Press), 103-113, (III) · Zbl 0302.49019 |
| [13] | Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl., 97, 151-201 (1983) · Zbl 0527.47037 |
| [14] | Browder, F. E., Coincidence theorems, minimax theorems, and variational inequalities, Contemporary Math., 26, 67-80 (1984) · Zbl 0542.47046 |
| [15] | Bellenger, J. C., Existence of maximizable quasiconcave functions on paracompact convex spaces, J. Math. Anal. Appl., 123, 333-338 (1987) · Zbl 0649.46006 |
| [16] | Simons, S., An existence theorem for quasiconcave functions with applications, Nonlinear Analysis, 10, 1133-1152 (1986) · Zbl 0616.47045 |
| [17] | Ding, X. P., Existence of maximizable quasi-concave functions on \(H\)-Spaces, Acta Math. Sinica, 36, 273-279 (1993), (In Chinese.) · Zbl 0822.46006 |
| [18] | Aubin, J. P., Mathematical Methods of Game and Economic Theory (1982), North-Holland: North-Holland Amsterdam |
| [19] | Shih, M. H.; Tan, K. K., Shapely selections and covering theorems of samples, (Lin, B. L.; Simons, S., Nonlinear and Convex Analysis (1987), Marcel Dekker: Marcel Dekker New York), 245-251 · Zbl 0646.52001 |
| [20] | Rudin, W., Functional Analysis (1973), McGraw-Hill: McGraw-Hill New York · Zbl 0253.46001 |
| [21] | Ha, C. W., Minimax and fixed point theorems, Math. Ann., 248, 73-77 (1980) · Zbl 0413.47042 |
| [22] | Browder, F. E., The fixed point theory of multivalued mappings in topological vector spaces, Math. Ann., 177, 283-301 (1968) · Zbl 0176.45204 |
| [23] | Ha, C. W., On a minimax inequality of Ky Fan, (Proc. Amer. Math. Soc., 99 (1987)), 680-682 · Zbl 0633.47037 |
| [24] | Reich, S., Fixed points in locally convex spaces, Math. Z., 125, 17-31 (1972) · Zbl 0216.17302 |
| [25] | Ko, H. M.; Tan, K. K., Coincidence theorems and matching theorems, Tamkang J. Math., 23, 297-309 (1992) · Zbl 0777.47034 |
| [26] | Shih, M. H.; Tan, K. K., Covering theorems of convex sets related to fixed point theorems, (Lin, B. L.; Simon, S., Nonlinear and Convex Analysis, Proceeding in Honor of Ky Fan (1987), Marcel Dekker: Marcel Dekker New York), 235-244 · Zbl 0637.47029 |
| [27] | Horvath, C. D., Constructibility and generalized convexity, J. Math. Anal. Appl., 156, 2, 341-357 (1991) · Zbl 0733.54011 |
| [28] | Lassonde, M., Marc Sur le principe KKM (French), C. R. Acad. Sci. Paris Sir. I Math., 310, 7, 573-576 (1990) · Zbl 0715.47038 |
| [29] | Deguire, P., Browder-Fan fixed point theorem and related results, Discuss. Math. Differential Incl., 15, 2, 149-162 (1995) · Zbl 0931.47041 |
| [30] | Deguire, P.; Lassonde, M., Marc Familles silectantes (French), Topol. Methods Nonlinear Anal., 5, 2, 261-269 (1995) · Zbl 0877.54017 |
| [31] | Deguire, P.; Yuan, X. Z., Maximal elements and coincident points for couple-majorized mappings in product topological vector spaces, Z. Anal. Anwendungen, 14, 4, 665-676 (1995) · Zbl 0848.47034 |
| [32] | Tarafdar, E.; Yuan, X. Z., A remark on coincidence theorems, (Proc. Amer. Math. Soc., 122 (1994)), 957-959, (3) · Zbl 0818.47056 |
| [33] | Dugundji, J., Topology (1966), Allyn and Bacon: Allyn and Bacon Boston, MA · Zbl 0144.21501 |
| [34] | Ding, X. P.; Tan, K. K., A minimax inequality with applications to existence of equilibrium points and fixed point theorems, Colloquium Math., 68, 233-247 (1992) · Zbl 0833.49009 |
| [35] | Park, S., Some coincidence theorems on acyclic multifunctions and applications to KKM theory, (Tan, K. K., Fixed Point Theory and Applications (1992), World Scientific: World Scientific Singapore), 248-278 · Zbl 1426.47005 |
| [36] | Tarafdar, E., A fixed point theorem in \(H\)-space and related results, Bull. Austral. Math. Soc., 42, 133-140 (1990) · Zbl 0714.47039 |
| [37] | Tan, K. K.; Yuan, X. Z., A minimax inequality with applications to the existence of equilibrium points, Bull. Austral. Math. Soc., 47, 483-503 (1993) · Zbl 0803.47059 |
| [38] | Zeidler, E., (Nonlinear Functional Analysis and its Applications: Fixed Point Theorems, Vol. I (1985), Springer-Verlag: Springer-Verlag New York) · Zbl 0583.47051 |
| [39] | Yuan, X. Z., Knaster-Kuratowski-Mazurkiewicz theorem, Ky Fan minimax inequalities and fixed point theorems, Nonlinear World, 2, 131-169 (1995) · Zbl 0923.47028 |
| [40] | Yuan, X. Z., The study of minimax inequalities and applications to economics and variational inequalities, Mem. Amer. Math. Soc., 132, 625, 140 (1998) · Zbl 0911.49005 |
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