×

Another form of KKM type theorem and its applications on generalized convex spaces. (English) Zbl 1108.47051

The authors formulate a version of the KKM theorem for \(G\)-convex spaces [cf.S. Park, Korean J.Comp.Appl.Math.7, 1–28 (2000; Zbl 0959.47035)]. The main result reads as follows. Theorem. Let \((X,D;\Gamma)\) be a \(G\)-convex space, \(Y\) a topological space, \(S:X\to Y\) and \(A:D\to Y\) maps, and \(K\) a nonempty compact subset of \(X\) such that \(S\) is upper semicontinuous on \(X\) and for each nonempty finite subset \(N\) of \(D\) there exists an \(L\) such that for every \(z\in L\setminus K\) there exists an \(x\in D\cap L\) satisfying \(S(z)\cap\overline{A(x)}=\emptyset\). Then either there exists an \(x_0\in K\) such that \(\overline{A(x)}\cap S(x_0)\not=\emptyset\) for all \(x\in D\), or there exists a finite set \(\{x_1,\dotsc,x_n\}\) in \(D\) and \(x\in\Gamma(x_1,\dotsc,x_n)\) such that \(S(x)\subset Y\setminus\bigcup_{i=1}^nA(x_i)\).
This reviewer is somewhat skeptical as to whether this is really interesting.

MSC:

47H10 Fixed-point theorems
47H04 Set-valued operators

Citations:

Zbl 0959.47035
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Knaster, B, Kuratowski, K, Mazurkiewicz S. Ein beweis des Fixpunktsatzes für n-dimensionale simplexe, Fund Math, 1929, 14:132–137. · JFM 55.0972.01
[2] Park S. Elements of the KKM theory for generalized convex spaces, Korean J Comp Appl Math, 2000, 7(1):1–28. · Zbl 0959.47035
[3] Fan Ky. A generalization of Tychonoff’s fixed point theorem Math Ann, 1961, 142:305–310. · Zbl 0093.36701 · doi:10.1007/BF01353421
[4] Lassonde M. On the use of KKM multimaps in fixed point theory and related topics, J Math Anal Appl, 1983, 97:151–201. · Zbl 0527.47037 · doi:10.1016/0022-247X(83)90244-5
[5] Horvath C D. Contractibility and generalized convexity, J. Math Anal Appl, 1991, 156:341–357. · Zbl 0733.54011 · doi:10.1016/0022-247X(91)90402-L
[6] Park S, Kim H J. Foundations of the KKM theory on generalized convex spaces, J. Math Anal Appl, 1997, 209:551–571. · Zbl 0873.54048 · doi:10.1006/jmaa.1997.5388
[7] Park S. New susclasses of generalized concex spaces. In: Cho Y J ed. Fixed Point Theory and Applications, New York: Nova Sci Publ, 2000, 91–98.
[8] Liu F C. On a form of KKM principle and supinfsup inequalities of von Neumann and Fan Ky type, J Math Anal Appl, 1991, 155:420–436. · Zbl 0734.47030 · doi:10.1016/0022-247X(91)90011-N
[9] Li H M, Ding X P. On versions of KKM principle in H-spaces and its appications, J Sichuan Normal Univ, 1993, 16(3):21–27.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.