×

A general coincidence theory for set-valued maps. (English) Zbl 0938.47036

The author derives coincidence theorems for multivalued maps admitting continuous selections by translating this problem into a fixed point problem for multivalued maps with acyclic values. His main fixed point result reads as follows: Let \(E\) be a Banach space and \(Q\ni 0\) a closed convex subset of \(E\). Choose a retraction \(r:E\to Q\) and let \(G\) be a map which assigns to each \(x\in Q\) a nonempty closed acyclic subset of \(E\). Assume that \(G(Q)\) is bounded and that \(G\circ r\) is condensing with closed graph. Assume moreover that for each sequence \((x_n,\lambda_n)\) in \(\partial Q\times[0,1]\) which converges to \((x,\lambda)\in\partial Q\times[0,1)\) with \(x\in\lambda G(x)\) there is an \(n_0\) such that \(\lambda_nG(x_n)\subset Q\) whenever \( n\geq n_0\). The conclusion is that \(G\) has a fixed point in \(Q\).
Reviewer: C.Fenske (Gießen)

MSC:

47H04 Set-valued operators
47H10 Fixed-point theorems
54C60 Set-valued maps in general topology
54C65 Selections in general topology
54H25 Fixed-point and coincidence theorems (topological aspects)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Aliprantis, C. D. and K. C. Border: Infinite Dimensional Analysis. Berlin: Springer- Verlag 1994. · Zbl 0839.46001
[2] Ben-El-Mechaiekh, H., Deguire, P. and A. Granas: Points fixes et coincidences pour lea applications multivoques. Part II: Applications de type 4 et . C.R. Acad. Sc. 295 (1982), 381 - 384. · Zbl 0525.47043
[3] Ben-El-Mechaiekh, H. and C. Isac: Generalized mtiltivalued variational inequalities. Pre- print.
[4] Ding, X. P.: A coincidence theorem involving contractible spaces. AppI. Math. Letters 10 (1997), 53 - 56. · Zbl 0879.54055 · doi:10.1016/S0893-9659(97)00034-7
[5] Ding, X. P., Kim, W. K. and K. K. Tan: A selection theorem and its applications. Bull. Austral. Math. Soc. 46 (1992), 205 - 212. · Zbl 0762.47030 · doi:10.1017/S0004972700011849
[6] Horvath, C.: Points fixes et coincidence, dons lea espaces topologiques compacts contrac- tiles. C.R. Acad. Sc. Paris 299 (1984), 519 - 521.
[7] Horvath, C.: Some results on multivalued mappings and inequalities without convexity. In: Nonlinear and Convex Analysis (ed.: B. L. Lin and S. Simons). New York: Marcel Dekker 1987, pp. 96 - 106. · Zbl 0619.55002
[8] Horvath, C.: Contractibility and general convexity. J. Math. Anal. AppI. 156 (1991), 341 - 357. · Zbl 0733.54011 · doi:10.1016/0022-247X(91)90402-L
[9] Fitzpatrick, P. M. and W. V. Petryshyn: Fixed point theorems for multivalued noncorapact acyclic mappings. Pac. J. Math. 54 (1974), 17 - 23. · Zbl 0312.47047 · doi:10.2140/pjm.1974.54.17
[10] Cranas, A. and F. C. Liu: Coincidences for set valued maps and minimax inequalities. J. Math. Pures AppI. 65 (1986), 119 - 148. · Zbl 0659.49007
[11] O’Regan, D.: Some fixed point theorems for concentrative mappings between locally convex spaces. Nonlin. Anal. 27 (1996), 1437 - 1446. · Zbl 0874.47035 · doi:10.1016/0362-546X(95)00130-N
[12] O’Regan, D.: Generalized multivalued quasi variational inequalities. Adv. Nonlin. Var. Inequ. 1(1998), 1 - 9. · Zbl 1007.49501
[13] O’Regan, D.: Coincidences for multivalued mappings and minimax inequalities. Comm. Appl. Anal. (to appear). · Zbl 0935.47037
[14] O’Regan, D.: Coincidences for admissible and 1V maps and minimax inequalities. J. Math. Anal. AppI. 220 (1998), 322 - 333. · Zbl 0909.54039 · doi:10.1006/jmaa.1997.5886
[15] [17] O’Regan, D.: Nonlinear alternatives for multivalued maps with applications to operator inclusions in abstract spaces. Proc. Amer. Math. Soc. (to appear).
[16] Park, S.: Generalized Leray-Schauder principles for compact admissible multifunctions. Top. Meth. Nonlin. Anal. 5 (1995), 271 - 277. · Zbl 0862.47035
[17] Su, C. H. and V. M. Sehgal: Some fixed point theorems for condensing multifunctions in locally convex spaces. Proc. Amer. Math. Soc. 50 (1975), 150 - 154. · Zbl 0326.47056 · doi:10.2307/2040531
[18] ’Fan, K. K.: Comparison theorems on minimax inequalities, variational inequalities and fixed point theorems. J. London Math. Soc. 28 (1983), 555 - 562. 1986. · Zbl 0497.49010 · doi:10.1112/jlms/s2-28.3.555
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.