Recent advances in the Lefschetz fixed point theory for multivalued mappings

Yıl 2021, Cilt: 4 Sayı: 2, 116 - 126, 30.06.2021

Lech Górnıewıcz

https://doi.org/10.53006/rna.941060

Öz

In 1923 S. Lefschetz proved the famous fixed point theorem which is now known as the Lefschetz fixed point theorem (comp. [5], [9], [20], [21]. The multivalued case was considered for the first time in 1946 by S. Eilenberg and D. Montgomery ([10]). They proved the Lefschetz fixed point theorem for acyclic mappings of compact ANR-spaces (absolute neighborhood retracts (see [4] or [13]) using Vietoris mapping theorem (see [4], [13], [16]) as the main tool. In 1970 Eilenberg, Montgomery's result was generalized for acyclic mappings of complete ANR-s (see [17]). Next, a class of admissible multivalued mappings was introduced ([13] or [16]).
Note that the class of admissible mappings is quite large and contains as a special case not only acyclic mappings but also infinite compositions of acyclic mappings. For this class of multivalued mappings several versions of the Lefschetz fixed point theorem were proved (comp. [11], [13]-15], [18], [19], [27]). In 1982 G. Skordev and W. Siegberg ([26]) introduced the class of multivalued mappings so-called now (1 − n)-acyclic mappings. Note that the class (1−n)-acyclic mappings contain as a special case n-valued mappings considered in [6], [12], [28]. We recommend [8] for the most important results connected with (1 − n)-acyclic mappings. Finally, the Lefschetz fixed point theorem was considered for spheric mappings (comp. [3], [2], [7], [23]) and for random multivalued mappings (comp. [1], [2], [13]). Let us remark that the main classes of spaces for which the Lefschetz fixed point theorem was formulated are the class of ANR-spaces ([4]) and MANR-spaces (multi absolute neighborhood retracts (see [27]). The aim of this paper is to recall the most important results concerning the Lefschetz fixed point theorem for multivalued mappings and to prove new versions of this theorem mainly for AANR-spaces (approximative absolute neighborhood retracts (see [4] or [13]) and for MANR-s. We believe that this article will be useful for analysts applying topological fixed point theory for multivalued mappings in nonlinear analysis, especially in differential inclusions.

Anahtar Kelimeler

multivalued mappings, absolute retracts, admissible mappigns, (1 − n)-acyclic mappings, spheric and random mappings, Lefschetz number., fixed point problem

Kaynakça

• J. Andres, L. Gorniewicz, On the Lefschetz fixed point theorem for radom multivalued mappings, Lib. Math. (N.S.) (2013), 69--78.
• J. Andres, L. Gorniewicz, Recent results on the topological fixed point theory of multivalued mappings: a survey, Fixed Point Theory Appl.184 (2015), 1--34.
• J. Andres, L. Gorniewicz, Lefschetz-type fixed point theorem for spheric mappings, Fixed Point Theory { 19} (2018), no.\ 2, 453--461.
• K. Borsuk, Theory of Retracts, PWN, Warszawa 1967.
• R. Brown, The Lefschetz Fixed Point Theorem, London 1971.
• R. Brown, The Lefschetz number of an $n$-valued multimap, Fixed Point Theory Appl. {2} (2007), 53--60.
• A. Dawidowicz, Spherical maps, Fund. Math. { 127} (1987), 187--196.
• Z. Dzedzej, Fixed Point Index for a class of nonacyclic multivalued maps, Dissertationes Math. {25} (1985).
• O. Ege, I. Karaca, Lefschetz fixed point theorem for digital images, Fixed Point Theory Appl. {253} (2013), 1--13.
• S. Eilenberg, D. Montgomerry, Fixed point theorems for multivalued transformations, Amer. J. Math. { 58} (1946), 214--222.
• G. Fournier, L. Gorniewicz, The Lefschetz fixed point theorem for some noncompact multivalued mappings, Fund. Math. { 94} (1977), 245--254.
• D.L. Goncalves, J. Guaschi, Fixed points of $n$-valued maps, the fixed point property and the case of surfraces - a braid approach, Indag. Math. {29} (2018), 91--124.
• L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings, second edition, Springer 2006.
• L. Gorniewicz, A Lefschetz-type fixed point theorem, Fund. Math. { 88} (1975), 103--115.
• L. Gorniewicz, The Lefschetz coincidence theorem, Lecture Notes in Math. { 886} (1981), 116--131.
• L. Gorniewicz, Homological methods in fixed point theory of multivalued mappings, Dissertationes Math. { 129} (1976), 1--66.
• L. Gorniewicz, A. Granas, Fixed point theorems for multivalued mappings of \rom{ANR}-s, J. Math. Pures Appl. { 49} (1970), 381--395.
• L. Gorniewicz, A. Granas, Some general theorems in coincidence theory, J. Math. Pures Appl. {60} (1981), 361--373.
• L. Gorniewicz, Danuta Rozploch-Nowakowska, The Lefschetz fixed point theory for morphisms in topological vector spaces, Topol. Methods Nonlinear Anal. {20} (2002), 215--233.
• A. Granas, J. Dugundji, Fixed Point Theory, Springer 2003.
• R. Brown, M. Furi, L. Gorniewicz and B. Jiang, Handbook of Topological Fixed Point Theory, eds, Springer 2005, 43--82; 217--264.
• J. Jezierski, An example of finitely valued fixed point free map, University Gdansk { 6} (1987), 87--92.
• D. Miklaszewski, A fixed point theorem for multivalued mappings wit non-acyclic values, Lecture Notes in Nonlinear Analysis {17} (2001), 125--131.
• B. O'Neill, A fixed point theorem for multivalued functions, Duke Math. J. {14} (1947), 689--693.
• D. Pastor, A remark on generalized compact maps, Stud. Univ. Zilina Math. Ser.{13} (2001), 147--155.
• H.W. Seigberg, G. Skordev, Fixed point index and chain approximation, Pacific J. Math. {102} (1982), 455-486.
• R. Skiba, M. Slosarski, On a generalization of absolute neighborhood retracts, Topology Appl. { 156} (2009), 697--709.
• F. Von Haesler, H.-O. Peitgen, G. Skordev, Lefschetz fixed point theorem for acyclic maps with multiplicity, Topol. Methods Nonlinear Anal. { 19} (2002), 339--374.