topological methods in nonlinear analysis
Topological analysis. From the basics to the triple degree for nonlinear Fredholm inclusions
This book focuses on one of the main topological tool of nonlinear analysis, the topological degree. One of the aims of the author is to provide a modern treatment of the topic, with the use of function triples, but he does not restrict himself to this aim. The book invites the reader to go on a journey from the basics of topology, through the fields of multivalued maps and Fredholm operators, up to constructions of topological degrees (both those which have been known for decades as well as more recent ones) which are presented in great detail.NEWLINENEWLINE The material is divided into three main parts. The first part gives a self-contained presentation of multivalued maps, measures of noncompactness, extension properties, and several other crucial topological tools. The material of this part, and consequently the rest of the book, is presented in a way well suited even for undergraduate students. It is worth pointing out that a comprehensive description of the set-theoretical axiomatic treatment is a very characteristic feature of the book. In the whole text, usage of the axiom of choice (AC) is always indicated. Special attention should be paid to Chapter~4 where, among properties of AE, ANE, AR and ANR spaces, a~detailed characterization of \(UV^\infty\) and \(R_\delta\) spaces is provided. These sets are crucial for further investigations. The so-called Vietoris* maps are defined in Chapter~5 and are used in the sequel to obtain a homotopy reduction in a construction of a degree. This notion is strictly motivated by \textit{W. Kryszewski} [Homotopy Properties of Set-Valued Mappings. Univ. N. Copernicus Publishing, Toruń (1997)], where some homotopic versions of the Vietoris theorem are obtained.NEWLINENEWLINE The second part of the book (Chapters 6--10) is headed for the Furi-Pera coincidence degree for Fredholm maps on Banach manifolds, but along the way familiarizes the reader with suitable information on Banach manifolds and elements of the analysis of Fredholm operators, especially an orientation of families of such operators. The Brouwer degree on manifolds is carefully presented in Chapter~9 as a background for Benevieri-Furi degrees. These degrees were presented first in [\textit{P. Benevieri} and \textit{M. Furi}, ``A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree theory'', Ann. Sci. Math. Qué. 22, No. 2, 131--148 (1998; Zbl 1058.58502)] and [\textit{P. Benevieri} and \textit{M. Furi}, ``A degree theory for locally compact perturbations of Fredholm maps in Banach spaces'', Abstr. Appl. Anal. 2006, Article 64764 (2006; Zbl 1090.47049)], but the author gives here a~modified construction of them, based on the reduction property of the Brouwer degree, which is also applied in the third part of the book to provide a degree theory for function triples. The coincidence degree described in Section 10.3 is defined for maps acting on a Banach manifold \(X\) with values in some Banach space \(Y\). A~construction of a coincidence degree in the case of a Banach manifold \(Y\) is left as an open problem by the author. Note that all Fredholm operators in the book are of index~0, but the author mentions, e.g., a paper by \textit{D. Gabor} and \textit{W. Kryszewski} [``A coincidence theory involving Fredholm operators of nonnegative index'', Topol. Methods Nonlinear Anal. 15, No. 1, 43--59 (2000; Zbl 0971.47046)], where Fredholm operators of nonnegative index were also examined.NEWLINENEWLINE The third part of the book is the central and also the most advanced one. The author constructs a topological tool (a~coincidence degree) to study solutions to the inclusion \(F(x)\in \varphi(\Phi(x))\) on an open subset of a Banach manifold, where \(F:\Omega\to Y\) is a Fredholm map of index 0, \(\varphi:\Gamma\to Y\) is a continuous map, and \(\Phi:\Omega\multimap \Gamma\) is an acyclic* multivalued map, that is, with \(UV^\infty\) values or with \(\dim \Phi(\Omega)<\infty\). The construction is given in three steps related to levels of generality. At first, the finite-dimensional case is described. Next, locally compact maps \(\varphi\circ\Phi\) with compact coincidence point sets are considered. At last, some classes of noncompact maps are studied. The author applies a fundamental set technique, reducing a problem to the previous one. A countable condensation used as a test for obtaining a convex-fundamental set is put as a final chord.NEWLINENEWLINE The book gives a very detailed study of different topological concepts that play important roles in a modern approach to nonlinear analysis problems, with special emphasis put on the study of problems involving set-valued maps and operators. It can surely be recommended to researchers and students interested in homotopy invariants and their applications in the theory of differential equations/inclusions.
scientific article; zbMATH DE number 6055190