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Topological fixed point principles for boundary value problems. (English) Zbl 1029.55002
Topological Fixed Point Theory and its Applications. 1. Dordrecht: Kluwer Academic Publishers. xv, 761 p. (2003).
This is an extremely valuable book written by two outstanding experts. On the onfe hand with a clever selection it could be used for graduate courses, on the other hand it is a comprehensive research monograph devoted to the application of topological fixed point theory to boundary value problems with special emphasis on the multivalued case (i.e., differential inclusions). The presentation is (in a restricted sense) self-contained – the reader is assumed to have a solid background in analysis (for example the Bochner integral is used without any explanation) and topology, but in most instances the authors develop their subject from scratch providing the necessary definitions and theorems in a strict and logical order. To be sure, not all theorems are supplied with proofs (and the decision which theorems deserve a proof seems a little bit arbitrary at times), but in most cases there are ample references to the literature (the list of references contains more than a thousand items). It is not quite understandable why the reader is sometimes left with phrases like “it is well-known that …” when it would have been easy to pinpoint an item in the list of references explaining the matter at hand.
The book is divided into three chapters (with the bulk of material being contained in Chapter 3) and three lengthy appendices. Each chapter is supplemented by extensive “remarks and comments” giving due credit to the authors’ sources and hints for further reading. Chapter 1 provides the necessary “Theoretical background”. Here, facts about ANR-spaces and multivalued mappings are recalled. There is a discussion of “admissible maps” as defined by the second author [Dissertationes Math., Warszawa 129 (1976; Zbl 0324.55002)]. A multivalued map $$\phi:X\multimap Y$$ is said to be admissible if there is a space $$\Gamma$$, a Vietoris map $$p:\Gamma\to X$$ and a continuous map $$q:\Gamma\to Y$$ such that $$\phi(x)=qp^{-1}(\{x\})$$ for every $$x\in X$$. There is a fairly extensive discussion of the Lefschetz fixed theory for admissible mappings using Leray-traces. After defining the fixed point index and the topological degree for admissible maps on retracts of open sets in locally convex spaces the authors turn to Nielsen theory. Here, the authors provide complete proofs since a reference to Nielsen theory for multivalued maps doesn’t seem to be readily available in the literature.
Chapter 2 is devoted to “General principles”, i.e., results from topological fixed point theory which can be obtained using the methods developed in the first chapter. Here, the authors deal with the topological structure and dimension of fixed point sets. ter discussing topological essentialities, the authors extend the Lefschetz and Nielsen theory developed in Chapter 1 to the relative case thus obtaining lower bounds for the number of coincidences for two maps. Nielsen theory is also invoked to prove periodic point theorems for multivalued mappings. Readers who don’t like homological methods will find the extensive discussion of approximation methods in this chapter particularly useful. Here, the authors explain with great care how to obtain fixed point results and how to define a topological degree for multivalued maps by approximation via single-valued maps. The chapter closes with a detailed discussion of continuation principles.
Chapter 3 is devoted to the main theme of the book, viz., “Applications to differential equations and inclusions”. It has 365 pages. The authors start by considering differential inclusions $$\dot{x}(t)\in F(t,x(t))$$ for almost all $$t$$ in an interval $$J$$, where $$x:J\to\mathbb{R}^n$$ is locally absolutely continuous and $$F:J\times\mathbb{R}^n\multimap\mathbb{R}^n$$ has nonempty compact convex values, $$F(t,\cdot)$$ is upper semicontinuous for all $$t\in J$$, and $$F(\cdot,x)$$ is measurable for all $$x\in\mathbb{R}^n$$. A boundary value problem is obtained if we require that $$x$$ belongs to a given subset $$S$$ of the space of locally absolutely continuous functions. An important rôle is played by the corresponding infinite dimensional problem where $$\mathbb{R}^n$$ is replaced by a Banach space $$E$$. Here, $$F$$ is again supposed to have nonempty compact convex values, $$F(t,\cdot)$$ is upper semicontinuous, and $$F(\cdot,x)$$ is strongly measurable on every compact subinterval of $$J$$. The solution is required to be locally absolutely continuous with $$\dot{x}(t)$$ existing for almost all $$t$$ in each subinterval $$[a,b]$$ such that $$\dot{x}\in L^1([a,b],E)$$. The results in this chapter require the Eilenberg-Montgomery theorem and Nielsen theory.
The second section studies the structure of the solution set. For example, if $$F:J\times\mathbb{R}^n\multimap\mathbb{R}^n$$ is as above and if $$F$$ has at most linear growth then the solution set of the initial value problem $$\dot{x}(t)\in F(t,x(t))$$, $$x(0)=x_0$$ is an $$R_\delta$$-set. Of course, there are much more advanced results in this section, e.g., theorems on the dimension of the solution set or on the structure of solution sets of functional differential inclusions.
The next section deals with the analogous problem for boundary value problems. In the fourth section the authors discuss Poincaré operators in the multivalued case, in particular for functional differential inclusions and for random systems.
Section 5 deals in extenso with existence results. It starts with the existence of bounded solutions for a semilinear differential inclusion by an intriguing application of a measure of compactness argument. Other results are obtained by applying the methods developed in the sections on the structure of the solution set; in particular, there is a wealth of results on the existence of periodic solutions. All of these results are, however, far too technical to be described even in a lengthy review.
In section 6, Nielsen theory is applied to obtain multiplicity results. There is a particularly interesting discussion of the existence of periodic solutions to a system of differential inclusions on the $$n$$-dimensional torus and to generalized local multivalued semi-processes. In the next section, the authors again obtain results on the existence of periodic solutions using Ważewski-type results.
Section 8 on bounded and guiding functions is the longest section of this book. If we again consider the boundary value problem $$\dot{x}(t)\in F(t,x(t))$$ for almost all $$t\in J=[a,b]$$, $$x\in S$$, then a family $$(K(t))_{t\in J}$$ of open sets in $$\mathbb{R}^n$$ is said to be a bound set for the problem if there is no solution $$x$$ such that $$x(t)\in\bar{K}(t)$$ for all $$t\in J$$ and $$x(\tau)\in\partial K(\tau)$$ for some $$\tau\in J$$. The authors obtain bounds for the Floquet boundary problem $$\dot{x}(t)\in F(t,x(t))$$ for almost all $$t\in[a,b]$$, $$x(b)=Mx(a)$$. This theory is then applied to obtain solutions to the Floquet boundary value problem. In the second part of this section the authors consider guiding potentials for locally Lipschitzian functions $$V:\mathbb{R}^n\to\mathbb{R}^n$$. $$V$$ is said to be a direct potential if there is an $$r_0>0$$ such that $$\inf\{\left<p,q\right>\mid p,q\in\partial V(x)\}\to 0$$ for $$\|x\|\geq r_0$$ where $$\partial V(x)$$ denotes the generalized gradient and $$\left<\cdot,\cdot\right>$$ is the inner product. If $$F:[0,a]\times\mathbb{R}^m\multimap\mathbb{R}^m$$ is a map with nonempty compact values then a direct potential is said to be a guiding function for $$F$$ if for every $$(t,x)\in[a,b]\times\mathbb{R}^n$$ with $$|x|\geq r_0$$ we have that $$\inf\{\left<p,q\right>\mid \;p\in F(t,x), q\in\partial V(x)\}\geq 0$$ (In the traditional approach, one requires $$V$$ to be $$C^1$$.) Assume that $$F$$ is as in the beginning of this chapter and that $$\sup\{|y|y\in F(t,x)\}\leq\mu(x)(1+|x|)$$ for all $$(t,x)\in[0,a]\times\mathbb{R}^n$$ with a nonnegative integrable function $$\mu$$. Assume that there is a guiding function $$V$$ for $$F$$ such that $$\deg(\partial V,B(0;r),0)\not=0$$ for $$r\geq r_0$$ then there is an $$a$$-periodic solution to $$\dot{x}(t)\in F(t,x(t))$$.
The authors also consider the case when the right hand side is not assumed to have convex values. In this case the solution set map is neither upper semicontinuous nor does it take closed values, so the authors have to develop a topological degree theory for the associated Poincaré operator which enables them to deduce the existence of periodic solutions even in this case.
Section 9 is about “infinitely many subharmonics”. Here, the authors deal with the generalization of Šarkovskiĭ type theorems. Easy examples show that the obvious generalizations to the multivalued case won’t work, but the authors provide the correct formulation and a wealth of counterexamples to seemingly obvious conjectures. They also go beyond Šarkovskiĭ’s result in that they consider two-dimensional problems. Following Boju Jiang [Contemp. Math. 152, 183-202 (1993; Zbl 0798.55001)] the authors use braid invariants in order to conclude the generic existence of $$k$$-orbits for infinitely many primes $$k$$ for an admissible mapping on a compact connected surface $$X$$ having at least three fixed points in the interior of $$X$$.
Before one starts reading section 10 on “almost periodic problems”, one should first consult Appendix 1 where a lot of instructive material is gathered comparing the various definitions of almost periodicity. The fixed point principles applied in section 10 are rather elementary, viz., the contraction mapping principle and fixed point theorems for condensing mappings in Fréchet spaces. The main tools are taken from the theory of almost periodic functions, and it is the authors’ objective to find conditions which ensure that bounded solutions to differential inclusions are almost periodic. This section is followed by a section containing a potpourri of results. Reading this section requires the appendices on “Derivo-periodic single-valued and multivalued functions” and on “Fractals and multivalued fractals”.
This book is no easy reading — it is written in a terse uncompromising no-nonsense style (there is just one funny misprint on p. 59, viz., a “contradiction with compact values”), there is almost no motivating text, but the exposition is always clear and reliable. On the other hand a reader seeking specific information will have serious difficulties which are mainly due to the poor subject index. For example, on p. 306 we meet the expression “regular exponential dichotomy”, but neither of these terms can be found in the index. The poor index, however, is more than compensated for by the excellent bibliography (which would be easier to use if the references were ordered according to the identifiers. For example, when trying to look up [KOZ1] one might find [KN] on p. 736 and start looking from here. But as a matter of fact, [KOZ1] is on p. 732. This is due to the fact that the references are ordered by authors’ names, so it would be easy to find [KOZ1] if one knew that the respective authors are Kamenskiĭ, Obuhovskiĭ and Zecca.) The printing is clear and pleasant and there are only a few typos. Some of these are simply due to TEXnical errors: on p. 110 some lines are marred on the left margin while one wonders why the second line in (H 2) on p. 531 is not italicized (solution: there is a mysterious “it” at the beginning of the line which obviously should have been $$\backslash$$it). But these are minor quibbles — on the whole the authors have done an admirable job bringing two notoriously difficult subjects (algebraic topology and boundary value problems for differential inclusions) together.
This reviewer cannot end the review without his “ceterum censeo”: As was indicated in the beginning this is a book of an encyclopedic nature which would be a must for every expert in the field if the publisher had not fixed a prohibitive price of approximately 0.30 EUR per page. So what service does the publisher provide in return? This reviewer suspects that the text has been directly produced from the TEX-file without any intervention of a copy editor. Such a copy editor would have been necessary to somewhat smoothen the English — both authors have mother tongues knowing neither definite nor indefinite articles, and it would have been an easy task to get the articles straight. Of course, the meaning of a sentence like “Every chapter concludes by the section with many remarks and comments” is understood at first reading, but “there always exists at most a countable infinite set” (p. 603) requires a moment’s reflection. A copy editor would perhaps also have insisted on a consistent spelling of names: we have Lipshitzian (p. 660) as well as Lipschitzian, Carathéodory as well as Caratéodory (p. 271), Borisovič is Borisovitch as well as Borisovich, and titles in French are lacking almost all accents.

##### MSC:
 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology 34A60 Ordinary differential inclusions 37B25 Stability of topological dynamical systems 47H05 Monotone operators and generalizations 55M20 Fixed points and coincidences in algebraic topology 28A80 Fractals 34B15 Nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 34K10 Boundary value problems for functional-differential equations 35B10 Periodic solutions to PDEs 35B15 Almost and pseudo-almost periodic solutions to PDEs 35B37 PDE in connection with control problems (MSC2000) 35G30 Boundary value problems for nonlinear higher-order PDEs