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Genericity in nonlinear analysis. (English) Zbl 1296.47002

Developments in Mathematics 34. New York, NY: Springer (ISBN 978-1-4614-9532-1/hbk; 978-1-4614-9533-8/ebook). xiii, 520 p. (2014).
Given a property that elements of a complete metric space \(X\) may have, it is natural to ask whether the property is generic, that is, whether “most” elements of \(X\) (in the sense of Baire category) possess the property. More specifically, a property is generic if the set of elements in \(X\) possessing the property contains a dense \(G_\delta\) subset of \(X\). As indicated by the title of the book, the authors consider generic results for a large variety of properties arising in nonlinear analysis.
In most instances, the authors take \(X\) to be a space of functions endowed with a complete metric, but the authors also obtain many results in the more general setting of complete hyperbolic spaces. The first chapter provides a short introduction to hyperbolic spaces and \(\sigma\)-porosity (a metric notion of a set that implies the set is of first category in \(X\)) as well as outlining some generic results that are to be discussed in future chapters. In order to illustrate the types of results to be considered, the authors prove a refinement of a classical result of F. S. De Blasi and J. Myjak [C. R. Acad. Sci., Paris, Sér. A 283, 185–187 (1976; Zbl 0332.47028)] that most mappings in the class of nonexpansive self-mappings of a closed, bounded, convex subset of a Banach space possess a unique fixed point which attracts all their iterates.
The result of De Blasi and Myjak is further generalized and developed in Chapter 2 where the authors establish, among many other results, that most mappings in a certain class of continuous mappings admit unique fixed points; and that, if \(K\) is a nonempty, closed, bounded, convex subset of a Banach space \(X\), most nonexpansive mappings \(A:K\to X\) with approximate fixed points admit a unique fixed point \(x_A\) such that the iterates \((A^nx)\) converge to \(x_A\) uniformly on \(K\). Chapter 2 also contains results concerning the stability of fixed points under small perturbations of mappings as well as results on classes of order-preserving mappings that arise in mathematical economics.
The class of contractive mappings, an important subclass of the class of nonexpansive mappings, is studied in Chapter 3. If \(\mathcal{A}\) denotes the set of nonexpansive self-mappings of a closed, bounded, convex subset \(K\) of a Banach space and \(h(A, B) = \sup\{ \| Ax - Bx\|: x\in K\}\) for \(A, B\in \mathcal{A}\), then \((\mathcal{A}, h)\) is a complete metric space. Although not the standard definition, the authors define a mapping \(A\in \mathcal{A}\) to be contractive if there exists a decreasing function \(\phi^A:[0,\text{diameter}(K)]\to [0,1]\) such that \(\phi^A(t)<1\) for all \(t\in (0, \text{diameter}(K)]\) and \(\| Ax - Ay\| \leq \phi^A(\| x-y\|)\, \| x-y\|\) for all \(x,y\in K\). This class of contractive mappings was introduced by E. Rakotch in [Proc. Am. Math. Soc. 13, 459–465 (1962; Zbl 0105.35202)] where he proved that each such contractive mapping \(A\) has a unique fixed point \(x_A\) in \(K\) and the sequence \((A^nx)\) converges to \(x_A\) uniformly on \(K\). The authors prove that the complement of the set of contractive mappings in \(\mathcal{A}\) is \(\sigma\)-porous.
In Chapter 7, the authors consider best approximation problems in general Banach spaces. Given a nonempty closed subset \(A\) of a Banach space \(X\) and a point \(x\in X\setminus A\), there is no guarantee that \(x\) has a closest point in \(A\). However, if \(X\) satisfies a nice geometric condition, such as uniformly convexity, the set of points in \(X\) having a unique nearest point in \(A\) is a dense \(G_\delta\) set in \(X\). In this chapter, the authors employ a new point of view to consider generic results for best approximation problems in Banach spaces. With \(S(X)\) denoting the set of nonempty closed subsets of \(X\), the authors equip \(S(X)\times X\) with a complete metric and show that the set of pairs \((A, x)\) in \(S(X)\times X\) for which \(x\) fails to have a unique best approximation in \(A\) is \(\sigma\)-porous.
Other chapters consider generic approaches to study the asymptotic behavior of trajectories of certain dynamical systems that originate in convex minimization problems (Chapter 4), mappings that are relatively nonexpansive with respect to Bregman distances (Chapter 5), convergence of infinite products of generic sequences of nonexpansive and uniformly continuous operators on closed, bounded, convex subsets of a Banach space (Chapter 6), descent methods (Chapter 8), existence of fixed points of certain set-valued mappings (Chapter 9), and the structure of minimal energy configurations in the Aubry-Mather theory (Chapter 10).
The book, which is self-contained and very well-written, is the first book to present an extensive collection of generic results in nonlinear analysis. As such, this book should prove useful to those currently doing research in nonlinear analysis and to those interested in beginning research in nonlinear analysis.

MSC:

47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
54H25 Fixed-point and coincidence theorems (topological aspects)
54E50 Complete metric spaces
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