The theory of multivalued maps which at its early stage was merely some “l’art pour l’art”, has now become a serious part of topology, functional analysis, operator theory, and measure theory. Moreover, multivalued maps have proved to be a useful tool in certain applications to game theory and mathematical economics. A particularly important field where such maps apply successfully are so-called differential inclusions (i.e., differential equations with multivalued right-hand side), a milestone in this area being K. Deimling’s monograph [“Multivalued differential equations” (De Gruyter Studies in Nonlinear Analysis and Applications 1) (1992; Zbl 0760.34002)]. Some minor survey articles and monographs have been devoted to the topological properties of multivalued maps [e.g., Yu. G. Borisovich, B. D. Gel’ man, A. D. Myshkis and V. V. Obukhovskij, J. Sov. Math. 24, 719–791 (1984; Zbl 0529.54013)], the fixed point theory of generalized contractions [e.g., I. A. Rus, “Generalized contractions and applications” (Cluj University Press, Cluj-Napoca) (2001; Zbl 0968.54029)], or the theory and applications of multivalued superposition operators [e.g., E. De Pascale, H. T. Nguyên, P. P. Zabrejko and the reviewer, “Multi-valued superpositions” (Diss. Math. 345) (1995; Zbl 0855.47037)].
In the present survey, monograph, the author discusses the fixed point problem \(x\in F(x)\), the surjectivity problem \(y\in F(x)\), the selection problem \(f(x)\in F(x)\), and, more generally, the coincidence problem \(F(x)\cap G(x)\neq\emptyset\) for multivalued maps \(F\) and \(G\) in metric spaces. In the first introductory chapter, some basic definitions and results on multivalued maps are collected without proofs on 40 pages. In particular, various notions of semicontinuity and measurability are discussed, including the Ryll-Nardzewski selection theorem.
The main part is the second chapter on operator inclusions of the types mentioned above. Here the author discusses three different topics, viz. conditions for the existence of continuous selections (such as the classical Martin selection theorem), fixed point and coincidence theorems for multivalued maps, and applications to differential or integral inclusions. The monograph closes with a detailed list of references which contains 273 items but, surprisingly, ignores some important references in the field.