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Fixed point theory in probabilistic metric spaces. (English) Zbl 0994.47077
Mathematics and its Applications (Dordrecht). 536. Dordrecht: Kluwer Academic Publishers. 262 p. EUR 102.00; $ 89.00; £62.00 (2002).
The authors present a detailed survey on the fixed point survey for single- and multivalued mappings in probabilistic metric spaces the main emphasis being on contraction mappings. The authors claim that “[t]he main text is self-contained”, but in several places (such as Theorems 3.13 and 3.14) the reader is simply referred to the literature.
The authors start in Chapter 1 with a lengthy discussion of triangular norms – a concept dating back to K. Menger [Proc. Nat. Acad. Sci. USA 28, 535-537 (1942; Zbl 0063.03886)]. A triangular norm is a mapping \(T\) from the closed unit square to the closed unit interval satisfying \(T(x,y)=T(y,x)\), \(T(x,T(y,z))=T(T(x,y),z)\), \(T(x,y)\leq T(x,z)\) whenever \(y\leq z\) and \(T(x,1)=x\) where \(x,y,z\in[0,1]\).
In Chapter 2, the authors turn to probabilistic metric spaces. A probabilistic metric space as defined by A. N. Sherstnev [Kazan Gos. Univ. Uch. Zap. 122, No. 4, 3-20 (1962; Zbl 0178.52404)] is a triple \((X,\mathcal{F},\tau)\) where \(S\) is a nonempty set \(S\), \(\mathcal{F}\) is a map assigning to \(p,q\in S\) an \(F_{p q}\) in the set \(\Delta^+\) of probability distributions with support on the positive half-axis, and a commutative and associative operation \(\tau:\Delta^+\times\Delta^+\to\Delta^+\) that is non-decreasing in each argument and satisfies \(\tau(H_0,\cdot)=\text{id}\) where \(H_0\) is the Dirac-distribution centered at \(0\). As to \(\mathcal{F}\) it is required that \(F_{p q}\not=H_0\) iff \(p\not=q\), \(F_{p q}=F_{q p}\), and \(F_{p r}\geq\tau(F_{p q},F_{q r})\) . (One thinks of \(F_{p q}(x)\) as the probability that the distance between \(p\) and \(q\) is less than \(x\).) A “Menger space” is obtained by letting \(\tau(F,G)(x)=\sup\{T(F(u),G(v))|\;u+v=x\}\) for some triangular norm \(T\). If we let \(\tau(F,G)=F*G\) (the convolution of \(F\) and \(G\)) we obtain a “Wald space” [A. Wald, Proc. Nat. Acad. Sci. USA 29, 196-197 (1943)]. In a similar way one defines a random normed space as a triple \((S,\mathcal{F},T)\) consisting of a (real or complex) vector space, a map \(\mathcal{F}:S\to\Delta^+\) (where we write \(\mathcal{F}(p)=:F_p\)) and a triangular norm \(T\) such that \(T(x,y)\geq\max(x+y-1,0)\), \(\lim_{x\to\infty}F_p(x)=1\), \(F_p(0)=0\), \(F_p=H_0\) iff \(p=0\), \(F_{\lambda p}(x)=F_p(\frac{x}{|\lambda|})\) for \(x>0\) and \(\lambda\not=0\), and \(F_{p+q}(x+y)\geq T(F_p(x),F_q(y))\) for \(x,y>0\) whenever \(p,q\in S\). If we let \(\mathcal{F}(p,q)=F_{p-q}\) we obtain a Menger space.
In Chapter 3 the authors turn to probabilistic contraction mappings: A mapping \(f:S\to S\) on a probabilistic metric space is said to be a probabilistic contraction if there is a \(k\in(0,1)\) such that \(F_{fp fq}(x)\geq F_{p q}(\frac{x}{k})\) whenever \(p,q\in S\) and \(x\in\mathbb{R}\). A sequence \((p_n)\) in a probabilistic metric space is said to be a Cauchy sequence if for every \(\varepsilon>0\) and \(\lambda\in(0,1)\) there is an \(n_0\) such that \(F_{p_{n+k} p_n}(\varepsilon)>1-\lambda\) whenever \(n\geq n_0\) and \(k\in\mathbb{N}\). V. M. Sehgal and A. T. Bharucha-Reid [Math. Systems Theory 6, 97-102 (1972; Zbl 0244.60004)] proved that a probabilistic contraction in a complete Menger space has a unique fixed point. Just as there are (literally) thousands of (more or less useless) generalizations of the contraction mapping principle there are at least as many in the probabilistic case, and the authors eagerly exploit this reservoir. In Chapter 4 the authors turn to multivalued probabilistic contractions more or less mimicking the transition from the single-valued to the multivalued in the classical “deterministic” case.
In Chapter 5, the authors deal with Hicks’ contraction principle [T. Hicks, Univ. v Novom Sadu, Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 13, 63-72 (1983; Zbl 0574.54044)]: Let \(S\) be a complete Menger space and \(f:S\to S\) a mapping such that there exists a \(k\in(0,1)\) satisfying \(F_{fp fq}(kt)>1-kt\) whenever \(t>0\) and \(p,q\in S\) are such that \(F_{p q}(t)>1-t\). Hicks (op. cit.) proved that in this situation, \(f\) has a unique fixed point. The authors present several variants of this fixed point theorem and their generalizations to the multivalued case.
In the last chapter the authors free themselves from contractivity assumptions when they turn to fixed point theorems of Brouwer and Schauder type. We describe a typical result in this chapter: Let \(A\) be a subset of a probabilistic metric space. The probabilistic diameter \(D_A\) of \(A\) is defined to be the function \(D_A:\mathbb{R}\to[0,\infty]\) given by \(D_A(x)=\sup_{u<x}\inf_{p,q\in A}F_{p q}(u)\). Assume that \(A\) is probabilistically bounded, i.e., \(\sup_{x\in\mathbb{R}}D_A(x)=1\). Then there is a function \(\alpha_A:\mathbb{R}\to[0,1]\) (serving as the probabilistic analogue of Kuratowski’s measure of noncompactness) where \(\alpha_A(x)\) is defined to be the supremum of all \(\varepsilon>0\) such that \(A\) can be covered by finitely many sets of probabilistic diameter at least \(\varepsilon\). Let then \((S,\mathcal{F},T)\) be a complete random normed vector space with continuous triangular norm \(T\), let \(A\subset S\) be closed convex and probabilistically bounded and \(f:A\to A\) a continuous map such that for \(C\subset A\) we have that \(\alpha_C\geq\alpha_{\text{cl }\text{conv} f(C)}\) only if \(\alpha_C=H_0\). If every closed and convex subset of \(A\) is admissible then \(f\) has a fixed point. {It may well be that the admissibility assumption turns out to be superfluous. The authors take great care to discuss admissibility and they even sketch T. Riedrich’s proof [Wiss. Z. TU Dresden 13, 1-6 (1964; Zbl 0158.13402)] that the space \(S([0,1])\) of (equivalence classes of) measurable functions on \([0,1]\) is admissible (i.e., each compact map can be approximated arbitrarily close by maps with finite dimensional range). The reason for the importance of this space in the context of random normed vector spaces is that one can make \((S,\mathcal{F},T)\) into a (generally non locally convex) topological vector space by introducing the F-norm \(\|x\|=\sup\{t|\;F_x(t)\leq 1-t\}\). In the meantime, however, R. Cauty [Fundam. Math. 170, No. 3, 231-246 (2001; Zbl 0983.54045)] has shown that Schauder’s fixed point theorem holds without the assumption of local convexity.} This chapter closes with a somewhat unexpected digression on degree theory which is nowhere used in the book and doesn’t seem to lend itself to a probabilistic generalization.
On the whole, the reviewer’s attitude towards this book is ambiguous: On the mathematical side, the presentation is very careful though it is difficult to start reading somewhere in the book due to the poor index. Moreover, the list of references contains a lot of minor typos (e.g., the author of ref. 214 is not Okan but Okon). More important, the book lacks any motivating ideas: for unprejudiced mathematicians it is difficult to understand why they should be interested in the plethora of generalizations of Banach’s contraction principle, so the authors should at least have given a small hint why the generalization of this concept to probabilistic spaces could be of interest. As a matter of fact, there is not a single application (not even a faked one) of a probabilistic fixed point theorem in the entire book. In the same way, new concepts in this book mostly lack any motivating text. Instead of explaining why they need a particular concept the authors usually just introduce a definition by phrases such as “In [n] the following definition is introduced … ”. With regard to the publisher there is always the same ceterum censeo: It is well known that copy editors tend to be badly paid; so when engaging authors with a mother tongue knowing neither definite nor indefinite articles the publisher (especially when demanding a price of 0.30 EUR per printed page!) should at least have engaged someone getting the articles straight and convincing the authors that they should refrain from coining new adjectives such as “maxitive”. Moreover, one cannot avoid the impression that large parts of the text have been written and read only by one of the authors — the quality of the English varies considerably, and there is no consistent scheme for the transliteration of words in cyrillic: Sometimes the authors use the scientific transliteration but some lines later one may find the same word transliterated as in a newspaper or in this Zbl. This may be a minor quibble, the main problem is the authors’ definition-theorem-proof style which doesn’t give any indication to the prospective readers as to why they should be interested in this book. So, at best, this may be a reference text for hardcore specialists.

47S50 Operator theory in probabilistic metric linear spaces
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
47H04 Set-valued operators
54-02 Research exposition (monographs, survey articles) pertaining to general topology
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
54E70 Probabilistic metric spaces
47S40 Fuzzy operator theory