Recent zbMATH articles in MSC 54C60https://zbmath.org/atom/cc/54C602024-03-13T18:33:02.981707ZWerkzeugSemi-linearity on spaces of set-valued functionshttps://zbmath.org/1528.540062024-03-13T18:33:02.981707Z"Apreutesei, Gabriela"https://zbmath.org/authors/?q=ai:apreutesei.gabriela"Croitoru, Anca"https://zbmath.org/authors/?q=ai:croitoru.ancaSummary: The aim of this paper is to offer a general theoretical frame for clustering validation problems. For this purpose we consider a near metric \(d_1\) and a nonnegative symmetrical application \(d_2\) on some spaces \(\mathcal{M}\) of set-valued functions and compare them. We also study the semi-linearity of the induced topologies \(\tau_1\) and \(\tau_2\) respectively, as well as the translated topology \(\tau_3\) of \(\tau_1\) on these spaces of set-valued functions and present sufficient conditions and characteristic properties for \(\tau_3\) to be semi-linear. We formulate some conditions which give approximation properties for addition on \(\mathcal{M}\times \mathcal{M}\) and for multiplication by scalars in special points of \(\mathbb{R}\times\mathcal{M}\).Directional derivatives for set-valued maps based on set convergenceshttps://zbmath.org/1528.540072024-03-13T18:33:02.981707Z"Durea, Marius"https://zbmath.org/authors/?q=ai:durea.mariusSummary: We explore the possibility to define and to meaningfully apply some new concepts of directional derivative which incorporate in their construction set convergences to set-optimization problems. We connect these new constructions with other directional derivatives for set-valued maps and we emphasize the flexibility and the potential applicability of this new approach. In this vein, we indicate a possible axiomatic perspective that allows one to significantly increase the number and (maybe) the efficiency of these derivatives when applied to concrete problems.Zero preservation for a family of multivalued functionals, and applications to the theory of fixed points and coincidenceshttps://zbmath.org/1528.540142024-03-13T18:33:02.981707Z"Fomenko, T. N."https://zbmath.org/authors/?q=ai:fomenko.tatyana-nikolaevna"Zakharyan, Yu. N."https://zbmath.org/authors/?q=ai:zakharyan.yu-nSummary: A theorem on the zero existence preservation for a parametric family of multivalued \(( \alpha, \beta )\)-search functionals on an open subset of a metric space is proved. Several corollaries on the existence preservation for preimages of a closed subspace, for coincidence points, and for common fixed points under the action of a parametric family (a number of families) of mappings are obtained. The notion of a Zamfirescu-type pair of mappings is introduced, and a coincidence theorem for such pairs of mappings is obtained. In addition, a theorem on the coincidence existence preservation for a parametric family of such pairs of mappings is obtained. The obtained results imply several well-known theorems.Erratum to: ``Zero preservation for a family of multivalued functionals, and applications to the theory of fixed points and coincidences''https://zbmath.org/1528.540152024-03-13T18:33:02.981707Z"Fomenko, T. N."https://zbmath.org/authors/?q=ai:fomenko.tatyana-nikolaevna"Zakharyan, Yu. N."https://zbmath.org/authors/?q=ai:zakharyan.yu-nErratum to the authors' paper [ibid. 102, No. 1, 272--275 (2020; Zbl 1528.54014)].Geometric progressions in distance spaces; applications to fixed points and coincidence pointshttps://zbmath.org/1528.540192024-03-13T18:33:02.981707Z"Zhukovskiy, Evgeny S."https://zbmath.org/authors/?q=ai:zhukovskiy.evgeny-sThe author investigates which analogues of Banach's and Nadler's fixed-point theorems and Arutyunov's coincidence-point theorem can be obtained for mappings defined on spaces \(X\) endowed with the generalized distance \(\rho_X\). This holds if each geometric progression with \(ratio < 1\) is convergent. Examples of spaces with and without this property are given. In particular, the required property holds in a complete \(f\)-quasimetric space \(X\), if the distance \(\rho X\) satisfies this inequality \(\rho_X(x, z) \leq \rho_X(x, y) + (\rho_X(y, z))^{\eta}, x, y, z\in X\), for some \(\eta \in (0, 1)\).
In the last part of the paper the author considers the following function \(f(r_1,r_2)=\max \{r_1^{\eta},r_2^{\eta}\}\) where \(\eta \in (0, 2^{-1}]\). He shows that for any \(\gamma > 0\) there exists an \(f\)-quasimetric space containing a geometric progression with ratio \(\gamma\) which is not a Cauchy sequence. For \(f\)-quasimetric spaces the ``zero-one law'' is discussed, which means that either each geometric progression with \(ratio< 1\) is a Cauchy sequence or, for any \(\gamma \in (0, 1)\), there exists a geometric progression with ratio \(\gamma\) that is not Cauchy.
Reviewer: Monica-Felicia Bota (Cluj-Napoca)