Recent zbMATH articles in MSC 39B62https://zbmath.org/atom/cc/39B622024-03-13T18:33:02.981707ZWerkzeugSupport theorems for generalized monotone functionshttps://zbmath.org/1528.260102024-03-13T18:33:02.981707Z"Bessenyei, Mihály"https://zbmath.org/authors/?q=ai:bessenyei.mihaly"Pénzes, Evelin"https://zbmath.org/authors/?q=ai:penzes.evelin\textit{S. Wąsowicz} [J. Math. Anal. Appl. 332, No. 2, 1229--1241 (2007; Zbl 1122.26024); J. Math. Anal. Appl. 365, No. 1, 415--427 (2010; Zbl 1188.26009)] wrote two papers, where he investigated generalized support-type properties of convex functions wrt. Chebyshev systems. Since that Bessenyei wondered whether or not such proerties hold also for convexity (another name generalized monotonicity) wrt. so-called Beckenbach interpolation falimies. In the present paper the complete solution is given.
Reviewer: Szymon Wąsowicz (Bielsko-Biała)Functional and differential inequalities and their applications to control problemshttps://zbmath.org/1528.340142024-03-13T18:33:02.981707Z"Benarab, S."https://zbmath.org/authors/?q=ai:benarab.sarra"Zhukovskaya, Z. T."https://zbmath.org/authors/?q=ai:zhukovskaya.zukhra-tagurovna|zhukovskaya.zukhra-tagirovna"Zhukovskiy, E. S."https://zbmath.org/authors/?q=ai:zhukovskiy.evgeny-s"Zhukovskiy, S. E."https://zbmath.org/authors/?q=ai:zhukovskiy.s-eSummary: We study boundary value and control problems using methods based on results on operator equations in partially ordered spaces. Sufficient conditions are obtained for the existence of a coincidence point for two mappings acting from a partially ordered space into an arbitrary set, an estimate for such a point is found, and corollaries about a fixed point for a mapping that acts in a partially ordered space and is not monotone are derived. The established results are applied to the study of functional and differential equations. For the Nemytskii operator in the space of measurable vector functions, sufficient conditions for the existence of a fixed point are obtained and it is shown that these conditions do not follow from the classical fixed point theorems. Assertions on the existence and estimates of the solution of the Cauchy problem are proved, and the solutions are given to a periodic boundary value problem and a control problem for systems of ordinary differential equations of the first order unsolved for the derivative of the desired vector function.A two-weight Sobolev inequality for Carnot-Carathéodory spaceshttps://zbmath.org/1528.350052024-03-13T18:33:02.981707Z"Alberico, Angela"https://zbmath.org/authors/?q=ai:alberico.angela"Di Gironimo, Patrizia"https://zbmath.org/authors/?q=ai:di-gironimo.patriziaSummary: Let \(X = \{X_1, X_2, \dots, X_m\}\) be a system of smooth vector fields in \(\mathbb{R}^n\) satisfying the Hörmander's finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Carathéodory space \(\mathbb{G}\) associated to system \(X\)
\[
\Bigg(\frac{1}{\int_{B_R} K(x)\; dx} \int_{B_R} |u|^t K(x) \; dx \Bigg)^{1/t} \leq C \, R \Bigg(\frac{1}{\int_{B_R} \frac{1}{K(x)} \; dx} \int_{B_R} \frac{|Xu|^2}{K(x)} \; dx \Bigg)^{1/2},
\]
where \(Xu\) denotes the horizontal gradient of \(u\) with respect to \(X\). We assume that the weight \(K\) belongs to Muckenhoupt's class \(A_2\) and Gehring's class \(G_{\tau}\), where \(\tau\) is a suitable exponent related to the homogeneous dimension.