Recent zbMATH articles in MSC 46Ghttps://www.zbmath.org/atom/cc/46G2021-07-26T21:45:41.944397ZWerkzeug\(\mu\)-absolute continuity and \(\mu\)-derivative, \(\mu\)-Newton-Leibniz formulahttps://www.zbmath.org/1463.280012021-07-26T21:45:41.944397Z"Sadigova, Sabina R."https://www.zbmath.org/authors/?q=ai:sadigova.sabina-rahib"Garayev, Tarlan Z."https://www.zbmath.org/authors/?q=ai:garaev.tarlan-zSummary: In this paper the concept of \(\mu\)-derivative at a point of some \(\mu\)-measurable function generated by some measure \(\mu\) in a measurable space is introduced. Following the classical case the concept of a \(\mu\)-absolute continuous function is defined on some interval, some of their properties are studied. Some classical facts are carried over to this case. The classical Newton-Leibniz formula is also generalized to the considered case.Maximal quasi-normed extension of quasi-normed latticeshttps://www.zbmath.org/1463.460012021-07-26T21:45:41.944397Z"Kusraev, Anatoliĭ Georgievich"https://www.zbmath.org/authors/?q=ai:kusraev.anatoly-georgievich"Tasoev, Batradz Botazovich"https://www.zbmath.org/authors/?q=ai:tasoev.batradz-botazovichSummary: The purpose of this article is to extend the Abramovich's construction of a maximal normed extension of a normed lattice to quasi-Banach setting. It is proved that the maximal quasi-normed extension \(X^\varkappa\) of a Dedekind complete quasi-normed lattice \(X\) with the weak \(\sigma \)-Fatou property is a quasi-Banach lattice if and only if \(X\) is intervally complete. Moreover, \(X^\varkappa\) has the Fatou and the Levi property provided that \(X\) is a Dedekind complete quasi-normed space with the Fatou property. The possibility of applying this construction to the definition of a space of weakly integrable functions with respect to a measure taking values from a quasi-Banach lattice is also discussed, since the duality based definition does not work in the quasi-Banach setting.Characterization and multiplicative representation of homogeneous disjointness preserving polynomialshttps://www.zbmath.org/1463.460102021-07-26T21:45:41.944397Z"Kusraeva, Zalina Anatol'evna"https://www.zbmath.org/authors/?q=ai:kusraeva.zalina-anatolevnaSummary: Let \(E\) and \(F\) be vector lattices and \(P: E\to F\) an order bounded orthogonally additive (i.e. \(|x|\wedge|y|=0\) implies \(P(x+y)=P(x)+P(y)\) for all \(x,y\in E\)) \(s\)-homogeneous polynomial. \(P\) is said to be disjointness preserving if its corresponding symmetric \(s\)-linear operator from \(E^s\) to \(F\) is disjointness preserving in each variable. The main results of the paper read as follows:
Theorem 3.9. The following are equivalent: (1) \(P\) is disjointness preserving; (2) \(\hat d^nP(x)(y)=0\) and \(Px\perp Py\) for all \(x,y\in E\), \(x\perp y\), and \(1\leq n<s\); (3) \(P\) is orthogonally additive and \(x\perp y\) implies \(Px\perp Py\) for all \(x,y\in E\); (4) there exist a vector lattice \(G\) and lattice homomorphisms \(S_1,S_2: E \to G\) such that \(G^{s\odot}\subset F$, $S_1(E)\perp S_2(E)\), and \(Px=(S_1x)^{s\odot}-(S_2x)^{s\odot}\) for all \(x\in E\); (5) there exists an order bounded disjointness preserving linear operator \(T:E^{s\odot}\to F\) such that \(Px=T(x^{s\odot})\) for all \(x\in E\).
Theorem 4.7. Let \(E\) and \(F\) be Dedekind complete vector lattices. There exists a partition of unity \((\rho_\xi)_{\xi\in\Xi}\) in the Boolean algebra of band projections \(\mathfrak P(F)\) and a family \((e_\xi)_{\xi\in\Xi}\) in \(E_+\) such that \(P(x)= o$-$\sum_{\xi\in\Xi}W\circ\rho_\xi S(x/e_\xi)^{s\odot}$ $(x\in E)\), where \(S\) is the shift of \(P\) and \(W:\mathscr F\to\mathscr F\) is the orthomorphism multiplication by \(o$-$\sum_{\xi\in\Xi}\rho_\xi P(e_\xi)\).Radon-Nikodym property.https://www.zbmath.org/1463.460322021-07-26T21:45:41.944397Z"Khurana, Surjit Singh"https://www.zbmath.org/authors/?q=ai:khurana.surjit-singhSummary: For a Banach space \(E\) and a probability space \((X,\mathcal{A},\lambda)\), a new proof is given that a measure \(\mu:\mathcal{A}\to E\), with \(\mu\ll\lambda\), has RN derivative with respect to \(\lambda\) iff there is a compact or a weakly compact \(C\subset E\) such that \(|\mu |_{C}:\mathcal{A}\to [0,\infty]\) is a finite valued countably additive measure. Here we define \(|\mu |_{C}(A)=\sup\{\sum_{k}|\langle\mu (A_{k}),f_{k}\rangle |\}\) where \(\{A_{k}\}\) is a finite disjoint collection of elements from \(\mathcal{A}\), each contained in \(A\), and \(\{f_{k}\}\subset E'\) satisfies \(\sup_{k}|f_{k}(C)|\leq 1\). Then the result is extended to the case when \(E\) is a Fréchet space.Topological vector space valued measures on topological spaceshttps://www.zbmath.org/1463.460432021-07-26T21:45:41.944397Z"Khurana, Surjit Singh"https://www.zbmath.org/authors/?q=ai:khurana.surjit-singhSummary: If \(X\) is a compact Hausdorff space space, \(E\) is a complete Hausdorff topological vector space and \(\mu : (C(X), {\| \cdot \|}) \rightarrow E\) a linear continuous exhaustive mapping, we first give a different proof that there is then a unique reqular, \(L^\infty\)-bounded, exhaustive \(E\)-valued Borel measure \(\mu\) on \(X\) such that \(\mu (f) = \int fd \mu\), for all \(f \in C(X)\). Then we consider \(X\) to be a completely regular Hausdorff space and prove the extension of Alexanderov's theorem: \(X\) is a completely regular Hausdorff space and \(\mu : C_b(X) \rightarrow E\) a linear, continuous, exhaustive mapping and \(\mathcal F\) is the algebra generated by zero-sets in \(X\). Then there exist a unique finitely additive, exhaustive measure \(\nu : F \rightarrow E\) such that
\begin{itemize}
\item[(i)] \(\nu\) is \(L^\infty\)-bounded i.e. the absolute convex hull of \(\nu (F) (\Gamma(\nu (F)))\) is bounded in \(E\);
\item[(ii)] \(\nu\) is inner regular by zero-sets and outer regular by positive-sets;
\item[(iii)] \(\int fd\nu = \mu(f)\), for all \(f \in C_b(X)\).
\end{itemize}Applications of anticompact sets to analogs of Denjoy-Young-Saks and Lebesgue theoremshttps://www.zbmath.org/1463.460682021-07-26T21:45:41.944397Z"Stonyakin, Fedor S."https://www.zbmath.org/authors/?q=ai:stonyakin.fedor-sergeevichSummary: We consider the problem of transfer of the Denjoy-Young-Saks theorem on derivates to infinite-dimensional Banach spaces and the problem of nondifferentiability of indefinite Pettis integral in infinite-dimensional Banach spaces. Our approach is based on the concept of an anticompact set proposed by us earlier. We prove an analog of the Denjoy-Young-Saks theorem on derivates in Banach spaces which have anticompact sets. Also in such spaces we obtain an analog of the Lebesgue theorem. This result states that each indefinite Pettis integral is differentiable almost everywhere in the topology of special Hilbert space generated by some anticompact set in the original space.Invertibility of linear relations generated by integral equation with operator measureshttps://www.zbmath.org/1463.470112021-07-26T21:45:41.944397Z"Bruk, Vladislav Moiseevich"https://www.zbmath.org/authors/?q=ai:bruk.vladislav-moiseevichSummary: We investigate linear relations generated by an integral equation with operator measures on a segment in the infinite-dimensional case. In terms of boundary values, we obtain necessary and sufficient conditions.
We consider integral equation with operator measures on a bounded closed interval in the infinite-dimensional case. In terms of boundary values, we obtain necessary and sufficient conditions under which these relations \(S\) possess the properties: \(S\) is closed relation; \(S\) is invertible relation; the kernel of \(S\) is finite-dimensional; the range of \(S\) is closed; \(S\) is continuously invertible relation and others. The results are applied to a system of integral equations becoming a quasidifferential equation whenever the operator measures are absolutely continuous as well as to an integral equation with multi-valued impulse action.Existence of countably many symmetric positive solutions for system of even order time scale boundary value problems in Banach spaceshttps://www.zbmath.org/1463.472282021-07-26T21:45:41.944397Z"Prasad, K. R."https://www.zbmath.org/authors/?q=ai:prasad.k-rama|prasad.kapula-rajendra"Khuddush, Md."https://www.zbmath.org/authors/?q=ai:khuddush.mdSummary: This paper establishes the existence and uniqueness of the solutions to the system of even order differential equations on time scales,
\[
\begin{aligned}
(-1)^nu^{(\Delta\nabla)^n}_1(t) =\omega_1(t)f_1(u_1(t),u_2(t)), \quad t\in[0,T]_\mathbb{T},\; n\in\mathbb{Z}^+,\\
(-1)^nu^{(\Delta\nabla)^m}_2(t) =\omega_2(t)f_2(u_2(t),u_2(t)), \quad t\in[0,T]_\mathbb{T}, \;m\in\mathbb{Z}^+,
\end{aligned}
\]
satisfying two-point Sturm-Liouville integral boundary conditions
\[
\begin{aligned}
\alpha_{i+1}u^{(\Delta\nabla)^i}_1(0)-\beta_{i+1}u^{(\Delta\nabla)^{i_\Delta}}_1 (0)=\int^T_0a_{i+1}(s)u^{(\Delta\nabla)^i}_1(s)\nabla s,\;0\le i\le n-1,\\
\alpha_{i+1}u^{(\Delta\nabla)^i}_1(T)-\beta_{i+1}u^{(\Delta\nabla)^{i_\Delta}}_1 (T)=\int^T_0a_{i+1}(s)u^{(\Delta\nabla)^i}_1(s)\nabla s,\;0\le i\le n-1,\\
\gamma_{j+1}u^{(\Delta\nabla)^j}_2(0)-\delta_{j+1}u^{(\Delta\nabla)^{j_\Delta}}_2 (0)=\int^T_0b_{j+1}(s)u^{(\Delta\nabla)^j}_2(s)\nabla s,\;0\le j\le m-1,\\
\gamma_{j+1}u^{(\Delta\nabla)^j}_2(T)-\delta_{j+1}u^{(\Delta\nabla)^{j_\Delta}}_2 (T)=\int^T_0b_{j+1}(s)u^{(\Delta\nabla)^j}_2(s)\nabla s,\;0\le j\le m-1,
\end{aligned}
\] by utilizing Schauder fixed point theorem. We also establish the existence of countably many symmetric positive solutions for the above problem by applying Hölder's inequality and Krasnoselskii's fixed point theorem.