Recent zbMATH articles in MSC 34Ghttps://zbmath.org/atom/cc/34G2022-07-25T18:03:43.254055ZWerkzeugExistence and uniqueness of mild solution for fractional-order controlled fuzzy evolution equationhttps://zbmath.org/1487.340032022-07-25T18:03:43.254055Z"Iqbal, Naveed"https://zbmath.org/authors/?q=ai:iqbal.naveed-h"Niazi, Azmat Ullah Khan"https://zbmath.org/authors/?q=ai:niazi.azmat-ullah-khan"Shafqat, Ramsha"https://zbmath.org/authors/?q=ai:shafqat.ramsha"Zaland, Shamsullah"https://zbmath.org/authors/?q=ai:zaland.shamsullahSummary: In this article, we investigated the existence and uniqueness of mild solutions for fractional-order controlled fuzzy evolution equations with Caputo derivatives of the controlled fuzzy nonlinear evolution equation of the form
\[
_0^c D_{\mathfrak{I}}^\gamma\mathfrak{x}(\mathfrak{I})=\alpha\mathfrak{x}(\mathfrak{I})+\mathfrak{P}(\mathfrak{I}, \mathfrak{x}(\mathfrak{I}))+\mathfrak{A}(\mathfrak{I})\mathfrak{W}(\mathfrak{I}), \quad \mathfrak{I}\in[0, T], \mathfrak{x}(\mathfrak{I}_0) = \mathfrak{x}_0,
\]
in which \(\gamma\in(0, 1)\), \(E^1\) is the fuzzy metric space and \(I=[0, T]\) is a real line interval. With the help of few conditions on functions \(\mathfrak{P}:I\times E^1\times E^1\longrightarrow E^1\), \(\mathfrak{W}(\mathfrak{I})\) is control and it belongs to \(E^1\), \(\mathfrak{A}\in F(I, L( E^1))\), and \(\alpha\) stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.Existence of solution for impulsive fractional differential equations via topological degree methodhttps://zbmath.org/1487.340152022-07-25T18:03:43.254055Z"Faree, Taghareed A."https://zbmath.org/authors/?q=ai:faree.taghareed-a"Panchal, Satish K."https://zbmath.org/authors/?q=ai:panchal.satish-kushabaAs it is known in the literature, there are two main interpretations of the impulses in fractional differential equations. In this paper, the authors use one of them to study existence results for the initial value problem for impulsive Caputo fractional differential equations in Banach space. It is studied the case when the order of fractional derivative is from the interval (0,1). The proofs are based on the application of topological degree method and fixed point theorem with some suitable conditions. An example is provided to illustrate the results. Unfortunately, in the example there are some typos: the absolute values have to be deleted in the statement of the problem.
Reviewer: Snezhana Hristova (Plovdiv)On the global asymptotic stability of time-invariant solutions of nonlinear differential equations of some class in a Hilbert spacehttps://zbmath.org/1487.341172022-07-25T18:03:43.254055Z"Ryazantseva, I. P."https://zbmath.org/authors/?q=ai:ryazantseva.i-pAuthor's abstract: In a Hilbert space, we consider third-order differential equations linear in derivatives with real-valued coefficients that are functions of the independent variable and with a term nonlinear in the unknown function defined by a strongly monotone operator satisfying the Lipschitz condition. For these equations, we demonstrate a method for establishing conditions sufficient for the global asymptotic stability of time-invariant solutions. For these equations, we give inequalities that permit estimating the rate of convergence of solutions to the time-invariant solution.
Reviewer: Alexander O. Ignatyev (Donetsk)Approximate controllability of second-order stochastic non-autonomous integrodifferential inclusions by resolvent operatorshttps://zbmath.org/1487.341182022-07-25T18:03:43.254055Z"Nirmalkumar, R."https://zbmath.org/authors/?q=ai:nirmalkumar.r"Murugesu, R."https://zbmath.org/authors/?q=ai:murugesu.rSummary: In this paper, we formulate a set of sufficient conditions for the approximate controllability for a class of second-order stochastic non-autonomous integrodifferential inclusions in Hilbert space. We establish the results with the help of resolvent operators and \textit{H. F. Bohnenblust} and \textit{S. Karlin}'s fixed point theorem [Contrib. Theory of Games, Ann. Math. Stud. 24, 155--160 (1950; Zbl 0041.25701)]is to prove the main result. An application is given to illustrate the main result.Approximate controllability of neutral integrodifferential inclusions via resolvent operatorshttps://zbmath.org/1487.341432022-07-25T18:03:43.254055Z"Tamilselvan, M."https://zbmath.org/authors/?q=ai:tamilselvan.mSummary: In this work, a set of sufficient conditions are established for the approximate controllability for neutral integrodifferential inclusions in Banach spaces. The theory of fractional power and \(\alpha\)-norm is used because of the spatial derivatives in the nonlinear term of the system. \textit{H. F. Bohnenblust} and \textit{S. Karlin}'s fixed point theorem [Contrib. Theory of Games, Ann. Math. Stud. 24, 155--160 (1950; Zbl 0041.25701)] is used to prove our main results. Further, this result is extended to study the approximate controllability for nonlinear functional control system with nonlocal conditions. An example is also given to illustrate our main results.Hyers-Ulam stability of non-linear Volterra integro-delay dynamic system with fractional integrable impulses on time scaleshttps://zbmath.org/1487.341722022-07-25T18:03:43.254055Z"Shah, Syed Omar"https://zbmath.org/authors/?q=ai:shah.syed-omar"Zada, Akbar"https://zbmath.org/authors/?q=ai:zada.akbarSummary: This manuscript presents Hyers-Ulam stability and Hyers-Ulam-Rassias stability results of non-linear Volterra integro-delay dynamic system on time scales with fractional integrable impulses. Picard fixed point theorem is used for obtaining existence and uniqueness of solutions. By means of abstract Grönwall lemma, Grönwall's inequality on time scales, we establish Hyers-Ulam stability and Hyers-Ulam-Rassias stability results. There are some primary lemmas, inequalities and relevant assumptions that helps in our stability results.Exponential decay of a first order linear Volterra equationhttps://zbmath.org/1487.350742022-07-25T18:03:43.254055Z"Conti, Monica"https://zbmath.org/authors/?q=ai:conti.monica-c"Dell'Oro, Filippo"https://zbmath.org/authors/?q=ai:delloro.filippo"Pata, Vittorino"https://zbmath.org/authors/?q=ai:pata.vittorinoSummary: We consider the linear Volterra equation of the first order in time \[\dot u(t)+\int_0^t g(s)A u(t-s) d s = 0\] where \(A\) is a positive bounded operator on a Hilbert space \(H\). The exponential decay of the related energy is shown to occur, provided that the kernel \(g\) is controlled by a negative exponential.Solvability of doubly nonlinear parabolic equation with \(p\)-Laplacianhttps://zbmath.org/1487.352362022-07-25T18:03:43.254055Z"Uchida, Shun"https://zbmath.org/authors/?q=ai:uchida.shunSummary: In this paper, we consider a doubly nonlinear parabolic equation \(\partial_t \beta (u) - \nabla \cdot \alpha (x, \nabla u) \ni f\) with the homogeneous Dirichlet boundary condition in a bounded domain, where \(\beta : \mathbb{R} \to 2^{\mathbb{R}}\) is a maximal monotone graph satisfying \(0 \in \beta (0)\) and \(\nabla \cdot \alpha (x, \nabla u)\) stands for a generalized \(p\)-Laplacian. Existence of solution to the initial boundary value problem of this equation has been studied in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on \(\beta\). However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for \(1 < p < 2\). Main purpose of this paper is to show the solvability of the initial boundary value problem for any \(p \in (1, \infty)\) without any conditions for \(\beta\) except \(0 \in \beta (0)\). We also discuss the uniqueness of solution by using properties of entropy solution.Strong dissipative hydrodynamical systems and the operator pencil of S. Kreinhttps://zbmath.org/1487.470262022-07-25T18:03:43.254055Z"Voytitsky, V. I."https://zbmath.org/authors/?q=ai:voytitsky.victor-ivanovichThis review article is devoted to the linear operator differential equation \(u''(t)+(A+iG)u'(t)+Bu(t)=f(t)\) provided strong dissipativeness and to spectral properties of the related operator pencil of the form \(L(\lambda)=I-\lambda S-\lambda^{-1}T\) in a Hilbert space. The paper deals with factorization problems and asymptotics for eigenvalues of the pencil. Such type of equations and pencils arise in hydrodynamics. The author considers applications to the following problems:
\begin{itemize}
\item normal oscillations of viscous liquid in an open vessel,
\item normal oscillations of heavy rotating liquid in an open vessel,
\item normal convective movements of heavy liquid in an open vessel,
\item normal oscillations of joined pendulums with cavities filled with viscous liquids.
\end{itemize}
Reviewer: Nikita V. Artamonov (Moskva)Optimal rates of decay for operator semigroups on Hilbert spaceshttps://zbmath.org/1487.470722022-07-25T18:03:43.254055Z"Rozendaal, Jan"https://zbmath.org/authors/?q=ai:rozendaal.jan"Seifert, David"https://zbmath.org/authors/?q=ai:seifert.david"Stahn, Reinhard"https://zbmath.org/authors/?q=ai:stahn.reinhardSummary: We investigate rates of decay for \(C_0\)-semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to say an upper bound which is also known to be a lower bound, under a comparatively mild assumption on the growth behaviour. This extends several statements obtained by \textit{C. J. K. Batty} et al. [J. Eur. Math. Soc. (JEMS) 18, No. 4, 853--929 (2016; Zbl 1418.34120)]. In fact, for a large class of semigroups our condition is not only sufficient but also necessary for this optimal estimate to hold. Even without this assumption we obtain a new quantified asymptotic result which in many cases of interest gives a sharper estimate for the rate of decay than was previously available, and for semigroups of normal operators we are able to describe the asymptotic behaviour exactly. We illustrate the strength of our theoretical results by using them to obtain sharp estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary.