Recent zbMATH articles in MSC 58https://zbmath.org/atom/cc/582021-01-08T12:24:00+00:00WerkzeugDynamic total slip-rate-dependent frictional contact problem for a nonlinear viscoelastic material with long memory.https://zbmath.org/1449.741552021-01-08T12:24:00+00:00"Brahim, Nouiri"https://zbmath.org/authors/?q=ai:brahim.nouiri"Benyattou, Benabderrahmane"https://zbmath.org/authors/?q=ai:benyattou.benabderrahmaneSummary: In this paper, we consider a mathematical model which describes the dynamic frictional contact between a deformable body and rigid foundation. We assume that the behavior of the body is described by a nonlinear viscoelastic constitutive law with long memory. The friction condition is modeled with a simplified version of Coulomb's law in which the normal stress is prescribed and the coefficient of friction depends on the total slip-rate. We present the classical formulation of the problem, and derive a variational formulation which consists a second order evolutionary quasi-variational inequality for the displacement field. Then, we establish the existence and uniqueness result of weak solution. The proof is based on the Faedo-Galerkin method and Banach's fixed point theorem. Finally, we show a convergence result when the relaxation coefficients of long memory tend to zero.The mechanism of the appearance of stochasticity in quantum mechanics.https://zbmath.org/1449.810282021-01-08T12:24:00+00:00"Samarin, Alekseĭ Yur'evich"https://zbmath.org/authors/?q=ai:samarin.aleksei-yurevichSummary: The technique for the dynamic description of interaction between the quantum particle and the measuring instrument is offered. This description allows to determine, that the statistic dispersion of measuring instrument characteristics is the cause of the results randomness of the quantum particle space localization. Space-time consideration of the macroscopic meter evolution, initiated by the quantum particle, allows to represent the mechanism of the appearance of probabilistic measure, expressed by the wave function modulus square.The Cauchy problems for dissipative hyperbolic mean curvature flow.https://zbmath.org/1449.580062021-01-08T12:24:00+00:00"Lv, Shixia"https://zbmath.org/authors/?q=ai:lv.shixia"Wang, Zenggui"https://zbmath.org/authors/?q=ai:wang.zengguiSummary: In this paper, we investigate initial value problems for hyperbolic mean curvature flow with a dissipative term. By means of support functions of a convex curve, a hyperbolic Monge-Ampère equation is derived, and this equation could be reduced to the first order quasilinear systems in Riemann invariants. Using the theory of the local solutions of Cauchy problems for quasilinear hyperbolic systems, we discuss lower bounds on life-span of classical solutions to Cauchy problems for dissipative hyperbolic mean curvature flow.Representation of Friedmann equation solution in form of generalized Dirichlet series.https://zbmath.org/1449.830112021-01-08T12:24:00+00:00"Kuryanovich, Èduard Anatol'evich"https://zbmath.org/authors/?q=ai:kuryanovich.eduard-anatolevichSummary: The cosmological Friedmann equation for the Universe, filled by scalar field with the quadratic potential, is reduced to the system of two first-order equations, one having the separable variables. The boundary-value problem with data at infinity is formulated for the second equation. The solution of this problem is represented in form of generalized Dirichlet series. The existence of classical solution in this form at the neighborhood of infinity is proved.The existence of triple classical solutions to impulsive problems with small non-autonomous perturbations.https://zbmath.org/1449.341002021-01-08T12:24:00+00:00"Liu, Jian"https://zbmath.org/authors/?q=ai:liu.jian.1"Zhao, Zengqin"https://zbmath.org/authors/?q=ai:zhao.zengqin"Yu, Wenguang"https://zbmath.org/authors/?q=ai:yu.wenguangSummary: We study the existence of solutions to nonlinear impulsive boundary value problems with small non-autonomous perturbations on the half-line. We show the existence of at least three distinct classical solutions by using variational methods and a three critical points theorem.The concavity of \(p\)-Renyi entropy power for the weighted doubly nonlinear diffusion equations on weighted Riemannian manifolds.https://zbmath.org/1449.580052021-01-08T12:24:00+00:00"Wang, Yuzhao"https://zbmath.org/authors/?q=ai:wang.yuzhao"Zhang, Huiting"https://zbmath.org/authors/?q=ai:zhang.huitingSummary: In this paper, we study the concavity of the entropy power on Riemannian manifolds. By using the nonlinear Bochner formula and Bakry-Emery method, we prove that \(p\)-Renyi entropy power is concave for positive solutions to the weighted doubly nonlinear diffusion equations on the weighted closed Riemannian manifolds with \(CD (-K, m)\) condition for some \(K \ge 0\) and \(m \ge n\), which generalizes the cases of porous medium equation and nonnegative Ricci curvature.Natural space of the micro-object.https://zbmath.org/1449.810272021-01-08T12:24:00+00:00"Samarin, Alekseĭ Yur'evich"https://zbmath.org/authors/?q=ai:samarin.aleksei-yurevichSummary: The immutability of classical dynamical laws for the microscopic object in the space which coordinate axes are the transition matrix elements of the corresponding coordinates is proved. It is stated that measurement is a micro-object localization process in the classical space when it interacts with the device. The description of the wave function reduction is obtained using the path integrals. The mechanism of the probability arising on measurement is offered, where the ``hidden parameter'' that is the cause of the measurement randomness of microscopic characteristics relates to the interaction process of classical instrument with micro-object. Both types of quantum mechanics processes -- evolution and reduction of the wave functions -- are described in a unified approach.Positive ground-state solution to singular Emden-Fowler equation with Dirichlet boundary value condition.https://zbmath.org/1449.340882021-01-08T12:24:00+00:00"Wang, Jia"https://zbmath.org/authors/?q=ai:wang.jia"Gao, Guifeng"https://zbmath.org/authors/?q=ai:gao.guifeng"Wang, Xinke"https://zbmath.org/authors/?q=ai:wang.xinke"Mao, Anmin"https://zbmath.org/authors/?q=ai:mao.anminSummary: In this paper, we consider the singular Emden-Fowler equation with Dirichlet boundary value condition. By using the Nehari method, we obtain the existence of positive ground-state solution and our work improves and extends some existing results.Gluing action groupoids: Fredholm conditions and layer potentials.https://zbmath.org/1449.354672021-01-08T12:24:00+00:00"Carvalho, Catarina"https://zbmath.org/authors/?q=ai:carvalho.catarina-a|carvalho.catarina-c"Côme, Rémi"https://zbmath.org/authors/?q=ai:come.remi"Qiao, Yu"https://zbmath.org/authors/?q=ai:qiao.yu.1|qiao.yu|qiao.yu.2Relevant subject of research, in the modern theory of the pseudodifferential operators, is the study of elliptic equations on non-compact manifolds, or manifolds with boundary and singularities, in particular conical domains. Pseudodifferential operators on Lie groupoids are intended to give a unified presentation of the problems, by recapturing as examples several results of the preceding literature, see \textit{V. Nistor} et al. [Pac. J. Math. 189, No. 1, 117--152 (1999; Zbl 0940.58014)]. In this line of ideas the authors of the present paper introduce a new class of groupoids, called boundary action groupoids, which are obtained by gluing reductions of action groupoids. The main result is a conditions for the Fredholm property (operators with parametrix and finite index). The condition involves standard ellipticity at the interior points, and invertibility of vector valued symbols at the singular points. Several applications are detailed.
Reviewer: Luigi Rodino (Torino)Wave equation of discrete spectrum states transition processes.https://zbmath.org/1449.810262021-01-08T12:24:00+00:00"Samarin, Alekseĭ Yur'evich"https://zbmath.org/authors/?q=ai:samarin.aleksei-yurevichSummary: For the dynamic description of object quantum transition, function of space coordinates matrix elements and time (transition wave function) is determined. It plays the role for transition analogous, to wave function for state. Matrix elements of coordinates and energy are offered in the form, allowing describe spatial distribution of system localization coordinates during its transition between states of a discrete spectrum. The differential equation for transition wave function is obtained. The time dependence of transition wave function is determined. Boundary conditions for a spatial part of transition wave function are established.Periodic solutions for a class of Kirchhoff-type differential systems.https://zbmath.org/1449.341242021-01-08T12:24:00+00:00"Zhang, Shengui"https://zbmath.org/authors/?q=ai:zhang.shenguiSummary: By using variational principle, the author studies periodic solutions for a class of superlinear Kirchhoff-type \(p (t)\)-Laplacian systems. Under the condition of no Ambrosetti-Rabinowitz-type growth, some results for the existence of periodic solutions are obtained by means of a variant mountain pass type theorem.Gradient estimates for a weighted nonlinear equation on complete noncompact manifolds.https://zbmath.org/1449.580042021-01-08T12:24:00+00:00"Li, Jing"https://zbmath.org/authors/?q=ai:li.jing.11"He, Guoqing"https://zbmath.org/authors/?q=ai:he.guoqing"Zhao, Peibiao"https://zbmath.org/authors/?q=ai:zhao.peibiaoSummary: \textit{B. Ma} et al. [Proc. Am. Math. Soc. 146, No. 11, 4993--5002 (2018; Zbl 1398.58007)] considered \(\Delta u+cu^{\alpha}=0 \; (\alpha < 0)\) with Ric\(_{ij}\) \(\ge K g_{ij}\), and obtained some gradient estimates. In the present paper, we investigate the weighted nonlinear equation \(\Delta_f u+cu^{\alpha}=0\) with Ric\(_{f}^N\) \(\ge-K\), where \(f\) is a smooth real-valued function on a complete noncompact Riemannian manifold \((M^n,g), \; \alpha> 0\) and \(c\) are two real constants, and we achieve some gradient estimates for positive solutions of this weighted nonlinear equation. The results posed in this paper can be regarded as a natural generalization of the above mentioned paper.Space of configurations and the special measures on it.https://zbmath.org/1449.280132021-01-08T12:24:00+00:00"Berezansky, Yu. M."https://zbmath.org/authors/?q=ai:berezanskii.yurii-makarovichThe author gives a novel exposition of the basic notions of analysis on configurations, including various topologizations and classes of measures.
Reviewer: Anatoly N. Kochubei (Kyïv)Existence of solutions for Schrödinger-Poisson system with asymptotically periodic terms.https://zbmath.org/1449.352112021-01-08T12:24:00+00:00"Wang, Da-Bin"https://zbmath.org/authors/?q=ai:wang.dabin"Ma, Lu-Ping"https://zbmath.org/authors/?q=ai:ma.luping"Guan, Wen"https://zbmath.org/authors/?q=ai:guan.wen"Wu, Hong-Mei"https://zbmath.org/authors/?q=ai:wu.hongmeiSummary: In this paper, we consider the following nonlinear Schrödinger-Poisson system \[ \begin{cases} -\Delta u + V(x)u+K(x)\phi u= f(x,u), \quad & x\in \mathbb{R}^3,\\ -\Delta \phi=K(x)u^{2}, & x\in \mathbb{R}^3, \end{cases} \] where \(V, K\in L^{\infty}(\mathbb{R}^3)\) and \(f:\mathbb{R}^3\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous. We prove that the problem has a nontrivial solution under asymptotically periodic case of \(V\), \(K\), and \(f\) at infinity. Moreover, the nonlinear term \(f\) does not satisfy any monotone condition.Self-adjointness, conservation laws and invariant solutions of the Buckmaster equation.https://zbmath.org/1449.580032021-01-08T12:24:00+00:00"Rashidi, Saeede"https://zbmath.org/authors/?q=ai:rashidi.saeede"Hejazi, Seyed Reza"https://zbmath.org/authors/?q=ai:hejazi.seyed-rezaSummary: The present paper considers the group analysis of extended (1 + 1)-dimensional Buckmaster equation and its conservation laws. Symmetry operators of Buckmaster equation are found via Lie algorithm of differential equations. The method of non-linear self-adjointness is applied to the considered equation. The infinite set of conservation laws associated with the finite algebra of Lie point symmetries of the Buckmaster equation is computed. The corresponding conserved quantities are obtained from their respective densities. Furthermore, the similarity reductions corresponding to the symmetries of the equation are constructed.Multiplicity of generalized Yamabe equations on Riemannian manifolds and applications to Emden-Fowler problems.https://zbmath.org/1449.530402021-01-08T12:24:00+00:00"Liao, Fangfang"https://zbmath.org/authors/?q=ai:liao.fangfang"Heidarkhani, Shapour"https://zbmath.org/authors/?q=ai:heidarkhani.shapour"Afrouzi, Ghasem A."https://zbmath.org/authors/?q=ai:afrouzi.ghasem-alizadeh"Roudbari, Sina Pourali"https://zbmath.org/authors/?q=ai:pourali-roudbari.sinaSummary: In this paper, we study the existence of multiple weak solutions for generalized Yamabe equations on Riemannian manifolds. As applications, we consider the Emden-Fowler equations involving sublinear terms at infinity.A shifted Chebyshev-tau method for finding a time-dependent heat source in heat equation.https://zbmath.org/1449.354582021-01-08T12:24:00+00:00"Akbarpour, Samaneh"https://zbmath.org/authors/?q=ai:akbarpour.samaneh"Shidfar, Abdollah"https://zbmath.org/authors/?q=ai:shidfar.abdollah"Saberi Najafi, Hashem"https://zbmath.org/authors/?q=ai:saberi-najafi.hashemSummary: This paper investigates the inverse problem of determining the time-dependent heat source and the temperature for the heat equation with Dirichlet boundary conditions and an integral over determination conditions. The numerical method is presented for solving the inverse problem. Shifted Chebyshev polynomial is used to approximate the solution of the equation as a base of the tau method which is based on the Chebyshev operational matrices. The main advantage of this method is based upon reducing the partial differential equation into a system of algebraic equations of the solution. Numerical results are presented and discussed.Description of discrete spectrum states transition processes.https://zbmath.org/1449.810302021-01-08T12:24:00+00:00"Samarin, A. Yu."https://zbmath.org/authors/?q=ai:samarin.aleksei-yurevichSummary: Matrix elements of transition are offered in the form, allowing to describe transition process of a quantum system between discrete spectrum states. It is shown, that alternativeness of possible paths of system movement allows to express matrix elements of coordinates and time functions by the corresponding functions of coordinates and time matrix elements. Dynamic relations between these functions of matrix elements during transition are established.Compatibility conditions on the reduced Poisson-Lie group.https://zbmath.org/1449.530342021-01-08T12:24:00+00:00"Aloui, Foued"https://zbmath.org/authors/?q=ai:aloui.foued"Zaalani, Nadhem"https://zbmath.org/authors/?q=ai:zaalani.nadhemSummary: Let \( (G, {\Lambda_G}, {{\langle, \rangle}_G})\) be a Poisson Lie-group equipped with a left invariant Riemannian metric and \(H\) a normal and closed coisotropic subgroup of \(G\). In this paper, we give necessary and sufficient conditions under which the compatibility conditions between the Poisson tensor \({\Lambda_G}\) and the metric \({{\langle, \rangle}_G}\) on \(G\) remain verified on the reduced Poisson-Lie group \( (G/H, {\Lambda_{G/H}}, {{\langle, \rangle}_{G/H}})\). In particular, if the immersed dual group \( (G/H)^*\) is totally geodesic in \({G^*}\), then these conditions are satisfied.Solution of the singular Cauchy problem for a general inhomogeneous Euler-Poisson-Darboux equation.https://zbmath.org/1449.353152021-01-08T12:24:00+00:00"Shishkina, Elina"https://zbmath.org/authors/?q=ai:shishkina.elina-leonidovnaSummary: In this paper, we solve Cauchy problem for a general form of an inhomogeneous Euler-Poisson-Darboux equation, where Bessel operator acts instead of the each second derivative. In the classical formulation, the Cauchy problem for this equation is not correct. However, for a specially selected form of the initial conditions, the equation has a solution. The general form of the Euler-Poisson-Darboux equation with such conditions we will call the singular Cauchy problem.Transportless conjugate gradient for optimization on Stiefel manifold.https://zbmath.org/1449.651172021-01-08T12:24:00+00:00"Figueroa, Edgar Fuentes"https://zbmath.org/authors/?q=ai:figueroa.edgar-fuentes"Dalmau, Oscar"https://zbmath.org/authors/?q=ai:dalmau.oscarSummary: In this paper, we focus on building an optimization scheme over the Stiefel manifold that maintains each iterate feasible. We focus on conjugate gradient methods and compare our scheme to the Riemannian optimization approach. We parametrize the Stiefel manifold using the polar decomposition to build an optimization problem over a vector space, instead of a Riemannian manifold. The result is a conjugate gradient method that averts the use of a vector transport, needed in the Riemannian conjugate gradient method. The performance of our method is tested on a variety of numerical experiments and compared with those of three Riemannian optimization methods.Multiple positive solutions to a \((2m)\)th-order boundary value problem.https://zbmath.org/1449.340702021-01-08T12:24:00+00:00"Boulaiki, Habiba"https://zbmath.org/authors/?q=ai:boulaiki.habiba"Moussaoui, Toufik"https://zbmath.org/authors/?q=ai:moussaoui.toufik"Precup, Radu"https://zbmath.org/authors/?q=ai:precup.raduSummary: The aim of the present paper is to study the existence, localization and multiplicity of positive solutions for a \((2m)\)th-order boundary value problem subject to the Dirichlet conditions. Our approach is based on critical point theory in conical shells and Harnack type inequalities.Existence of periodic solutions for a class of damped vibration problems.https://zbmath.org/1449.341162021-01-08T12:24:00+00:00"Chen, Mengxi"https://zbmath.org/authors/?q=ai:chen.mengxi"Wang, Zhiyong"https://zbmath.org/authors/?q=ai:wang.zhiyong.1|wang.zhiyong.2|wang.zhiyongSummary: This paper is dedicated to study the periodic solutions for a class of damped vibration problems. By virtue of an auxiliary function, we obtain some new superquadratic growth and asymptotically quadratic growth conditions. Using the minimax methods in critical point theory, we establish some existence results, which unify and generalize some known results in the literature.Generalized Han-Liu-Zhang's anomaly cancellation formulas.https://zbmath.org/1449.580022021-01-08T12:24:00+00:00"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.5|wang.yong.7|wang.yong.10|wang.yong.3|wang.yong.8|wang.yong.6|wang.yong.1|wang.yong|wang.yong.9|wang.yong.2"Wu, Tong"https://zbmath.org/authors/?q=ai:wu.tongSummary: In [in: Frontiers in differential geometry, partial differential equations and mathematical physics. In memory of Gu Chaohao. Hackensack, NJ: World Scientific. 87--104 (2014; Zbl 1299.81039)], \textit{F. Han} et al. gave a anomaly cancellation formula which generalized the Green-Schwarz formula and the Schwartz-Witten formula. In this paper, we give two generalized Han-Liu-Zhang formulas and also a Han-Liu-Zhang formula in odd dimension. By studying modular invariance properties of some characteristic forms, some new anomaly cancellation formulas in odd dimension are given.Phase transitions on \(C^*\)-algebras arising from number fields and the generalized Furstenberg conjecture.https://zbmath.org/1449.460572021-01-08T12:24:00+00:00"Laca, Marcelo"https://zbmath.org/authors/?q=ai:laca.marcelo"Warren, Jacqueline M."https://zbmath.org/authors/?q=ai:warren.jacqueline-mFor an algebraic number field \(K\) with ring of integers \(O_K\), the (multiplicative) monoid \(O_K^{\times}\) of non-zero integers action on the (additive) group \(O_K\) gives rise to the semi-direct product \(O_K\rtimes O_K^{\times}\), called here the ``affine'' or ``\(ax+b\)'' monoid of algebraic integers in \(K\). The Toeplitz-like \(C^*\)-algebra generated by the left regular representation of the \(ax+b\) monoid acting by isometries on \(\ell^2(O_K\rtimes O_K^{\times})\) was studied by \textit{J. Cuntz} et al. [Math. Ann. 355, No. 4, 1383--1423 (2013; Zbl 1273.22008)], who analysed the equilibrium states of the time evolution on this \(C^*\)-algebra determined by the absolute norm, and characterized the simplex of KMS equilibrium states of this dynamical system for any inverse temperature \(\beta\in(0,\infty]\).
In the paper under review, the low-temperature range of the classification of KMS equilibrium states is studied, using the parametrization in terms of tracial states of direct sums of group \(C^*\)-algebras. Because of the action of units arising here, a higher-dimensional version of Furstenberg's seminal conjecture on rigidity for probability measures on the circle invariant under the multiplicative action of a non-lacunary semigroup of integers [\textit{H. Furstenberg}, Math. Syst. Theory 1, 1--49 (1967; Zbl 0146.28502)] enters the picture. The main results classify the behaviours arising in terms of the ideal class group, the degree, and the unit rank of \(K\), and an explicit description of the primitive ideal space of the associated transformation group \(C^*\)-algebra for number fields of unit rank at least \(2\) that are not complex multiplication fields.
Reviewer: Thomas B. Ward (Leeds)Noncommutative geometry and structure of space-time.https://zbmath.org/1449.580012021-01-08T12:24:00+00:00"Chamseddine, Ali H."https://zbmath.org/authors/?q=ai:chamseddine.ali-hSummary: I give a summary review of the research program using noncommutative geometry as a framework to determine the structure of space-time. Classification of finite noncommutative spaces under few assumptions reveals why nature chose the Standard Model and the reasons behind the particular form of gauge fields, Higgs fields and fermions as well as the origin of symmetry breaking. It also points that at high energies the Standard Model is a truncation of Pati-Salam unified model of leptons and quarks. The same conclusions are arrived at uniquely without making any assumptions except for an axiom which is a higher form of Heisenberg commutation relations quantizing the volume of space-time. We establish the existence of two kinds of quanta of geometry in the form of unit spheres of Planck length. We provide answers to many of the questions which are not answered by other approaches, however, more research is needed to answer the remaining challenging questions.Arising of atoms and molecules stable states due to luminous electrons exchange coupling.https://zbmath.org/1449.810292021-01-08T12:24:00+00:00"Samarin, A. Yu."https://zbmath.org/authors/?q=ai:samarin.aleksei-yurevichSummary: The influence of the process of exchange interaction during the lifetime of luminous atoms and molecules was studied. The opportunity of stationary states existence of interacting luminous atoms and molecules was established. With the help of Feynman integrals paths method, the mechanism of occurrence of stationary states was determined, on condition of exchange interaction taking place. The way of definition of specific conditions at which such states occur is demonstrated. Dependence of a matrix element of potential accounting exchange interaction on coordinates of electrons and their wave functions was obtained.The almost Einstein operator for \((2,3,5)\) distributions.https://zbmath.org/1449.530332021-01-08T12:24:00+00:00"Sagerschnig, Katja"https://zbmath.org/authors/?q=ai:sagerschnig.katja"Willse, Travis"https://zbmath.org/authors/?q=ai:willse.travisA \((2,3,5)\) distribution on a 5-manifold is a \((2,3)\)-plane distribution which can be viewed as maximally non-integrable. This distribution induces a conformal structure of signature \((2,3)\). In the paper the corresponding tractor calculus is developed. The paper is motivated by the question whether the conformal structure contains an Einstein metric.
Reviewer: Hans-Bert Rademacher (Leipzig)Discrete energy behavior of a damped Timoshenko system.https://zbmath.org/1449.652572021-01-08T12:24:00+00:00"Sabrine, Chebbi"https://zbmath.org/authors/?q=ai:sabrine.chebbi"Makram, Hamouda"https://zbmath.org/authors/?q=ai:makram.hamoudaSummary: In this article, we consider a one-dimensional Timoshenko system subject to different types of dissipation (linear and nonlinear damping). Based on a combination between the finite element and the finite difference methods, we design a discretization scheme for the different Timoshenko systems under consideration. We first come up with a numerical scheme to the free-undamped Timoshenko system. Then we adapt this numerical scheme to the corresponding linear and nonlinear damped systems. Interestingly, this scheme reaches to reproduce the most important properties of the discrete energy, namely we show for the discrete energy the positivity, the energy conservation property and the different decay rate profiles. We numerically reproduce the known analytical results established on the decay rate of the energy associated with each type of dissipation.A note on the reversibility of Finsler manifolds.https://zbmath.org/1449.530192021-01-08T12:24:00+00:00"Yin, Songting"https://zbmath.org/authors/?q=ai:yin.songtingSummary: For a Finsler manifold with the weighted Ricci curvature bounded from below, we give Cheng type and Mckean type comparison theorems for the first eigenvalue of Finsler Laplacian. When the weighted Ricci curvature is nonnegative, we also obtain Calabi-Yau type volume growth theorem. These generalize and improve some recent literatures. Especially, by using the relationship of the counterparts between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility.Classic theorem by Lyapunov for differential equations in Hilbert spaces.https://zbmath.org/1449.350402021-01-08T12:24:00+00:00"Vavilov, S. A."https://zbmath.org/authors/?q=ai:vavilov.sergey-a"Fedotova, V. S."https://zbmath.org/authors/?q=ai:fedotova.v-sSummary: A theorem analogical to Lyapunov Classic Theorem is formulated for differential equations in Hilbert spaces. Example from the theory of partial differential equations is presented. The result automatically demonstrates the well-know conditions of continuum existence for periodic solutions of ordinary differential equations systems. Moreover, by applying the topological degree theory, these conditions can be set as less rigid than those formulated in Hopf Bifurcation Theory.