Ultimate Boundedness in the Sense of Poisson of Solutions to Systems of Differential Equations and Lyapunov Functions

Mathematical Notes - The notions of different types of boundedness in the sense of Poisson of solutions to systems of differential equations are introduced. Sufficient conditions are obtained for...     

Uniform Boundedness in the Sense of Poisson of Solutions of Systems of Differential Equations and Lyapunov Vector Functions

We introduce several generalizations of the properties of equiboundedness and uniform boundedness of solutions of ordinary differential systems, which are united by the common names of equiboundedness in the sense of Poisson and uniform boundedness in the sense of Poisson. For each of the above-introduced properties, we use the method of Lyapunov vector functions to obtain sufficient criteria for the system to have a certain property. In terms of the upper Dini derivative of the Lyapunov function given by a system, several criteria are established for the solutions of this system to have the relevant type of uniform boundedness in the sense of Poisson.     

Vector Lyapunov Functions and Ultimate Poisson Boundedness of Solutions of Systems of Differential Equations

Various forms of uniform-ultimate Poisson boundedness of solutions and of ultimate Poisson equiboundedness of solutions are introduced. Sufficient conditions for various forms of uniform-ultimate Poisson boundedness and of ultimate Poisson equiboundedness of solutions are obtained by using the method of vector Lyapunov functions.     

Poisson Total Boundedness of Solutions of Systems of Differential Equations and Lyapunov Vector Functions

We introduce the notions of Poisson total boundedness of solutions, partial Poisson total boundedness of solutions, and partial Poisson total boundedness of solutions with partly controlled initial conditions. We use the Lyapunov vector function method to obtain sufficient conditions for the Poisson total boundedness of solutions, the partial Poisson total boundedness of solutions, and the partial Poisson total boundedness of solutions with partly controlled initial conditions. As a consequence, we obtain sufficient conditions for the above-mentioned kinds of Poisson total boundedness of solutions based on the Lyapunov function method.

     

Lyapunov Functions,Krasnosel’skii Canonical Domains,and the Existence of Poisson Bounded Solutions

Mathematical Notes - The notions of Poisson boundedness and Poisson partial boundedness of solutions of systems are introduced. Based on the Lyapunov function method and...     

一类不连续非自治系统的一致最终有界性

主要讨论右端不连续的非自治系统在Filippov解意义下的一致最终有界性问题.首先给出不连续系统全局强一致最终有界的定义,并得到了不连续系统全局一致强最终有界的Lya- punov定理.最后给出了在一类带有不连续摩擦项的力学系统中的应用.     

一类不连续时滞系统的一致最终有界性

主要讨论不连续的时滞自治系统,在Filippov解意义下的一致最终有界性问题．基于Lyapunov-Krasovskii泛函给出了全局强一致最终有界的Lyapunov定理,并将其应用到一类带有不连续摩擦项的时滞力学系统．     

Analysis of the set of trajectories of nonlinear dynamics: Stability and boundedness of motions

For a set of differential equations with the Hukuhara derivative, we obtain sufficient conditions for various types of boundedness of trajectories and stability of the set of stationary solutions. To this end, we use scalar and vector Lyapunov functions constructed on the basis of an auxiliary matrix-valued function.     

Partial total boundedness of solutions to systems of differential equations with partly controlled initial conditions

The notions of partial total boundedness of solutions with partially controlled initial conditions and of partial total equiboundedness of solutionswith partially controlled initial conditions are introduced. The direct Lyapunov method and the method of Lyapunov vector functions are used to obtain sufficient conditions for these types of boundedness of the solutions.     

An existence theorem for the unmodified Vlasov Equation

The initial value problem for the modified Vlasov equation with a mollification parameter δ > 0, as introduced by Batt, has a unique global solution in the weak sense whenever f₀ ε L¹ and f₀ ≧ 0 λ-a.e. Assuming boundedness of f₀ and boundedness of the kinetic energy, it is shown that, as δ → 0, there are subsequences δ_n → 0 such that the corresponding solutions converge weakly in the measure-theoretical sense. The limits are shown to be global weak solutions of the initial value problem for Vlasov's equation, and these solutions are seen to be weakly continuous with respect to t. For the plasma physical case, boundedness of the kinetic energy is a consequence of energy conservation.     

Dissipative martingale solutions of the stochastically forced Navier–Stokes–Poisson system on domains without boundary

We construct solutions to the randomly-forced Navier–Stokes–Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability space and the stochastic driving force are also unknowns of the problem. Additionally, these solutions dissipate energy, satisfies a relative energy inequality in the sense of Dafermos (1979) and satisfy a renormalized form of the continuity equation in the sense of DiPerna and Lions (1989).     

Well-posedness for parametric quasivariational inequality problems and for optimization problems with quasivariational inequality constraints

In this article, the concepts of well-posedness and well-posedness in the generalized sense are introduced for parametric quasivariational inequality problems with set-valued maps. Metric characterizations of well-posedness and well-posedness in the generalized sense, in terms of the approximate solutions sets, are presented. Characterization of well-posedness under certain compactness assumptions and sufficient conditions for generalized well-posedness in terms of boundedness of approximate solutions sets are derived. The study is further extended to discuss well-posedness for an optimization problem with quasivariational inequality constraints.     

Stochastic averaging principles for multi-valued stochastic differential equations driven by poisson point Processes

The purpose of this article is to investigate an averaging principle for multi-valued stochastic differential equations (MSDEs) driven by Poisson point processes. The solutions to MSDEs driven by Poisson point processes can be approximated by solutions to averaged MSDEs in the sense of both convergence in mean square and convergence in probability. Finally, an example is presented to illustrate the averaging principle.     

Doubly perturbed neutral diffusion processes with Markovian switching and Poisson jumps

In this work, we will establish new results on the boundedness, existence and uniqueness of the solutions to a class of doubly perturbed neutral stochastic functional equations with Markovian switching and Poisson jumps. Most of the existing results use a singly perturbed model and a global Lipschitz condition, but in this work the doubly perturbed model has been studied and the coefficients of equations are non-Lipschitz. So some known results have been improved and generalized.     

Poisson stable motions of monotone nonautonomous dynamical systems

In this paper, we study the Poisson stability(in particular, stationarity, periodicity, quasiperiodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations(ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above mentioned Poisson stable trajectories under some suitable conditions, which is interesting in its own right for monotone dynamics.     

On Derivation and Classification of Vlasov Type Equations and Equations of Magnetohydrodynamics. The Lagrange Identity,the Godunov Form,and Critical Mass

The derivation of the Vlasov–Maxwell and the Vlasov–Poisson–Poisson equations from Lagrangians of classical electrodynamics is described. The equations of electromagnetohydrodynamics (EMHD) type and electrostatics with gravitation are obtained. We obtain and compare the Lagrange equalities and their generalizations for different types of the Vlasov and EMHD equations. The conveniences of writing the EMHD equations in twice divergent form are discussed. We analyze exact solutions to the Vlasov–Poisson–Poisson equations with the presence of gravitation where we have different types of nonlinear elliptic equations for trajectories of particles with critical mass m ² = e ²/G, which has an obvious physical sense, where G denotes the gravitation constant and e is the electron charge. As a consequence we have different behaviors of particles: divergence or collapse of their trajectories.     

Boundedness and Nonemptiness of the Efficient Solution Sets in Multiobjective Optimization

For a given multiobjective optimization problem, we study recession properties of the sets of efficient solutions and properly efficient solutions. We work out various consequences based on the obtained recession properties, including a characterization for the boundedness and nonemptiness of the set of (properly) efficient solutions when the problem is a convex problem. We also show that the boundedness and nonemptiness of the set of efficient solutions is equivalent to that of the set of properly efficient solutions under an additional mild condition. Finally, we provide some new verifiable necessary conditions for the nonemptiness of the set of efficient solutions in terms of the associated recession functions and recession cones.     

Higher-order derivatives of Lyapunov functions and partial boundedness of solutions with partially controllable initial conditions

Mathematical Notes - Certain sufficient criteria for the types of partial boundedness of solutions with partially controllable initial conditions are obtained in terms of higher-order derivatives...     

Sharp Growth Estimates for Modified Poisson Integrals in a Half Space

For continuous boundary data, including data of polynomial growth, modified Poisson integrals are used to write solutions to the half space Dirichlet and Neumann problems in Rⁿ. Pointwise growth estimates for these integrals are given and the estimates are proved sharp in a strong sense. For decaying data, a new type of modified Poisson integral is introduced and used to develop asymptotic expansions for solutions of these half space problems.     

Partial uniform boundedness of solutions of systems of differential equations with partly controlled initial conditions

We introduce the notion of partial uniform boundedness of solutions with partially controlled initial conditions in the general case, that is, the case in which the part of variables with respect to which the boundedness of solutions is studied is a subset of the part of variables with respect to which the initial conditions are controlled. We obtain a criterion for the partial uniform boundedness of solutions with partly controlled initial conditions. We introduce the notion of partial total boundedness of solutions with partly controlled initial conditions. We obtain a sufficient condition for the partial total boundedness of solutions with partly controlled initial conditions.