Vector Lyapunov functions andp-th-order conditional stability and boundedness

Summary In this paper we introduce the concept of p-th-order conditional stability and boundedness for ordinary differential equations in Banach spaces. Using vector Lyapunov functions and the comparison technique we give sufficient conditions for p-th-order conditional stability and boundedness properties of the abstract differential equation. Our results generalize in many ways various known results of stability and boundedness. Entrata in Redazione il 23 luglio 1977.     

滞后型脉冲泛函微分方程有界性的Lyapunov逆定理

本文通过建立滞后型脉冲泛函微分方程饱和解的存在唯一性定理,在广义常微分方程与滞后型脉冲泛函微分方程等价的基础上,研究了滞后型脉冲泛函微分方程关于一致有界性的Lyapunov逆定理.     

On stability and L p solutions of ordinary differential equations

Summary The usual definition of the stability of a solution of a system of ordinary differential equations is extended by introducing two positive control functions. These functions are used to control the rate of growth of the in?tial position of the solution and the rate of growth of the solution. Definitions and results are also given for the corresponding analogues of boundedness, weak boundedness, and uniform properties of the sotions of differential equations. The problem of determining when solutions of certain linear and weakly nonlinear differential equations lie in a modified Lp-space is also considered. This research was supported by the National Science Foundation under grant GP-8921. Entrata in Redazione il 13 maggio 1969.     

Periodic solutions in n dimensions and Volterra equations

If a nonlinear autonomous n-dimensional system of ordinary differential equations has a bounded solution with a certain uniform stability property, this solution approaches an almost periodic solution with the same stability property. (More precisely, the almost periodic solution is in the set of ω-limit points of the given solution.) If the bounded solution has, in addition to the uniform stability property, an asymptotic stability property, then the solution approaches a periodic solution with the same stability properties. Practical (i.e., computable) sufficient conditions for boundedness of solutions are obtained. The results are applied to generalized Volterra equations.     

Solution Estimates for Nonlinear Nonautonomous Evolution Equations in Spaces with Normalizing Mappings

Nonlinear nonautonomous evolution equations in a space with a normalizing mapping (a generalized norm) are considered. Solution estimates are established. In particular cases these estimates generalize the Wazewski and Lozinskii estimates from the theory of ordinary differential equations. By the obtained estimates, the following problems are investigated: asymptotic stability, boundedness of solutions, input-output stability, existence of periodic solutions. Applications to integro-differential equations are discussed.     

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.     

Convergence to equilibria in scalar nonquasimonotone functional differential equations

We consider a class of scalar functional differential equations generating a strongly order preserving semiflow with respect to the exponential ordering introduced by Smith and Thieme. It is shown that the boundedness of all solutions and the stability properties of an equilibrium are exactly the same as for the ordinary differential equation which is obtained by “ignoring the delays”. The result on the boundedness of the solutions, combined with a convergence theorem due to Smith and Thieme, leads to explicit necessary and sufficient conditions for the convergence of all solutions starting from a dense subset of initial data. Under stronger conditions, guaranteeing that the functional differential equation is asymptotically equivalent to a scalar ordinary differential equation, a similar result is proved for the convergence of all solutions.     

Stability and boundedness of solutions of Stieltjes Differential Equations

Dynamical equations on time scales are formulated by means of Stieltjes differential equations, which, depending on the time integrator, include ordinary differential equations and difference equations as well as mixtures of both. Explicit conditions for the boundedness and stability of solutions are presented here for linear and nonlinear Stieltjes differential equations. In addition, the continuous dependence of solutions on the time integrator is established by means of a Gronwall-like inequality for equations with different time integrators.     

积分方程的周期解与一致最终有界性

讨论了具有无限时滞的Ｖｏｌｔｅｒｒａ积分方程的周期解和一致最终有界性．把泛函微分方程中的一个著名定理,即一致有界性和一致最终有界性保证周期解的存在性,推广到积分方程．     

Numerical solutions of index-1 differential algebraic equations can be computed in polynomial time

The cost of solving an initial value problem for index-1 differential algebraic equations to accuracy ɛ is polynomial in ln(1/ɛ). This cost is obtained for an algorithm based on the Taylor series method for solving differential algebraic equations developed by Pryce. This result extends a recent result by Corless for solutions of ordinary differential equations. The results of the standard theory of information-based complexity give exponential cost for solving ordinary differential equations, being based on a different model.     

SpaceC g and uniform behavior of solutions of functional differential equations with infinite delay

To study the uniform behavior of solutions to functional differential equations with infinite delay, this paper introduces a new spaceC _g of the initial functions with a new norm |·| _g . Corresponding definitions ofg-uniform boundedness (g-UB) andg-uniform ultimate boundedness (g-UUB), and theorems ensuringg-UB org-UUB are given. Partially supported by the Science Foundation of Academia Sinica     

Method of lines for physiologically structured models with diffusion

We deal with a size-structured model with diffusion. Partial differential equations are approximated by a large system of ordinary differential equations. Due to a maximum principle for this approximation method its solutions preserve positivity and boundedness. We formulate theorems on stability of the method of lines and provide suitable numerical experiments.     

BOUNDEDNESS OF IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

In this paper, we study the boundedness of a class of impulsive functional differential equations with infinite delays. We establish a uniform boundedness theorem and a uniformly ultimate boundedness theorem, which show that a certain impulsive perturbation can make an unbounded system into uniformly bounded, even uniformly ultimate bounded.     

Symmetry Analysis of Differential Equations Using MATHLIE

A general procedure for solving ordinary differential equations of arbitrary order is discussed. The method used is based on symmetries of a differential equation. The known symmetries allow the derivation of first integrals of the equation. The knowledge of at least r symmetries of an ordinary differential equation of order n with r n is the basis for deriving the solution. Our aim is to show that Lie's theory is a useful tool for solving ordinary differential equations of higher orders. Bibliography: 12 titles.     

Periodic Solutions for Functional Differential Inclusion with Nonconvex Right Hand Sides

In this paper we present a general existence result of periodic solutions for functional differential inclusions with nonconvex right hand sides, by using the asymptotic fixed point theory. In our result, the uniform boundedness and ultimate boundedness are only assumed to the solutions with bounded initial functions. On the other hand, the dissipativity is sought on a suitable bounded convex subset of the state space of solutions. This becomes difficult for the systems with infinite delay since in this case the subset is probably not forward invariant for the orbits of solutions. These are also considerable even for the usual functional differential equations with infinite delay. As an application, we answer an open problem on the existence of an equilibrium state for multivalued permanent systems.     

Solving differential equations using modified Picard iteration

Many classes of differential equation are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and engineering and include non-linear as well as linear differential equations. Examples involving partial as well as ordinary differential equations are presented. The method is easy to implement on a computer and the solutions so obtained are essentially power series. With its conceptual clarity (differential equations are integrated directly), its uniform methodology (the overall approach is the same in all cases) and its straightforward computer implementation (the integration and iteration procedures require only standard commercial software), the modified Picard methods offer obvious benefits for the teaching of differential equations as well as presenting a basic but flexible tool-kit for the solution process itself.     

Reaction-Diffusion Equations with Polynomial Drifts Driven by Fractional Brownian Motions

A reaction-diffusion equation on [0, 1] ^d with the heat conductivity κ > 0, a polynomial drift term and an additive noise, fractional in time with H > 1/2, and colored in space, is considered. We have shown the existence, uniqueness and uniform boundedness of solution with respect to κ. Also we show that if κ tends to infinity, then the corresponding solutions of the equation converge to a process satisfying a stochastic ordinary differential equation.     

Uniform ultimate boundedness of nonlinear nonstationary systems

A certain class of nonlinear, nonstationary systems of differential equations is studied. It is assumed that the right-hand sides of the equations under consideration are homogeneous functions of order smaller than one with respect to the phase variables. The purpose of this paper is to obtain sufficient conditions for the uniform ultimate boundedness of systems of this form. A method for constructing nonstationary Lyapunov functions is suggested and applied to prove that the asymptotic stability of the zero solution of the corresponding averaged system implies the uniform ultimate boundedness of the initial nonstationary system. Classes of perturbations that do not violate uniform ultimate boundedness, even in the case where the order of the perturbations exceeds the homogeneity order of the unperturbed equations, are described. Unlike in previous works, where the results are based on the averaging method, the presence of a small parameter on the right-hand sides of the equations under examination is not assumed. Dissipativity is ensured at the expense of homogeneity orders.     

Boundedness and stability criteria for linear ordinary differential equations of the second order

We establish some correlations for solutions of ordinary differential equations and the imaginary part of the complex solution of the corresponding Riccati equation. On the basis of these correlations and the I. M. Sobol’ theorem we prove some new stability and boundedness criteria for linear equations of the second order.