Regular variation on measure chains

In this paper we show how the recently introduced concept of regular variation on time scales (or measure chains) is related to a Karamata type definition. We also present characterization theorems and an embedding theorem for regularly varying functions defined on suitable subsets of reals. We demonstrate that for a “reasonable” theory of regular variation on time scales, certain additional condition on a graininess is needed, which cannot be omitted. We establish a number of elementary properties of regularly varying functions. As an application, we study the asymptotic properties of solution to second order dynamic equations.     

The exact analytic solutions of a nonlinear differential iterative equation

This paper is concerned with a second-order nonlinear iterated differential equation of the form _c0x^″(z)+_c1x^′(z)+_c2x(z)=x(p(z)+bx(z))+h(z)$c_0{x}^{″}\left(z\right)+c_1{x}^{\prime }\left(z\right)+c_{2}x\left(z\right)=x(p\left(z\right)+bx\left(z\right))+h\left(z\right)$. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only the general case |β|≠1$\left|\beta \right|\ne 1$, but also the critical case |β|=1$\left|\beta \right|=1$, especially when β$\beta$ is a root of unity. Furthermore, the exact and explicit analytic solution of the original equation is investigated for the first time. Such equations are important in both applications and the theory of iterations.     

Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales

In this paper, by using critical point theory, we establish the existence of weak solutions of the two-point boundary value problem for a second-order dynamic equation on an arbitrary time scale T$\mathbb{T}$, so that the well known case of differential dynamic systems (T=R)$(\mathbb{T}=\mathbb{R})$ and the recently developed case of discrete dynamic systems (T=Z)$(\mathbb{T}=\mathbb{Z})$ are unified. To the best of our knowledge, this is the first time that boundary value problems of dynamic equations on time scales have been dealt with by using critical point theory.     

A time-varying analogue of Jensen's formula

A generalization of Jensen's formula to operators associated with time-varying systems is presented. Poles and zeros of the transfer function are replaced by elements of the spectrum of two weighted shifts, one associated with the system, the other with its inverse. Exponential dichotomies, together with two operator factorizations play a key role.This work was supported in part by NSF grant ECS-9800057.     

Nonlinear perturbations of nonuniform exponential dichotomy on measure chains

This paper focuses on nonlinear perturbations of flows in Banach spaces, corresponding to a nonautonomous dynamical system on measure chains admitting a nonuniform exponential dichotomy. We first define the nonuniform exponential dichotomy of linear nonuniformly hyperbolic systems on measure chains, then establish a new version of the Grobman-Hartman theorem for nonuniformly hyperbolic dynamics on measure chains with the help of nonuniform exponential dichotomies. Moreover, we also construct stable invariant manifolds for sufficiently small nonlinear perturbations of a nonuniform exponential dichotomy. In particular, it is shown that the stable invariant manifolds are Lipschitz in the initial values provided that the nonlinear perturbation is a sufficiently small Lipschitz perturbation.     

A definition of spectrum for differential equations on finite time

Hyperbolicity of an autonomous rest point is characterised by its linearization not having eigenvalues on the imaginary axis. More generally, hyperbolicity of any solution which exists for all times can be defined by means of Lyapunov exponents or exponential dichotomies. We go one step further and introduce a meaningful notion of hyperbolicity for linear systems which are defined for finite time only, i.e. on a compact time interval. Hyperbolicity now describes the transient dynamics on that interval. In this framework, we provide a definition of finite-time spectrum, study its relations with classical concepts, and prove an analogue of the Sacker-Sell spectral theorem: For a d-dimensional system the spectrum is non-empty and consists of at most d disjoint (and often compact) intervals. An example illustrates that the corresponding spectral manifolds may not be unique, which in turn leads to several challenging questions.     

A new criterion for global robust stability of interval delayed neural networks

A novel criterion for the global robust stability of Hopfield-type interval neural networks with delay is presented. An example showing the effectiveness of the present criterion is given.     

Periodic solutions for dynamic equations on time scales

Massera type criteria are established for the existence of periodic solutions of linear and nonlinear dynamic equations on time scales. Some interesting properties of the exponential function on time scales are presented. Furthermore, a sufficient condition guaranteeing the boundedness of the solutions of the equation is presented.     

One-dimensional p-Laplacian with a strong singular indefinite weight, I. Eigenvalue

In this paper, we prove the existence of eigenvalues for the problem

     

Existence of positive solutions for nonlinear three-point problems on time scales

In this paper, by using fixed point theorems in cones, we study the existence of at least one, two and three positive solutions of a nonlinear second-order three-point boundary value problem for dynamic equations on time scales. As an application, we also give some examples to demonstrate our results.     

Stability radii of positive linear Volterra-Stieltjes equations

We study stability radii of linear Volterra-Stieltjes equations under multi-perturbations and affine perturbations. A lower and upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer functions. Then, the class of positive linear Volterra-Stieltjes equations is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices. As direct consequences of the obtained results, we get some results on robust stability of positive linear integro-differential equations and of positive linear functional differential equations. To the best of our knowledge, most of the results of this paper are new.     

A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation

We obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution.We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct. Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point.     

On the stability of the translation equation and dynamical systems

In this paper we prove the stability of the functional equation F(s,F(t,x))=F(s+t,x) in the class of functions F:R×I→I, which are continuous with respect to each variable, and where I⊂R is a real interval. We also discuss the stability in the sense of Hyers-Ulam of dynamical systems on I. We show some properties of δ-approximate solutions of the translation equation on a real interval.