On index-2 linear implicit difference equations

This paper deals with an index-2 notion for linear implicit difference equations (LIDEs) and with the solvability of initial value problems (IVPs) for index-2 LIDEs. Besides, the cocycle property as well as the multiplicative ergodic theorem of Oseledets type are also proved.     

An asymptotic result and a stability criterion for linear nonautonomous delay difference equations

Linear delay difference equations with variable coefficients and constant delays are considered. By the use of an appropriate solution of the so called generalized characteristic equation, an asymptotic result is obtained and a stability criterion is established.     

Initial-value problems for linear partial difference equations

The method presented in [4] for the solution of linear difference equations in a single variable is extended to some equations in two variables. Every linear combination of a given functionf and of its partial differences can be obtained by the discrete convolution product off by a suitable functionl (which depends on the considered linear combination), and we want to solve in a convolutional form difference equations in the whole plane. However, the convolution of two functions may not be possible if their supports contain half straight lines with opposite directions. To avoid this, we take four sets of functions corresponding to the quadrants such thatl belong to every set, every set endowed with the convolution and with the usual addition is a ring, and there is an inverse ofl in each of the four rings. This is attained by taking, for each ring, a set of functions whose supports belong to suitable cones. After choosing such rings, a very natural initial-value first-order Cauchy Problem (in partial differences) is reduced to a convolutional form. This is done either by a direct method or by introducing the forward difference functions _i f(i=1,2) in a general way depending on the shape of the support off so that Laplace-like formulas with initial and final values) hold. Applications to difference equations in the whole plane and to partial differential problems are made.     

Eigenvalue comparisons for second order difference equations with Neumann boundary conditions

The boundary value problem for second order difference equation

     

A Laplace-like method for solving linear difference equations

This paper is concerned with the problem of solving linear difference equations of ordern with constant coefficients and with given initial conditions in which the variable runs not only through the integers but over . The main idea is the introduction of a suitable commutative ring of functions with discrete convolution as multiplication rule which works, although it is not a field. The existence of inverses is studied and, after the introduction of suitable functions, the problem is reduced by means of a Laplace-like relation to an algebraic equation. Examples of application are given. Finally some remarks make the connection with the Operational Calculus of Mikusinski and with the Operational Calculus of Fenyö. The advantages of this method lie in the fact that it is applicable to functions others than the step functions, in its simplicity from the theoretical point of view, in its usefulness even when computation is required and in its formal similarity to the classical treatment of linear differential equations with constant coefficients.     

Eigenvalue comparisons for a class of boundary value problems of second order difference equations

We consider the boundary value problem for second order difference equation

     

Applying carvalho's method to find periodic solutions of difference equations

In this paper we apply the method initially developed in [1] for differential-difference equations, to the case of difference equations, in order to find 2 and 3-periodic solutions of some equations that often appear in the literatures as are for instance the case of Applications 2,5 which are examples of population growth models, and Application 4, which is a standard example of nonlinear higher order scalar difference equation depending on two parameters (see, Kocik and Ladas [3]).     

Periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay

The existence of periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay is obtained by using stability properties of a bounded solution.     

On asymptotic equivalence of linear and quasilinear difference equations

The asymptotic equivalence of systems of difference equations of linear and quasilinear type is investigated. The first result on the asymptotic equivalence of linear systems is a discrete analog of an improved version of the Levinson's well-known theorem on asymptotic equivalence of linear differential equations, while the second one providing conditions for asymptotic equivalence of linear and quasilinear systems is related to that of Yakubovich in differential equations case.     

Homogeneous polynomials as Lyapunov functions in the stability research of solutions of difference equations

The linear autonomous system of difference equations x(n+1)=Ax(n) is considered, where is a real nonsingular k×k matrix. In this paper it has been proved that if W(x) is any homogeneous polynomial of m-th degree in x, then there exists a unique homogeneous polynomial V(x) of m-th degree such that ΔV=V(Ax)-V(x)=W(x) if and only if where are the eigenvalues of the matrix A. The theorem on the instability has also been proved.     

A link between the Perron-Frobenius theorem and Perron’s theorem for difference equations

Let A_n,n∈N, be a sequence of k×k matrices which converge to a matrix A as n→∞. It is shown that if x_n,n∈N, is a sequence of nonnegative nonzero vectors such that

     

On the computation of parameter derivatives of solutions of linear difference equations

A method is given to compute the parameter derivatives of recessive solutions of second-order inhomogeneous linear difference equations. The case of difference equations in which all solutions have the same rate of growth is also discussed.     

Self-adjoint extensions for second-order symmetric linear difference equations

In this paper, self-adjoint extensions for second-order symmetric linear difference equations with real coefficients are studied. By applying the Glazman-Krein-Naimark theory for Hermitian subspaces, both self-adjoint subspace extensions and self-adjoint operator extensions of the corresponding minimal subspaces are completely characterized in terms of boundary conditions, where the two endpoints may be regular or singular.     

Estimation for the region of attraction of nonlinear difference equations

Summary. We consider nonlinear systems of difference equations with and , where A is any matrix, is a continuous vector-function, and is a numeral parameter. The spectrum of A belongs to the unit circle . We give the estimations for the region of attraction and the speed of convergence solutions to the zero solution of the systems. We indicate a set M such that for solutions of the system with parameter the limit is true. Received April 7, 2000 / Revised version received January 8, 2001 / Published online November 15, 2001     

On the exponential dichotomy of linear difference equations

We consider a system of linear difference equationsx ⁿ⁺¹ =A (n)xⁿ in anm-dimensional real or complex spaceVsum with detA(n) = 0 for some or alln εZ. We study the exponential dichotomy of this system and prove that if the sequence {A(n)} is Poisson stable or recurrent, then the exponential dichotomy on the semiaxis implies the exponential dichotomy on the entire axis. If the sequence {A (n)} is almost periodic and the system has exponential dichotomy on the finite interval {k, ...,k +T},k εZ, with sufficiently largeT, then the system is exponentially dichotomous onZ.     

On global asymptotic stability of nonlinear higher-order difference equations

In this paper, we generalize the main theorem of Liz and Ferreiro [E. Liz, J.B. Ferreiro, A note on the global stability of generalized difference equations, Appl. Math. Lett. 15 (2002) 655–659] and some other global stability results for nonautonomous higher-order difference equations to the case when contraction-type steps are incorporated together with the steps when the difference sequence can increase.