Mihály Pituk

More on Poincaré’s and Perron’s Theorems for Difference Equations∗

Consider the scalar kth order linear difference equation: x(n + k) + pi(n)x(n + k - 1) + … + pk(n)x(n) = 0 where the limits qi=limn→∞Pi(n) (i=1,…,k) are finite. In this paper, we confirm the conjecture formulated recently by Elaydi. Namely, every nonzero solution x of (*) satisfies the same asymptotic relation as the fundamental solutions described earlier by Perron, ie., 𝛠= lim supn→∞ |x(n)| is equal to the modulus of one of the roots of the characteristics equation χ k + q 1χ k−1+…+qk=0. This result is a consequence of a more general theorem concerning the Poincaré difference system x(n+1)=[A+B(n]x(n), where A and B(n) (n=0,1,…) are square matrices such that ‖B(n)‖ →0 as n → ∞. As another corollary, we obtain a new limit relation for the solutions of (*).

A criterion for the exponential stability of linear difference equations

Abstract We give an affirmative answer to a question formulated by Aulbach and Van Minh by showing that the linear difference equation x n +1 = A n x n , for n ∈ N in a Banach space B is exponentially stable if and only if for every f = {fn}n=1∞ ∈ lp( N B), where I x n +1 = A n x n + f n , for n ∈ N , x 1 = 0 is bounded on N .

Boundedness and Stability for Higher Order Difference Equations

Sufficient conditions are given under which the higher order difference equation x n+1= f(x n,x n-1,...,xn-k ), n=0,1,2,... generates an order preserving discrete dynamical system with respect to the discrete exponential ordering. It is shown that under the above monotonicity assumption the boundedness of all solutions as well as the local and global stability of an equilibrium hold if and only if they hold for the much simpler first order equation x n+1=h(x n ), where h(x)=f(x,x,…,x). As an application, a second order nonlinear difference equation from macroeconomics and a discrete analogue of a model of haematopoiesis are discussed.

Asymptotic Expansions for Higher-Order Scalar Difference Equations

We give an asymptotic expansion of the solutions of higher-order Poincaré difference equation in terms of the characteristic solutions of the limiting equation. As a consequence, we obtain an asymptotic description of the solutions approaching a hyperbolic equilibrium of a higher-order nonlinear difference equation with sufficiently smooth nonlinearity. The proof is based on the inversion formula for the z -transform and the residue theorem.

Global asymptotic stability in a perturbed higher-order linear difference equation

Abstract In this note, we give a sufficient condition for the global asymptotic stability of the zero solution of the difference equation χ(n+1)= ∑ i=0 k p 1 (n)χ(n-i)+f(n,χ(n),χ(n-1),…,χ(n-1)), n=0,1,… where k and l are nonnegative integers, the coefficients pi(n) are real numbers, and the nonlinearity f satisfies the growth condition |f(n,χ0,χ1,…χl|≤q 0 ≤i≤lmax|χ i |, for n = 0,1… and χ i ϵR 0≤i≤, where q is a constant. The stability condition is formulated in terms of the fundamental solution of the unperturbed equation y(n+1)= ∑ i=0 k p i (n)y(n-1).

Asymptotic Behavior of a Poincaré Difference Equation

In this paper, we describe the asymptotic behavior of the solutions of the difference equation provided that the characteristic equation of the corresponding equation with constant coefficients has a dominant root and the sequences are in l 2 and of bounded variation. The main tools in the proof are perturbation theorems on asymptotic constancy and stability of the solutions, decomposition in the variation-of-constants formula and estimates on the complementary spaces.

Asymptotic estimates and exponential stability for higher-order monotone difference equations

Asymptotic estimates are established for higher-order scalar difference equations and inequalities the right-hand sides of which generate a monotone system with respect to the discrete exponential ordering. It is shown that in some cases the exponential estimates can be replaced with a more precise limit relation. As corollaries, a generalization of discrete Halanay-type inequalities and explicit sufficient conditions for the global exponential stability of the zero solution are given.

The limits of the solutions of a nonautonomous linear delay difference equation

Abstract Consider the system of linear delay difference equations where the coefficients A j ( n ) are square matrices and k j and l j are nonnegative integers. In this note, we show that if the coefficients are “small”, then every solution of the above equation tends to a constant vector as n → ∞ and the value of the limit can be characterized by a special solution of the matrix equation and the initial conditions.

Nonoscillatory solutions of a second-order difference equation of Poincaré type☆

Abstract Poincare’s classical theorem about the convergence of ratios of successive values of solutions applies if the characteristic roots of the associated limiting equation are simple and have different moduli. In this work, it is shown that for the nonoscillatory solutions the conclusion of Poincare’s theorem is also true in the case where the limiting equation has a double positive characteristic root.

Convergence to equilibria in scalar nonquasimonotone functional differential equations

Abstract 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.