Hideaki Matsunaga

Global attractivity for a nonlinear difference equation with variable delay

Abstract In this paper, we give sufficient conditions under which every solution of the nonlinear difference equation with variable delay x ( n + 1) − x ( n ) + p n f ( x ( g ( n ))) = 0, n = 0, 1, 2, … tends to zero as n → ∞. Here, p n is a nonnegative sequence, f : R → R is a continuous function with xf ( x ) > 0 if x ≠ 0, and g : N → Z is nondecreasing and satisfies g ( n ) ≤ n for n ≥ 0 and lim n →∞ g ( n ) = ∞.

Exact stability criteria for delay differential and difference equations

Abstract For linear delay differential and difference equations with one coefficient matrix A , we give explicit necessary and sufficient conditions for the asymptotic stability of the equations in terms of tr A , det A and the delay parameter. We also summarize some exact stability criteria for certain classes of linear delay differential and difference equations.

A note on asymptotic stability of delay difference systems

For the linear delay difference system, where is a real constant matrix and is a nonnegative integer, we present an explicit necessary and sufficient condition for the asymptotic stability of the zero solution of this system in terms of,, and the delay.

DELAY-DEPENDENT AND DELAY-INDEPENDENT STABILITY CRITERIA FOR A DELAY DIFFERENTIAL SYSTEM

For a linear delay differential system with two coefficients and one delay, we establish some necessary and sufficient conditions on the asymptotic stability of the zero solution, which are composed of delay-dependent and delay-independent stability criteria. On the former criterion, the range of the delay is explicitly given.

Formal adjoint equations and asymptotic formula for solutions of Volterra difference equations with infinite delay

For linear Volterra difference equations with infinite delay, we obtain an explicit asymptotic representation formula of solutions by means of the formal adjoint equation associated with a certain bilinear form.

Classification of global behavior of a system of rational difference equations

Abstract This paper deals with a system of rational difference equations x n + 1 = a y n + b c y n + d , y n + 1 = a x n + b c x n + d , n = 0 , 1 , 2 , … , where a , b , c , d are real numbers with c ≠ 0 and a d − b c ≠ 0 . We establish a representation formula of solutions of the system and classify global behavior of solutions when no initial values belong to the forbidden set of the system.

Oscillation Criteria for a Difference System with Two Delays

The oscillation of all solutions of a linear autonomous difference system with two delays is studied. Explicit necessary and sufficient conditions in terms of the coefficient matrix and the delays are established, which are some extensions of the previous results. As an application, we can completely classify the oscillation and the asymptotic stability of a delay difference system.

Asymptotic Representation of Solutions of Linear Autonomous Difference Equations

We establish an explicit asymptotic representation formula for solutions of linear autonomous difference equations with infinite delay. As an application, we investigate the limit of solutions of a certain delay difference equation in the critical case where the equation loses its asymptotic stability.

Effect of off-diagonal delay on the asymptotic stability for an integro-differential system

Abstract Our concern is to solve the stability problem for a linear integro-differential system with distributed delay in the off-diagonal terms. Some new necessary and sufficient conditions are established for the zero solution of the system to be asymptotically stable. The proof of our main theorem is given by a careful analysis of the locations of roots of the associated characteristic equation.