Tadayuki Hara

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 ) = ∞.

A necessary and sufficient condition for oscillation of the generalized Liénard equation

SummaryIn this paper we consider the Liénard system x′= y − F(x), y′= − g(x) and give a necessary and sufficient condition under which all solutions oscillate.

Nonlinear oscillations of second order differential equations of Euler type

We consider the nonlinear equation t2x′′ + g(x) = 0, where g(x) satisfies xg(x) > 0 for x 6= 0, but is not assumed to be sublinear or superlinear. We discuss whether all nontrivial solutions of the equation are oscillatory or nonoscillatory. Our results can be applied even to the case g(x) x → 1 4 as |x| → ∞, which is most difficult.

Continuation results for differential equations by two Liapunov functions

SummaryIn this paper we give conditions under which all solutions of a system of differential equations are continuable in the future. We use two Liapunov functions which are not radially unbounded for fixed t. The results are more flexible than the work done by Conti and Strauss. Our theorems are applied to the Liénard equation without the assumption that xg(x)>0 if x ≠ 0.

On the uniform ultimate boundedness of the solutions of certain third order differential equations

in which a, b, c, p, f, g and h are continuous functions depending only on the arguments shown and f,, g, and h’(x) exist and are continuous for all x and y. Equations (1.1) and (1.2) have been the object of a good bit of research over the past several years (cf. [2-IO]). Many of these authors considered the boundedness of solutions of (1.1) and (1.2) in the case a(t) = b(t) = c(t) = 1, assuming one of the following conditions on p(t) and p(t, x, y, z):

STABILITY OF A 2-DIMENSIONAL NEURAL NETWORK WITH TIME DELAYS

A 2-dimensional neural network with time delayed connections between neurons is considered. Based on the construction of Liapunov functionals, we obtain sufficient criteria to ensure local and global asymptotic stability of the equilibrium of the neural network.

An application of the Hurwitz theorem to the root analysis of the characteristic equation

In this work we show that a classical result of A. Hurwitz is still very effective in studying the root analysis of the characteristic equation for a linear functional differential equation. A conjecture was made by Funakubo et al. (2006) [3] regarding the asymptotic stability condition of the zero solution of a linear integro-differential equation of Volterra type. We applied the Hurwitz theorem to the characteristic equation in question and showed the existence of a root with positive real part and solved the conjecture. The Hurwitz theorem is expected to work well for the root analysis in critical cases.

On the boundedness of solutions of perturbed linear systems

Y’ = A (0 Y + W), W,) where x, y, g, h are n-vectors, A(t) is a continuous n X n matrix for t > 0, g(t, y) is continuous for 12 0, y E R”, and h(t) is continuous for t > 0. Strauss and Yorke [8,9] have studied the perturbing uniform asymptotically stable systems, and Furuno and Hara [4] have shown some more detailed results. Bernfeld [ 1 ] and Lovelady [6] have studied the perturbing uniformly bounded and uniformly ultimately bounded systems. On the other hand Coppel [2,3] has studied the boundedness of solutions of (PL,) from the point of view of the dichotomy theory. Lovelady [5] referred to the connections between the perturbation problem and the dichotomy theory. Here we shall give some further results on the boundedness of solutions of perturbed linear systems. This paper is much influenced by Strauss and Yorke [9]. The purpose of this paper is to prove theorems on the perturbation from (L) to (PL) and (PL,) of uniform boundedness (Theorem 3.1), uniform boundedness and ultimate boundedness (Theorem 4.1), and uniform boundedness and uniform ultimate boundedness (Theorem 5.1). Let G, be the class of functions g(t, y) such that I] g(t, r)l] 0 and II y]] > R, where j: y(t) dt r,, > 0 and

Non-continuation results for differential equations by several Liapunov functions☆

Abstract Non-continuability of solutions of ordinary differential equations on ¦t0, ∞) is investigated by use of several Liapunov functions (Theorem 2.1). Some examples are given in Section 3. Necessary and sufficient conditions for continuation of solutions of a system x′ = y ψ(x) , y′ = −g(x)h(y) are given.