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Showing papers on "Asymptotology published in 1970"


Journal ArticleDOI
TL;DR: In this paper, it was shown that for a nonlinear one-dimensional differential delay equation a(t) = F;(t, x1(.)) (DDE) one can frequently determine (almost by inspection) if the 0 solution is asymptotically stable and give a region of attraction.

146 citations






Journal ArticleDOI
TL;DR: In this paper, it was shown that (1.3) is necessary and sufficient for the boundedness of all solutions of the Ricatti equation with initial conditions in the first quadrant (third quadrant) will intersect the positive (negative) x-axis.
Abstract: is necessary and sufficient for the global asymptotic stability of the equilibrium point (0, 0) of (1.2). The idea of the proof is to show that (1.3) is necessary and sufficient for the boundedness of all solutions of (1.2). In doing this the crucial step is determining whether a solution with initial conditions in the first quadrant (third quadrant) will intersect the positive (negative) x-axis. In this paper we investigate global asymptotic stability of the origin (0, 0) of (1.2) for the case when a ? 1. In addition to the above idea, the main feature of our approach is to reduce (1.2) to a first order generalized Ricatti equation and then determine conditions under which it has and does not have positive solutions for large values of the independent variable. Using a different method, Willet and Wong [4] have considered this same problem. Their approach is more general and (1.2) is considered as a special case. If a _ 1, they obtain a necessary and sufficient condition (which is the same as Burton's for cx 1, however, it will be shown here that their condition is included in one of those obtained here.

25 citations




Journal ArticleDOI
TL;DR: In this article, the authors formulated the definition of the asymptotic expansion of a generalized function depending on a parameter and proved a number of theorems about the properties of such expansions and operations.
Abstract: The definition is formulated of the asymptotic expansion of a generalized function depending on a parameter. A number of theorems are proved about the properties of asymptotic expansions and operations on them, in particular, theorems on differentiation and integration. For generalized functions of the formf (x)eixt,f (x) ɛS', t → ±∞ the relation is investigated between the singularity carrierf and the carrier of coefficient functionals.

8 citations


Journal ArticleDOI
TL;DR: In this article, the majorant function that is used in connection with the comparison technique is usually assumed to be non-increasing in the dependent variable, however, properties of the asymptotic manifold are discussed here under the opposite monotonicity assumption.
Abstract: The asymptotic properties of the solutions of a linear homogeneous system of differential equations determine, under suitable restrictions, the asymptotic properties of a set of solutions of a nonlinear perturbation of this linear equation. The comparison principle is used here to generate an asymptotic manifold of the perturbed equation. The majorant function that is used in connection with the comparison technique is usually assumed to be nondecreasing in the dependent variable. However, properties of the asymptotic manifold are discussed here under the opposite monotonicity assumption, namely, that the majorant function is nonincreasing in the dependent variable. This type of majorant, function arises, for example, in certain gravitation problems. The main result on the structure of asymptotic manifolds which have an asymptotic uniformity is that solutions close to the manifold are either in the manifold or do not exist in the future.

5 citations


01 Aug 1970
TL;DR: In this paper, a sequence of general experiments is considered over a k-dimensional parameter, and conditions of local asymptotic normality (LAN) of the families of distributions are proved under the sole condition of continuity of Fisher's information.
Abstract: : A sequence of general experiments is considered over a k-dimensional parameter. Under conditions of local asymptotic normality (LAN) of the families of distributions, we prove that, from the point of view of the local asymptotic minimax, there is a lower bound, which may be obtained only if the estimator has certain linear relation to the derivative of the likelihood function. This entails asymptotic normality with Fisher's variance. Conditions LAN are proved under the sole condition of continuity of Fisher's information. (Author)