A simple proof of the Grobman–Hartman theorem for nonuniformly hyperbolic flows

We give a simple and direct proof of the Grobman–Hartman theorem for nonautonomous differential equations obtained from perturbing a nonuniform exponential dichotomy. In particular, we do not need to pass through discrete time and obtain the result as a consequence of a corresponding result for maps. To the best of our knowledge, this is the first direct approach for nonuniform exponential dichotomies. We also show that the conjugacies are continuous in time and Hölder continuous in space. In addition, we describe the dependence of the conjugacies on the perturbation, and we obtain a reversibility result for the conjugacies of reversible differential equations. We emphasize that the additional work required to consider nonuniform exponential dichotomies is substantial.     

The Morse–Witten complex via dynamical systems

Given a smooth closed manifold M, the Morse–Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in Weber [Der Morse–Witten Komplex, Diploma Thesis, TU Berlin, 1993] and is based on tools from hyperbolic dynamical systems. For instance, we apply the Grobman–Hartman theorem and the λ-lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.     

Hartman–Wintner’s theorem and its applications

A theorem of Hartman–Wintner enables us to find a nonzero homogeneous polynomial approximate to the difference of two solutions of an elliptic partial differential equation of second order. This theorem plays a crucial role in the study of an umbilical point on each of a special Weingarten surface, a surface with constant anisotropic mean curvature and a Willmore surface. In the present paper, we will survey the roles of Hartman–Wintner’s theorem on these surfaces.     

Nonlinear equations with essentially infinite-dimensional differential operators

We consider nonlinear differential equations and boundary-value problems with essentially infinite-dimensional operators (of the Laplace–Lévy type). An analog of the Picard theorem is proved.     

常微分方程双曲不动点结构稳定性的证明（英文）

In ordinary differential equations, structural stability of hyperbolic fixed points is a classical result, but the proof of this result in [2] has same small mistake. In this paper,we will correct the above mistake by using the Hartman theorem and its idea.     

Some remarks on Nagumo’s theorem

We provide a simpler proof for a recent generalization of Nagumo’s uniqueness theorem by A. Constantin: On Nagumo’s theorem. Proc. Japan Acad., Ser. A 86 (2010), 41–44, for the differential equation x′ = f(t, x), x(0) = 0 and we show that not only is the solution unique but the Picard successive approximations converge to the unique solution. The proof is based on an approach that was developed in Z. S. Athanassov: Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Jap. 35 (1990), 351–367. Some classical existence and uniqueness results for initial-value problems for ordinary differential equations are particular cases of our result.     

On generalized solutions of differential equations with operator coefficients

We prove a theorem on the smoothness of generalized solutions of differential equations with operator coefficients. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 715–720, May, 2006.     

Class and rank of differential modules

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class – a substitute for the length of a free complex – and on the rank of a differential module in terms of invariants of its homology. These results specialize to basic theorems in commutative algebra and algebraic topology. One instance is a common generalization of the equicharacteristic case of the New Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning complexes over commutative noetherian rings, and of a theorem of G. Carlsson on differential graded modules over graded polynomial rings.     

Operator approach to the Cauchy-Kovalevskaya theorem

For operator differential equations in a Banach space, we present the conditions for initial data which are necessary and sufficient for the Cauchy problem to have a solution in the class of analytic, entire, or exponential-type entire vector functions. In the case where an operator differential equation is a system of partial differential equations, the sufficient condition obtained coincides with the well-known Cauchy-Kovalevskaya theorem on the solvability of the Cauchy problem in the class of analytic functions. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 7–12, April–June, 1998.     

The Hodge–de Rham decomposition theorem and an application to a partial differential equation

The Hodge–de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a novel way.     

Reducibility of a linear system of differential equations with odd almost periodic coefficients

A theorem on the Kolmogorov reducibility of a system of ordinary differential equations with odd almost periodic coefficients is proved. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 109–119, July, 1996.     

Some remarks on linear functional differential inequalities of hyperbolic type

We prove that, for the validity of a certain theorem on differential inequalities for a linear functional differential equation of hyperbolic type {fx327-01} with a negative linear operator {fx327-02}, it is necessary that ℓ be an (a, c)-Volterra operator. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 283–292, February, 2008.     

On the Independence of Hartman Sequences

 Given and , we define by setting if and only if , where denotes the fractional part of α, i.e. α is considered as an element of the torus . If the topological boundary of A has Haar measure 0, then is called a Hartman sequence, which is a generalisation of Kronecker and Beatty sequences. In this article we answer a question of Winkler by showing explicitly for which sets , and vectors , we have . The main tool of the proof is Weyl’s theorem on uniform distribution.     

Homoclinic and Periodic Orbits Arising Near the Heteroclinic Cycle Connecting Saddle-focus and Saddle Under Reversible Condition

In this paper, we study the dynamical behavior for a 4-dimensional reversible system near its heteroclinic loop connecting a saddle-focus and a saddle. The existence of infinitely many reversible 1-homoclinic orbits to the saddle and 2-homoclinic orbits to the saddle-focus is shown. And it is also proved that, corresponding to each 1-homoclinic （resp. 2-homoclinic） orbit F, there is a spiral segment such that the associated orbits starting from the segment are all reversible 1-periodic （resp. 2-periodic） and accumulate onto F. Moreover, each 2-homoclinic orbit may be also accumulated by a sequence of reversible 4-homoclinic orbits.     

A finitely additive version of kolmogorov’s law of the iterated logarithm

In their paperSome finitely additive probability (to appear in Ann. Probability), Roger A. Purves and William D. Sudderth introduced the measurable strategy idea. In this paper, we first generalize the measurable strategy idea to the more general sigma-fields of subsets ofX and prove an important theorem. Then, based on this theorem, we state and prove a finitely additive version of Kolmogorov’s law of the iterated logarithm and a finitely additive version of Hartman and Wintner’s law of the iterated logarithm in a finitely additive setting. This research was written with the partial support of the U.S. Army Grant DA-ARO-D-31-124-70-G-102.     

On the Independence of Hartman Sequences

 Given and , we define by setting if and only if , where denotes the fractional part of α, i.e. α is considered as an element of the torus . If the topological boundary of A has Haar measure 0, then is called a Hartman sequence, which is a generalisation of Kronecker and Beatty sequences. In this article we answer a question of Winkler by showing explicitly for which sets , and vectors , we have . The main tool of the proof is Weyl’s theorem on uniform distribution. Received 3 November 2000; in final form 24 April 2001     

Dynamic realization games in newsvendor inventory centralization

Consider a set N of n (> 1) stores with single-item and single-period nondeterministic demands like in a classic newsvendor setting with holding and penalty costs only. Assume a risk-pooling single-warehouse centralized inventory ordering option. Allocation of costs in the centralized inventory ordering corresponds to modelling it as a cooperative cost game whose players are the stores. It has been shown that when holding and penalty costs are identical for all subsets of stores, the game based on optimal expected costs has a non empty core (Hartman et al. 2000, Games Econ Behav 31:26–49; Muller et al. 2002, Games Econ Behav 38:118–126). In this paper we examine a related inventory centralization game based on demand realizations that has, in general, an empty core even with identical penalty and holding costs (Hartman and Dror 2005, IIE Trans Scheduling Logistics 37:93–107). We propose a repeated cost allocation scheme for dynamic realization games based on allocation processes introduced by Lehrer (2002a, Int J Game Theor 31:341–351). We prove that the cost subsequences of the dynamic realization game process, based on Lehrer’s rules, converge almost surely to either a least square value or the core of the expected game. We extend the above results to more general dynamic cost games and relax the independence hypothesis of the sequence of players’ demands at different stages.     

On the Existence of Periodic Solutions for a Kind of Second Order Neutral Functional Differential Equation

By means of the continuation theorem of coincidence degree theory, some new results on the non-existence, existence and unique existence of periodic solutions for a kind of second order neutral functional differential equation are obtained.     

Resonance elliptic variational inequalities with discontinuous nonlinearities of linear growth

We consider resonance elliptic variational inequalities with second-order differential operators and discontinuous nonlinearities of linear growth. The theorem on existence of a strong solution is proved. The initial-value problem is reduced to the problem of existence of a fixed point for a compact multivalued mapping and then the existence of this point is established by the Leray–Schauder method.