On forced fast oscillations for delay differential equations on compact manifolds

We prove an existence result for forced oscillations of delay differential equations on compact manifolds with nonzero Euler-Poincaré characteristic. When the period is smaller than the delay we need the asymptotic fixed point index theory for C¹ maps due to Eells and Fournier, and Nussbaum.     

Stability and fixed points of point-dissipative systems

It is known that if T: X → X is completely continuous or if there exists an n₀ > 0 such that Tⁿ⁰ is completely continuous, then T point dissipative implies that there is a maximal compact invariant set which is uniformly asymptotically stable, attracts bounded sets, and has a fixed point (see Billotti and LaSalle [Bull. Amer. Math. Soc.6 1971]). The result is used, for example, in studying retarded functional differential equations, or parabolic partial differential equations. This result has been extended by Hale and Lopes [J. Differential Equations13 1973]. They get the result that if T is an α-contraction and compact dissipative then there is a maximal compact invariant set which is uniformly asymptotically stable, attracts neighborhoods of compact sets, and has a fixed point. The above result requires the stronger assumption of compact dissipative. The principal result of this paper is to get similar results under the weaker assumption of point dissipative. To do this we must make additional assumptions. We will show these assumptions are naturally satisfied by stable neutral functional differential equations and retarded functional differential equations with infinite delay. The result has applications to many other dynamical systems, of course.     

Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems

We prove the existence of periodic solutions in a compact attractor of (R⁺)ⁿ for the Kolmogorov system x′_i = x_if_i(t, x₁, …, x_n), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.     

A condition on delay for differential equations with discrete state-dependent delay

Parabolic differential equations with discrete state-dependent delay are studied. The approach, based on an additional condition on the delay function introduced in [A.V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. 70 (11) (2009) 3978–3986] is developed. We propose and study an analogue of the condition which is sufficient for the well-posedness of the corresponding initial value problem on the whole space of continuous functions C. The dynamical system is constructed in C and the existence of a compact global attractor is proved.     

Asymptotic estimates and blow-up theory for critical equations involving the p-Laplacian

We prove the SH₁^p—theory for critical equations involving the p-Laplace operator on compact manifolds. We also prove pointwise estimates for these equations.     

Attractivity properties of α-contractions

It is known that if T:X → X is completely continuous where X is a Banach space, then point dissipative and compact dissipative are equivalent, and imply the existence of a maximal compact invariant set which is uniformly asymptotically stable and attracts bounded sets uniformly. If T is an α-contraction, it is not known whether point dissipative and compact dissipative are equivalent. However, it is known that if T is an α-contraction and compact dissipative, then there exists a maximal compact invariant set which is uniformly asymptotically stable and attracts a neighborhood of any compact set uniformly. In this paper we show that for most practical examples which give rise to α-contraction, point dissipative and compact dissipative are equivalent. For example, we show this is true for stable neutral functional differential equations, retarded functional differential equations of infinite delay, and strongly damped nonlinear wave equations. We conjecture that this should be true for almost any practical application which gives rise to an α-contraction.     

Method of limiting equations for the stability analysis of equations with infinite delay in the Carathéodory conditions: I

We consider systems of nonautonomous nonlinear differential equations with infinite delay. We introduce Carathéodory type conditions for the right-hand side in an equation, which permit one, on the one hand, to cover a fairly broad class of systems and, on the other hand, include the right-hand side in a compact function space and construct the so-called limiting equations. In the investigation, we use the construction of admissible spaces with fading memory, which permits one to obtain constructive results for the class of equations under study.     

The attractor of the delayed functional-differential diffusion equation

We consider a class of functional-differential diffusion equations with delay and spatial rotation that arise in nonlinear optics. The existence of a compact global attractor is established for the corresponding discrete dynamic systems. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 147–152.     

Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities

In this work, we study the existence of bounded and almost automorphic solutions for evolution equations in Banach spaces. We suppose that the linear part is the infinitesimal generator of a compact C₀-semigroup of bounded linear operators and the nonlinear part is an almost automorphic function with respect to the second argument. We give sufficient conditions ensuring the existence of an almost automorphic solution when there is at least one bounded solution on R⁺. We use the subvariant functional method to show that every K-minimizing mild solution is compact almost automorphic. Applications are provided for both heat and wave equations with nonlinearities in several functional spaces.     

FBSDEs with time delayed generators: L-solutions, differentiability, representation formulas and path regularity

We extend the work of Delong and Imkeller (2010) [6] and [7] concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general L^p-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in L^p. We introduce decoupled systems of SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic L²-path regularity to delay FBSDEs.     

Almost automorphic solutions to logistic equations with discrete and continuous delay

We obtain sufficient conditions for the existence and uniqueness of a positive compact almost automorphic solution to a logistic equation with discrete and continuous delay. Moreover, we provide a counterexample to some results in literature which deal with the uniqueness of almost periodic solutions to logistic type equations.     

Integrated semigroups and linear partial differential equations with delay

We study existence and uniqueness of solutions for linear partial differential equations with delay in L^p-spaces using an approach of Batkai and Piazzera and a recent perturbation result for integrated semigroups. We apply our result to an equation with delay in the highest-order derivatives.     

Semigroup Properties and the Crandall Liggett Approximation for a Class of Differential Equations with State-Dependent Delays

We present an approach for the resolution of a class of differential equations with state-dependent delays by the theory of strongly continuous nonlinear semigroups. We show that this class determines a strongly continuous semigroup in a closed subset of C^0, 1. We characterize the infinitesimal generator of this semigroup through its domain. Finally, an approximation of the Crandall-Liggett type for the semigroup is obtained in a dense subset of (C, ‖·‖_∞). As far as we know this approach is new in the context of state-dependent delay equations while it is classical in the case of constant delay differential equations.     

Bounded mild solutions of perturbed Volterra equations with infinite delay

We study existence and regularity of bounded mild solutions on the real line to perturbed integral equations with infinite delay in the space of almost periodic functions (in the Bohr sense), the space of compact almost automorphic functions, the space of almost automorphic functions and the space of asymptotically almost automorphic functions.     

Negatively invariant sets of compact maps and an extension of a theorem of Cartwright

Let T be a C¹ map from an open subset of a separable Hilbert space into the Hilbert space, and Γ a negatively invariant compact set, that is T(Γ) ? Γ. Suppose the derivative of T for x?Γ is a uniform contraction on a subspace of finite codimension. Then the topological dimension of Γ is finite. This result may be used to show that for certain delay differential equations and partial differential equations, any almost periodic solution has only finitely many rationally independent frequencies, thus extending results of Cartwright for ODE's.     

On the ratio of different norms of caloric functions and related eigenfunctions of an integral operator

We investigate the ratio of L 1 and L 2 norms of the Cauchy problem solutions of heat equations with compact support initial data.The related asymptotic behavior of the eigenvalues and eigenfunctions of certain integral operators is obtained.     

Existence of some neutral partial differential equations with infinite delay

This study intends to investigate a class of quasi-linear partial neutral functional differential equations with infinite delay. We assume that the linear part generates an analytic compact semigroup and the nonlinear part satisfies certain conditions. A sufficient condition is given to ensure the existence of mild and classical solutions. Finally, an example is given to illustrate our abstract results.     

PERIODICITY IN FUNCTIONAL DIFFERENTIAL EQUATIONS

ＰＥＲＩＯＤＩＣＩＴＹＩＮＦＵＮＣＴＩＯＮＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＺｈａｎｇＳｈｕｎｉａｎ(张书年)（ＳｈａｎｇｈａｉＪｉａｏｔｏｎｇＵｎｉｖｅｒｓｉｔｙ，上海交通大学，邮编：２００２４０）Ａｂｓｔｒａｃｔ：Ａｃｒｉｔｅｒｉｏｎｆ...     

Nonlocal Cauchy problem for delay integrodifferential equations of Sobolev type in Banach spaces

In this paper, we prove the existence of mild and strong solutions of nonlinear time varying delay integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. The results are obtained by using the theory of compact semigroups and Schaefer's fixed-point theorem.     

The first-approximation stability of a nonstationary system with delay

We study the exponential stability of a nonlinear system of differential equations with constant delay such that the right-hand side of one of its subsystems contains the multiplier e ^t . We obtain a sufficient condition for the first-approximation stability of this system.