Autonomous and non-autonomous attractors for differential equations with delays

Abstract

The asymptotic behaviour of some types of retarded differential equations, with both variable and distributed delays, is analysed. In fact, the existence of global attractors is established for different situations: with and without uniqueness, and for both autonomous and non-autonomous cases, using the classical notion of attractor and the recently new concept of pullback one, respectively.

Attractivity for neutral functional differential equations

“…Furthermore, the asymptotic behaviour of non-autonomous ordinary differential equations is studied in [17]. Ideas for non-autonomous functional differential equations are presented in [1,3,4,5]. Because of various possible reasons, the development of ideas for the family of equations complementary to (2) is much slower.…”

“…Namely, it intends to extend the findings of [1] and [5] on the existence of pullback attractors for delay differential equations to NFDEs of form (3). The rest of the paper is organised as follows.…”

“…The basic theory of neutral functional differential equations (see [8]) implies, under standard assumptions, the existence of the unique solution of (3) on [s−r, ∞). That is, if an initial function φ ∈ C is prescribed at the initial time s ∈ R then there is an x(·; s, φ) which satisfies (3) and, in addition, the initial condition x s (·) = φ, in other words,…”

Asymptotic behaviour of monotone multi-valued dynamical systems

“…Using again (9) or (8) one can see that after t 0 either the two components x i (t) remain di¤erent for all t, or for some t 1 > t 0 we have again x (t 1 ) = e x (t 1 ). Indeed, if x 1 (t 1 ) = e x 1 (t 1 ) ; x 2 (t 1 ) > e x 2 (t 1 ) and x i (t) > e x i (t) ; for t 2 [t 1 ; t 1 ), i = 1; 2 (for instance), then…”

Existence and regularity of pullback attractors for an incompressible non-Newtonian fluid with delays

“…Actually, the theory of pullback attractors has been shown to be very useful in the understanding of the dynamics of non-autonomous and random dynamical systems, including those with delays (see e.g. [1,19,21,22,23,48,50,51]). There are several versions of the concept of pullback attractors; see [48] for details.…”