Almost Automorphic and Almost Periodic Dynamics for Quasimonotone Non-Autonomous Functional Differential Equations

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

“…, n for a certain δ > 0, then we could directly apply Lemma 7.17 in [14] which affirms that in a column dominant family of linear almost periodic systems, the null solution is exponentially stable as t → ∞. However, our requirement in hypothesis (a5) is not so restrictive, and we have to follow another argument, based on the existence of a so-called strong super-equilibrium (see Novo et al [33] for the introduction of this concept in the field of non-autonomous FDEs).…”

“…And the same happens, by linearity, with any of the constant maps given by λ1 for any λ > 0, so that we have a family of strong super-equilibria approaching 0. In this situation, we can apply Theorem 5.3 in [33] also in this linear context of cooperative ODEs to conclude that the null solution of z ′ (s) = A(ω·s) T z(s) determines a unique attractor as s → ∞, which is uniformly asymptotically stable.…”

A note on equilibrium of eventually strongly monotone skew-product semiflows

“…It is helpful to note that, as shown in [5], these notions induce minimal sets which are almost automorphic extension of the base flow. Since ordered minimal sets provide a global picture of the dynamics (see [6]), it is natural to ask how two continuous equilibria can be guaranteed to be ordered?…”

“…The concepts of equilibria and semi-equilibria turn out to be of prime importance in the study of monotone skew-product semiflows (see [1][2][3][4][5][6] and the references therein). Note that for autonomous and periodic systems these concepts are well-known (see [7]), while our definitions for skew-product semiflows are motivated by the corresponding concepts for random dynamical systems (see [1][2][3]).…”

Necessary and sufficient conditions for the existence of equilibrium in abstract non-autonomous functional differential equations

“…A good investigation of such sets is undoubtedly necessary for a better understanding of the flows or semiflows they describe (see [20,31,32] for recent developments). Especially when the coefficients of the non-autonomous equation exhibit some recurrent variation in time t, its solutions define a skew-product semiflow in a natural way which permits the analysis of the dynamical properties of the trajectories using methods of topological dynamics and ergodic theory [25,28,30,32,35].…”