Convergence in monotone and uniformly stable skew-product semiflows with applications

Abstract

The 1-covering property of omega limit sets is established for monotone and uniformly stable skew-product semiflows with the componentwise separating property of bounded and ordered full orbits. Then these results are applied to study the asymptotic almost periodicity of solutions to almost periodic reaction-di¤usion equations and di¤eren-tial systems with time delays. The earlier convergence results for autonomous and periodic monotone systems are generalized to the almost periodic case without the strong monotonicity assumption.

Translation-invariant monotone systems II: Almost periodic/automorphic case

“…Applying this abstract result to various almost periodic differential equations, one gets that if each solution has the compact property and is uniformly stable, then every solution for systems considered is asymptotically almost periodic. Novo, Obaya and Sanz [35] have shown that the omega limit set is a uniformly stable minimal set which admits a fiber distal flow extension if its semiorbit is uniformly stable, which not only extends the classical extension result in Shen and Yi [39], but also drops the distal assumption in the abstract result in [29]. Later, together with their topological result with this Lyapunov integer-valued function method, they have investigated a series of functional differential equations with infinite delay and obtained one covering property of the base space ( [35,33]), some of which are even new in autonomous/periodic cases.…”

“…This technique was further exploited in the multiple equilibria/fixed points case to prove global convergent results in monotone (without stronger notion) dynamical systems with every equilibrium/fixed point stable [25,26,27], sublinear nonlinearity [24] and possessing a positive gradient first integral [28]. Now this method has even been developed to study skew-product monotone semiflows without the stronger notion (see [29]). Using this technique on each fibre together with the theory on topological dynamical systems, it is shown that every omega limit set for uniformly stable monotone skew-product semiflows is topologically conjugate to the base space flow if the base space flow is minimal and distal and the state space has lattice structure with an additional precompact assumption.…”

“…The purpose of this paper is to prove that every omega limit set for a positively translation-invariant monotone skew-product is conjugate to its minimal base flow. The main tool used here is included in [29,35,39].…”

On the global attractivity of monotone random dynamical systems

“…This result has been frequently applied in the study of modelling biology, chemistry and control theory (see [5,15,16,25] and references therein) when they lack of stronger monotonicity. The same version of result for skew-product monotone semiflow is contained in [14].…”

“…As Sontag et al pointed out in [15, p. 297], checking this condition in practice is often not so easy, or even worse: a system may be monotone but fail to satisfy the stronger notion (see, e.g., [5,15,16,25] and many references therein). The second author has introduced an integer-valued function to get rid of the irreducibility condition and proved the global convergence for various monotone systems, for example, systems for every equilibrium being stable [9], sublinearity [10], systems possessing a first integral with positive gradient [11]; this technique has even been applied to the study of skew-product monotone systems without stronger notion (see [14] and [17]). This idea was first formed by the second author in [12] to solve a global stability conjecture for three dimensional cooperative systems without irreducibility.…”