Bifurcations in non-autonomous scalar equations

In a previous paper we introduced various definitions of stability and instability for non-autonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations.     

Stability for non-autonomous linear evolution equations with L p -maximal regularity

We study stability and integrability of linear non-autonomous evolutionary Cauchy-problem $$(P),\left\{ \begin{gathered} \dot u(t) + A(t)u(t) = f(t) t - a.e. on [0,\tau ] \hfill \\ u(0) = 0, \hfill \\ \end{gathered} \right.$$ where A: [0, τ] → L(X,D) is a bounded and strongly measurable function and X, D are Banach spaces such that . Our main concern is to characterize L ^p -maximal regularity and to give an explicit approximation of the problem (P).     

Operator splitting for non-autonomous evolution equations

We establish general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tools are evolution semigroups allowing the direct application of existing results on autonomous problems. The results obtained are illustrated by the example of an autonomous diffusion equation perturbed with time dependent potential. We also prove convergence rates for the sequential splitting applied to this problem.     

Maximal space regularity for abstract linear non-autonomous parabolic equations

The linear non-autonomous evolution equation $\text{u′(t) ? A(t) u(t) = ?(t), t ∈ [0, T]}$, with the initial datum u(0) = x, in the space C([0, T], E), where E is a Banach space and {A(t)} is a family of infinitesimal generators of bounded analytic semigroups is considered; the domains D(A(t)) are supposed constant in t and possibly not dense in E. Maximal regularity of the strict and classical solutions, i.e., regularity of u′ and A(·)u(·) with values in the interpolation spaces D_A(0)(θ, ∞) and D_A(0)(θ) between D(A(0)) and E, is studied. A characterization of such spaces in a concrete case is also given.     

GLOBAL ATTRACTIVITY OF LINEAR NON－AUTONOMOUS NEUTRAL DIFFERENTIAL－DIFFERENCE EQUATIONS

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Matrix semigroups whose ring commutators have real spectra are realizable

We introduce the numerical spectrum \(\sigma _n(A)\subseteq {\mathbb {C}}\) of an (unbounded) linear operator A on a Banach space X and study its properties. Our definition is closely related to the numerical range W(A) of A and always yields a superset of W(A). In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, \(\sigma _n(A)\) is always closed, convex and contains the spectrum of A. In the paper we strongly emphasise the connection of our approach to the theory of \(C_0\)-semigroups.     

A Note on non-autonomous scalar functional differential equations with small delay

We prove that the minimal sets in the skew-product semiflows generated from a non-autonomous scalar functional differential equation with a small delay are all almost automorphic extensions of the base. This result is not true for arbitrary delay equations. The point is that, for a small delay, so-called special solutions exist and permit us to tackle the problem by means of some related scalar ODE's for which the study is much simpler. To cite this article: A.I. Alonso et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Oscillation for linear non-autonomous systems of difference equations with continuous arguments

In this paper, we establish sufficient conditions for the oscillation of the linear non-autonomous systems of difference equations with continuous arguments

     

Existence and sharp regularity results for linear parabolic non-autonomous integro-differential equations

We study existence, uniqueness and maximal regularity of the strict solutionu∈C ¹([0,T],E) of the integro-differential equation \(u'(t) - A(t)u(t) - \int {_0^1 } B(t,s)u(s)ds = f(t),t \in [0,T],\) with the initial datumu(0)=x, in a Banach spaceE, {itA(itt)}_f∈|0,1| is a family of generators of analytic semigroups whose domainsD _A(t) are not constant int as well as (possibly) not dense inE, whereas {itB(itt)}_0≦11≦T is a family of closed linear operators withD _B(t,s) ?D _A(s) ∨t∈[s, T]. We prove necessary and sufficient conditions for existence of the strict solution and for Hölder continuity of its derivative; well-posedness of the problem with respect to the Hölder norms is also shown.     

Solving linear ordinary differential equations by exponentials of iterated commutators

Summary A sequence of transformations of a linear system of ordinary differential equations is investigated. It is shown that these transformations produce new systems which represent progressively smaller perturbations of the original set of equations.The transformations are implemented as a basis of a numerical method. Order, stability and error control of this method are analyzed. Numerical examples demonstrate the potential of this approach.     

Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations

In this paper, we first introduce the concepts and properties of the square-mean weighted pseudo almost automorphy and the square-mean bi-almost automorphy for a stochastic process. With these preliminary settings and by virtue of the theory of the semigroups of the operators, the Banach fixed point theorem and the stochastic analysis techniques, we investigate the well-posedness of the square-mean weighted pseudo almost automorphic solutions for a general class of non-autonomous stochastic evolution equations that satisfy either global or only local Lipschitz condition. Moreover, we estimate the boundedness of attractive domain for the case where the only local Lipschitz condition is taken into account. Finally, we provide two illustrative examples to show the practical usefulness of the analytical results that we establish in the paper.     

Maximal regularity for non-autonomous stochastic evolution equations in UMD Banach spaces

A non-autonomous stochastic linear evolution equation in UMD Banach spaces of type 2 is considered. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to a stochastic partial differential equation.     

One-parameter semigroups for linear evolution equations

No abstract. October 10, 2000     

Well-posedness of linear partial differential equations with unbounded delay operators

In this paper we show the well-posedness of the following constant delay equation:

     

Third-order scalar evolution equations with conservation laws

Associated to the family of third-order quasilinear scalar evolution equations is the geometry of point transformations. This geometry provides a framework from which to study the structure of conservation laws of the equation, and to study the special nature of the geometry of those equations which do possess conservation laws. There is an easy and obvious normal form for equations which possess at least one conservation law. The geometric structure of the equation gives rise to a simple yet much less obvious normal form for equations which possess at least two conservation laws.     

Asymptotic properties of solutions to linear non-autonomous neutral differential equations

In this article we study asymptotic properties of solutions to first order linear neutral differential equations with variable coefficients and constant delays. Results are stated in terms of the solution to a characteristic equation. By doing this, we extend some of the results obtained for delay equations in [J.G. Dix, Ch.G. Philos, I.K. Purnaras, An asymptotic property of solutions to linear non-autonomous delay differential equations, Electron. J. Differential Equations 2005 (2005) 1-9] to neutral equations.