Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. II

In this paper we continue the research started in a previous paper, where we proved that the linear differential equation

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The existence of a weighted mean for almost periodic functions

In a recent paper by Liang et al. (2010) [1], the original question which is that of the existence of a weighted mean for almost periodic functions was raised. In particular, they showed through an example that there exist weights for which a weighted mean for almost periodic functions may or may not exist. In this note we give some sufficient conditions which do guarantee the existence of a weighted mean for almost periodic functions, which will then coincide with the classical Bohr mean. Moreover, we will show that under those conditions, the corresponding weighted Bohr transform exists.     

Stability, resonance and Lyapunov inequalities for periodic conservative systems

This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.     

On perturbations of quasiperiodic Schrödinger operators

Using relative oscillation theory and the reducibility result of Eliasson, we study perturbations of quasiperiodic Schrödinger operators. In particular, we derive relative oscillation criteria and eigenvalue asymptotics for critical potentials.     

Fixed point theorems for mixed monotone operators with PPF dependence

In this paper, fixed point theorems for mixed monotone operators are investigated by weakening the requirements in the contractive assumption and strengthening the metric space utilized with a partial order, taking the domain space C[[a,b],E]$C[[a,b],E]$ different from the range E$E$ and using the method of reflection operators.     

Local spectral properties of reflectionless Jacobi, CMV, and Schrödinger operators

We prove that Jacobi, CMV, and Schrödinger operators, which are reflectionless on a homogeneous set E (in the sense of Carleson), under the assumption of a Blaschke-type condition on their discrete spectra accumulating at E, have purely absolutely continuous spectrum on E.     

Borg’s criterion for almost periodic differential equations

Borg’s criterion is used to prove the existence of an exponentially asymptotically stable periodic orbit of an autonomous differential equation and to determine its domain of attraction. In this article, this method is generalized to almost periodic differential equations. Both sufficient and necessary conditions are obtained for the existence of an exponentially stable almost periodic solution. The condition uses a Riemannian metric, and an example for the explicit construction of such a metric is presented.     

Some existence results for a periodic problem with non-smooth potential

In this paper, two existence results for a class of second order periodic boundary value problems with non-smooth potential are obtained. We extend the Castro-Lazer-Thews reduction method to non-smooth functionals, the obtained result is then exploited to prove the existence of a nontrivial solution. Furthermore, we prove the existence of multiple solutions by using a multiplicity result based on local linking.     

Fredholm operators,evolution semigroups,and periodic solutions of nonlinear periodic systems

Let X be a Banach space and L the generator of the evolution semigroup associated with the τ  -periodic evolutionary process _{{U(t,s)}t≥s}${\{U(t,s)\}}_{t\ge s}$ on the space _Pτ(X)$P_{\tau }(X)$ of all τ-periodic continuous X  -valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp=−?F(p,?)$Lp=-?F(p,?)$ under the condition that 1 is a normal eigenvalue of the monodromy operator U(τ,0)$U(\tau ,0)$. The proof is based on a new decomposition of the space _Pτ(X)$P_{\tau }(X)$ by constructing a right inverse of L.     

Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media

The Ginzburg-Landau-Allen-Cahn equation is a variational model for phase coexistence and for other physical problems. It contains a term given by a kinetic part of elliptic type plus a double-well potential. We assume that the functional depends on the space variables in a periodic way.We show that given a plane with rational normal, there are minimal solutions, satisfying the following properties. These solutions are asymptotic to the pure phases and are separated by an interface. The convergence to the pure phases is exponentially fast. The interface lies at a finite distance M from the chosen plane, where M is a universal constant. Furthermore, these solutions satisfy some monotonicity properties with respect to integer translations (namely, integer translations are always comparable to the function).We then show that all the interfaces of the global periodic minimizers satisfy similar monotonicity and plane-like properties.We also consider the case of possibly irrationally oriented planes. We show that either there is a one parameter family of minimizers whose graphs provide a field of extremals or there are at least two solutions, one which is a minimizer and another one which is not. These solutions also have interfaces bounded by a universal constant, they enjoy monotonicity properties with respect to integer translations and the nonminimal solutions are trapped inside a gap of the lamination induced by the minimizers.     

Representation theorems for Sobolev spaces on intervals and multiplicity results for nonlinear ODEs

We present some topological isomorphisms between Wm,p((0,1)) and Lp((0,1))×Rm. Combined with some arguments from Fourier analysis, we obtain explicit Schauder bases for Wm,p((0,1)) and some of its subspaces. In addition, we apply such results to treat some boundary value problems involving nonlinear elliptic differential equations.     

Singular functions and relative index for elliptic corner operators

We show an explicit link between the nature of a singular point and the behaviour of the coefficients of the equation, under which formal asymptotic expansions are still available. We also derive a general relative index theorem for elliptic operators. To the memory of Lamberto Cattabriga     

Maximal parabolic regularity for divergence operators including mixed boundary conditions

We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.     

The transformation operator for Schrödinger operators on almost periodic infinite-gap backgrounds

We investigate the kernels of the transformation operators for one-dimensional Schrödinger operators with potentials, which are asymptotically close to Bohr almost periodic infinite-gap potentials.     

Zero separation results for solutions of second order linear differential equations

The oscillation of solutions of f^″+Af=0$f^{″}+Af=0$ is discussed by focusing on four separate situations. In the complex case A$A$ is assumed to be either analytic in the unit disc D$\mathbb{D}$ or entire, while in the real case A$A$ is continuous either on (−1,1)$(-1,1)$ or on (0,∞)$(0,\infty )$. In all situations A$A$ is expected to grow beyond bounds that ensure finite oscillation for all (non-trivial) solutions, and the separation between distinct zeros of solutions is considered.     

Dynamics of a class of ODEs more general than almost periodic

A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C₀-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations.     

Traveling waves of a curvature flow in almost periodic media

In a plane media with almost periodic vertical striations, we study a curvature flow and construct two kinds of traveling waves, one having a straight line like profile and the other having a V shaped profile. For each of the first-kind traveling waves, its profile is the graph of a function whose derivative is almost periodic. For each of the second-kind traveling waves, its profile is like a pulsating cone, whose two tails approach asymptotically the profiles of the first-kind traveling waves. Also we consider a homogenization problem and provide an explicit formula for the homogenized traveling speed.     

Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem.     

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.