In this work we consider the Cauchy problem associated with dissipative perturbations of infinite-dimensional Hamiltonian systems. we describe abstract conditions under which the problem is locally and globally well posed. Moreover, we establish the existence of global attractor. Finally, we present several applications of the theory.

This paper describes the characterization of optimal constants for some coercivity inequalities in W1,p(Ω), 1 < p < ∞. A general result involving inequalities of p-homogeneous forms on a reflexive Banach space is first proved. The constants are shown to be the least eigenvalues of certain eigenproblems with equality holding for the corresponding eigenfunctions. This result is applied to three different classes of coercivity results on W1,p(Ω). The inequalities include very general versions of the Friedrichs and Poincaré inequalities. Scaling laws for the inequalities are also described.

In this paper, we consider a ring of neurons with self-feedback and delays. As a result of our approach based on global bifurcation theorems of delay differential equations coupled with representation theory of Lie groups, the coexistence of its asynchronous periodic solutions (i.e. mirror-reflecting waves, standing waves and discrete waves), bifurcated simultaneously from the trivial solution at some critical values of the delay, will be established for delay not only near to but also far away from the critical values. Therefore, we can obtain wave solutions of large amplitudes. In addition, we consider the coincidence of these periodic solutions.

By using a monotonic functional on a suitable matrix space, some new oscillation criteria for self-adjoint matrix Hamiltonian systems are obtained. They are different from most known results in the sense that the results of this paper are based on information only for a sequence of subintervals of [t0, ∞), rather than for the whole half-line. We develop new criteria for oscillations involving monotonic functionals instead of positive linear functionals or the largest eigenvalue. The results are new, even for the particular case of self-adjoint second-differential systems which can be applied to extreme cases such as

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

In this paper, we consider a ring of neurons with self-feedback and delays. As a result of our approach based on global bifurcation theorems of delay differential equations coupled with representation theory of Lie groups, the coexistence of its asynchronous periodic solutions (i.e. mirror-reflecting waves, standing waves and discrete waves), bifurcated simultaneously from the trivial solution at some critical values of the delay, will be established for delay not only near to but also far away from the critical values. Therefore, we can obtain wave solutions of large amplitudes. In addition, we consider the coincidence of these periodic solutions.

This paper deals with eigenvalue problems of the formwhere 0 < σ < τ and V(x) is such that the spectrum of −u″ consists of eigenvalues λ1, λ2,…situated below the continuous spectrum [Λ,+∞[.

We analyse the existence of (multiple) solutions for λ < λ1 as well as for λ > λ1 when λ is in a spectral lacuna.

The existence of solutions depends on the weight of μ > 0. Moreover, when λ increases (while μ is kept fixed), some solutions are lost when crossing eigenvalues.

The above results are derived with the help of an abstract approach based on variational techniques for multiple solutions. This approach can even be applied to a wider class of problems, the one presented herein being only a model problem.

We establish existence results for a class of semilinear elliptic equations with exponential nonlinearity by studying a suitable eigenvalue problem. We also establish a uniqueness result for those equations by making use of the implicit function theorem.

We give a simple explicit description of the norm in the complex interpolation space (Cp[Lp(M)], Rp[Lp(M)])θ for any von Neumann algebra M and any 1 ≤ p ≤ ∞.

This paper deals with eigenvalue problems of the form where 0 < σ < τ and V(x) is such that the spectrum of −u″ consists of eigenvalues λ1, λ2,…situated below the continuous spectrum [δ,+∞[.

We analyse the existence of (multiple) solutions for λ < λ1 as well as for λ > λ1 when λ is in a spectral lacuna.

The existence of solutions depends on the weight of μ > 0. Moreover, when λ increases (while μ is kept fixed), some solutions are lost when crossing eigenvalues.

The above results are derived with the help of an abstract approach based on variational techniques for multiple solutions. This approach can even be applied to a wider class of problems, the one presented herein being only a model problem.

We establish existence results for a class of semilinear elliptic equations with exponential nonlinearity by studying a suitable eigenvalue problem. We also establish a uniqueness result for those equations by making use of the implicit function theorem.

This paper describes the characterization of optimal constants for some coercivity inequalities in W1,p(Ω), 1 < p < ∞. A general result involving inequalities of p-homogeneous forms on a reflexive Banach space is first proved. The constants are shown to be the least eigenvalues of certain eigenproblems with equality holding for the corresponding eigenfunctions. This result is applied to three different classes of coercivity results on W1,p(Ω). The inequalities include very general versions of the Friedrichs and Poincaré inequalities. Scaling laws for the inequalities are also described.

We discuss some ill-posedness results for solutions arising from the Picard iterations algorithm (i.e. the Banach fixed-point theorem) in the case of the nonlinear heat equation, the viscous Hamilton–Jacobi equation, the convection–diffusion equation and the incompressible Navier–Stokes system.

This note extends the results in ‘Optimal coercivity inequalities in W1,p(Ω)’ (G. Auchmuty, Proc. R. Soc. Edinb. A 135, 915–933.) describing the dependence of the optimal constant in the p-version of Friedrichs' inequality on the boundary integral term. In particular, it is shown that this constant is continuous, increasing, concave and increases to the optimal constant for the Dirichlet problem as s → ∞.

Let $X$, $Y$ be compact Hausdorff spaces and let $T:C(X,\mathbb{R})\to C(Y,\mathbb{R})$ be an invertible linear operator. Non-standard analysis is used to give a new intuitive proof of the Amir–Cambern result that if $\|T\|\hskip1pt\|T^{-1}\|\lt2$, then there is a homeomorphism $\psi:Y\to X$. The approach provides a proof of the following representation theorem for such near-isometries:

$$ Tf=(T1_X)(f\circ\psi)+Sf, $$

with $\|S\|\leq2(\|T\|-(1/\|T^{-1}\|))$, so $\|S\|\lt\|T\|$. If $\|T\|\hskip1pt\|T^{-1}\|=1$, then $S=0$, giving the well-known representation for isometries.

It is shown that given a uniform algebra $A$ and a peak point $x_0$ for $A$, every function in $A$ can be expressed as a linear combination of two functions in $A$ that peak at $x_0$.

Universal Taylor series are defined on simply connected domains, but they do not exist on an annulus. Instead we introduce universal Laurent or Laurent–Faber series on finitely connected domains in $\mathbb{C}$. These are generic universalities. Furthermore, we study some properties of universal Laurent series on an annulus.

Let $A$ be a commutative comodule algebra over a commutative bialgebra $H$. The group of invertible relative Hopf modules maps to the Picard group of $A$, and the kernel is described as a quotient group of the group of invertible group-like elements of the coring $A\otimes H$, or as a Harrison cohomology group. Our methods are based on elementary $K$-theory. The Hilbert 90 theorem follows as a corollary. The part of the Picard group of the coinvariants that becomes trivial after base extension embeds in the Harrison cohomology group, and the image is contained in a well-defined subgroup $E$. It equals $E$ if $H$ is a cosemisimple Hopf algebra over a field.

In this paper we study a non-standard eigenvalue problem which arises in the context of a thermal wave propagation problem, and some generalizations thereof. The eigenvalue distribution is fully explored and useful bounds on the location of the eigenvalues are obtained.