The paper deals with equivalent norms and Schauder bases in anisotropic Besov spaces where 1≦p <∞, = (s1,…, sn), and 0 < sk <1.

In the Inverse Spectral Theorem in the form given by Levitan and Gasymov, necessary and sufficient conditions are given for a non-decreasing function, p(ℷ), to be the spectral function of a Sturm–Liouville problem. In these conditions, p(ℷ) is compared with the spectral function for the particular Strum–Liouville problem

If the method of Levitan and Gasymov's proof is slightly adapted, the necessary and sufficient conditions can be stated in a more general form in which p(ℷ) is compared with the spectral function for any problem of the form

where hO is real and qo(x) locally integrable.

Given any countably infinite set of isloated points on the ℷ -axis, it is shown that there is a continuous q(x) such that these points constitute exactly the point-continuous spectrum for the equation yn″(x) + (ℷ —q(x))y(x) = 0(0≦x<∞) with some homogenous boundary condition at x = 0. This extends a result given by Eastham and McLeod for countably infinite sets of isolated points on the positive ℷ-axis.

In the Hilbert space framework, we give some results concerning the behaviour when t goes to infinity for solutions of equations of the form:

A is assumed to be a maximal monotone operator and F(t) is a periodic function.

When F = 0, under a compactness assumption for trajectories of (1), we give the complete description of the asymptotic behaviour, e.g. every trajectory is asymptotic to an almost-periodic solution of (1). When F ≠ cst, the compactness hypothesis being too restrictive, we concentrate our efforts on the case of the equation:

with Dirichlet boundary condition) and get weak convergence to particular solutions of the equation when β is either univalued or strictly monotone. The methods used in these cases seem of general interest for hyperbolic equations of dissipative type with periodic forcing term.

This paper investigates the effect on the structure of a band of imposing conditions on the congruence or right congruence lattice of the band. Bands whose congruence lattices are distributive or Boolean are characterized, as are bands whose right congruence lattices are Boolean.

This paper studies the stability regions associated with the multi-parameter system

where the functions qr(xr), ars(xr) are periodic and the system is subjected to periodic or semi-periodic boundary conditions.

In 1975 K. W. Brodie and W. N. Everitt dealt with integral inequalities of the form

in the two cases T = ℝ, T = ℝ+, and obtained the best possible constants KT(μ) for all μ ε ℝ. The proof was not elementary, but an elementary proof was given in 1977 by E. T. Copson. This note shows how Copson's method can be greatly simplified so as to obtain the results in a more straightforward manner.

We prove existence and uniqueness results for the solution of nonlinear elliptic boundary value problems, where the linear part of the equation is given by a second-order elliptic operator not in divergence form.

Consider mild solutions on the real line of non-homogeneous differential equations in a Banach space: u′(t) = Au(t) + f(t), where A is the infinitesimal generator of a C0-semigroup.

We prove an existence result for optimal solutions (as defined in the text) in reflexive spaces and an uniqueness fact in uniformly convex B-spaces.

Let I be the group of rotations of the circle and suppose that a map f from I into I is “almost linear”. More precisely, let ε be a positive real number and suppose that d(f(x)+f(y), f(x + y))<ε for all x and y in I; f is said to be an ε-homomorphism. If ε is sufficiently small, then a homomorphism h from I into I can be constructed such that d(f(x), h(x))≦ε for all x in I. This result is refined and generalized in several ways.

This paper deals with a class of interpolation series of the form

called R-series. It is equiconvergent with the Dirichlet series

If the nth Legendre polynomial for the interval (0,1) is denoted by (−1)nLn(t), then the bilinear formula

serves as generating function for the Rn(z). It also leads in easy steps to R-series expansions for rational functions.

Lagrange [7] has shown that a function holomorphic and of finite rate of growth in a right half-plane can be expanded in an R-series whose abscissa of convergence is limited by the rate of growth of f(z). The converse problem is attacked in Theorem 2 below where it is shown that

We give a survey of the current state of knowledge on the Arens second dual of a Banach algebra, including some simplified proofs of known results, some new results, some open problems and a full bibliography of the subject.

This paper is devoted to the study of some non linear Schrödinger equations in two dimensions, arising in non linear optics; in particular, it is concerned with solutions to the Cauchy problem. The problem of global existence and regularity of the solutions, the asymptotic behaviour of global solutions, and the blow-up of non global solutions are studied.

We consider the expansion of a function in Lr (the class of measurable functions whose rth powers are Lebesgue integrable over some interval) in terms of the eigenfunctions arising from a singular Sturm-Liouville problem defined over an infinite or semi-infinite interval. We show that if l ≦ r ≦ inline1 or if r ≧ 4 there exists f in Lr whose eigenfunction expansion is divergent in the rth mean sense, and that the terms of the series form an unbounded sequence in Lr The result extends some work of Askey and Wainger concerning the Hermite series expansions of functions in Lr(–∞, ∞).