The classical inequality ‖y′‖2≦K‖y‖ ‖y″‖ and its higher order analogues are extended from the Lp spaces to the weighted spaces, for appropriate w.

An elementary matrix has ones down the main diagonal and at most one element off the diagonal that differs from zero. We study the subgroups of SL2ℤ generated by sets of elementary matrices. Specifically, we give a stringent condition that the entries of a matrix belonging to such a group must satisfy.

We consider ordinary linear differential systems of first order with distributional coefficients and distributional nonhomogeneous terms. Firstly the coefficients are assumed to be functions, secondly to be first order distributions (i.e. first derivatives of functions which are integrable or of bounded variation), and thirdly to be distributions of higher order.

Large time behaviour of solutions to a damped quasi-linear wave equation are studied. Conditions are obtained which guarantee the global existence of a classical solution. The asymptotic behaviour of this solution is studied in the case of a unique equilibrium solution and in the case of multiple equilibria. The results are applied to various special examples.

Let H be a Hilbert space in which a symmetric operator S with a dense domain Ds is given and let S have a finite deficiency index (r, s). This paper contains a necessary and sufficient condition for validity of the following inequalities of Kolmogorov type

and a method for calculating the best possible constants Cn,m(S).

Moreover, let φ be a symmetric bilinear functional with a dense domain Dφ such that Ds ⊂ Dφ and φ(f, g) = (Sf, g) for all f ∈ Ds, g ∈ Dφ. A necessary and sufficient condition for validity of the inequality

as well as a method for calculating the best possible constant K are obtained. Then an analogous approach is worked out in order to obtain the best possible additive inequalities of the form

The paper is concluded by establishing the best possible constants in the inequalities

where T is an arbitrary dissipative operator. The theorems are extensions of the results of Ju. I. Ljubič, W. N. Everitt, and T. Kato.

We study the boundary-value problem, for (λ/k,ψ),

Here ∆ denotes the Laplacian, H is the Heaviside step function and one of A or k is a given positive constant. We define

and usually omit the subscript. Throughout we are interested in solutions with ψ>0 in Ω and hence with λ/=0.

In the special case Ω = B(0, R), denoting the explicit exact solutions by ℑe, the following statements are true, (a) The set Aψ, issimply-connected, (b) Along ℑe, the diameter of Aψ tendsto zero when the area of Aψ, tends to zero.

For doubly-symmetrised solutions in domains Ω such as rectangles, it is shown that the statements (a) and (b) above remain true.

Valdivia (1978) introduced the class of suprabarrelled spaces, and (1979) deduced some uniform boundedness properties for scalar valued exhausting additive set functions on a σ-algebra from the suprabarrelledness of certain spaces. In this paper, it is shown that those uniform boundedness properties hold for G-valued exhausting additive set functions, G being a commutative topological group, on a larger class of Boolean algebras. Such properties are proved in Valdivia (1979) by means of duality theory arguments and ‘sliding hump’ methods, whereas here they are derived from the Baire category theorem. This generalization enables us to find a wide class of compact topological spaces K such that the subspaces of C(K) which satisfy a mild property are suprabarrelled.

A necessary and sufficient condition is given for the uniform exponential stability of certain autonomous differential–difference equations whose phase space is a Hilbert space. It is shown that this property is preserved when the delays depend homogeneously on a nonnegative parameter.

This paper deals with initial value problems in ℝ2 which are governed by a hyperbolic differential equation containing a small positive factor ε in front of the second order part of the differential operator. Pointwise a priori estimates for the solution are established by means of an energy integral method. With the help of these estimates it is shown that the solution admits an asymptotic expansion into powers of ε which is uniformly valid in compact subsets of ℝ2. All results are obtained under the assumption that the characteristics of the first order part of the differential operator satisfy a so-called “timelike” condition. A discussion of the concepts “timelike” and “spacelike” is given.

Given the linear hyperbolic evolution equation (P0) on a reflexive Banach space, we present a new method for an existence proof of unbounded solutions admitting an exponential growth rate as time tends to infinity. Utilizing abstract Wiener—Hopf techniques, an operational calculus is developed for the construction of the resolving operator associated with the problem under consideration. The results are based upon the fundamental hypothesis that the spectral set of the time-independent mapping A is contained in the interior of a parabola. The distance of the focus from the vertex of this parabola turns out to be a measure for the growth rate. Applicability of the results is shown in the case where A is a non-symmetric perturbation of a self-adjoint partial differential operator.

For an ordinary differential operation Lλ of order 2N which depends differentiably on a parameter λ, we study the differentiability with respect to λ of all solutions to Lλf = 0 which are in L2[a,∞). Applications to spectral theory are given, including a formula for the rate of change with respect to the end-point a of the spectrum of the weighted eigenvalue problem Lf = λwf, f∈L2[a,∞), f[i](a) = 0 for i ≦N − 1. The weight w may be a function or an operator. The formula seems new even when w = 1.

Structure conditions on a strongly nonlinear operator A(u) are given, under whichthe initial-Dirichlet-boundary value problem for has weak solutions.

The investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroup S we study the minimum good congruence γ such that S/γ is adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.

Given a Banach space X, we investigate the behaviour of the metric projection PF onto a subset F with a bounded complement.

We highlight the special role of points at which d(x, F) attains a maximum. In particular, we consider the case of X as a Hilbert space: this case is related to the famous problem of the convexity of Chebyshev sets.

For the Cauchy problem for strictly hyperbolic systems with general eigenvalues, we obtain existence of global smooth solutions under certain conditions on the composition of the eigenvalues and the initial data; on the other hand, we give a sufficient condition which guarantees that singularities of the solution must occur in a finite time and describe certain applications. The present paper includes the corresponding results in earlier papers by several authors as special cases.

This paper studies the equation

where the differential operator Ln is defined by

and a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case where

where N is an integer with l≦N≦n–1.