We consider ihe differential expression M[y]: = −y″ + qy on [0, ∞) where q_∈ Lp [0, ∞) for some p ≧ 1. It is known that M, together with the boundary conditions y(0) = 0 or y′(0) = 0, defines linear operators on L2 [0, ∞). We obtain lower bounds for the spectra of these operators. Our bounds depend on the Lp norm of q_ and extend results of Everitt and Veling.

Asymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.

We characterize the finite projective double Stone algebras, and also make some observations concerning infinite projective double Stone algebras.

Let Ω = ℝN or Ω be a bounded regular open set of ℝN and let γ(x, S): Ω × ℝ → ℝ be a continuous nondecreasing function in s, measurable in x, such that γ(x, 0) = 0 almost everywhere. We solve, for f ∈ L1(Ω), the problem (P): −Δu + γ(., u) = f in Ω, u = 0 on ∂Ω. (In fact, for this result, instead of assuming that γ is nondecreasing in s we need only that γ(x, s)s≧0.) We deduce an ‘almost’ necessary and sufficient condition on , in order that (P) has a solution. Roughly speaking, this condition is f = −ΔV + g, with g ∈ L1(Ω) and γ(., V) ∈ L1(Ω)

Denote by (αj, βj), j = 1, 2, … an infinite set of disjoint open intervals on the half-line (0, ∞). Suppose that the following conditions are fulfilled:

With the aid of the first two trace formula presented earlier by the author, we prove in this paper that there exists a function q, defined on the whole real line, such that for the Schrödinger equation −y″ + q(x)y = λy (−∞<x<∞), the intervals (αj, βj) are spectrum lacunae. As an example, we consider the case when the intervals (αj, βj) are adjacent intervals of the Cantor trinary perfect set.

Let Sturm–Liouville problems

with continuous coefficients and appropriate boundary conditions, be coupled by the eigenvalue λ = (λ1, … λk). When k = 1, there are various oscillation, perturbation and comparison theorems concerning existence and continuous or monotonic dependence of eigenvalues, eigenfunctions and their zeros (i.e. focal points).

We attempt a unified theory for such results, valid for general fc, under conditions known as "left" and “right” definiteness. A representative result may be stated loosely as follows: if LD holds then (elementwise) monotonic dependence of p, q and the matrix [ars] forces monotonic dependence of λ. LD is a generalisation of the “polar” case for k = 1, and was originally conceived for a quite different purpose, viz. completeness of eigenfunctions via elliptic partial differential equation theory.

We deal with the asymptotic behaviour, as t→∞, of complex-valued solutions of nonlinear differential equations

Upper bounds for ∣x(l)(t)∣, 0≦j≦n, are obtained by obtaining upper bounds for solutions u(t) of Bihari-type integral inequalities of the form

Let ψ ∈ C2[0,1] be a positive function on (0, 1]. Under certain assumptions on ψ, the set

is a pseudoconvex domain with C2-boundary, for which it is possible to construct a Henkin-type operator Hψ = Kψ + Bψ solving in Dψ. The operator Bψ, is L∞-continuous because it has a Riesz potential type kernel, while the L∞-continuity of Kψ depends on the flatness of ψ at 0. Our main result states that Kψ is continuous from L∞(∂Dψ) into L∞(Dψ) if and only if

This note gives a simple proof of uniqueness for positive solutions of certain non-linear boundary value problems on ℝ+ which are typified by the equation

with boundary conditions u′(0) = u(+∞) = 0. In the autonomous case (r ≡ 1), this is easy to see, by quadrature. The proof here supposes r to be non-increasing on ℝ+.

In this paper, we introduce a new class of solutions of reaction-diffusion systems, termed directional wave front solutions. They have a propagating character and the propagation direction selects some distinguished boundary points on which we can impose boundary conditions. The Neumann and Dirichlet problems on these points are treated here in order to prove some theorems on the existence of directional wave front solutions of small amplitude, and to partially establish their asymptotic behaviour.

We give an inequality for the concentration function of a sum X1 + … + Xn of independent random variables when Xv has a finite absolute moment of order kv (2 < kv ≦ 3). It is an extension of somewhat similar inequalities found earlier by Offord and by the author in the case of finite third-order absolute moments.

We extend several known properties of the Dirichlet index to the case of minimal conditions and prove that the index is invariant under positive t bounded perturbations of the associated quadratic form. The index is also shown to be minimal for fourth order operators with certain growth conditions on some of their coefficients.

Accurate computation of the evolution of initially localized disturbances in compressible parallel flows is a tedious task requiring superposition of a large number of Fourier modes with differing temporal growth rates. An alternative approximate method, similar to that developed by Craik (1981, 1982) for viscous incompressible flows, is presented here. This involves asymptotic evaluation, by the saddle point method, of a double Fourier integral representation of the disturbance, with the actual dispersion relation replaced by a simpler analytic expression containing several parameters which may be adjusted to approximate the flow under investigation. Limiting cases yield informative results in simple closed form: these exemplify the possible shapes into which the disturbance may evolve. In particular, ‘splitting’ of the disturbance into two dominant regions is demonstrated.

For β > β0: = 1 −[(n − 2)/2]2 and n ≧ 2, it was recently shown by Simon that the self-adjoint operator associated with −Δ + βr−2 in L2(ℝn) has domain H2(ℝn) ∩D(r−2) the constant β0 being the best possible. An alternative proof of this result is given.

This paper considers the asymptotic form, as λ tends to infinity in sectors omitting the real axis, of the matrix Titchmarsh-Weyl coefficient M(λ) for the fourth order equation y(4) + q(x)y = λy, where q(x) is real and locally absolutely integrable. By letting M0(λ) denote the m-coefficient for the Fourier case y(4) = λy, the asymptotic formula M(λ) = M0(λ) + 0(1) is established.

We define spaces of functions on an interval naturally related to functional differential equations of retarded type in the sense that exponential solutions of such equations form a uniform basis (or Riesz basis) in these spaces. Embedding theorems relating these spaces to the Sobolev spaces are established using a result on Carleson's measure defined on a half plane.