In this paper I extend the analysis of regular problems containing the eigenvalue parameter in the boundary conditions given by Walter (1973) and myself (1977) to singular problems which involve the eigenvalue parameter linearly in a regular or a limit-circle boundary condition at the left endpoint. The formulation of the limit-circle boundary conditions follows that given in another paper by the present author in 1977, and has the advantage that a λ-dependent boundary condition at a regular endpoint becomes a special case of a λ-dependent boundary condition at a limit-circle endpoint. The simplicity of the spectrum is also built into the formulation given, and the spectral function is shown to have bounded total variation over (−∞, ∞) which is known in terms of the parameters of the λ-dependent boundary condition independently of the limit-circle/limit-point classification at the right endpoint. The theory is applied to the constant coefficient equation in [0, ∞) and the Bessel equation of order zero in (0, ∞), explicit formulae for the spectral function being obtained in each case. Finally, the question is posed as to whether the classical Weyl theory for problems not involving λ in the boundary conditions can also be formulated so as to involve spectral functions having bounded total variation.

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.

We investigate the existence, uniqueness and regularity of solutions to the linear differential equation Lu = f under nonlinear mixed boundary conditions on domains with singular boundary points.

An exponential form of the Riccati transformation is used to establish zeros of solutions of a class of characteristic initial value problems.

A quadratic functional Q is considered which is defined by an integral on a subset of functions in a weighted Hilbert space. The functional Q is minimized subject to the Dirichlet index of the associated differential operator being minimal. The infimum of Q is shown to be the least point in the spectrum of a certain self-adjoint operator which arises as a Friedrichs extension.

A new method is developed for obtaining the asymptotic form of solutions of the fourth-order differential equation

where m, n are integers and 1 ≦ m, n ≦ 2. The method gives new, shorter proofs of the well-known results of Walker in deficiency index theory and covers the cases not considered by Walker.

A monotone iteration scheme for the solution of the initial boundary problems associated with a system of semilinear parabolic differential equations has been developed that does not require the nonlinearities to be quasimonotone. The class of equations to which this scheme applies includes physical models that describe combustion processes involving Arrhenius reaction terms.

The nonlinear operator equation (N), , describes both semilinear elliptic boundary value problems (P) and their natural discretisations (Ph). (Here is a positive compact linear operator and [t]+ ≡ (t + |t|)/2 for all t ∊ ℝ.) It is proved that, for q ≧ 0 (q ≢ 0), in a Banach lattice E the equation (N) has an unbounded continuum ℱ of nontrivial solutions (λ, ψ) ∊ ℝ × E bifurcating from infinity at (λ1, ∞) (here is the first (positive) eigenvalue of ). All nontrivial solutions (λ, ψ) have λ ≧ λ1, and if maps a smaller cone Ks, into itself then there is a ℱs ⊂ℝ×Ks with similar properties to ℱ. The existence theory for (N) is applied to problems (P) and (Ph) which are defined on the simply-connected domain Ω. It is shown that the projection of ℱ on the λ-axis is either unbounded if the continuous function q > 0 except on a set of measure zero in Ω, or is bounded if q ≡ 0 on a subdomain of Ω. If is the second eigenvalue for in (Ph) then there is at most one nontrivial solution for each λ satisfying λ1 < λ < λ2; in the corresponding uniqueness result for (P) q is restricted to being strictly positive somewhere on ∂Ω Additional properties for the solutions of (P) and (Ph) are also proved.

This paper continues the study of the boundary value problem, for (λ, ψ)

Here δ denotes the Laplacian, k is a given positive constant, and λ1 will denote the first eigenvalue for the Dirichlet problem for −δ on Ω. For λ ≦ λ1, the only solutions are those with ψ = 0. Throughout we will be interested in solutions (λ, ψ) with λ > λ1 and with ψ > 0 in Ω.

In the special case Ω = B(0, R) there is a branch ℱe, of explicit exact solutions which bifurcate from infinity at λ = λ1 and for which the following conclusions are valid, (a) The set Aψ,

is simply-connected, (b) Along ℱe, ψm → k, ‖ψ‖1 → 0 and the diameter of Aψ tends to zero as λ → ∞, where

Here it is shown that the above conclusions hold for other choices of Ω, and in particular, for Ω = (−a, a)×(−b, b). (Existence is settled in Part I, and elsewhere.)

The results of numerical and asymptotic calculations when Ω = (−a, a)×(−b, b) are given to illustrate both the above, and some limitations in the conclusions of our analysis.

We give a simple description of the wave operators appearing in the Lax-Phillips scattering theory. This is used to derive a relation between the scattering matrix and a kind of time delay operator and to characterize all scattering systems having the same scattering operator.

If ℍ is a rational indefinite quaternion algebra and T is an order of ℍ, let G+(T) denote the subgroup of units u ∊ T with norm n(u) = + 1. For a certain class of orders, Eichler has determined the measure μ(G+(T)) of G+(T), viewed as a Fuchsian group. This is extended to arbitrary orders by methods depending only upon classical number-theory and group theory. As an application, an estimate of the magnitude of a small non-trivial solution of the diophantine Pellian equation

is supplied and the restrictions on the integers D and P associated with previous work on this question are eliminated.

For the investigation of certain kinds of reflexivity in topological vector spaces, Sova has shown that the ideas of panneaux and of espaces pannelés are as crucial as are those of barrels and of barrelled spaces. The objectives of this note are: (i) to establish the existence of (concentrated) barrels which are not panneaux, and (ii) to observe the equivalence of the conditions “pannelé” and “barrelled” for metrizable spaces, and the failure of this equivalence for general spaces.

Suppose that f: ℝ×ℂN→ℂN is holomorphic in z and continuous in t, and that Φ: ℂN×ℂN→ℂN is holomorphic. Boundary value problems of the form

are considered. The particular interest is in the structure and topological properties of the set of solutions. The paper is motivated by the corresponding properties of the set of periodic solutions of ż = f(t, z) when f is periodic in t. Consideration of this complex equation gives information about the periodic solutions of the real equation ẋ = f(t, x).

The solutions of stochastic differential equations are used to construct Markov processes on the Banach manifold C(S, M) of continuous maps from a compact metric space S into a smooth complete finite dimensional Riemannian manifold M. In the special case where S is a single point the construction gives a large class of diffusion processes on the manifold M, including (under certain curvature conditions) the Brownian motion process.

It is shown that the elements of the closed operator algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients with a fixed finite set of points of discontinuity can be written as the sum of a singular integral operator, a compact operator, and generalized Mellin convolutions. Their Gohberg-Krupnik symbol is given in terms of the Mellin transform. This gives an explicit construction of an operator with prescribed Gohberg—Krupnik symbol.

A characterization of the symmetric group of degree 9 is given in terms of the centralizer of an element of order 3.

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.

Let N(u, g, h) denote the number of subgroups of index u, genus g and parabolic class number h in the classical modular group. In 1974, Petersson proved that, for g = 0 and h = 1, 2, N(u, g, h) grows exponentially with u. We obtain a formula valid for all g and h. From this we derive an asymptotic estimate, showing that Petersson's result gives the correct type of growth. The proofs use coset diagrams.