In this paper a spectral theory for completely coupled linear operator systems is developed. These systems take the form

where Ak, Bk are n × n matrices with operator entries. Λ is an n × n matrix with complex scalar entries and xk is an n × 1 column vector. The main result is a Parseval equality and expansion theorem.

In this paper the boundary behaviour of the solution of Dirichlet's problem for the equation Δu = f in domains with C2,α-boundary is studied. Without using Schauder's a priori estimates, we prove the following theorem, which is indispensable in Schauder's method to solve Dirichlet's problem for a general linear elliptic equation.

Theorem: Let (f, g)∈ C0,α (Ḡ)×C2,α (∂G) and u ∈ C0.0(Ḡ) ∩C2,α(G) be a solution of the problem Δu = f, u|∂G = g. Then u ∈ C2,α(Ḡ).

We present an example (section 3) which serves two purposes: it is a counterexample to a conjecture of Weidmann (see section 1) and it clarifies the relations between different properties related to A-compactness (see section 2).

The semilinear elliptic differential operator L[u] = Δu + c(x, u) is studied and sufficient conditions are derived for all solutions of uL[u] ≦ 0 with suitable boundary conditions to be oscillatory in unbounded domains of Rn. Here, unbounded domains to be considered are cones, strips and cylinders in Rn. The results are based on the conditions for the non-existence of positive solutions of ordinary differential inequalities.

We study the convergence to the stationary state for the parabolic equation u, = uxx + F(u). There exist wave-type solutions u(x, t) = φ(x − ct) for a continuum of velocities c. In the asymptotic behavior of this equation was investigated for a step function as initial data. In this paper we obtain the asymptotic behavior for a large class of monotone initial data.

All solutions with initial data in this class evolve to wave-type solutions, where the rate of decay of the initial data determines the asymptotic speed.

Conditions are obtained for certain elliptic balls in ℂn+1 to have empty intersection with the Nyquist set of a vector polynomial G(z). Such conditions are shown to yield explicit criteria for the existence of periodic solutions of non-autonomous scalar differential equations of the form A* G(D)y = p.

This paper is devoted to the study of the weak respectively strong convergence of solutions of a variational inequality, with nonlinear partial differential operators of the generalized divergence form and of semimonotone type, under a perturbation of the domain of definition. In this study we use abstract convergence theorems given by Stroescu and Vivaldi, convergence concepts defined according to Stummel and compactness theorems of the natural imbedding of the Cartesian product of Sobolev spaces into the direct sum of Lp spaces, also by Stummel.

Two closely related classes X(n) and Y(n) of two generator two relation groups are studied. The group presentations arise from an investigation of a Fibonacci type group of order 1512. The Reidemeister-Schreier algorithm is used to show that the groups X(n) are finite and not metabelian. The orders of these groups are determined and shown to be divisible by powers of Fibonacci numbers or by powers of Lucas numbers. In addition these groups add to the relatively few examples of non-metabelian two generator two relation groups whose orders are known precisely.

We show that if L is a bounded (semi-)lattice and f is a range-closed idempotent residuated mapping on L then f is multiplicative if and only if it is decreasing. We then use this to prove that a bounded (semi-)lattice is implicative if and only if it can be coordinatised by a weakly multiplicative (right) Baer semigroup.

We construct a fundamental solution for the n dimensional time independent anisotropic neutron transport equation. This is an operator valued distribution G(x) with a singularity at the origin. By estimating G(x) we are able to construct smooth solutions to the transport equation. We are also able to derive in a straightforward fashion results of Birkhoff and Abu-Shumays on the existence of harmonic solutions to the isotropic transport equation. When n = 1, G(x) is a function which is continuous except at x = 0. We show that the classical formula for the jump of G(x) at the origin is equivalent to the completeness of Case's full range eigenfunction expansion.

The object of this paper is to demonstrate, that with the open mapplng theorem of S.Banach one can prove very easily the following estimate

for all u ∈ C2,α and 0 ≤ t ≤ 1, if one knows, that for all bounded G ⊂ Rn, with boundary ∂G ∈ C2,α and for all (f, g) ∈ C0,α × C2,α (∂G) Dirichle's problem Δu = f, u|∂G = g has a solution u ∈ C2,α. This estimate can be used to solve Dirichle's problem for a general linear elliptic equation by Schauder's method.

In previous papers [11,12,13], certain spaces of generalized functions were studied from the point of view of fractional calculus. In this paper, we show how a Hankel transform Hv of order v can be defined on for all complex numbers v except for those lying on a countable number of lines of the form Re v = constant in the complex v-plane. The mapping properties of Hv on are obtained. Various connections between Hv (or modifications of Hv) and operators of fractional integration are examined.

This paper presents two inequalities satisfied by any electromagnetic radiation field which is defined in the region outside a smooth, bounded surface and which varies harmonically with time. These inequalities provide an upper bound on the error introduced into the radiation pattern of the field when the field is approximated in numerical calculations. All constants which appear in the inequalities and the error estimate are computable.

The Epstein operator is defined by

where x = (x1, …, xn) ∈ Rn, y ∈ R,

and H, K, L, M are real constants such that c2(y) > 0. The operator arises in the study of acoustic wave propagation in plane-stratified fluids with sound speed c(y) at depth y. In this paper it is shown that A defines a selfadjoint operator in the Hilbert space ℋ = L2(Rn + 1c−2(y) dx dy) where dx = dx1 … dxn. The spectral family of A is constructed, the spectrum is shown to be continuous and an eigenfunction expansion for A is given in terms of families of improper eigenfunctions.

Here the Riemann boundary value problem-well known in analytic function theory as the problem to find entire analytic functions having a prescribed jump across a given contour-is solved for solutions of a pseudoparabolic equation which is derived from the complex differential equation of generalized analytic function theory. The general solution is given by use of the generating pair of the corresponding class of generalized analytic functions which gives rise to a representation for special bounded solutions of the pseudoparabolic equation. These solutions are obtained by linear integral equations one of which is given by a development of the generalized fundamental kernels of generalized analytic functions and which leads to a Cauchy-type integral representation. The bounded solutions are needed to transform the general boundary value problem (of non-negative index) with arbitrary initial data into a homogeneous problem which can easily be solved by the Cauchy-type integral (if the index is zero).

Conditions are obtained which ensure that the maximal operator generated by the formally self-adjoint second-order differential expression

in L2(Rn), n ≥ 1 has the Dirichlet and separation properties.

M. G. Krein's method of directing functional is generalized in a straight-forward way to symmetric linear relations. Applications to Stieltjes differential boundary problems are indicated.