Let d denote the dimension of the vector space consisting of all solutions of the equation − (p(t)y′)′ + q(t)y = 0, a ≤ t < ∞; that lie in the function space L2[a, ∞). By means of certain bounds on the solutions of this equation, sufficiency criteria are obtained for the cases d = 0 and d = 2.

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.

We consider the equation Lφ − λpφ + γqφ2 = f on a bounded domain in Rn with homogeneous Neumann-Dirichlet boundary conditions. L is a negative definite uniformly elliptic differential operator, while, p, q and f are positive functions. We show that there exists exactly one positive solution for each λ ∈ R and γ > 0. This solution can be analytically continued throughout Re γ > 0: it is a Laplace transform of a positive measure. The measure is bounded prior to the bifurcation point of the associated “homogeneous” equation and unbounded after. Noting that any Laplace transform of positive measure has associated with it a natural sequence of Tchebycheff systems, it now follows that one can obtain monotonically converging upper and lower bounds which are provided by the generalized Padé approximants generated from the Tchebycheff systems.

This paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expression

on [0,∞), together with a real homogeneous boundary condition at t = 0.

In this paper, the electrostatic potential of a point charge in a Reisser-Nordström gravitational field is found in closed form by using the theory of Hadamard's elementary solution of a partial differential equation of elliptic type.

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.

This paper is concerned with integral inequalities of the form

where p, q are real-valued coefficients, with p and w non-negative, on the compact interval [a, b] and D is a linear manifold of functions so chosen that all three integrals are absolutely convergent.

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.

We consider the nonconvolution initial value problem

where μ is a small positive parameter, b(t, s) is a given real kernel, and F, g are given real functions. For the convolution case b(t,s) = a(t − s). Lodge, McLeod, and Nohel recently established many qualitative properties of the solution of (+); we extend their results to the general nonconvolution problem. In particular, conditions are given that ensure that the solution of (+) decreases to a limiting value α(μ) > 1 as t → ∞.

It has been known for some time that certain least-squares problems are “ill-conditioned”, and that it is therefore difficult to compute an accurate solution. The degree of ill-conditioning depends on the basis chosen for the subspace in which it is desired to find an approximation. This paper characterizes the degree of ill-conditioning, for a general inner-product space, in terms of the basis.

The results are applied to least-squares polynomial approximation. It is shown, for example, that the powers {1, z, z2,…} are a universally bad choice of basis. In this case, the condition numbers of the associated matrices of the normal equations grow at least as fast as 4n, where n is the degree of the approximating polynomial.

Analogous results are given for the problem of finite interpolation, which is closely related to the least-squares problem.

Applications of the results are given to two algorithms—the Method of Moments for solving linear equations and Krylov's Method for computing the characteristic polynomial of a matrix.

This paper obtains, under certain general conditions on the coefficient q, a best-possible upper bound on the real parameter λ for the differential equation

to have a non-trivial solution in the integrable-square space L2 (a, ∞).

The inequality considered in this paper is

where N is the real-valued symmetric differential expression defined by

General properties of this inequality are considered which result in giving an alternative account of a previously considered inequality

to which (*) reduces in the case p = q = 0, r = 1.

Inequality (*) is also an extension of the inequality

as given by Hardy and Littlewood in 1932. This last inequality has been extended by Everitt to second-order differential expressions and the methods in this paper extend it to fourth-order differential expressions. As with many studies of symmetric differential expressions the jump from the second-order to the fourth-order introduces difficulties beyond the extension of technicalities: problems of a new order appear for which complete solutions are not available.

We consider the Friedrichs extension A of a minimal Sturm-Liouville operator L0 and show that A admits a Schrödinger factorization, i.e. that one can find first order differential operators Bk with where the μk are suitable numbers which optimally chosen are just the lower eigenvalues of A (if any exist). With the help of this theorem we derive for the special case L0u = −u″ + q(x)u with q(x) → 0 (|x| → ∞) the inequality

σd(A) being the discrete spectrum of A. This inequality is seen to be sharp to some extent.

The asymptotic behaviour of semigroups of nonlinear contractions which have a set of fixed points containing a ball of finite codimension is studied. It is shown that the ω-limit sets of such semigroups are finite dimensional tori, and that an analogue of the classical Kronecker-Weil theorem holds for such semigroups.

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.

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.

We study the initial value problem for the nonlinear Volterra integrodifferential equation

where μ > 0 is a small parameter, a is a given real kernel, and F, g are given real functions; (+) models the elongation ratio of a homogeneous filament of a certain polyethylene which is stretched on the time interval (— ∞ 0], then released and allowed to undergo elastic recovery for t > 0. Under assumptions which include physically interesting cases of the given functions a, F, g, we discuss qualitative properties of the solution of (+) and of the corresponding reduced problem when μ = 0, and the relation between them as μ → 0+, both for t near zero (where a boundary layer occurs) and for large t. In particular, we show that in general the filament does not recover its original length, and that the Newtonian term —μy′ in (+) has little effect on the ultimate recovery but significant effect during the early part of the recovery.

We consider the eigenfunction expansions associated with a symmetric differential operator M[·] of order 2n with coefficients defined on an open interval (a, b). Each singular endpoint of (a, b) is assumed to be of limit-n type. A direct convergence theory is established for the eigenfunction series expansion of a function y in a set Termwise differentiation of the series is established for the derivatives of order up to n. For O ≤ i ≤ n − 1, the i-fold differentiated series converges absolutely and uniformly to y(i) on compact intervals; the n−fold differentiated series converges to yn in the mean. The expansion theory is valid also when an essential spectrum is present. An explicit formula is given for the calculation of the spectral matrix.

If G is a group, then G is amenable as a semigroup if and only if l1(G), the group algebra, is amenable as an algebra. In this note, we investigate the relationship between these two notions of amenability for inverse semigroups S. A complete answer can be given in the case where the set Es of idempotent elements of S is finite. Some partial results are obtained for inverse semigroups S with infinite Es.

This paper considers semilinear elliptic boundary value problems of the form

where the partial derivative ∂f/∂u is bounded above by the least eigenvalue of the linear elliptic operator L. Existence and uniqueness of solutions is proved by using monotone operator theory and sub and supersolution techniques.