Let P(x) and Q(x) be n × n matrices whose entries involve a single given function q(x). Under suitable conditions, estimates

of Liouville-Green type are obtained for the components yν(x) of solutions of the system

In the case n = 2, the estimate is applied to systems arising from fourth-order equations and a generalization of a result of Eastham [2] is obtained.

Suppose that f(t, λ) is, for fixed t in an open interval (a, b), a regular (or analytic or holomorphic) function of λ, when λ lies in a domain D of the complex plane. We assume that f is integrable with respect to t for fixed λ ∈ D, and consider the parametric integral

By hypothesis F(λ) exists for λ ∈ D and Professor W. N. Everitt has asked whether F(λ) is necessarily regular as a function of λ.

Sufficient conditions are developed for asymptotic stability of the autonomous linear functional differential equation of retarded type. If the asymptotic stability of

implies the asymptotic stability of

then these conditions are also necessary. Necessary and sufficient conditions are developed for the largest cone in the region of stability. These results are illustrated with the example

A description is given of the rational functions A(X), B(X) over a field Ω for which A(B(X)) is an nth power in Ω(X).

All the Fibonacci groups in the family F(2, n) have been either fully identified or determined to be infinite, bar one, namely F(2, 9). Using computer-aided techniques it is shown that F(2, 9) has a quotient of order 152.5741, and an explicit matrix representation for a quotient of order 152.518 is given. This strongly suggests that F(2, 9) is infinite, but no proof of such a claim is available.

An incident wave creates a discontinuity in the acceleration of the shock front. The amplitudes of the reflected and transmitted waves are also determined. Special attention is given to the case of the weak shocks and the characteristic shocks.

The integral inequalities with which this paper is concerned are

and

where K(μ) and K1(μ) are positive numbers which depend on μ.

It is shown that each of these inequalities is valid only when μ = 0. Results relating to the corresponding inequalities on the extended interval (−∞, ∞) are also given.

The solutions of the differential equation Lny + p(x)y = 0, where Lny = ρn(ρn−1 … (ρ1(ρ0y)′)′ …)′ and p(x) is of one sign, are classified according to their behaviour as x → ∞. The solution space is decomposed into disjoint, non-empty sets Sk, 0≦K≦n, such that (−1)n−kp(x)≦0. We study the growth properties and the density of the zeros of the solutions which belong to the different sets Sk, the structure of the sets and its connection with (k, n − k)-disfocality.

We give in this note a second order singular differential expression of the form Lf = −f″ + qf on [0, ∞) that satisfies the Dirichlet condition but that is not bounded below.

This paper is concerned with the structure of M = Maps(G), the near-ring of all mappings from a group G to itself which commute with a group S* of automorphisms of G. Here S is S* together with the zero endomorphism. Necessary and sufficient conditions on the pair (G, S) are obtained for M to be (i) regular, (ii) unit regular, (iii) an equivalence near-ring. These conditions take a very simple form. In the case (iii), the two-sided M-subgroups of M are determined. The next result shows that under suitable conditions, M is a simple near-ring. A definition of transitivity is given for subnear-rings of M, and some properties of transitive near-rings are proved. Finally two examples are given to show that all the classes of near-rings considered are distinct.

Let B be a complete Boolean algebra of projections on a complex Banach space X and let (B) denote the closed algebra of operators generated by B in the norm topology. It is shown that there is a complex Hilbert space H, a complete Boolean algebra B0 of self-adjoint projections on H, and an algebraic isomorphism of B onto B. This isomorphism is bicontinuous when B and B are endowed with the norm topologies, the weak operator topologies or the ultraweak operator topologies. It is also bicontinuous on bounded sets with respect to the strong operator topologies on B and B. As an application, it is shown that the weak and ultraweak operator topologies in fact coincide on B.

This paper deals mainly with showing that meromorphic solutions of Poincaré functional equations cannot satisfy algebraic differential equations. Results of this type for entire solutions have previously been obtained by H. Wittich.

It is established that under certain restrictions the solution u of the equation uxy − gu = 0 satisfying u(x, 0) = p(x) and u(0,y) = q(y) in ([0, = ∞) × [0, ∞), changes sign in where (X, Y) is any point in the relevant region.

Conditions for the existence of anti-isomorphisms and anti-automorphisms of completely 0-simple semigroups are derived and a method is given for the enumeration of finite 0-simple and simple semigroups up to isomorphism and anti-isomorphism.

Formulae are derived for the first two approximations to the radiation pattern of a penetrable body in the long wavelength limit. They are expressed in terms of a solution of Laplace's equation satisfying certain boundary conditions. Various inequalities and principles which help fix this solution are given.

By adapting its well-known proof, the Poincaré–Bendixson theorem, on the existence of periodic orbits of plane autonomous systems, is extended to vector differential equations of the form f(D)x + bφ(g(D)x) = 0. The only restrictions placed on the vector function φ(y) are that its Jacobian matrix should be continuous and lie within a suitably chosen ellintic ball.

A non-linear spectral theory is developed which includes the spectral theory of linear operators and the theory of (maximal) monotone operators. In this nonlinear theory certain polytone operators will play the role of the linear or monotone operators. The concept of λ-polytonicity allows the characterization of regular points in terms of maximality. Furthermore, properties of the spectrum of non-linear operators are discussed in terms of the corresponding properties of their linearizations and vice versa.

A stability condition is derived for solutions of the Von Kárman-Batchelor equations for the flow of a viscous, incompressible fluid between rotating, coaxial (infinite) disks. The rigid motion solution which arises when the angular velocities of the disks are equal is stable with respect to perturbations which go to zero sufficiently rapidly at infinity, for all values of the Reynolds number. If the angular velocities are sufficiently close the stability condition derived applies to perturbations whose “deformation energy” is sufficiently confined to a “core” region.

Let L be a formally self-adjoint linear differential operator of order m with strictly positive leading coefficient and let m = 2n + 1 if m is odd, m = 2n if m is even. Let y1, y2,…, yn be n given mutually conjugate solutions of Ly = 0 on I, where I is some interval, whose Wronskian is non-zero on I. Then L = (−1)nQ*Q or L = (−1)nQ*DQ where Q is a differential operator of order n, Q* is the adjoint operator and D denotes differentiation. This fact is used to construct further solutions yn+1,−, ym of Ly = 0 so that y1,…, ym is a basis for the solutions of Ly = 0 and for which yi and yn+j are mutually conjugate if i ≠ j. If y1 ≠ 0 on I the degree of L may be lowered by 2 to obtain a formally self-adjoint operator L1 for which mutually conjugate solutions are constructed. If this process is continued a factorization result is obtained which is related to a result of Pólya.

The theories of fractional calculus and of the Hankel transform developed in for the spaces F′p, μ of generalised functions are used to study distributional analogues of dual integral equations of Titchmarsh type. These are shown to have infinitely many solutions in F′p, μ under very general conditions on the parameters involved. These results are used to study the corresponding classical problem in weighted Lp spaces. Existence and uniqueness of classical solutions are investigated and examples given of both uniqueness and non-uniqueness for the classical problem.