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.

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).

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 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 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.

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.

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.

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 paper is concerned with a Three-dimensional theory of non-linear magnetosonic waves in a turbulent plasma. A perturbation method is used that allows us to obtain a transport equation, like Burgers equation, but with a variable coefficient.

In 1932, Hardy and Littlewood proved the inequalities

The best possible values of the constants K1 and K2 being 1 and 4 respectively. The object of this paper is to prove analogous results for infinite series in which the derivative of the real function f is replaced by the finite difference

of a sequence {an} of real numbers.

Let F be any closed subset of ℝN. Stein's regularized distance is a smooth (C∞) function, defined on the complement cF, that approximates the distance from F of any point x ∈ cF in the manner shown by the inequalities (*) in the Introduction below. In this paper we use a method different from Stein's to construct a one-parameter family of smooth approximations to any positive Lipschitz continuous function, with the effect that the constants in (*) can be made arbitrarily close to 1. It is shown that partial derivatives of order two or more, while necessarily unbounded, are best possible in order of magnitude.

Let T be a hyponormal operator on a Hilbert space, so that T*T – TT*≧ 0. Let T have the Cartesian representation T = H + iJ where H has the spectral family {Et} and suppose that EtJ − JEt is compact for almost all t on a Borei set α satisfying E(α) = I. The principal result (Theorem 3) is that under these hypotheses T must be normal. In case T is hyponormal and essentially normal some sufficient conditions are given assuring that, for a fixed t, EtJ − JEt is compact.

We seek non-trivial solutions (u,λ)∈C1([0,1])×[0,∞ with u(x)≧0 for all x ∈[0,1], of the nonlinear eigenvalue problem –u″(x)=λf(u(x)) for x ∈ (0,1) and u(0)=u(1)=0,where f:[0,∞)→[0,∞) is such that f(p) = 0, for p ∈ [0,1), and f(p) = K(p), for p ∈ (1,∞), and K: [1, ∞)→(0, ∞) is assumed to be twice continuously differentiable. (The value ƒ(1) is only required to be positive.)

Existence and multiplicity theorems are given in the cases where ƒ is asymptotically sub-linear and ƒ is asymptotically super-linear. Moreover if strengthened assumptions are made on the growth of the non-linear term ƒ we obtain the precise number of non-trivial solutions for given values of λ ∈ [0, ∞).

The theory of singular left-definite canonical eigenvalue problems treated by Nieβen and Schneider in is generalized to arbitrary λ∈(ℂ\ℝ∪{0}. In this enlarged theory the Green's matrix of the problem is evaluated and a natural analogue of the Titchmarsh-Kodaira formula is proved. This formula permits the explicit computation of the spectral matrix playing the main role in the expansion theorems of this theory.