Sufficient conditions are given which ensure nonexistence of spherically symmetric entire solutions of Δpu = f(u), p ≧ 2. Sufficient conditions for existence of spherically symmetric entire solutions of Δpu = f(r, u) are also given.

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.

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.

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.

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.

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.

In the Hilbert space framework, we give some results concerning the behaviour when t goes to infinity for solutions of equations of the form:

A is assumed to be a maximal monotone operator and F(t) is a periodic function.

When F = 0, under a compactness assumption for trajectories of (1), we give the complete description of the asymptotic behaviour, e.g. every trajectory is asymptotic to an almost-periodic solution of (1). When F ≠ cst, the compactness hypothesis being too restrictive, we concentrate our efforts on the case of the equation:

with Dirichlet boundary condition) and get weak convergence to particular solutions of the equation when β is either univalued or strictly monotone. The methods used in these cases seem of general interest for hyperbolic equations of dissipative type with periodic forcing term.

In this paper we study the wave equation, in particular the propagation of discontinuities. Two problems are considered: diffraction of a normally incident plane pulse by a plane screen and diffraction of a spherical wave by the same screen. It is shown that when an incident wave front strikes the edge of the screen a diffracted wave front is produced. The discontinuities are precisely computed in a neighbourhood of the edge for a small time interval after the arrival of the incident wave front and a theorem of Hörmander on the propagation of singularities is used to obtain a globalresult.

The interior initial-boundary value problem for the wave equation in m ≧ 1 space dimension is considered for vanishing boundary values. Certain regularity, dependent on m, is required for the solution and additionalboundary conditions, the number of which being also dependent on m, are imposed on the given right hand side. Emphazising the case m = 3, the Rothe method is applied after the problem has been rewritten as a hyperbolic first order evolution problem for m + 1 unknown functions. The sequence of discrete solutions obtained is shown to be discretely convergent to the continuous solution in the sense of uniform convergence if the solution of the continuous problem is assumed to exist. A priori estimates are derived both for the discrete solutions and the continuous solution.

Given any countably infinite set of isloated points on the ℷ -axis, it is shown that there is a continuous q(x) such that these points constitute exactly the point-continuous spectrum for the equation yn″(x) + (ℷ —q(x))y(x) = 0(0≦x<∞) with some homogenous boundary condition at x = 0. This extends a result given by Eastham and McLeod for countably infinite sets of isolated points on the positive ℷ-axis.

The following theorem is proved: Let S(t), t≧0 be a dynamical system in an infinite dimensional Banach space X such that S(t) = S1(t)+S2(t) for t≧0, where (1) uniformly in bounded sets of x in X, and (2) S2(t) is compact for t sufficiently large. Then, if the orbit {S(t)x: t ≧0} of x ∈ X is bounded in X, it is precompact in X. Applications are made to an age dependent population model, a non-linear functional differential equation on an infinite interval, and a non-linear Volterra integrodifferential equation.

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, ∞).

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.

If the diffusion matrix coefficient of an Itô stochastic differential equation is everywhere non-singular, then the corresponding Chapman-Kolmogorov semi-group may be defined on L∼(Rn), the space of Lebesgue equivalence classes of essentially bounded Borei measurable functions. However, if the diffusion matrix is singular at some points of Rn, it is not clear that this can always be done. We show that in certain situations it is possible to do so.

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.

Consider mild solutions on the real line of non-homogeneous differential equations in a Banach space: u′(t) = Au(t) + f(t), where A is the infinitesimal generator of a C0-semigroup.

We prove an existence result for optimal solutions (as defined in the text) in reflexive spaces and an uniqueness fact in uniformly convex B-spaces.

The construction of the approximate solution within a rectangle of a singular elliptic problem is discussed. It is found that, provided the boundary data satisfy certain continuity conditions at the corners of the rectangle, ordinary boundary layers and parabolic boundary layers only are necessary to describe the solution. A correction term, however, has to be added to the solution if the continuity conditions on the boundary data are not satisfied.

One considers the linear differential systems where is a (not necessarily diagonal) matrix and one relates the computation of a general multiplicity defined from this system to the corresponding multiplicity of some eigenvalues of . Then applying these conclusions, one gives simple conditions ensuring the existence of odd or even periodic solutions for systems having the form .