We study certain perturbations of the differential equation Δu − u + up = 0 on all of n-dimensional Euclidean space. Conditions are obtained which ensure the existence of a solution to the perturbed equation near a given solution to the unperturbed equation. We have to overcome degeneracy of the unperturbed solution and lack of smooth dependence on the perturbation parameter. An abstract version of the argument is sketched in a functional-analytic setting related toequivariant bifurcation theory. We consider also a smooth perturbation with several parameters and study the singularities of the mapping which maps each solution to its associated parameters.

This paper is devoted to the existence of periodic solutions for the scalar forced Lienard differential equation

The key assumptions relate the asymptotic behaviour as x →± ∞of g(t; x)/x to the “critical values” of the positively 1-homogeneous problem

No condition on f, except continuity, is assumed. Our approach is based on Leray–Schauder degree techniques and a priori estimates.

We give examples of symplectic manifolds which are also non-trivial principal torus-bundles with Lagrangian fibres. These bundles are examples of spaces with an obstruction to the global existence of action-angle variables.

Let us consider equations in the form

where Λ is an open subset of a normed space. For any fixed λ ∊ Λ, T, L(λ,.) and M(λ,.) are mappings from the closure D0 of a neighbourhood D0 of the origin in a Banach space X into another Banach space Y with T(0) = L(λ, 0) = M(λ, 0) = 0. Let λ be a characteristic value of the pair (T, L) such that T − L(λ,.) is a Fredholm mapping with nullity p and index s, p> s≧ 0. Under sufficient hypotheses on T, L and M, (λ, 0) is a bifurcation point of the above equations. Some well-known results obtained by Crandall and Rabinowitz [2], McLeod and Sattinger [5] and others will be generalised. The results in this paper are extensions of the results obtained by the author in [7].

We prove the global existence and uniqueness of the solution of the initial and boundary value problem for the equation

by using the classical Galerkin method when the forcing term and the initial data are in some sense small. The asymptotic behaviour of the solution as t → ∞ is also considered.

This paper is concerned with the problem of showing uniform stability and equiasymptotic stability of thezero solution of functional differential equations with either finite or infinite delay. The investigations are based on Liapunov's direct method and attention is focused on those equations whose right-hand sides are unbounded for bounded state variables.

The set of effective elasticity tensors for all two-dimensional mixtures of two isotropic incompressible elastic materials taken in prescribed proportion is described. In two dimensions the effective tensors are completely characterised by bounds on their eigenvalues.

We consider the class of Noetherian UFN-rings, that is, Noetherian prime rings such that every non-zero ideal contains a non-zero normal element and such that the monoid of non-zero normal elements is a unique factorisation monoid. We ask whether this class of rings, a generalisation of the commutative Noetherian unique factorisation domains (UFD), is closed under polynomial extensions. The general question is apparently difficult and remains open. However, we give positive answers in special cases, in particular, for algebras over an infinite field and for domains.

Given a set valued mapping ∑: ℝ → ∑n, we prove a closure property with respect to -convergence for the differential inclusion

under very mild assumptions on ∑.

In [2] Atkinson considers the asymptotic form of the eigenvalues of the linear differentialequation

where a < 0 < b and y satisfies appropriate conditions at a andb. In particular Atkinson considers where q is singular at 0. In this case q(x) = xK, his results cover the case l ≦ K < . We extend Atkinson's results to cover more singular q, in the power case 1 ≦K <

A necessary and sufficient condition for the boundedness of the generalised Stieltjes transformation on weighted Lebesgue spaces is obtained and applied to extend the Hilbert double series theorem.

In this paper the authors investigate two classes of triple series equations of trigonometric type. Closed form solutions are given for eight triple series in each class.

In this paper we study the global existence and nonexistence of spatially periodic solutions to the initial value problem of the Ginzburg–Landau equation.

The Ambarzumyan theorem connecting the Sturm–Liouville problem and the corresponding problem associated with the Fourier differential equation is extended to a class of second order matrix differential systems.

This paper presents new comparison and uniqueness results for the solutions of parabolic quasilinear boundary value problems with (and without) obstacles. A stability result in L1(Ω) yields the asymptotic stabilisation in this space, when t → ∞) towards the corresponding elliptic problem.

In this paper we extend the notion of equichordal curve to closed simple curves in ℝn. Although it is not known if an equichordal curve can have more than one fulcrum, we show that, for plane curves, any fulcrum is inside the curve. We establish connections with the theories of transnormal and self-parallel curves and lower bounds for the length and chordal area are obtained. Such bounds are the best possible.

When a rigid body performs a rotation in a fluid, the system of governing equations consists of conservation of linear momentum of the fluid and conservation of angular momentum of the rigid body. Since the torque at the interface involves the drag due to the fluid flow, the conservation of angular momentum may be viewed as a boundary condition for the field equations of fluid motion. The familiar no-slip condition becomes an additional equation in the system which not only governs the fluid motion, but also the motion of the rigid body. The unknown functions in the system of equations are the velocity field and the pressure field of the fluid motion and the angular velocity of the rigid body.

In this paper we obtain existence and uniqueness results for the steady state problem in which a rigid body rotates about an axis of symmetry in a viscous incompressible fluid.

Certain permutation representations of the Hall–Janko group J2 are studied. These representations are of interest in connection with the problem of whether J2 can act as astrongly irreducible collineation group of a finite projective plane.

Existence as well as uniqueness theorems under certain growth conditions are given for entire solutions to linear elliptic equations in the n-dimensional space (2≦n). Introducing a proper norm, the proofs are based on a priori estimates. These estimates could be used to solve nonlinear equationsin the space, but the conditions on the nonlinearity have to be strong as the a priori estimates applyonly to classical, not to weak, solutions.

This paper discusses the relationship between the uniqueness and the ordering of strictly positive solutions of elliptic predator-prey interacting systems. If (ū, v) and (u#, v#) are two such solutions with ū ≧ u# or v ≧ v#, then ū ≡= u#, v ≡v#. When the positive solutions are numerically close to the extreme case, the solution is unique.