In the Banach space of real sequences which converge to zero with the supremum norm, we construct a parallelisable dynamical system with uniformly-bounded trajectories.

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

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.

The purpose of this paper is to give universal bounds for the order of the Postnikov k-invariants of infinite loop spaces. This is done by giving universal bounds for the order of the k-invariants of m-connected r-fold loop spaces in dimensions ≦ r + 2m. An application of the result provides information on the Hurewicz homomorphism between the algebraic K-theory of aring and the homology of its general linear group.

A 3-dimensional autonomous ordinary differential equation is studied which models certain cellular biochemical reactions. Extended Poincaré-Bendixson theory is used to obtain algebraic conditions on the parameters which are sufficient for the existence of at least one stable closed trajectory. Similar conditions are also obtained for the absence of chaos and for the global convergence of solutions to a critical point.

The equilibrium equations for elastic deformations of an infinite strip are considered. Under the assumption of sufficiently small strains along the whole body, it is shown that all solutions lie on a six-dimensional manifold. This is achieved by rewriting the field equations as a differential equation in a function spaceover the cross-section, the axial variable taken as time. Then the theory of centre manifolds for elliptic systems applies. Thus the local Saint-Venant's problem is solved. Moreover, the structure of the finite-dimensional solution space is analysed to reveal exactly the two-dimensional rod equations of Kirchhoff. The constitutive relations for this rod model are calculated in a mathematically rigorous way out of the constitutive law of the material forming the strip.

In this paper, we study problems of the form:

We obtain some existence and regularity results when Ω is either a ball or an annulus, without convexity hypothesis on g. We then apply these results to some shear problems in nonlinear elasticity.

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.

A Levinson theorem is proved for a Dirac system with one singular endpoint. The number ofbound state is expressed in terms of the change in asymptotic phase of an appropriate solution and in terms of factors whose values depend on the presence of half-bound states. The behaviour of the asymptotic phase is used to determine the asymptotic behaviour of the Titchmarsh-Weyl m-function.

In this paper we establish the best possible value of the constant K in the inequality ∥f ′ ∥2 ≦ K ∥f∥ ∥f″∥ for functions f that are defined and twice continuously differentiable on a compact interval [a, b] ⊂ ℝ and whose first derivative vanishes at some point in [a, b].

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 ∑.

Let ℤ(G) be the integral group ring of the infinite dihedral group G; the aim is to calculate Autℤ(G), the group of Z-linear automorphisms of ℤ(G). It is shown that Aut ℤ(G), the subgroup of Aut *ℤ(G) consisting of those automorphisms that preserve elementwise the centre of ℤ(G), is a normal subgroup of Aut *ℤ(G) of index 2 and that Aut*ℤ(G) may be embedded monomorphically into M2(ℤ[t]), the ring of 2 × 2 matrices over a polynomial ring ℤ[t] From this embedding and by the Noether—Skolem Theorem it is shown that Inn (G), the group of inner automorphisms of ℤ(G) induced by the units of ℤ(G), is a normal subgroup of Aut*ℤ(G)/ such that Aut*ℤ(G)/Inn (G) is isomorphic to the Klein four-group.

In this paper we study some properties of a semilinear elliptic eigenvalue problem with nondefinite right-hand side. In the first part we show that every solution will have its maximum in some specified interval J. If the domain is inside a cone in ℝN with N > 1, then J is strictly smaller than in the one-dimensional case. In the second part we show, for bounded domains, that if the maximum is inside some subinterval of J, then for any eigenvalue there will be at most one solution.

The even and odd Hilbert transformations, H+ and H−, are known to be bounded from the power-weighted space ℒμp to itself for −1 < μ 1 and 0 < μ < 2 respectively. We show that they fail to be surjective on ℒ0,p and ℒ1, p respectively, and we characterise the spaces H+(ℒ0, p), H- (ℒ1, p) and find inverses for H+, H− on them.

The paper deals with smooth nonlinear ODE systems in ℝn, ẋ = f(x), such that the derivative f′(x) has a matrix representation of Jacobi type (not necessarily symmetric) with positive off diagonal entries. A discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system ẏ = f′(x(t))y. Two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system; it is also proved that the nonwandering points are critical points. A new class of Morse–Smale systems in ℝn is then explicitly constructed.

We treat the time-harmonic Maxwell equations in an exterior domain with prescribed boundary data [n, E] in the Sobolev space of square integrable tangential fields with square integrable surface divergence. By using boundary integral equation methods, existence and uniqueness results are established. Furthermore, we investigate the completeness of electric and magnetic dipoles distributed on an inner surface in this Sobolev space.

The equation studied here is Lny + p(x)y = 0, where Ln is a disconjugate differential operator and p(x) is of a fixed sign. We define a basis of the solution space and order its elements according to their relative magnitudes near infinity. Our method is independent of the possible oscillation or nonoscillation of the solutions and it is achieved by utilising the fact that some minors of the Wronskian never vanish.

Semilinear elliptic partial differential systems of second order with weak coupling are considered in exterior domains Ω ⊆ ℝN, N≧3. Conditions on the nonlinearities are given which guarantee the existence of solutions u with positive components in Ω such that u|∂Ω = 0 and u(x)→0 uniformly as |x|→∞. Asymptotic decay estimates for the solutions are established, including an exponential decay law under extra hypotheses.

We consider a class of non-self adjoint multipoint eigenvalue problems. Using necessary conditions for the regularity of these problems, we obtain a theorem on the expansion of certain functions into a series of eigen- and associated functions.