The real difference equation an+2 − (λ|an+1| + μan+1) + an = 0 may be interpreted as a dynamical system Φ:(an, an+1) ↦ (an+1, an+2) acting in the plane. The set ΛP of points (λ, μ) for which the mapping Φ is periodic has a rich structure. In this paper, we derive some geometric properties of ΛP (for example, we show that it is unbounded and uncountable), and we derive criteria for Φ to be periodic. We also investigate when Φ is conjugate to a rotation of the plane, and we describe how the rotation numbers of the corresponding circle maps Φ/|Φ| are related to the structure of ΛP.

Using the viscosity-capillarity admissibility criterion for shock waves, we solve the Riemann problem for the system of conservation laws

where σ(u) = u3 − u. This system is hyperbolic at (u, v) unless . We find that the Riemann problem has a unique solution for all data in the hyperbolic regions, except for a range of data in the same phase (i.e. on the same side of the nonhyperbolic strip). In the nonunique cases, there are exactly two admissible solutions. The analysis is based upon a formula describing all saddle-to-saddle heteroclinic orbits for a family of cubic vector fields in the plane.

As a generalisation of the well-known result of Perron and Frobenius, it was shown by Rothblum [13] and independently by Richman and Schneider [12] that every nonzero matrix with non-negative entries has a basis of the root space corresponding to the maximal eigenvalue, represented by root vectors with non-negative entries. Krein and Rutman [9] showed that a positive compact nonquasinilpotent operator on a Banach lattice has a positive eigenvector corresponding to its spectral radius. As an extension of both results, we give sufficient conditions on such an operator in order that its spectral subspace corresponding to its spectral radius has a basis made exclusively of positive root vectors.

In [3], uniform asymptotic expansions were derived for solutions of the oblate spheroidal wave equation (z2 − 1)d2p/dz2 + 2zdp/dz − (λ + μ2/(z2 − 1)) p = 0, for the case where the parameter μ is real and non-negative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u → ∞, uniform asymptotic expansions were derived involving elementary, Airy and Bessel functions, these being valid in certain subdomains of the complex z plane. In this paper the complementary case, where λ is real and negative, is considered. Asymptotic expansions are derived which are valid in certain subdomains of the half-plane |arg (z)| ≦ π/2, uniformly valid for u → ∞ with λ /u2 fixed and negative, and 0 ≦ μ/u ≦ − ½λ /u2 − δ, where δ is an arbitrary positive constant. Explicit error bounds are available for all the approximations.

The problem of approximation of attractors for semidynamical systems (SDSs) in a metric space is studied. Let some (exact) SDS possessing an attractor M be inaccurately defined, i.e. let another (approximate) SDS, which is close in some sense to the exact one, be given. The problem is to construct a set , which is close to M in the Hausdorff metric.

The suggested procedure for constructing is finite, which makes it possible to use it in computations. The results obtained are suitable for numerical approximation of attractors for a rather large class of semidynamical systems, including ones generated by the Lorenz equations and the Navier–Stokes equations.

This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.

It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around singularities, and their bifurcations. Besides, it is proved that such a realisation can always be achieved with FDEs whose nonlinearities are defined in terms of a finite number of delayed values of the solutions.

Existence results are established for second-order boundary value problems for ordinary differential equations on non-compact intervals.

The Weierstrass necessary condition for a multidimensional basic variational problem due to Hadamard is strengthened.

We show the existence of a timelike periodic trajectory for a time-dependent Lorentzian metric on ℝn × ℝ.

Weighted Poincaré inequalities are established in any bounded domain D in ℝn (n ≧ 2), and their connection with the Minkowski content of ∂D is explored.

In this paper we analyse the change of stability of Schrödinger semigroups with indefinite potentials when a coupling parameter varies. Generically, the change of stability takes place at a principal eigenvalue associated with the problem. The uniqueness of the principal eigenvalue is shown for several classes of potentials.

It is shown that Hankel transforms of functions on certain weighted LP spaces satisfy Lipschitz and integral Lipschitz conditions. In particular, Fourier-cosine and Fourier-sine transforms satisfy such Lipschitz conditions on such spaces.

We derive asymptotic formulae for the distribution functions of the real parts of the eigenvalues of an oblique derivative problem involving an indefinite weight function.