The objective is to derive a variation of constants formula for systems of functional differential equations (or delay differential equations) coupled with functional equations (or difference equations). The difficulties arise because of the constraints imposed by the functional equations.

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.

A representation theory for the Hankel transformation Hv, defined for suitable f and for v > −1, by

on the spaces ℒμ, p defined by the norm

is developed. Using this, representation theories are developed for some related transformations.

A class of nonlinear Hill's equations on ℝ is considered, where the nonlinearity is concentrated on a compact interval [−N, N]. For values of the parameter λ not in the spectrum of the linearised equation (which is purely continuous) an equivalent nonlinear Sturm–Liouville problem on [−N, N] with parameter-dependent boundary conditions at x = ± N is given. Extending this problem to all real values of the parameter in a suitable way makes it possible to prove the existence of unbounded solution components for both the extended Sturm–Liouville problem and the original problem. The complicated structure of the extended problem results in new phenomena. For example, the number of zeros of different functions in the same solution component may be different.

We present new explicit solutions to some classes of quasilinear evolution equations arising in different applications, including equations of the Boussinesq type:

and quasilinear heat equations:

The method is based on construction of finite-dimensional linear functional subspaces which are invariant with respect to spatial operators having quadratic nonlinearities. The corresponding nonlinear evolution equations on invariant subspaces are shown to be equivalent to finite-dimensional dynamical systems. Examples of two-, three- and five- dimensional invariant subspaces are given. Some generalisations to N-dimensional quadratic operators are also considered.

We study the asymptotic behaviour as t → ∞ of the solution u = u(x, t) ≧ 0 to the quasilinear heat equation with absorption ut = (um)xx − f(u) posed for t > 0 in a half-line I = { 0 < x < ∞}. For definiteness, we take f(u) = up but the results generalise easily to more general power-like absorption terms f(u). The exponents satisfy m > 1 and p >m. We impose u = 0 on the lateral boundary {x = 0, t > 0}, and consider a non-negative, integrable and compactly supported function uo(x) as initial data. This problem is equivalent to solving the corresponding equation in the whole line with antisymmetric initial data, uo(−x) = −uo(x).

In this paper, using a recent generalisation of Morse Theory, we study the existence of periodic solutions of the Lagrangian equation (1.1) with subquadratic potential and asymptotically flat, nonconstant, time-dependent metric on ℝN. In Section 3, we get also an ‘alternative result’ about the minimal period or the existence of infinitely many solutions.

I discuss implications of the rank-one convexity condition in a particular case.

In this paper, we use the Lyapunov–Schmidt reduction and the S1 × S1-index which is due to Chenkui Zhong to prove that any exact symplectic diffeomorphisms on T2k × CPn × CPm have at least 1 + min {m, n} fixed points.

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.

We state a necessary and sufficient polyconvexity condition in R2X2 for functions of class C1. This condition is applied to f(X) = |X|2(|X|2 −2 det X) for obtaining a convex representation in R2X2 x R.

We define a quantity called the reduced C* exponential rank rcel (A) of a C*-algebra A, which satisfies rcel (A) ≦ cel (A). We show that rcel (A) = ∞ whenever A has two distinct normalised traces which agree on K0(A), and we prove a partial converse. This gives some understanding of why cel (A) = π cer (A) for some C*-algebras A but not for others. We also characterise rcel (A) as the supremum of the rectifiable distances from unitaries in the identity component of the unitary group to the commutator subgroup of this component.

We treat several classes of Riemannian manifolds whose shape operators of geodesic spheres or Jacobi operators share some properties with the ones on symmetric spaces.

Let 1 < p, q < ∞. It is shown for complex scalars that there are no nontrivial M-ideals in ℒ(Lp[0, 1]) if p ≠ 2, and is the only nontrivial M-ideal in .

Let R1(x),…, Rd(x) be rational functions in Iℚ(x), such that 1, R1(x),…, Rd(x) are linearly independent over Iℚ. For almost all primes p, their mod p reductions, are well-defined rational functions over Fp and are linearly independent over Fp We show that asymptotically the points

are uniformly distributed in [0, l)d.

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.

We give an explicit expression for the quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function in terms of the singular values. This envelope is also the convex, polyconvex and rank 1 convex envelope of the Saint Venant–Kirchhoff stored energy function. Moreover, it coincides with the Saint Venant–Kirchhoff stored energy function itself on, and only on, the set of matrices whose singular values arranged in increasing order are located outside an ellipsoid. It vanishes on, and only on, the set of matrices whose singular values are less than 1. Consequently, a Saint Venant–Kirchhoff material can be compressed under zero external loading.

This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in LP(Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L∞. In the case of the interior control, we also prove approximate controllability in C0(Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.

Cardinal interpolation by integer translates of shifted three-directional box splines is studied. It is shown that, for arbitrary orders, k, l, m ∈ N of the directional vectors, this problem is correct if and only if the shift vector is taken from the hexagonal shift region (modulo translation with respect to the lattice Z2). This confirms a conjecture of S. D. Riemenschneider [9], and settles the problem studied in [5] for the special case k = l = m in full generality. The method of proof is from homotopy theory.

Let φ:ℝ→ℝ be defined by φ(s) = |s|p−2s, with p > 1 a fixed number. We extend Sturm Comparison Theorem of the linear differential equation

to the nonlinear differential equation

by using the Wirtinger inequality. A Lyapunov inequality and some oscillation criteria of (E) are also given.