In this paper we give some results on the existence of periodic solutions to the second order Hamiltonian system:

where and Ω is an open set of ℝn with non-empty bounded complement ℝn\Ω; we suppose V(t, x) is periodic in t, V(t, x)→ + ∞ as x → ∂Ω and V is super (or sub)-quadratic as |x| → + ∞.

Using explicit equations for Jacobi vector fields on a Sasakian space form, we characterise such spaces by means of the shape operator of small geodesic spheres.

Let Ω be a bounded domain in ℝ2. The study, begun in Keady [13], of the boundary-value problem, for (λ/k, ψ),

is continued. Here Δ denotes the Laplacian, H is the Heaviside step function and one of λ or k is a given positive constant. The solutions considered always have ψ > 0 in Ω and λ/k > 0, and have cores

In the special case Ω = B(0, R), a disc, the explicit exact solutions of the branch τe have connected cores A and the diameter of A tends to zero when the area of A tends to zero. This result is established here for other convex domains Ω and solutions with connected cores A.

An adaptation of the maximum principles and of the domain folding arguments of Gidas, Ni and Nirenberg [9] is an important step in establishing the above result.

We generalise the Paley–Wiener closedness theorem and apply it to a class of time periodic Hamiltonians to show that all solutions to the corresponding Schrodinger equation decay.

Let denote the semigroup, under composition, of all entire functions of a complex variable and for , let Γ(f) denote the Schützenberger group of the ℋ-class of which contains f. The main results of this paper completely describe Γ(P) where P is a polynomial function. Γ(P) is trivial if P is a constant function. If Deg (P) (the degree of P) is one, then Γ(P) is isomorphic to the group of all matrices of the form where a and b are complex numbers with a ≠ 0. If Deg (P) > 1, then Γ(P) is isomorphic to ℂM, the multiplicative group of nonzero complex numbers if and only if P(z) = α(z +β)n + γ where α ≠ 0. If P is not of this form and Deg (P) > 1, then Γ (P)is a finite cyclic group and its order can be determined by defining (where αn−1 and α n are the coefficients in P of zn−1 and zn respectively) and then inspecting the coefficients of Q.

Transmission problems for the vector Helmholtz equation are considered. By using boundary integral equation methods, existence and uniqueness theorems in the form of Fredholm's alternative are established.

In this paper, a counterexample is given to the H1-boundedness of solutions to a sequence of systems of linear partial differential equations uniformly satisfying a strict Legendre–Hadamard condition and whose coefficients depend on one direction only. This counterexample is relevant for the theory of homogenisation of laminated elastic materials.

Weakly coupled semilinear parabolic systems of the form with homogeneous boundary conditions are studied. The nonlinear function g: C([−r, 0] × Ω ℝn) → ℝn is assumed to be locally Lipschitz continuous with r≧0 a given real number and Ω ⊂ ℝm a bounded domain, , ut for t ≧ 0 is denned by ut (σ, ξ) = u(t + σ, ξ), − r ≦σ ≦ 0 ξ ∊Ω and A is a uniformly elliptic second order diagonal operator. Let u be a bounded classical solution. We first establish precompactness results for the orbit of u in several function spaces. Using these results and assuming that a Liapunov function V is known for the corresponding ordinary functional differential equation ż =g(zt), we then show under some general conditions that the limit set ω+ (as t→∞) of u consists of spatially homogeneous functions only. Moreover, ω+ is invariant with respect to z = g(z,) and V = 0 on ω+. The proof uses a Liapunov function for the full system whichis obtained from V via a simple construction (cf. (3.3)). The theory is illustrated with an example.

The nonlinear evolution problem [Bu(t)]′ = A(t, Bu)u + f(t, Bu) with B a constant linear operator and A = A(t, Bu) a time-dependent nonlinear operator from one Banach space to another, is studied. Existence and uniqueness results are obtained by making use of the theory of B-evolutions and the fractional powers of A and B. Two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions.

The semilinear damped wave equations

subject to homogeneous Neumann boundary conditions, admit spatially homogeneous solutions (i.e. u(x, t) = u(t)). In order that every solution tends to a spatially homogeneous one, we look for conditions on the coefficients a and d, and on the Lipschitz constant of f with respect to u.

A solution is given to an open question of Blyth and Varlet concerning bistable subvarieties of MS-algebras.

Periodic solutions of certain one-dimensional non-autonomous differential equations are investigated (equation (1.4)); the independent variable is complex. The motivation, which is explained in the introductory section, is the connection with certain polynomial two-dimensional systems. Several classes of coefficients are considered; in each case the aim is to estimate the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients. In particular, we need to know when there is a full neighbourhood of periodic solutions. We give a number of sufficient conditions and investigate the implications for the corresponding two-dimensional systems.

The characteristic polynomial of a finite multigraph G is expressed in terms of characteristic polynomials oflocal modifications of G. The resulting formula is used to investigate the largest eigenvalues of certain theta graphs.

The purpose of this paper is to give a sharp bound for the order of instability of an unstable hyperplane of a rank 2 stable reflexive sheaf E on ℙn, in terras of the Chern classes of E; and a sharp boundfor the order of instability of an unstable hyperplane of a rank 2 nonstable reflexive sheaf E on ℙn, in terms of the Chern classes of E and the order of nonstability of E.

The deficiency indices (mean deficiency index) and the essential spectrum for a class of odd order ordinary differential expressions are determined. The considered expressions are relatively bounded or relatively compact perturbations of symmetric expressions with odd order terms having as coefficients real powers of the independent variable.

A simple method is given for the construction of real symmetric differential expressions that are not in the limit-point case but have the real half-line as essential spectrum.

Let mt(ω) be the range of a standard brownian bridge on R with ω(0) = ω(t) = 0 and let µt(ω) be the corresponding Wiener measure. We determine the asymptotic behaviour for large t of ∫ e−G(mt(ω)) µt(dω) for an increasing convex function G.

Let E1 and E2 be real normed linear spaces such that the dimension of any of them is at least 2. We prove that the norms in E1 × E2 which verify a simple property of monotonicity with regard to the initial norms in E1 and E2 are the only norms in E1 × E2 which preserve best linear approximations, in the sense that ifyk ∊ Lk is best approximation to xk from the linear subspace Lk, (k = 1,2), then (y1, y2) is best approximation to (x1, x2) from L1 × L2.

We consider the boundary value problem

where Ω ⊂ ℝn is a bounded domain, n≧3, 2* = 2n/(n − 2) is the critical exponent for the Sobolev embedding and λ is a real positive parameter. We prove the existence of infinitely many solutions of (*) when Ω exhibits suitable symmetries.

We consider the second order, linear differential equation

in the case where, roughly, q ∊ L1 [0, ∞). We devise a representation for the spectral function, τ(t), associated with (*) which is valid for t sufficiently large. Our results are the best possible.