For Q ∊ L2(0, 1) we investigate the set Γ ∊ ℝ2 of pairs (α β,) for which the problem

has a nontrivial solution which has exactly one zero in (0,1) and is positive near x = 0. We show that Γ is stable in a certain sense under small perturbations of Q. The dependence of Γ upon Q is illustrated by an example.

In this paper we begin a systematic study of smooth tight maps with singularities of surfaces into E3, particularly C∞-stable maps. The class of C∞-stable tight maps of surfaces into E3 is much larger and richer than the class of C∞ (or even topological) tight immersions. We describe the general structure of C∞-stable tight maps of surfaces into E3. We show that, given any integer n ≧ 2 and a compact surface X other than the sphere or the projective plane, there is a C∞ -stable tight map X → E3 with exactly n topcycles. This is very different from the situation for tight topological immersions, where Cecil and Ryan have shown that the number α(f) of topcycles of a map f: X → E3, X a compact surface other than S2, satisfies the bound 2 ≦ α(f) ≦ 2 − Euler number of X. We prove also an analogue of the Cecil–Ryan result for C∞-stable maps.

A class of scalar periodic differential equations x ]f(x, t, A) with regularisable infinity in which the Riccati equation is included, can be treated as autonomous systems on the torus T2. Through this geometric interpretation, the properties of the set of bifurcation points of λ (∊R) are studied. When the results obtained are applied to the Riccati equation, they can match the well-known properties of the spectral set of the corresponding Hill's equation.

We discuss the symmetric solutions of the semilinear elliptic equation Δu + λ(u+ u|u|p−1) = 0, u|∂B = 0 (*), where B is the unit ball in ℝ3. The value of p is taken close to 5, the critical Sobolev exponent for ℝ3. An asymptotic description of the solutions of (*) with large norm is obtained. This predicts a fold bifurcation if p > 5 and the structure of this bifurcation is studied in the limit p – 5→ 0. We find good agreement between the asymptotic description and some numerical calculations. These results are illuminated by recasting the problem (*) in the form of a dynamical system by means of a suitable change of variables. When |p – 5|≪1 and ∥u ≫1, the transformed solutions of (*) are also solutions of a perturbed Hamiltonian system and we study the behaviour of these solutions by using Melnikov methods.

The paper considers the small time evolution of the interface which appears in the boundary value problem

A diffusion dominated small time outer expansion is not uniformly valid when x = O(t½ log t) and has to be supplemented by an inner expansion valid in a region where diffusion and absorption are of equal importance. For p < 1 a zero order inner solution is constructed, which is consistent with a moving interface where u is zero, and the small time development of this is given.

These results are first derived in a heuristic way using the method of matched expansions. However, an important aim of the paper is to give rigorous proofs, being one of the very few cases in nonlinear diffusion where this has been done.

We consider initial boundary value problems for the equations of linear thermoelasticity in both bounded and unbounded domains and for both nonhomogeneous and anisotropic media. For bounded domains, it is shown that the unique solution of the problem is time-asymptotically equal to the solution of a particular initial boundary value problem which is obtained from a natural decomposition of the original initial data and which represents a (in general non-vanishing) time harmonic part. For the unbounded case similar results are obtained, but now in the sense of weak convergence which lead to the result of local energy decay: the solution tends to zero in every compactum.

The best constants in Landau's inequality for the difference operator in the classical sequence spaces lp are known explicitly only for p = 1, 2, ∞. This is true in both the infinite N = (0, 1, 2, …) and biinfinite Z= (… − 1, 0, 1, …) cases. It is known that there are no extremals when p = 2 in both the infinite and biinfinite cases. Also, it is known that there are extremals when p = ∞ in the biinfinite case. Here we prove that there are no extremals in the other three cases where the best constants are known explicitly. The proofs for these three cases are quite different from each other.

This paper proves the existence of (at least) one solution of the following equation:

Here, is an elliptic operator of Leray-Lions type acting from into Lp′(0, T; W−1.p′ (Ω)), (1/p + 1/p′ = 1) and |F(u, ∇u)| ≧b(|u|)(l + |≧u|P). There are no smoothness assumptions on the bounded open set Ω; the operator and the nonlinearity F(u, ∇u) are denned in terms of Carathéodory functions. These points are the most characteristic features of this paper.

Assuming the existence of upper and lower solutions allows us to obtain L∞(Q)-estimates. An estimate is then proved. The final step is to prove the strong convergence in of the approximations. This proof relies on the method introduced by L. Boccardo, F. Murat and J. P. Puel for elliptic and parabolic problems of this type.

Let G be a group and let A be a fixed point free group of automorphisms of G. It is shown that the centraliser near-ring MA(G) has at most one nontrivial ideal. Conditions on the pair (A, G) are given which force MA(G) to be simple. It is shown that if a nonsimple near-ring MA(G) exists, then A and G have unusual properties.

We prove that a free discrete transformation group on a compact connected space X is completely determined by the way C(X) lies inside the corresponding crossed product C*-algebra.

In this paper we consider differential-boundary operators T over a finite interval depending on a complex parameter. A differential-boundary operator admits boundary conditions in the differential part. The boundary part contains multipoint boundary conditions and integral conditions. For Birkhoff-regular boundary conditions we prove that every Lp -function is expansible into a series with respect to the eigenfunctions and the associated functions of the differential-boundary operator. Here the Birkhoff-regularity only depends on the boundary conditions at the endpoints of the interval, i.e. T is Birkhoff-regular if and only if T0 is Birkhoff-regular where T0 arises from T by omitting the boundary part in the differential equations, the interior point boundary conditions and the integral condition.