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.

Let us consider equations in the form

where Λ is an open subset of a normed space. For any fixed λ ∊ Λ, T, L(λ,.) and M(λ,.) are mappings from the closure D0 of a neighbourhood D0 of the origin in a Banach space X into another Banach space Y with T(0) = L(λ, 0) = M(λ, 0) = 0. Let λ be a characteristic value of the pair (T, L) such that T − L(λ,.) is a Fredholm mapping with nullity p and index s, p> s≧ 0. Under sufficient hypotheses on T, L and M, (λ, 0) is a bifurcation point of the above equations. Some well-known results obtained by Crandall and Rabinowitz [2], McLeod and Sattinger [5] and others will be generalised. The results in this paper are extensions of the results obtained by the author in [7].

We prove the global existence and uniqueness of the solution of the initial and boundary value problem for the equation

by using the classical Galerkin method when the forcing term and the initial data are in some sense small. The asymptotic behaviour of the solution as t → ∞ is also considered.

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.

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 ∑.

In this paper the authors investigate two classes of triple series equations of trigonometric type. Closed form solutions are given for eight triple series in each class.

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 presents new comparison and uniqueness results for the solutions of parabolic quasilinear boundary value problems with (and without) obstacles. A stability result in L1(Ω) yields the asymptotic stabilisation in this space, when t → ∞) towards the corresponding elliptic problem.

Certain permutation representations of the Hall–Janko group J2 are studied. These representations are of interest in connection with the problem of whether J2 can act as astrongly irreducible collineation group of a finite projective plane.

This paper discusses the relationship between the uniqueness and the ordering of strictly positive solutions of elliptic predator-prey interacting systems. If (ū, v) and (u#, v#) are two such solutions with ū ≧ u# or v ≧ v#, then ū ≡= u#, v ≡v#. When the positive solutions are numerically close to the extreme case, the solution is unique.

Let m(λ) denote the Titchmarsh–Weyl m-function associated with the differential equation

with Neumann boundary condition at 0. In the case q ∊ CM[0, ε) for some integer M and ε > 0 we prove the following result (Theorem 2.1): If for some integer ŋ ≧ 3 we know that Im {λ(ŋ + l)/2m0(λ)} = 0 on the ray λ = |λ| eiπ/ŋ, then q(v) (0) = 0 for all v in the set {0, 1, … M} for Which ≢ ŋ − 2 mod ŋ.

Comparison principles for systems of reaction–diffusion equations in unbounded domains and coupledvia both reaction and diffusion terms are considered. Applications are made to the FitzHugh–Nagumo equations and models of coupled nerve fibres.

We give a new method for calculating the Γ-limit functional encountered in the problems of homogenisation. We use the Legendre–Lagrange transform in the convex analysis and regularisation method to obtain the explicit expression of the Γ-limit functional. The result can be applied to some nonlocal function spaces.

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].

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.

In this paper we apply a theorem of Khelemskiĭ and Sheĭnberg, characterising amenability by means of bounded approximate identities, to weighted discrete convolution algebras. In doing this we give a condition for a weighted discrete convolution algebra to have a bounded approximate identity. Under the condition that the semigroup (S,.) is one-sided cancellative, we prove that, if some weighted discrete convolution algebra on S is amenable, then (S,.) is actually a group. We further characterise all amenable weighted discrete convolution algebras on groups, thus extending a well-known theorem of B. E. Johnson [9].