Functions of bounded kth variation and absolutely kth continuous functions are considered on sets and various properties are studied.

In this paper, we consider Ishikawa's iteration scheme to compute fixed points of real Lipschitz functions. Two general convergence theorems are obtained. Our results generalise the result of Hillam.

Uniqueness and regularity of periodic solutions to the semilinear dissipative wave equation with small parameter ∈ > 0,

are investigated when g(u) has a certain ‘critical’ nonlinearity.

We apply Straßmann's theorem to p–adic power series satisfying linear differential equations with polynomial coefficients and note that our approach leads to our estimating the number of integer zeros of polynomials on a given interval and thence to an investigation of the number of p–adic small values of a function on such an interval, that is, of the number of solutions of a congruence modulo pr.

Let G be a group and k an integer greater than 1. We say that G has the square property of k-sets and we write G ∈ DS(k) if |X2| < k2 for any subset X of G of order k. The groups in DS(2) are exactly the Dedekind groups, as Freiman showed. The class .DS(3) has been recently studied by Berkovich, Freiman and Praeger. In this paper we complete the classification of DS(3)-groups by characterising finite 2-groups of exponent 4 in DS(3).

Let X be a real Banach space with a uniformly convex dual, X*, and let C be a nonempty closed convex and bounded subset of X. Let T: C → C be a strongly accretive and a continuous mapping. For any f ∈ C, let S: C → C be defined by Sx = f + x – Tx for each x ∈ C. Then, the iteration process xo ∈ C,

under suitable conditions on the real sequence converges strongly to a solution of the equation Tx = f in C. Furthermore, if T is strongly accretive and Lipschitz with Lipschitz constant L ≥ 1 then the iteration process x0 ∈ C,

under suitable conditions on the real sequences and converges strongly to a solution of the equation Tx = f in C. Explicit error estimates are obtained.

Let α > 0 and let f[α](z) be the αth fractional derivative of an analytic function f on the unit disc D. In this paper we show that f ∈ BMOA if and only if |f[α](z)|2 (l - |z|2)2α−1dA(z) is a Carleson measure and f ∈ VMOA if and only if |f[α](z)|2 (1 − |z|2)2α−1dA(z) is a vanishing Carleson measure, where A denotes the normalised Lebesgue measure on D. Hence a significant extension of familiar characterisations for analytic functions of bounded and vanishing mean oscillation is obtained.

In this paper, we obtain a version of the sliding plane method of Gidas, Ni and Nirenberg which applies to domains with no smoothness condition on the boundary. The method obtains results on the symmetry of positive solutions of boundary value problems for nonlinear elliptic equations. We also show how our techniques apply to some problems on half spaces.

For any α ∈ (0, 1), examples of C∞ functions fα: [0, 1] → [0, 1] with zero topological entropy and possessing a δ−scrambled set of Lebesgue measure α are given. This answers a question posed by Smítal.

The differentiability, of a specified strength, of a convex function at a point, is shown to be characterised by the convergence of subdifferentials in the appropriate topology on the dual space. This is used to prove that if each gauge is densely differentiable then so is each convex function. The generic version of this is equivalent to a conjecture which, for Gateaux differentiability and Banach spaces, is the long standing open question of whether X × ℝ is Weak Asplund whenever X is. Some progress is made towards a resolution.

In this note, we shall prove a new variational inequality in non-compact sets and as an application, we prove a generalisation of the Schauder-Tychonoff fixed point theorem.

We prove that some metric inequalities imply weak or weak-star normal structure. In particular, we prove that every ω*-compact convex set in the space C1(lp, lq) of nuclear operators from lp into lq, (1 < p, q < ∞, 1/p + 1/q = 1) has the weak* normal structure. This generalises a recent result of C. Lennard.

In this paper a formula is obtained for the entries of the diagonal factor in the U D L factorisation of an invertible operator matrix in the case when its inverse has a chordal graph. As a consequence, in the finite dimensional case a determinant formula is obtained in terms of some key principal minors. After a cancellation process this formula leads to a determinant formula from an earlier paper by W.W. Barrett and C.R. Johnson, deriving in this way a different and shorter proof of their result. Finally, an algorithmic method of constructing minimal vertex separators of chordal graphs is presented.

We study the asymptotic behaviour as t → ∞ of solutions of the initial-boundary value problem vt = G(u, v), ut = uxx + F(u, v), and t > 0, x ∈ ℝ or x ∈ ℝ+ for a wide class of initial and boundary values, where F and G are smooth functions so that the system has three rest points.

We obtain new sufficient conditions for the oscillation of all solutions of the neutral differential equation with variable coefficients

where P, Q, R ∈ C([t0, ∞), R+), r ∈ (0, ∞) and τ, σ ∈ [0, ∞). Our results improve several known results in papers by: Chuanxi and Ladas; Lalli and Zhang; Wei; Ruan.

Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie group G and for any strictly positive integer k there exist non located and non trivially decomposable representations of order k of the nuclear Lie group (M;G) of all the G-valued test-functions on a given paracompact manifold M.

It is known that the maximal ideals in the rings C(X) and C*(X) of continuous and bounded continuous functions on X, respectively, are in one-to-one correspondence with βX. We have shown previously that the same is true for any ring A(X) between C(X) and C*(X). Here we consider the problem for rings A(X) contained in C*(X) which are complete rings of functions (that is, they contain the constants, separate points and closed sets, and are uniformly closed). For every noninvertible f ∈ A(X), we define a z–filter ZA(f) on X which, in a sense, provides a measure of where f is ‘locally invertible’. We show that the map ZA generates a correspondence between ideals of A(X) and z–filters on X. Using this correspondence, we construct a unique compactification of X for every complete ring of functions. Each such compactification is explicitly identified as a quotient of βX. In fact, every compactification of X arises from some complete ring of functions A(X) via this construction. We also describe the intersections of the free ideals and of the free maximal ideals in complete rings of functions.

We consider the sequence Λ = {0 < λ2 < λ2 < …}, for which λn → +∞. We denote by PD(Λ) the class of Dirichlet's series having the form F(s) = defined in the half plan Re s > 0 converging absolutely and Re F ≥ 0. If N0 = {0, 1,2, …} then the class PD(N0) coincides with the Caratheodory's class P. In this paper some classical results holding for the class P are generalised in any class PD(Λ). In special cases for the sequence Λ extreme problems are examined in the class PD(Λ).

The authors give affirmative answers to two questions of H. Neumann and Wiegold on linked embeddings of groups.

The Hecke groups

are Fuchsian groups of the first kind. In an interesting analogy to the use of ordinary continued fractions to study the geodesics of the modular surface, the λ-continued fractions (λF) introduced by the first author can be used to study those on the surfaces determined by the Gq. In this paper we focus on periodic continued fractions, corresponding to closed geodesics, and prove that the period of the λF for periodic has nearly the form of the classical case. From this, we give: (1) a necessary and sufficient condition for to be periodic; (2) examples of elements of ℚ(λq) which also have such periodic expansions; (3) a discussion of solutions to Pell's equation in quadratic extensions of the ℚ(λq); and (4) Legendre's constant of diophantine approximation for the Gq, that is, γq such that < γq/Q2 implies that P/Q of “reduced finite λF form” is a convergent of real α ∉ Gq(∞).