Conditions, generalizing the usual Liouville–Green conditions, are obtained under which the equation {r(x)y′}′ + q(x)y = 0, where q and r are positive and have derivatives of a sufficiently high order, has solutions of the form y ∼ (rq)–¼e10. 0 ∼ f (q/r)½ dx as x →∞. A related result is obtained for a system of two first-order equations y′ = A(x)y, where the eigenvalues of A are pure imaginary.

This paper studies how, in a difference equation, uniform asymptotic stability for commensurable delays implies uniform asymptotic stability globally in the delays. A result is also given on renorming the space C= C([ – ∥r∥, 0]; ℂn so that a difference operator has norm less than one if it is uniformly asymptotically stable globally in the delays.

Let F(X, Y, Z) be a non-singular quadratic form with rational coefficients. The curve EF(x2, y2, z2) = 0 is of genus 3. A procedure is described for deciding whether there is an effective divisor on E of degree 3 defined over the rationals. There is such a divisor if and only if there is a point on E defined over some algebraic number field of odd degree. An example is constructed for which there is no such divisor although (i) there are points on E defined over all p-adic fields and over the reals and (ii) there are infinitely many rational points on each of the three curves F(X, y2, z2) = 0, F(x2, Y, z2) = 0 and F(x2, y2, Z) = 0.

A sufficient condition on the angles of a bounded open subset Ω of ℝn is given for the best possible regularity of a solution to a class of parabolic problems with non-linear mixed boundary conditions.

Extensions of the integral version of Hardy's Inequality were given by Kadlec and Kufner (1967) and by Copson (1976). This paper provides several levels of further generalization of their results, obtained mostly by specializing four main inequalities. Most of the inequalities have the form ∥Kf∥ ≤ C ∥f∥, where K is an integral transform and ∥.∥ is a generalized Lp-norm; some have the inequality sign reversed. Best possible constants C are obtained in several cases, under mild extra hypotheses.

The aim of the present paper is to establish some new integral inequalities of Opial type in two independent variables. Our results are the two independent variable generalizations of some of the inequalities recently established by the present author and in special cases yield the two independent variable analogue of the Opial inequality and its generalization given by G. S. Yang.

A weak (enthalpy) formulation of the problem of a free boundary moving in the thermal concentrated capacity is given. The problem is to solve the heat equation in a given domain, while on a part of the boundary of this domain the solution (or rather its trace) solves a Stefan problem with forced convection. The existence of a global weak solution is proved by the method of finite differences. Some regularity is obtained from this proof, and the continuity of the temperature is proved. The uniqueness, which is related to the existence of mushy regions, is discussed. A classical enthalpy formulation is conjectured.

We give some results on existence and uniqueness for the solution of elliptic boundary value problems of type

when the βi are not necessarily smooth.

Let D be a compact convex planar domain containing the origin, the boundary of which is of class C∞ and has finite non-vanishing curvature throughout. For the number A(i) of lattice points in the “blown up” domain √tD, the estimate

is established. This is a generalization of Hardy's classical result for the circle problem. The proof is based on asymptotic formulae for certain exponential integrals due to E. Hlawka.

We consider a method for determining the asymptotic solution to a sufficiently wide class of ordinary linear homogeneous differential equations in a sector of a complex plane or of a Riemann surface for large values of the independent variable z. The main restriction of the method is the condition that the coefficients in the equation should be analytic and single-valued functions in the sector for | z | ≫ 1 possessing the power order of growth for |z| → ∞. In particular, the coefficients can be any powerlogarithmic functions. The equation

can be taken as a model equation. Here ai are complex numbers, aij are real numbers (i = 1,2,…, n; j = 0, 1, …, m) and ln1 Z≡ln z, lnsz= lnlnS−1z = S = 2, … It has been shown that the calculation of asymptotic representations for solution to any equation in the class considered may be reduced to the solution of some algebraic equations with constant coefficients by means of a simple and regular procedure. This method of asymptotic integration may be considered as an extension (to equations with variable coefficients) of the well known integration method for linear differential equations with constant coefficients. In this paper, we consider the main case when the set of all roots of the characteristic polynomial possesses the property of asymptotic separability.

A nonlinear boundary value problem (P) having positive parameters L and a is considered. We associate with it a family of perturbed problems () affected by the presence of a barrier parameter γ related to L and a. There is a critical value L*(a) of the parameter L such that for L >L*(a), (P) has no regular solution. Then some natural extensions of (P), solutions of a free boundary value problem, arise as singular limits of ().

We use a modified form of the Stieltjes inversion formula due to Atkinson together with some recent results on the asymptotic form of Titchmarch–Weyl m-functions to obtain an asymptotic expression for the spectral functions τ(t) associated with the differential equation