We consider the possibility of solving semilinear elliptic boundary value problems in unbounded domains. We first treat the case when the non-linear terms are independent of terms involving gradients. Using a monotone iteration scheme, we show that the existence of a weak subsolution v and a weak supersolution w ≧ v, implies the existence of a weak solution u, and v ≦ u ≦ w. We also state conditions which guarantee the existence of a solution when only a subsolution is known to exist. Next, we suppose the non-linear terms can depend on gradient terms. Using a method developed in [4], based on perturbation theory of maximal monotone operators, we prove the existence of a H2(Ω) solution lying between a given H2(Ω) subsolution v and a given H2(Ω) supersolution w ≧ v.

Hilbert boundary value problems for a half-space are considered for analytic representations of Schwartz distributions: given data g ∈D'(ℛ) and a coefficient x we seek functions F(z) analytic for Jmz≠0 whose limits exist in D'(ℛ) and satisfy F+—XF– = g on an open subset U of the real line R. U is the complement of a finite set which contains the singular support and the zeros of X·X and its reciprocal satisfy certain growth conditions near the boundary points of U. Solutions F(z) are shown to exist, and their general form is determined by obtaining a suitable factorisation of x.

Some continuous dependence theorems are presented for classical solutions of initial-boundary value problems in viscoelastic materials which occupy the exterior of a bounded domain in Euclidean three space. Two broad classes of anisotropic viscoelastic materials are considered. The theorems are proved by a combination of the Protter and the Graffi methods.

The existence of a variational solution is shown for the strongly non-linear elliptic boundary value problem in unbounded domains. The proof is a generalisation to Orlicz-Sobolev space setting of the idea introduced in [15] for the equations involving polynomial non-linearities only.

The Stieltjes transformation is extended to generalised functions both by the direct approach and the method of adjoints, and the resulting extensions are correlated. Inversion formulae are developed, as is the application of fractional integration to these transforms. An integral transformation with a hypergeometric kernel is also briefly considered.

Using the theory of the weighted Sobolev space H1,2(μ), on a bounded domain Ω, in Rn, the existence and regularity of solutions u in K to the variational inequality

is established for various convex subsets K of H1, 2(μ). The growth conditions imposed on the functions A and B give the differential inequality degenerate elliptic structure, extending the results on regularity for inequalities of elliptic type.

It is established that under certain restrictions the solution u of the characteristic initial value problem uxy+g(x, y)u = 0, u(x, 0) = p(x) and u(0, y) = q(y), where p(x) > 0 and q(y) > 0, in [0, ∞) x [0, ∞) changes sign along a monotonic decreasing curve which is asymptotic to the axes.

Dual extremum principles characterising the solution of initial value problems for the heat equation are obtained by imbedding the problem in a two-point boundary-value problem for a system in which the original equation is coupled with its adjoint. Bounds on quantities of interest in the original initial value problem are obtained. Such principles are examples of ones which can be obtained for a general class of linear operators on a Hilbert space.

In this paper it is shown that the analysis of Titchmarsh's book [32] for regular Sturm-Liouville problems on a finite closed interval carries over readily to regular problems involving the eigenvalue parameter in the boundary condition at one end-point. The manner in which this type of problem is associated with a self-adjoint operator in Hilbert space has recently been pointed out by Walter in [36], and his operator-theoretic formulation is adopted here. The use of the eigenfunction expansion is illustrated by applying it to solve a heat-conduction problem for a solid in contact with a fluid.

Sufficient conditions are given to insure that all solutions of a perturbed non-linear second-order differential equation have certain integrability properties. In addition, some continuability and boundedness results are given for solutions of this equation.

It is shown that Ф = | grad u |2–uΔu, where u is a solution of Δ2u+pf(u) = 0 in D, assumes its maximum value on the boundary of D. This principle leads one to a lower bound on the first eigenvalue in the non-linear Dirichlet eigenvalue problem and to the non-existence of solutions to this non-linear partial differential equation subject to certain zero boundaryconditions.

In 1932, Hardy and Littlewood [1] proved the inequality

The constant 4 is best possible; equality occurs when f(x) = A Y(Bx), where

y(x) = e−½x sin (x sin y−y) (y = ⅓π), (x ≧ o)

and A and B (>0) are constants. In [2], three proofs are given. The inequality has also been discussed in [3, 4]. A very elementary proof in which the function Y(x) emerges naturally is given in this paper.

This paper gives simple proofs of two integral inequalities recently proved by Brodlie and Everitt[1].

In a previous paper [2], a theory of fractional integration was developed for certain spaces Fp,μ of generalised functions. In this paper we extend this theory by relaxing some of the restrictions on the various parameters involved. In particular we show how a generalised Erdelyi-Kober operator can be defined on Fʹp,μ for 1 ≦ p ≦ ∞ and for all complex numbers μ except for those lying on a countable number of lines of the form Re μ = constant in the complex μ-plane. Mapping properties of these generalised operators are obtained and several applications mentioned.