Functional and functional differential equations in a Banach space X are related to systems of operators A(t) in C = C(−r, 0; X), given by

Conditions are sought on F such that A(t) generates an evolution system U(t, s)φThis system gives the segments of solution for φ in a certain domain which is determined.

Sufficient conditions are given which ensure nonexistence of spherically symmetric entire solutions of Δpu = f(u), p ≧ 2. Sufficient conditions for existence of spherically symmetric entire solutions of Δpu = f(r, u) are also given.

It is shown that there exists an uncountable number of rings R with the properties (1) if r∈R then 2r = 0 and r2 = 0, (2) if r ∈ R then there exist s, t in R with r = st.

We show the existence of weak solutions of nonlinear parabolic partial differential equations in unbounded domains, provided that a variant of the Leray-Lions conditions is satisfied.

A collisionless gas flows through a finite rectangular duct which reflects molecules diffusely. The transmission probability Q of the duct involves the solution of a pair of coupled integral equations. Complementary variational principles have been employed which supply upper and lower bounds to Q. Numerical calculations have been made for a variety of duct shapes and compared, where appropriate, to those of other authors.

One considers the linear differential systems where is a (not necessarily diagonal) matrix and one relates the computation of a general multiplicity defined from this system to the corresponding multiplicity of some eigenvalues of . Then applying these conclusions, one gives simple conditions ensuring the existence of odd or even periodic solutions for systems having the form .

This is a contribution to the general problem of determining the extent to which the geometry of a riemannian manifold is determined by properties of its geodesic spheres. In particular we show that total umbilicity of geodesic spheres determines riemannian manifolds of constant sectional curvature; quasi-umbilicity of geodesic spheres determines Kähler and nearly-Kähler manifolds of constant holomorphic sectional curvature; and the condition that geodesic spheres have only two different principal curvatures, one having multiplicity 3, determines manifolds locally isometric to the quaternionic projective spaces. The use of Jacobi vector fields leads to a unified treatment of these different cases.

Some results are given for Weyl's m-coefficient in the algebra B = C(K). Included are a discussion of a theorem of Hille, a generalisation of Weyl's L.P./L.C. classification, and an example.

A quasi-periodic boundary value problem for the Helmholtz equation in an unbounded domain is considered. This problem arises from scattering of plane waves by periodic structures.

Existence and uniqueness theorems are proved, and the continuation of the resolvent of this problem to a Riemannian surface is constructed. This construction makes no use of the continuation of the resolvent kernel but runs along the following lines:

First a family of differential operators is defined, which is holomorphic in a generalized sense. Then, using a result from analytic perturbation theory about families of operators with compact resolvent, it is shown that the family of inverses of these differential operators gives the desired continuation.

In this work we present an algorithm for computing an integrable function almost everywhere on (0,1) when its moments are known. The method is based on the use of certain delta-shaped sequences, and can be adjusted to take advantage of the local smoothness of the function.

As an application, we give an algorithm for the pointwise inversion of the Laplace transform which utilizes the values of the image function at equidistant points.

A new class of irregular boundary value problems—non-regular in the sense of Birkhoff—is studied. This class of strongly irregular problems includes the class of boundary value problems with irregular decomposing boundary conditions. For each strongly irregular problem we can find a problem with irregular decomposing boundary conditions so that we have equiconvergence with respect to Riesz typical means of the eigenfunction expansions arising from these two problems of an arbitrary summable function.

We consider the formally self-adjoint 2mth-order elliptic differential operator in ℝn given by where lt is an operator of order t, and establish conditions under which the operator on is essentially self-adjoint in L2. A feature is that the major conditions have to be imposed only in an increasing sequence of annular regions surrounding the origin.

We consider the formally self-adjoint 2mth-order elliptic differential operator in ℝn given by where lt is an operator of order t, and pt ≧0 for t ≧1 and establish conditions under which the operator on is essentially self-adjoint in L2. A feature is that the major conditions (including the positivity of the coefficients) have to be imposed only in an increasing sequence of annular regions surrounding the origin.