Conditions are obtained which ensure that the maximal operator generated by the formally self-adjoint second-order differential expression

in L2(Rn), n ≥ 1 has the Dirichlet and separation properties.

We give some results on a boundary-value problem for an ordinary differential equation whose coefficients are in the B*-algebra C(K), where K is a compact metric space. We deduce the existence of a countable number of eigenvalues and corresponding eigenfunctions, the latter being complete in a certain sense. There follows an expansion result and some remarks on a self-adjoint realisation of the associated differential operator.

Within recent years considerable attention has been devoted to extensions of the classical Sturmian theory of real linear homogeneous differential equations of the second order. In particular, such extensions have included not only self-adjoint systems of differential equations, but also higher order self-adjoint differential and integro-differential equations. For problems in these latter categories, however, only limited attention has been given to detailed application of the general oscillation and comparison criteria. The present paper is devoted to this area, and, in particular, it is shown how existing criteria may be exploited to obtain comparison theorems between equations of different orders. Although the presented results have ready extensions to vector differential and integro-differential equations of higher order, [see, for example, 5,6,7], for simplicity attention is restricted to scalar equations. Section 2 is devoted to the statement of known general criteria of oscillation for self-adjoint equations of higher order, with special applications of these criteria presented in Section 3. Finally, Section 4 sketches the framework of corresponding applications for self-adjoint higher order integro-differential equations.

In this paper the boundary behaviour of the solution of Dirichlet's problem for the equation Δu = f in domains with C2,α-boundary is studied. Without using Schauder's a priori estimates, we prove the following theorem, which is indispensable in Schauder's method to solve Dirichlet's problem for a general linear elliptic equation.

Theorem: Let (f, g)∈ C0,α (Ḡ)×C2,α (∂G) and u ∈ C0.0(Ḡ) ∩C2,α(G) be a solution of the problem Δu = f, u|∂G = g. Then u ∈ C2,α(Ḡ).

The paper is concerned with giving sufficient conditions that in the non-linear boundary-value problem

there should be no secondary bifurcation, i.e. that, given a branch of solutions (u, λ) bifurcating from the trivial solution, there should be no further bifurcation on that branch. Sufficient conditions on G are given which include, for example, Kolodner's problem of the motion of a heavy rotating string.

This paper studies a linked system of second order ordinary differential equations

where xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.