The following results are obtained for symmetric differential expressions of arbitrary order r ≧ 1 with matrix coefficients on the half-line, with a non-negative (and possibly identically degenerate) weight matrix W(t), and with a spectral parameter λ: upper and lower bounds for the deficiency indices N(λ) are found; it is proved that N(λ) is independent of λ for Im λ< 0and for Im λ>0; under very general conditions it is proved that the maximum possible values of the deficiency indices in the half-planesIm λ≷0 can only be attained simultaneously; sufficient conditions for first-order expressions to be quasi-regular are derived; and it isshown that a symmetric system of any order reduces to a canonical, first-order system.

Examples are constructed, and the case of the whole line is also touched on.

A proof is given of a formula connecting Weyl's function m(λ) with a spectral integral, in the setting of the singular theory of the Sturm-Liouville differential equation. In addition, a relatively short treatment is given of the argument introducing the function. The theory applies to both the limitpoint and limit-circle cases.

By a new method it is proved that a non-linear elliptic boundary value problem of rather general type admits a weak solution lying between a given weak lower solution ϕ and a given weak upper solution ψ≧ϕ

Non-linear integral operators are studied under perturbations of the integrals, the domains of integration, the kernels and the inhomogeneous terms. The main theorem establishes the local solubility of the perturbed integral equations, the biconvergence of their solutions and associated two-sided error estimates. This theorem is of interest even for the special class of linear Fredholm integral equations.

A necessary and sufficient condition is developed for determination of the uniform stability of a class of non-autonomous linear differential-difference equations. This condition is the analogue of the Liapunov criterion for linear ordinary differential equations.

Let S be a compact subset of the open unit disc in C. Associate to S the set

Let R(X) be the uniform algebra on X generated by the rational functions which are holomorphic near X. It is shown that the spectrum of R(X) is determined in a simple wayby the potential-theoretic properties of S. In particular, the spectrum of R(X) is X if and only if the functions harmonic near S are uniformly dense in the continuous functions on S. Similar results can be obtained for other subsets of C2 constructed from compact subsets of C.

We study the asymptotic behaviour for ɛ→+0 of the solution Φ of the elliptic boundary value problem

is a bounded domain in ℝ2, 2 is asecond-order uniformly elliptic operator, 1 is a first-order operator, which has critical points in the interior of , i.e. points at which the coefficients of the first derivatives vanish, ɛ and μ are real parameters and h is a smooth function on . We construct firstorder approximations to Φ for all types of nondegenerate critical points of 1 and prove their validity under some restriction on the range of μ.

In a number of cases we get internal layers of nonuniformity (which extend to the boundary in the saddle-point case) near the critical points; this depends on the position of the characteristics of 1 and their direction. At special values of the parameter μ outside the range in which we could prove validity we observe ‘resonance’, a sudden displacement of boundary layers; these points are connected with the spectrum of the operator ɛ2 + 1 subject to boundary conditions of Dirichlet type.

Let d(L) denote the deficiency indices (which are equal) of a formally symmetric differential expression L with real coefficients. Then it is shown that

(a)if k>1, d(Lk)−d(Lk−1) ≤ order L, and

(b)if j ≥ k > 1, d(Lj)−d(Lj−1) ≥ d(Lk)−d(Lk−1).

This paper, following Stratford [9], considers a boundary layer developing with uniform pressure when x < x0, and with large and constant when x > x0. The analysis improves or extends the work of Stratford as follows.

(a) For large values of inner and outer asymptotic expansions are derived and matched—Stratford considered only the inner solution, his outer boundary condition being determined by physical arguments.

(b) The skin-friction is determined as a series in powers of , whereas Stratford's, solution was obtained by approximate methods. In the limit as λ→∞ the predicted separation position, ξ = 0.09766, is probably correct to3 significant figures, at least, and differs little from Stratford's value.

(c) This solution further shows that for finite λ, separation occurs when

ξ = 0.09766-0.00014λ−3.

Although the coefficient 0.00014 is not wholly reliable, clearly the position of separation varies little with λ unless λ−3 is very much greater than unity.

(d) The present solution also yields the displacement and momentum thicknesses. When x > x0 the displacement thickness is

where B1(ξ) is given as a slowly converging seriesin powers of ξ. It is shown how to improve convergence, using certain properties of the flow close to the separation position, and thus to sum the series.

Likewise

so that when λ is large the momentum thickness changes even less than does the displacement thickness.

Consider the multiparameter system

where ut is an element of a separable Hilbert space Hi, i = 1, …, n. The operators Sij are assumed to be bounded symmetric operators in Hi and Ai is assumed self-adjoint. In addition consider the operator equation

where B is densely defined and closed in a separable Hilbert space H and Tj, j = 1, …, n is a bounded operator in H. The problem treated in this paper is to seek an expression for a solution v of (**) in terms of the eigenfunctions of the system (*).

The main result of this paper is to establish the completeness of the eigenfunctions for the multiparameter eigenvalue problem defined by the system of ordinary differential equations

0 ≤ x, ≤ 1, r = 1, …, k, subject to the Sturm-Liouville boundary conditions

r = 1, …, k. In addition it is assumed that the coefficients ars of the spectral parameters λs, satisfy the ellipticity condition , s = 1, …, k, for all xrɛ[0, 1], r = 1, …, k, and some real k-tuple μ1, …, μk and where is the co-factor of asr in the determinant . The theory developed here contrasts with the results known when …k is assumed non-vanishing for all xrɛ[0,1].

An explicit characterisation is given of those sequences of positive integers which occur as the deficiency indices associated with the sequence of powers of some formally symmetric 2nth order real differential expression on [0, ∞) which is regular at 0.

The paper discusses the asymptotic behaviour of solutions of the functional differential equation

where a is a complex constant, 0<λ<1, and b is a constant such that Re b = 0, but b ≠ 0.

Optimal switching curves are investigated for some second-order control systems with random disturbances, saturating control, and a mean-square-error performance index.

It is first shown that the optimal switching curves are asymptotic at infinity to the optimal switching curves for corresponding deterministic systems.

The systems are then discretised in time and state, and ordinary dynamic programming is applied to the resulting Markov chain models. The discretisation techniques are described in some detail.

The same techniques are applied to a deterministic system for which the optimal switching curve is known, and it is found that the resulting discretisation errors are considerable.

Thus it turns out that ordinary discrete dynamic programming gives only a rough approximation to the optimal switching curve. It seems desirable to tackle the equations of continuous dynamic programming by techniques more refined than crude discretisation. One such technique is suggested.

This paper develops the general theory of the propagation of Lipschitz discontinuities in first- and second-order partial derivatives of the initial data for a conservative quasi-linear hyperbolic system with discontinuous coefficients. After establishing that such weak discontinuities propagate along characteristics the appropriate transport equations are derived. The effect on this form of wave propagation of the strong discontinuity associatedwith the discontinuous coefficients is then studied and the transmission and reflection characteristics of the resulting waves are analysed. In conclusion, an application of this general theory is made to the propagation of plane shear waves through two different continuous hyperelastic solids to determine the transmitted and reflected waves.

It is shown that three independent axioms uniquely determine the topological degree of set-valued maps of the form I – G, where G is a convex-valued, limit compact map. This extends earlier work of Amann and Weiss, Nussbaum, and others, in that, apart from dealing with set-valued maps, a larger class of maps is considered even in the single-valued case.

The paper derives some new Bessel function inequalities These are used to determine conditions for the absence of roots of the transcendental equation F(z)Zv(z)+Zv′(z) = 0, where Zv(z) is a Bessel function, in certain regions of the v and z-plane

The paper gives asymptotic estimates of the form

as λ→∞ for the length l(μ)of a gap, centre μ in the essential spectrum associated with second-order singular differential operators. The integer r will be shown to depend on the differentiability properties of the coefficients in the operators and, in fact, r increases with the increasing differentiability of the coefficients. The results extend to all r ≧ – 2 the long-standing ones of Hartman and Putnam [10], who dealt with r = 0, 1, 2.

For a non-linear functional differential equation from population biology, a result on asymptotic stability is obtained by investigating the zeros of the characteristic equation of the linearised functional differential equation.

The simultaneous integral equations of Noble and Cooke, for inter alia the Dirichlet problem for an annular disc, are transformed into equations closely analogous to those recently given by Clements and Love for the corresponding Neumann problem. The transformed equations are relatively simple, and do not involve any artificial preliminary dissection of the known functions. They are also uncoupled, and admit iterative solution for virtually all radius ratios.

In the case of the conducting annular disc, iteration of these equations isused to obtain two series for the capacity, an interim one with all terms positive, and a final one with all terms after the first negative. The final series is more rapidly convergent as well as neater. It is used to provide a family of inequalities which enclose the capacity and determine it to high accuracy with very few terms, failing only when the radius ratio is close to 1.