John V. Baxley

Nonlinear boundary value problems for shallow membrane caps, II

Abstract Suppose a shallow membrane cap, with an undeformed shape described in cylindrical coordinates by z = C(1−rγ) (where 0⩽r⩽1 and γ>1), is subjected to a uniform vertical pressure P. If the resulting deformed shape is radially symmetric, then under certain assumptions, the radial stress Sr satisfies the ordinary differential equation r 2 S r ″ + 3rS r ′ = λ 2 r 2y−2 2 + βvr 2 S r − r 2 8S r 2 , for 0 0 (if the boundary stress S is specified) or S r ′ (1)+(1 − v)S r (1) = ⌜ (if the boundary displacement ⌜ is specified). Here v(0⩽v 1, a radially symmetric solution Sr(r), positive for 0 S⩽ 1 (4βv) or ⌜ (1−v) ⩽ 1 (4βv) , the solution is unique. In the case γ⩽ 4 3 , if λ is fairly large, it may happen that Sr(r) → 0 as r → 0. In all other cases, Sr(r) has a positive limit as r → 0. Rather detailed information on the behavior of solutions Sr(r) is provided. Conditions are obtained which guarantee monotonicity of Sr. In any case, Sr has at most one critical point and is monotone in some neighborhood of r = 0. A computational algorithm, making use of the qualitative behavior of Sr, is discussed and some numerical results are included.

Existence and uniqueness for two-point boundary value problems

Boundary value problems associated with y ″ = f ( x , y , y ′) for 0 ≦ x ≦ 1 are considered. Using techniques based on the shooting method, conditions are given on f ( x , y , y ′) which guarantee the existence on [0, 1] of solutions of some initial value problems. Working within the class of such solutions, conditions are then given on nonlinear boundary conditions of the form g ( y (0), y ′(0)) = 0, h ( y (0), y ′(0), y (1), y ′(1)) = 0 which guarantee the existence of a unique solution of the resulting boundary value problem.

THE FRIEDRICHS EXTENSION OF CERTAIN SINGULAR DIFFERENTIAL OPERATORS, II

We study the Friedrichs extension for a class of 2nth order ordinary differential operators. These selfadjoint operators have compact inverses and the central problem is to describe their domains in terms of boundary conditions.

Eigenvalues of singular differential operators by finite difference methods, II☆

Note is made of an earlier paper which defined finite difference operators for the Hilbert space L2(m), and gave the eigenvalues for these operators. The present work examines eigenvalues for higher order singular differential operators by using finite difference methods. The two self-adjoint operators investigated are defined by a particular value in the same Hilbert space, L2(m), and are strictly positive with compact inverses. A class of finite difference operators is considered, with the idea of application to the theory of Toeplitz matrices. The approximating operators consist of a good approximation plus a perturbing operator.

Nonlinear second order boundary value problems: Continuous dependence and periodic boundary conditions

ZusammenfassungZwei-punkt Randwertaufgaben für die Differentialgleichung (*)y″=f(x,y,y′),a≤x≤b, sind betrachtet. Ziemlich gewöhnliche hinreichende Bedingungen sind gegeben für die Existenz und Einzigkeit einer Lösung Φ(x,α,β) der Gleichung (*) und die Randbedingungeng(y(a), y′(a))=α, h(y(b), y′(b))=β, und für die stegige Abhängigkeit der beiden Φ′(x, α, β) von (x, α, β). Diese Ergebnisse haben dann zur Folge Existenz- und Einzig- keitsätze für die verallgemeinerte periodische Randwertaufgabe, die aus der Gleichung (*) und den Randbedingungeny(a)=y(b)+G (y′(a), y(b), H(y′(a), y′(b))=0 bestehen.

On the Weyl spectrum of a Hilbert space operator

Using the perturbation definition of the Weyl spectrum, conditions are given on a closed (possibly unbounded) linear operator T in a Hilbert space which allow the Weyl spectrum to be characterized as a subset of the spectrum of T.

Positive solutions of singular nonlinear boundary value problems

For a class of problems of the form y″=g(t,y)f(y′),0 −∞, and find asymptotic formulae for y(t) and y′(t) as t→1−. We particularly include cases where the nonlinearity causes singularities at the endpoints of (0,1).

Extreme eigenvalues of Toeplitz matrices associated with certain orthogonal polynomials

Previous results of the author and I. I. Hirschman on the asymptotic behavior of the extreme eigenvalues of truncated Toeplitz matrices associated with Laguerre polynomials are extended using methods, initiated by S. V. Parter, involving the study of related finite difference operators. The same technique also leads to the corresponding results, already obtained by Hirschman, in the case of the Jacobi polynomials.

Some Properties of R. G. D. Allen's Treatment of Kalecki's 1935 Model of Business Cycles

More than sixty-five years have passed since Michal Kalecki (1935) published one of the first formal mathematical models of business cycles. His paper presents a closed-form analytic solution. This characteristic, among others, sets Kaleckis work apart from that of contemporary literary business cycle theorists such as Friedrich A. Hayek (1935) and John Maynard Keynes (1936).

Some Partial Differential Operators with Discrete Spectra

We study selfadjoint realizations of the formal differential operator Tu = m −1 [(p 1 ,u x ) x + (p 2 u y ) y ] in the weighted Hilbert space L m 2 (Ω) where Ω is the square domain (0,1) × (0,1). Assuming m, p 1 , p 2 are positive and reasonably smooth and that singularities of T occur only along the boundaries x=0 or y=0, a variety of strictly positiveselfadjoint realizations of T are constructed, each of which, with a further integrability condition on the coefficients, has a discrete spectrum.