Calvin D. Ahlbrandt

Partial differential equations on time scales

Discrete and continuous formulations of partial differential operators are unified by a time scale formulation of partial differential operators. Results include an Euler-Lagrange equation for double integral variational problems on time scales and a Picone identity which implies a Sturm-Picone comparison theorem for second-order elliptic PDEs on time scales.

The Effect of Variable Change on Oscillation and Disconjugacy Criteria with Applications to Spectral Theory and Asymptotic Theory

This study of the effect of variable change on differential operators was motivated by recent papers of Tipler [ 131 and Hinton and Lewis [37] which appeared in the same issue of the Journal of D@‘erential Equations. Tipler gave a geometric proof [ 13, p. 1671 of the following result of Hawking and Penrose (see [29, 301). Let F(t) be a real valued continuous, nonnegative, function on (-W, co) such that F(b) > 0 for some point b. Then there exists a pair of conjugate points on (-co, co) for the differential equation

The (n, n)-disconjugacy of a 2nth order linear difference equation

Abstract We give a formulation of generalized zeros and (n, n) disconjugacy for even order formally self-adjoint scalar difference equations. Positive definiteness of an associated quadratic functional is shown to imply (n, n) disconjugacy. This study is motivated by the open question of existence of a Reid roundabout theorem for singular discrete linear Hamiltonian systems.

Floquet Theory for Time Scales and Putzer Representations of Matrix Logarithms

Dedicated to Allan Peterson on the occasion of his 60th birthday. A Floquet theory is presented that generalizes known results for differential equations and for difference equations to the setting of dynamic equations on time scales. Since logarithms of matrices play a key role in Floquet theory, considerable effort is expended in producing case-free exact representations of the principal branch of the matrix logarithm. Such representations were first produced by Putzer as representations of matrix exponentials. Some representations depend on knowledge of the eigenvalues while others depend only on the coefficients of the characteristic polynomial. Logarithms of special forms of matrices are also considered. In particular, it is shown that the square of a nonsingular matrix with real entries has a logarithm with real entries.

Discrete variational inequalities

The inequalities considered are for discrete quadratic functionals motivated by the theory of the second variation for discrete variational problems. They are discrete versions of classical integral inequalties associated with quadratic functionals. Connections are made between positive definiteness of discrete quadratic functionals and discrete analogues of Jacobi conditions and Sturmian theorems for associated three term recurrence relations

Variable Change for Sturm-Liouville Differential Expressions on Time Scales

- {K_n}{y_{n + 1}} + {B_n}{y_n} - K_{n - 1}^T{y_{n - 1}} = 0.

Discrete versions of continuous isoperimetric problems

These equations arise in the study of symplectic continued fractions.

Discrete Linear Hamiltonian Systems

Abstract In a series of papers starting with a 1959 paper in J. Math. & Mechanics [1], W. T. Reid presented Sturmian theory and asymptotic behavior for generalized differential systems. These systems were equivalent to “a type of linear vector Riemann-Stieltjes integral equation.” Reids primary result was his “Roundabout Theorem” for this generalized setting. As he pointed out, if the measure is piecewise constant, then results for difference equations ensue. The objectives of this study are 1. (i)to interpret Reids results for both Jacobi and Riccati difference equations and 2. (ii)to compare those results with subsequent studies of difference equations based on discrete variational theory.