Jerry Ridenhour

Lyapunov Inequalities for Time Scales

The theory of time scales has been introduced in order to unify discrete and continuous analysis. We present a Lyapunov inequality for Sturm-Liouville dynamic equations of second order on such time scales, which can be applied to obtain a disconjugacy criterion for these equations. We also extend the presented material to the case of a general linear Hamiltonian dynamic system on time scales. Some special cases of our results contain the classical Lyapunov inequalities for differential equations as well as only recently developed Lyapunov inequalities for difference equations.

The (2,2)-disconjugacy of a fourth order difference equation

Let a fourth order linear difference operator L be defined by for t in a discrete interval [a + 2,b + 2] where b - a is a nonegative integer. Solutions of Ly = 0 are then determined on the discrete interval [a,b + 4]. Defining generalized zeros appropriately, the equation Ly = 0 is (2,2)-disconjugate on [a,b + 4] provided no nontrivial solution of Ly = 0 has two distinct generalized zeros, each of order two or more, in [a,b + 4]. Letting q–(t) = max{−q(t),0}, the main theorem gives a condition on q(t), in the form of a strict inequality that q–(t) summed over the interval [a + 2b + 2] satisfies, which guarantees that Ly = 0 is (2,2)-disconjugate on [a,b + 4]. An example is given showing that the inequality is sharp. The proof of the main result utilizes an appropriately defined quadratic form.

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.

A disconjugacy criterion of W. T. Reid for difference equations

Our main result is a disconjugacy criterion for the swlfadjoint vector difference equation Ly(t) A [P(t 1 )Ay(t 1)] + Q(t)y(t) = 0. This result is the analogue of a famous result of W. T. Reid for the corresponding differential equations case. Unlike the differential equations case we will see there is an exceptional case in which, as we will show by counterexample, the conclusion of the main result is no longer valid. A disfocality criterion is also given. We believe these results are new even in the scalar case. We are concerned with the n-dimensional second order selfadjoint vector difference equation Ly(t) A[P(t 1)Ay(t 1)] + Q(t)y(t) = 0, where P(t) is an n x n Hermitian matrix function on the integer interval [a, b+ 1] _ {a, a+ 1, ... , b+ l} with P(t) > 0 (positive definite) in [a, b+ 1] and Q(t) is an n x n Hermitian matrix function on [a + 1, b + 1]. Solutions of the equation Ly(t) = 0 are defined on the integer interval [a, b + 2]. If y(t) is a complex vector solution of Ly(t) = 0 on [a, b + 2], then y*(t)P(t l)Ay(t 1) Ay*(t l)P(t l)y(t) = c on [a + 1, b + 2] for some constant c. If c = 0, we say y(t) is a prepared solution of Ly(t) = 0. Hence, if y(t) is a prepared solution of Ly(t) = 0, then (1) y*(t)P(t 1)Ay(t 1) = Ay* (t 1)P(t 1)y(t) on [a + 1, b + 2] (so y*(t)P(t l)Ay(t 1) is real-valued on [a + I, b + 2]). It follows from (1) that (2) y*(t 1)P(t 1)y(t) = y*(t)P(t 1)y(t 1) on [a + 1, b + 2] (so y*(t l)P(t l)y(t) is real-valued on [a + I, b + 2]). We now define what we mean by a generalized zero of a nontrivial prepared solution y(t) of Ly(t) = 0. The definition is relative to the fixed interval [a, b + 2] and the left endpoint a is treated separately. In particular, we say y(t) has a generalized zero at a if and only if y(a) = 0, while we say y(t) has a Received by the editors August 4, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 34C10; Secondary 34A30.

Atkinson's superlinear oscillation theorem for matrix difference equations

The concept of a generalized zero of a prepared solution of a superlinear matrix difference equation is introduced. Riccati techniques are used to establish necessary and sufficient conditions for all prepared solutions to be oscillatory.

Stability of equilibria in quantitative genetic models based on modified-gradient systems

ABSTRACT Motivated by questions in biology, we investigate the stability of equilibria of the dynamical system which arise as critical points of f, under the assumption that is positive semi-definite. It is shown that the condition , where is the smallest eigenvalue of , plays a key role in guaranteeing uniform asymptotic stability and in providing information on the basis of attraction of those equilibria.