Allan Peterson

Dynamic equations on time scales: a survey

The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of differential and difference equations. In this paper we give an introduction to the time scales calculus. We also present various properties of the exponential function on an arbitrary time scale, and use it to solve linear dynamic equations of first order. Several examples and applications, among them an insect population model, are considered. We then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients. Finally, we consider self-adjoint equations and, more generally, so-called symplectic systems, and present several results on the positivity of quadratic functionals.

Three positive fixed points of nonlinear operators on ordered banach spaces

We generalize the fixed-point theorem of Leggett-Williams, which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a Banach space. We then show how to apply our theorem to prove the existence of three positive solutions to a second-order discrete boundary value problem. @ 2001 Elsevier Science Ltd. All rights reserved.

Positive solutions for a nonlinear differential equation on a measure chain

We are concerned with proving the existence of positive solutions of general two point boundary value problems for the nonlinear equation Lx(t) := -[r(t)x^@D(t)]^@D=/tf(t, x(t)). We will use fixed point theorems concerning cones in a Banach space. Important results concerning Greens functions for general two point boundary value problems for Lx(t) := -[r(t)x^@D(t)]^@D=0 will also be given.

OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR DYNAMIC EQUATIONS ON TIME SCALES

By means of generalized Riccati transformation techniques and generalized exponential functions, some oscillation criteria are given for the nonlinear dynamic equation \[ (p(t)x^{\Delta} (t))^{\Delta}+q(t)(f\circ x^{\sigma})=0 \] on time scales. The results are also applied to linear and nonlinear dynamic equations with damping, and some sufficient conditions are obtained for the oscillation of all solutions.

Discrete fractional calculus

Preface.- 1. Basic Difference Calculus.- 2. Discrete Delta Fractional Calculus and Laplace Transforms.- 3. Nabla Fractional Calculus.- 4. Quantum Calculus.- 5. Calculus on Mixed Time Scales.- 6. Fractional Boundary Value Problems.- 7. Nonlocal BVPs and the Discrete Fractional Calculus.-Solutions to Selected Problems.- Bibliography.- Index.

Comparison theorems for linear dynamic equations on time scales

We obtain several comparison theorems for second order linear dynamic equations on a time scale. These results extend comparison theorems for the continuous case and provide some new results in the discrete case, as well as other more general situations.

Boundedness and oscillation for nonlinear dynamic equations on a time scale

We obtain some boundedness and oscillation criteria for solutions to the nonlinear dynamic equation (p(t)x Δ (t)) Δ + q(t)(f o x σ ) = 0, on time scales. In particular, no explicit sign assumptions are made with respect to the coefficient q(t). We illustrate the results by several examples, including a nonlinear Emden-Fowler dynamic equation.

Oscillation criteria for second-order matrix dynamic equations on a time scale

We obtain oscillation criteria for a second-order self-adjoint matrix differential equation on a measure chain in terms of the eigenvalues of the coefficient matrices and the graininess function. We illustrate our results with some nontrivial examples.

Oscillation criteria for nonlinear damped dynamic equations on time scales

Abstract We present new oscillation criteria for the second order nonlinear damped delay dynamic equation ( r ( t ) ( x Δ ( t ) ) γ ) Δ + p ( t ) ( x Δ σ ( t ) ) γ + q ( t ) f x ( τ ( t ) ) = 0 . Our results generalize and improve some known results for oscillation of second order nonlinear delay dynamic equation. Our results are illustrated with examples.

Three positive solutions to a discrete focal boundary value problem

Abstract We are concerned with the discrete focal boundary value problem Δ3x(t − k) = f(x(t)), x(a) = Δx(t2) = Δ2x(b + 1) = 0. Under various assumptions on f and the integers a, t2, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results we use fixed point theorems concerning cones in a Banach space.