CLICK HERE for some pictures courtesy of Roman Hilscher

Participating speakers were as follows:

• Martin Bohner, University of Missouri, Rolla, Missouri
• Roman Hilscher, Michigan State University, East Lansing, Michigan
• Jo Hoffacker, University of Nebraska, Lincoln, Nebraska
• Billur Kaymakcalan, Georgia Southern University, Statesboro, Georgia
• Bonita Lawrence, University of South Carolina, Beaufort, South Carolina
• Ronald Mathsen, North Dakota State University, Fargo, North Dakota

Titles and Abstracts of the talks were as follows:

• Martin Bohner: Laplace Transform and Z-Transform: Unification and Extension. We start this talk by recalling the concepts of the Laplace transform and Z-transform as methods to solve higher order differential and difference equations with constant coefficients. Then we introduce the Laplace transform for an arbitrary time scale. Two particular choices of time scales, namely the reals and the integers, yield the concepts of the classical Laplace transform and of the classical Z-transform. Other choices of time scales yield new concepts of our Laplace transform, which can be applied to find solutions of higher order linear dynamic equations with constant coeffficients. We present several useful properties of our Laplace transform and offer formulas for the Laplace transforms of many elementary functions, among them results for the convolution of two functions on a time scale, which is introduced in this talk as well.
• Roman Hilscher: A Wirtinger type inequality and nonoscillation of dynamic equations on time scales. Wirtinger type inequalities play an important role in the investigation of nonoscillatory properties of higher order Sturm-Liouville differential equations. Recently, discrete versions of such Wirtinger type inequalities have been derived and used to obtain parallel nonoscillation results for higher order Sturm-Liouville difference equations. We derive a similar Wirtinger type inequality on an arbitrary time scale and prove the nonoscillation of certain second order dynamic equations. However, strong limitations occur when trying to apply the result to higher order dynamic equations.
• Jo Hoffacker: Green's function and eigenvalue comparisons for focal boundary value problems on time scales. The Green's function and its sign are developed for a nth order focal boundary value problem. This is then used to do eigenvalue comparisons for other focal BVPs.
• Billur Kaymakcalan: Coupled Solutions and Monotone Iterative Techniques for some Nonlinear Initial Value Problems on Time Scales. A unified approach to monotone iterative technique concerning the existence of coupled maximal and minimal solutions for dynamic equations on time scales is developed in the case of the nonlinear term involved admitting a splitting of a difference of two monotone functions. Relevance of such problems in mathematical biology using both the differential and difference form of the Logistic equation is pointed out as an example and further features of the technique as well as other examples are presented in the proceeding talk by the joint author.
• Bonita Lawrence: Monotone Iterative Techniques for Dynamic Equations on Periodic Time Scales. Using the Monotone Iterative Technique introduced in the previous talk by the joint author we present further features of this method and use them to analyze the boundedness behavior of some nonlinear dynamic equations on periodic time scales arising from mathematical biology.
• Ronald Mathsen: Factoring Linear Delta-Differential Operators and Oscillation of Neutral Delta-Differential Operators on Time Scales. We introduce nth order Delta-differential operators on time scales and show Abel's Theroem. Then we construct both the Polya factorization and the Trench Factorization of disconjugate operators. Next we discuss the extension to time scales of oscillation results for neutral differential delay equations and develop two oscillation theorems which are new even when the time scale is the reals. We conclude with examples illustrating these results.