Dynamic Equations on Time Scales

An introduction to measure chain (time scale) theory with emphasis on its usefulness in allowing for the simultaneous development of differential equations and difference equations. Motivating the subject is the notion that dynamic equations on measure chains can build bridges between continuous and discrete mathematics. The study of measure chain theory has also led to applications in the study of insect population models, neural networks, heat transfer, epidemic models. This book may stimulate the development of new kinds of equations with potentially new applications. Includes examples, exercises, solutions.

"This would be an excellent book to use in a topics course on dynamic equations on time scales at the advanced undergraduate level and/or beginning graduate level."

Zentralblatt Math

"The monograph under review comes at an excellent time in the rapid development of dynamic equations on time scales. Both authors are authorities in this field of study and they have produced an excellent introduction to it. Much of the material is accessible to upper-level undergraduate mathematics majors, and yet, the results and the techniques are pertinent to active researchers in the area."

Mathematical Reviews

Autorentext
An introduction to measure chain (time scale) theory with emphasis on
its usefulness in allowing for the simultaneous development of
differential equations and difference equations. Motivating the
subject is the notion that dynamic equations on measure chains can
build bridges between continuous and discrete mathematics. The study
of measure chain theory has also led to applications in the study of
insect population models, neural networks, heat transfer, epidemic
models. This book may stimulate the development of new kinds of
equations with potentially new applications. Includes examples,
exercises, solutions.

Klappentext

On becoming familiar with difference equations and their close re­ lation to differential equations, I was in hopes that the theory of difference equations could be brought completely abreast with that for ordinary differential equations. [HUGH L. TURRITTIN, My Mathematical Expectations, Springer Lecture Notes 312 (page 10), 1973] A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. [E. T. BELL, Men of Mathematics, Simon and Schuster, New York (page 13/14), 1937] The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD thesis [159] in 1988 (supervised by Bernd Aulbach) in order to unify continuous and discrete analysis. This book is an intro­ duction to the study of dynamic equations on time scales. Many results concerning differential equations carryover quite easily to corresponding results for difference equations, while other results seem to be completely different in nature from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice, once for differential equa­ tions and once for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which is an arbitrary nonempty closed subset of the reals.

Inhalt

1. The Time Scales Calculus.- 1.1. Basic Definitions.- 1.2. Differentiation.- 1.3. Examples and Applications.- 1.4. Integration.- 1.5. Chain Rules.- 1.6. Polynomials.- 1.7. Further Basic Results.- 1.8. Notes and References.- 2. First Order Linear Equations.- 2.1. Hilger's Complex Plane.- 2.2. The Exponential Function.- 2.3. Examples of Exponential Functions.- 2.4. Initial Value Problems.- 2.5. Notes and References.- 3. Second Order Linear Equations.- 3.1. Wronskians.- 3.2. Hyperbolic and Trigonometric Functions.- 3.3. Reduction of Order.- 3.4. Method of Factoring.- 3.5. Nonconstant Coefficients.- 3.6. Hyperbolic and Trigonometric Functions II.- 3.7. Euler-Cauchy Equations.- 3.8. Variation of Parameters.- 3.9. Annihilator Method.- 3.10. Laplace Transform.- 3.11. Notes and References.- 4. Self-Adjoint Equations.- 4.1. Preliminaries and Examples.- 4.2. The Riccati Equation.- 4.3. Disconjugacy.- 4.4. Boundary Value Problems and Green's Function.- 4.5. Eigenvalue Problems.- 4.6. Notes and References.- 5. Linear Systems and Higher Order Equations.- 5.1. Regressive Matrices.- 5.2. Constant Coefficients.- 5.3. Self-Adjoint Matrix Equations.- 5.4. Asymptotic Behavior of Solutions.- 5.5. Higher Order Linear Dynamic Equations.- 5.6. Notes and References.- 6. Dynamic Inequalities.- 6.1. Gronwall's Inequality.- 6.2. Holder's and Minkowski's Inequalities.- 6.3. Jensen's Inequality.- 6.4. Opial Inequalities.- 6.5. Lyapunov Inequalities.- 6.6. Upper and Lower Solutions.- 6.7. Notes and References.- 7. Linear Symplectic Dynamic Systems.- 7.1. Symplectic Systems and Special Cases.- 7.2. Conjoined Bases.- 7.3. Transformation Theory and Trigonometric Systems.- 7.4. Notes and References.- 8. Extensions.- 8.1. Measure Chains.- 8.2. Nonlinear Theory.- 8.3. Alpha Derivatives.- 8.4. Nabla Derivatives.- 8.5. Notes and References.- Solutions to Selected Problems.