Advances in Dynamic Equations on Time Scales

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Martin Bohner, Allan C. Peterson
Springer Science & Business Media, Dec 6, 2002 - Mathematics - 348 pages
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The development of time scales is still in its infancy, yet as inroads are made, interest is gathering steam. Of a great deal of interest are methods being intro duced for dynamic equations on time scales, which now explain some discrepancies that have been encountered when results for differential equations and their dis crete counterparts have been independently considered. The explanations of these seeming discrepancies are incidentally producing unifying results via time scales methods. The study of dynamic equations on time scales is a fairly new subject, and research in this area is rapidly growing. It has been created in order to unify continuous and discrete analysis, and it allows a simultaneous treatment of dif ferential and difference equations, extending those theories to so-called dynamic equations. An introduction to this subject is given in Dynamic Equations on Time Scales: An Introduction with Applications (MARTIN BOHNER and ALLAN PETER SON, Birkhauser, 2001 [86]). The current book is designed to supplement this introduction and to offer access to the vast literature that has already emerged in this field. It consists of ten chapters, written by an international team of 21 experts in their areas, thus providing an overview of the recent advances in the theory on time scales. We want to emphasize here that this book is not just a collection of papers by different authors.
 

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Contents

Introduction to the Time Scales Calculus
1
12 Differentiation
2
13 Mean Value Results
4
14 Integration
7
15 The Regressive Group
10
16 Alpha Dynamic Equations
12
Some Dynamic Equations
17
22 Linear Equations
19
632 The Quasilinearization Method
181
633 A Note on Reversed Order Upper and Lower Solutions
186
Positive Solutions of Boundary Value Problems
189
722 Eigenvalue Problems
191
73 Existence of at Least One Solution
194
732 Intervals of Eigenvalues
210
74 Existence of at Least Two Solutions
225
742 The AveryHenderson Fixed Point Theorem and Applications
229

23 Euler Equations
23
24 Logistic Equations
30
25 The Regressive Vector Space
34
26 Bernoulli Equations
38
27 Riccati Equations
40
28 Clairaut Equations
43
Nabla Dynamic Equations
47
32 The Nabla Exponential Function
49
33 Examples of Exponential Functions
55
34 Nonhomogeneous First Order Linear Equations
58
35 Wronskians
61
36 Nabla Hyperbolic and Trigonometric Functions
65
37 Reduction of Order
70
38 Nabla Riccati Equations
73
310 Polynomials and Taylors Formula
79
Second Order SelfAdjoint Equations with Mixed Derivatives
85
43 Second Order Linear Dynamic Equations
92
44 Abels Formula and Reduction of Order
96
45 Oscillation and Disconjugacy
100
46 The Riccati Equation
108
Riemann and Lebesgue Integration
117
52 The Riemann Delta and Nabla Integrals
118
53 Properties of the Riemann Integral
127
54 The Fundamental Theorem of Calculus
137
55 Mean Value Theorems for Integrals
142
56 Improper Integrals
145
562 Examples
149
563 Improper Integrals of Second Kind
155
57 The Lebesgue Delta and Nabla Integrals
157
Lower and Upper Solutions of Boundary Value Problems
165
62 Separated Boundary Value Problems
166
622 The Quasilinearization Method
169
623 Mixed Derivative Problems
175
63 Periodic Boundary Value Problems
177
75 Existence of at Least Three Solutions
235
751 The LeggettWilliams Fixed Point Theorem
236
752 More General Triple Fixed Point Theorems
240
753 Applications to Boundary Value Problems
241
Disconjugacy and Higher Order Dynamic Equations
251
82 Initial Value Problems
252
83 Generalized Zeros of Higher Order
254
84 Wronskian Determinants
256
85 Interpolating Families of Functions
257
86 Disconjugacy
258
87 A Trench Factorization and Principal Solutions
264
88 A Boundary Value Problem and Greens Function
267
89 Monotone Methods
271
810 Open Problems
272
Boundary Value Problems on Infinite Intervals A Topological Approach
275
93 Applications to Boundary Value Problems
279
94 Systems on Infinite Intervals
285
Symplectic Dynamic Systems
293
1022 Symplectic Dynamic System
294
1023 Conjoined Bases
295
103 Vector Solutions and Generalized Zeros
297
104 Riccati Operators Quadratic Functionals and Picones Identity
299
1042 Quadratic Functionals
300
1043 Picones Identity
304
105 First Results on Positive Definiteness
308
106 Sturmian Theorems
316
107 Quadratic Functionals with Boundary Conditions
318
1072 Separated Boundary Conditions
321
108 Roundabout Theorem
328
109 Flow of Symplectic Systems
329
1010 Prufer Transformation
331
Bibliography
335
Index
345
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