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Theory Of Causal Differential Equations [Kõva köide]

(The Catholic Univ Of America, Usa), (Suny At Geneseo, Usa), (Florida Inst Of Technology, Usa), (Illinois Wesleyan Univ, Usa)
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Teised raamatud teemal:
The theory of causal differential equations (CDE) includes several types of dynamic systems such as ordinary differential equations, functional differential equations, integro-differential equations (with or without memory), differential equations with anticipation and retardation. This is the first book which describes the theory of CDE as an independent discipline, incorporating the recent general theory of CDE and introducing several new ideas. This book is a timely introduction to the subject in a more generalised frame work. The present monograph collects recent works in this broad area and provides possible extensions to other dynamic systems involving causal operators such as CDE in abstract spaces, CDE with memory, CDE with fractional derivatives and causal set differential equations-giving initial apparatus, for further study in this important branch of nonlinear analysis.
Preface v
Preliminaries
1(22)
Introduction
1(1)
Causal Operators
1(5)
Known Basic Results
6(3)
Fixed Point Theorems and Auxiliary Results
9(3)
Preliminaries in a Banach Space
12(3)
Directional Derivatives
15(2)
Measure of Noncompactness
17(1)
The Measure of Nonconvexity
18(4)
Notes and Comments
22(1)
Basic Theory
23(50)
Introduction
23(1)
Causal Functional and Differential Inequalities
24(3)
Existence and Extremal Solutions
27(5)
Global Existence
32(3)
Existence and Uniqueness
35(5)
Nagumo-Type Conditions
40(3)
Continuous Dependence Relative to Initial Data
43(4)
Existence of Euler Solutions
47(4)
Flow Invariance
51(2)
Systems of Causal Differential Inequalities
53(2)
Nonlinear Variation of Parameters
55(4)
Integral Equations of Sobolev Type
59(6)
Differential Equations of Sobolev Types
65(5)
Notes and Comments
70(3)
Theoretical Approximation Methods
73(32)
Introduction
73(1)
Method of Lower and Upper Solutions
74(1)
Monotone Iterative Technique
75(4)
Generalized Monotone Iterative Technique
79(8)
Monotone Technique for PBVPs
87(4)
The Method of Quasilinearization (MQL)
91(3)
Extension of Quasilinearization
94(5)
Newton's Method Versus Quasilinearization
99(3)
Notes and Comments
102(3)
Stability Theory
105(32)
Introduction
105(1)
Comparison Theorems via Lyapunov Functions
106(5)
Definitions of Stability and Boundedness
111(3)
Definitions Relative to Two Measures
114(2)
Stability Criteria-Lyapunov Functions
116(2)
Stability Criteria-Lyapunov Functionals
118(2)
Lyapunov Functions on Product Spaces
120(3)
Stability in Terms of two Measures
123(2)
Vector Lyapunov Functions
125(3)
Cone-valued Lyapunov Functions
128(8)
Notes and Comments
136(1)
Miscellaneous Topics in Causal Systems
137(66)
Introduction
137(1)
Causal Set Differential Equations
138(8)
Comparison Results and Stability Theory
146(11)
Causal Differential Equations in a Banach Space
157(6)
Global Existence and Inequalities in Cones
163(6)
Fractional Causal Differential Equations
169(6)
Causal Differential Equations with Memory
175(6)
Causal Differential Systems with Retardation and Anticipation
181(7)
Monotone Iterative Technique
188(5)
Neutral Differential Equations with Causal Operators on a Semi-Axis
193(8)
Notes and Comments
201(2)
Bibliography 203(4)
Subject Index 207