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Theory of Causal Differential Equations / by V. Lakshmikantham, S. Leela, Zahia Drici, F. A. McRae
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Atlantis Studies in Mathematics for Engineering and Science ; 5 | SpringerLink BücherPublisher: Amsterdam : Atlantis Press, 2010Publisher: World Scientific, 2010Description: Online-Ressource (XI, 208p, digital)ISBN:- 9789491216251
- 515.352
- 515.35
- QA372
Contents:
Summary: Preliminaries -- Basic Theory -- Theoretical ApproximationMethods -- Stability Theory -- Miscellaneous Topics in Causal Systems.Summary: The problems of modern society are both complex and inter-disciplinary. Despite the - parent diversity of problems, however, often tools developed in one context are adaptable to an entirely different situation. For example, consider the well known Lyapunov’s second method. This interesting and fruitful technique has gained increasing signi?cance and has given decisive impetus for modern development of stability theory of discrete and dynamic system. It is now recognized that the concept of Lyapunov function and theory of diff- ential inequalities can be utilized to investigate qualitative and quantitative properties of a variety of nonlinear problems. Lyapunov function serves as a vehicle to transform a given complicated system into a simpler comparison system. Therefore, it is enough to study the properties of the simpler system to analyze the properties of the complicated system via an appropriate Lyapunov function and the comparison principle. It is in this perspective, the present monograph is dedicated to the investigation of the theory of causal differential equations or differential equations with causal operators, which are nonanticipative or abstract Volterra operators. As we shall see in the ?rst chapter, causal differential equations include a variety of dynamic systems and consequently, the theory developed for CDEs (Causal Differential Equations) in general, covers the theory of several dynamic systems in a single framework.PPN: PPN: 1651281688Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Atlantis Studies in Mathematics for Engineering and Science; Preface; Contents; Chapter 1: Preliminaries; 1.1 Introduction; 1.2 Causal Operators; 1.3 Known Basic Results; 1.4 Fixed Point Theorems and Auxiliary Results; 1.5 Preliminaries in a Banach Space; 1.6 Directional Derivatives; 1.7 Measure of Noncompactness; 1.8 The Measure of Nonconvexity; 1.9 Notes and Comments; Chapter 2: Basic Theory; 2.1 Introduction; 2.2 Causal Functional and Differential Inequalities; 2.3 Existence and Extremal Solutions; 2.4 Global Existence; 2.5 Existence and Uniqueness; 2.6 Nagumo-Type Conditions
2.7 Continuous Dependence Relative to Initial Data2.8 Existence of Euler Solutions; 2.9 Flow Invariance; 2.10 Systems of Causal Differential Inequalities; 2.11 Nonlinear Variation of Parameters; 2.12 Integral Equations of Sobolev Type; 2.13 Differential Equations of Sobolev Type; 2.14 Notes and Comments; Chapter 3: Theoretical Approximation Methods; 3.1 Introduction; 3.2 Method of Lower and Upper Solutions; 3.3 Monotone Iterative Technique; 3.4 GeneralizedMonotone Iterative Technique; 3.5 Monotone Technique for PBVPs; 3.6 The Method of Quasilinearization (MQL)
3.7 Extension of Quasilinearization3.8 Newton'sMethod Versus Quasilinearization; 3.9 Notes and Comments; Chapter 4: Stability Theory; 4.1 Introduction; 4.2 Comparison Theorems via Lyapunov Functions; 4.3 Definitions of Stability and Boundedness; 4.4 Definitions Relative to Two Measures; 4.5 Stability Criteria-Lyapunov Functions; 4.6 Stability Criteria-Lyapunov Functionals; 4.7 Lyapunov Functions on Product Spaces; 4.8 Stability in Terms of two Measures; 4.9 Vector Lyapunov Functions; 4.10 Cone-valued Lyapunov Functions; 4.11 Notes and Comments
Chapter 5: Miscellaneous Topics in Causal Systems5.1 Introduction; 5.2 Causal Set Differential Equations; 5.3 Comparison Results and Stability Theory; 5.4 Causal Differential Equations in a Banach Space; 5.5 Global Existence and Inequalities in Cones; 5.6 Fractional Causal Differential Equations; 5.7 Causal Differential Equations with Memory; 5.8 Causal Differential Systems with Retardation and Anticipation; 5.9 Monotone Iterative Technique; 5.10 Neutral Differential Equations with Causal Operators on a Semi-Axis; 5.11 Notes and Comments; Bibliography; Subject Index
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