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This book provides a systematic treatment of the Volterra integral equation by means of a modern integration theory which extends considerably the field of differential equations. It contains many new concepts and results in the framework of a unifying theory. In particular, this new approach is suitable in situations where fast oscillations occur.
Contents:
- Introduction
- Kapitza's Pendulum and a Related Problem
- Elementary Methods: Averaging
- Elementary Methods: Internal Resonance
- Strong Riemann-Integration of Functions of a Pair of Coupled Variables
- Generalized Ordinary Differential Equations: Strong Riemann-Solutions (Concepts)
- Functions ψ1, ψ2
- Strong Riemann-Solutions of Generalized Differential Equations: A Survey
- Approximate Solutions: Boundedness
- Approximate Solutions: A Lipschitz Condition
- Approximate Solutions: Convergence
- Solutions
- Continuous Dependence
- Strong Kurzweil Henstock-Integration of Functions of a Pair of Coupled Variables
- Generalized Differential Equations: Strong Kurzweil Henstock-Solutions
- Uniqueness
- Differential Equations in Classical Form
- On a Class of Differential Equations in Classical Form
- Integration and Strong Integration
- A Class of Strong Kurzweil Henstock-Integrable Functions
- Integration by Parts
- A Variant of Gronwall Inequality
- Existence of Solutions of a Class of Generalized Ordinary Differential Equations
- A Convergence Process as a Source of Discontinuities in the Theory of Differential Eequations
- A Class of Strong Riemann-Integrable Functions
- On Equality of Two Integrals
- A Class of Generalized Ordinary Differential Equations with a Restricted Right Hand Side
Readership: Graduate students and mathematicians in differential equations; physicists and engineers interested in classical mechanics.
