Author | Hsieh, Po-Fang. author |
---|---|
Title | Basic Theory of Ordinary Differential Equations [electronic resource] / by Po-Fang Hsieh, Yasutaka Sibuya |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1999 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1506-6 |
Descript | XI, 469 p. online resource |
I. Fundamental Theorems of Ordinary Differential Equations -- I-1. Existence and uniqueness with the Lipschitz condition -- I-2. Existence without the Lipschitz condition -- I-3. Some global properties of solutions -- I-4. Analytic differential equations -- Exercises I -- II. Dependence on Data -- II-1. Continuity with respect to initial data and parameters -- II-2. Differentiability -- Exercises II -- III. Nonuniqueness -- III-l. Examples -- III-2. The Kneser theorem -- III-3. Solution curves on the boundary of R(A) -- III-4. Maximal and minimal solutions -- III-5. A comparison theorem -- III-6. Sufficient conditions for uniqueness -- Exercises III -- IV. General Theory of Linear Systems -- IV-1. Some basic results concerning matrices -- IV-2. Homogeneous systems of linear differential equations -- IV-3. Homogeneous systems with constant coefficients -- IV-4. Systems with periodic coefficients -- IV-5. Linear Hamiltonian systems with periodic coefficients -- IV-6. Nonhomogeneous equations -- IV-7. Higher-order scalar equations -- Exercises IV -- V. Singularities of the First Kind -- V-1. Formal solutions of an algebraic differential equation -- V-2. Convergence of formal solutions of a system of the first kind -- V-3. TheS-Ndecomposition of a matrix of infinite order -- V-4. TheS-Ndecomposition of a differential operator -- V-5. A normal form of a differential operator -- V-6. Calculation of the normal form of a differential operator -- V-7. Classification of singularities of homogeneous linear systems -- Exercises V -- VI. Boundary-Value Problems of Linear Differential Equations of the Second-Order -- VI- 1. Zeros of solutions -- VI- 2. Sturm-Liouville problems -- VI- 3. Eigenvalue problems -- VI- 4. Eigenfunction expansions -- VI- 5. Jost solutions -- VI- 6. Scattering data -- VI- 7. Reflectionless potentials -- VI- 8. Construction of a potential for given data -- VI- 9. Differential equations satisfied by reflectionless potentials -- VI-10. Periodic potentials -- Exercises VI -- VII. Asymptotic Behavior of Solutions of Linear Systems -- VII-1. Liapounoffโs type numbers -- VII-2. Liapounoffโs type numbers of a homogeneous linear system -- VII-3. Calculation of Liapounoffโs type numbers of solutions -- VII-4. A diagonalization theorem -- VII-5. Systems with asymptotically constant coefficients -- VII-6. An application of the Floquet theorem -- Exercises VII -- VIII. Stability -- VIII- 1. Basic definitions -- VIII- 2. A sufficient condition for asymptotic stability -- VIII- 3. Stable manifolds -- VIII- 4. Analytic structure of stable manifolds -- VIII- 5. Two-dimensional linear systems with constant coefficients -- VIII- 6. Analytic systems in ?n -- VIII- 7. Perturbations of an improper node and a saddle point -- VIII- 8. Perturbations of a proper node -- VIII- 9. Perturbation of a spiral point -- VIII-10. Perturbation of a center -- Exercises VIII -- IX. Autonomous Systems -- IX-1. Limit-invariant sets -- IX-2. Liapounoffโs direct method -- IX-3. Orbital stability -- IX-4. The Poincarรฉ-Bendixson theorem -- IX-5. Indices of Jordan curves -- Exercises IX -- X. The Second-Order Differential Equation $$\frac{{{d̂2}x}}{{d{t̂2}}} + h(x)\frac{{dx}}{{dt}} + g(x) = 0 $$ -- X-1. Two-point boundary-value problems -- X-2. Applications of the Liapounoff functions -- X-3. Existence and uniqueness of periodic orbits -- X-4. Multipliers of the periodic orbit of the van der Pol equation -- X-5. The van der Pol equation for a small ?> 0 -- X-6. The van der Pol equation for a large parameter -- X-7. A theorem due to M. Nagumo -- X-8. A singular perturbation problem -- Exercises X -- XI. Asymptotic Expansions -- XI-1. Asymptotic expansions in the sense of Poincarรฉ -- XI-2. Gevrey asymptotics -- XI-3. Flat functions in the Gevrey asymptotics -- XI-4. Basic properties of Gevrey asymptotic expansions -- XI-5. Proof of Lemma XI-2-6 -- Exercises XI -- XII. Asymptotic Expansions in a Parameter -- XII-1. An existence theorem -- XII-2. Basic estimates -- XII-3. Proof of Theorem XII-1-2 -- XII-4. A block-diagonalization theorem -- XII-5. Gevrey asymptotic solutions in a parameter -- XII-6. Analytic simplification in a parameter -- Exercises XII -- XIII. Singularities of the Second Kind -- XIII-1. An existence theorem -- XIII-2. Basic estimates -- XIII-3. Proof of Theorem XIII-1-2 -- XIII-4. A block-diagonalization theorem -- XIII-5. Cyclic vectors (A lemma of P. Deligne) -- XIII-6. The Hukuhara-Turrittin theorem -- XIII-7. An n-th-order linear differential equation at a singular point of the second kind -- XIII-8. Gevrey property of asymptotic solutions at an irregular singular point -- Exercises XIII -- References