PageHeader : Funkcialaj Ekvacioj, 1 (195S), 77?132
Title : Non-linear Ordinary Differential Equations
Title : with Periodic Coefficients
AuthorInfo : By Yasutaka SIBUYA
AuthorInfo : (Ochanomizu University)
section : Introduction
section : CONTENTS.
section : CHAPTER I. LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH BOUNDED COEFFICIENTS CONTAINING A FINITE NUMBER OF PARAMETERS
subsection : 1. Differential equations with bounded coefficients
subsubsection : 1. 1. Formal reduction.
subsection : 1. 2. Convergence of the series $Q(t, a)$ .
subsection : 2. Differential Equations with periodic coefficients
subsubsection : 2. 1. Formal reduction.
subsubsection : 2. 2. Convergence of the series $Q(t, a)$ .
subsubsection : 2. 3. Equations of Mathieu.
subsubsection : 2. 4. Remarks.
section : CHAPTER II. FORMAL REDUCTION OF NON-LINEAR ORDbNARY DIFFERENTIAL EQUATIONS WITH BOUNDED COEFFICIENTS
subsection : 3. Formal reduction
subsubsection : 3. 1. The case of bounded coefficients.
subsubsection : 3. 2. The case of almost periodic coefficients.
subsubsection : 3. 3. The case of periodic coefficients.
subsection : 4. Classical results
subsubsection : 4. 1. Lemmas. Now we shall prove the following two lemmas:
subsubsection : 4. 2. Classical results.
section : CHAPTER III. BOUNDED SOLUTIONS OF NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH ALMOST PERIODIC COEFFICIENTS
subsection : 5. Remarks on the solutions of the differential equations (3. 1. 5)
subsubsection : 5. 1. The case of bounded coefficient.
subsubsection : 5. 2. The case of almost periodic coefficients.
subsection : 6. Bounded solutions of the system (3. 1. 1) with almost periodic $\mathrm{c}\dot{\mathrm{o}}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$
subsubsection : 6. 1. Solutions bounded in [ $0,$ $+\infty)$ .
subsubsection : 6. 2. Almost periodic solution.
subsubsection : 6. 3. Remarks on the general solutions.
subsubsection : 6. 4. The case $\lambda_{i}$ are purely imaginary.
section : CHAPTER IV. NON-LINEAR ORDINARY DIFFERENTIAL EQUATION WITH BOUNDED COEFFICIENTS CONTAINING A FINITE NUMBER OF PARAMETERS
subsection : 7. Preliminary remarks
subsubsection : 7. 1. Reduction of problem.
subsection : 8. Formal reduction
subsubsection : 8. 1. The case of bounded coefficients.
subsubsection : 8. 2. Remarks on the case of almost periodic coefficients.
subsection : 9. Convergence of formal solutions
subsubsection : 9. 1. Preliminary remarks.
subsubsection : 9. 2. Convergence of the formal solution (9. 1. 21).
section : CHAPTER V. PF.RIODIC SOLUTIONS
subsection : 10. Formal calculation
subsubsection : 10. 1. Auxiliary equations.
subsubsection : 10. 2. Convergence of the formal series.
subsection : 11. Conditions of periodicity
subsubsection : 11. 1. Main theorem.
subsubsection : 11. 2. Differential equations containing parameters.
subsection : 12. Application of formal reductions
subsubsection : 12. 1. Formal solutions.
subsubsection : 12. 2. Criterion for convergence.
subsubsection : 12. 3. Applications of the criterion.
subsubsection : 12. 4. Remarks on existence of periodic solutions.
subsubsection : 12. 5. The case of algebraic functions.
subsection : 13. Autonomous systems
subsubsection : 13. 1. Preliminary remarks.
subsubsection : 13. 2. The case $\omega_{0}=0$ .
subsubsection : 13. 3. The case $\omega_{0}\neq 0$ .
subsubsection : 13. 4. Convergence of the formal solution.
section : References
subsection : Supplement to References