Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations

Title page
Preface
Basic Notation
Chapter I. LINEAR DIFFERENTIAL EQUATIONS
1. Equations Having Properties A and B
1.1. Classification of nonoscillatory solutions
1.2. Conjugate points of the equation (1.1)
1.3. Necessary and sufficient conditions for the equation (1.1) to have properties A and B
1.4. Integral criteria for the equation (1.1) to have properties A and B
1.5. On general equations
Notes
2. Oscillatory and Nonoscillatory Equations
2.1. Some auxiliary assertions
2.2. Oscillation and nonoscillation criteria
2.3. On third, fourth and sixth order equations
Notes
3. Oscillation Properties of Solutions of Equations with Strongly Oscillating Coefficients
3.1. Zeros of solutions on a finite interval
3.2. Oscillation of all solutions of equations with strongly oscillating coefficients
Notes
4. The Subspace of Solutions Vanishing at Infinity
4.1. Some integral identities and inequalities
4.2. On functions satisfying the conditions (4.19)
4.3. On a related problem
4.4. Theorems on the dimension of the space V^{(n,σ)} (p₀, . .. , p_{n-l})
4.5. Theorems on the dimension of the space V{(n,σ)}(p)
4.6. On solutions of second order equations
Notes
5 . Bounded and Unbounded Solutions
5.1. Inequalities of Kolmogorov-Gorny type
5.2. Lemmas on the solvability of related boundary value problems
5.3. On the space B{(n)}(p)
5.4. Existence theorems for unbounded oscillatory solutions
5.5. On second order equations
Notes
6. Asymptotic Formulas
6.1. Statement of the main theorem
6.2. Auxiliary assertions
6.3. Proof of the main theorem
6.4. Equations with almost-constant coefficients
6.5. Equations with asymptotically small coefficients
6.6. Equations asymptotically equivalent to two-term ones
Notes
Chapter II. QUASILINEAR DIFFERENTIAL EQUATIONS
7. Statement of the Problem. Auxiliary Assertions
7.1. Linear equations having the Levinson property
7.2. On a system of nonlinear integral equations
8. The Family of L_h Type Solutions of the Equation (7.1)
8.1. Necessary and sufficient conditions for the existence of L_h type solutions
8.2. Asymptotic representations for solutions of a related differential inequality
8.3. Stability of the farnily of L_h type solutions
Notes
9. L⁰_h, L^∞_h and L_h Type Equations
9.1. Lemmas on integral inequalities
9.2. L⁰_h type equations
9.3. L^∞_h type equations
9.4. L_h type equations
Notes
Chapter III. GENERAL NONLINEAR DIFFERENTIAL EQUATIONS
10. Theorems on the Classification of Equations with Respect to Their Oscillation Properties
10.1. Comparison theorem
10.2. Equations having property A or B
10.3. Equations having property A_k , B_k, A*_ k , or B*_k
Notes
11. Singular Solutions
11.1. Nonoscillatory first kind singular solutions
11.2. Nonoscillatory second kind singular solutions
11.3. Nonexistence theorem for singular solutions
Notes
12. Fast Growing Solutions
12.1. Existence theorem
12.2. Asymptotic estimates
Notes
13. Kneser Solutions
13.1. Existence theorem
13.2. Kneser solutions vanishing at infinity
Notes
14. Proper Oscillatory Solutions
14.1. Existence theorems
14.2. Proper oscillatory solutions vanishing at infinity
Notes
Chapter IV. HIGHER ORDER DIFFERENTIAL EQUATIONS OF EMDEN-FOWLER TYPE
15. Oscillatory Solutions
15.1. Classification of equations with respect to their oscillation properties
15.2. Existence of proper oscillatory solutions
15.3. Proper oscillatory solutions vanishing at infinity
15.4. Oscillatory first kind singular solutions
Notes
16. Nonoscillatory Solutions
16.1. Kneser solutions
16.2. Solutions with power asymptotics
16.3. Fast growing solutions
16.4. Second kind singular solutions
Notes
Chapter V. SECOND ORDER DIFFERENTIAL EQUATIONS OF EMDEN-FOWLER TYPE
17. Existence Theorems for Proper and Singular Solutions
17.1. Existence of proper solutions
17.2. Existence of nonoscillatory singular solutions
17.3. Existence of oscillatory singular solutions
Notes
18. Oscillation and Nonoscillation Criteria for Proper Solutions
18.1. Oscillation of all proper solutions
18.2. Existence of at least one oscillatory proper solution
18.3. Nonoscillation of all proper solutions
Notes
19. Unbounded and Bounded Solutions. Solutions Vanishing at Infinity
19.1. Bounded solutions
19.2. Solutions vanishing at infinity
19.3. Unbounded oscillatory solutions
Notes
20. Asymptotic Formulas
20.1 Asymptotic representations for oscillatory solutions
20.2. Auxiliary assertions
20.3. Asymptotic representations for nonoscillatory solutions (the case of a negative coefficient)
20.4. Asymptotic representations for nonoscillatory solutions (the case of a positive coefficient)
20.5. Asymptotic representations for nonoscillatory singular solutions
Notes
REFERENCES
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