Date

2013

Author

Ertem, Türker

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In almost all works in the literature there are several results showing asymptotic relationships between the solutions of x′′ = f (t, x) (0.1) and the solutions 1 and t of x′′ = 0. More specifically, the existence of a solution of (0.1) asymptotic to x(t) = at + b, a, b ∈ R has been obtained. In this thesis we investigate in a systematic way the asymptotic behavior as t → ∞ of solutions of a class of differential equations of the form (p(t)x′)′ + q(t)x = f (t, x), t ≥ t_0 (0.2) and (p(t)x′)′ + q(t)x = g(t, x, x′), t ≥ t_0 (0.3) by the help of principal u(t) and nonprincipal v(t) solutions of the corresponding homogeneous equation (p(t)x′)′ + q(t)x = 0, t ≥ t_0. (0.4) Here, t_0 ≥ 0 is a real number, p ∈ C([t_0,∞), (0,∞)), q ∈ C([t_0,∞),R), f ∈ C([t_0,∞) × R,R) and g ∈ C([t0,∞) × R × R,R). Our argument is based on the idea of writing the solution of x′′ = 0 in terms of principal and nonprincipal solutions as x(t) = av(t) + bu(t), where v(t) = t and u(t) = 1. In the proofs, Banach and Schauder’s fixed point theorems are used. The compactness of the operator is obtained by employing the compactness criteria of Riesz and Avramescu. The thesis consists of three chapters. Chapter 1 is introductory and provides statement of the problem, literature review, and basic definitions and theorems. In Chapter 2 first we deal with some asymptotic relationships between the solutions of (0.2) and the principal u(t) and nonprincipal v(t) solutions of (0.4). Then we present existence of a monotone positive solution of (0.3) with prescribed asimptotic behavior. In Chapter 3 we introduce the existence of solution of a singular boundary value problem to the Equation (0.2).

Subject Keywords

Dynamics., Differential equations, Integral equations, Asymptotic expansions.

URI

http://etd.lib.metu.edu.tr/upload/12615405/index.pdf
https://hdl.handle.net/11511/22280

Collections

Graduate School of Natural and Applied Sciences, Thesis