In almost all works in the literature there are several results showing asymptotic relationships between the solutions of
x&prime / &prime / = f (t, x) (0.1)
and the solutions 1 and t of x&prime / &prime / = 0. More specifically, the existence of a solution of (0.1) asymptotic to x(t) = at + b, a, b &isin / R has been obtained.
In this thesis we investigate in a systematic way the asymptotic behavior as t &rarr / &infin / of solutions of a class of differential equations of the form
(p(t)x&prime / )&prime / + q(t)x = f (t, x), t &ge / t_0 (0.2)
and
(p(t)x&prime / )&prime / + q(t)x = g(t, x, x&prime / ), t &ge / t_0 (0.3)
by the help of principal u(t) and nonprincipal v(t) solutions of the corresponding homogeneous equation
(p(t)x&prime / )&prime / + q(t)x = 0, t &ge / t_0. (0.4)
Here, t_0 &ge / 0 is a real number, p &isin / C([t_0,&infin / ), (0,&infin / )), q &isin / C([t_0,&infin / ),R), f &isin / C([t_0,&infin / ) × / R,R) and g &isin / C([t0,&infin / ) × / R × / R,R).
Our argument is based on the idea of writing the solution of x&prime / &prime / = 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&rsquo / 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).
| Identifer | oai:union.ndltd.org:METU/oai:etd.lib.metu.edu.tr:http://etd.lib.metu.edu.tr/upload/12615405/index.pdf |
| Date | 01 January 2013 |
| Creators | Ertem, Turker |
| Contributors | Zafer, Agacik |
| Publisher | METU |
| Source Sets | Middle East Technical Univ. |
| Language | English |
| Detected Language | English |
| Type | Ph.D. Thesis |
| Format | text/pdf |
| Rights | Access forbidden for 1 year |
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