ASYMPTOTIC THEORY FOR A CLASS OF FOURTH - ORDER DIFFERENTIAL EQUATIONS

Abstract

Abstract : An asymptotic theory is developed for a class of fourth-order differential equations. Under a general condition on the coefficients of the differential equation, we obtain the forms of the asymptotic solutions. In this thesis, we consider the asymptotic form of a fundamental set of solutions of the fourth-order differential equation (2.1.1), which is (p0(x) y"(x))" + (p1(x) y'(x))' - p2(x) y(x) = 0 as x→∞. We shall consider the case where p1 is small compared to p1 and p2, in the sense that p1 = 0{(p0 p2)1/2} p2 as x→∞ In this case, all the solutions have a similar exponential factor as given in Theorem 2.3.1 in chapter 2 (§3). In chapter 1 (§3), we introduce the fundamental theorem of Eastham, which, subject to a dichotomy condition on A and to the smallness condition on R, is that, subject to a dichotomy condition on A and to the smallness condition on R. Also, in chapter 1 (§4), we use the recent asymptotic theorem of Eastham, which, although a consequence of the basic Levinson theorem, is in a form suitable for many applications. One of those applications is to differential equations of the second order, which is given in chapter 1 (§5). The methods we use to obtain the solutions of (2.1.1) in the above case are given in chapter 2. Also, in chapter 2 (§3) we use Eastham's asymptotic theorem [7, 8] to obtain the main asymptotic result. Two examples of (2.1.1) are considered at the end of chapter 2 (§4). Finally, in chapter 2 (§4) we discuss the relationship of our result to the asymptotic theory of fourth-order differential equations based on Eastham's result [5], Al-Hammadi's result [2], and Hinton's result [10].