Positive Solutions of Fourth Order Thomas-Fermi Type Differential Equations in the Framework of Regular Variation
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Positive Solutions of Fourth Order Thomas-Fermi Type Differential Equations in the Framework of Regular Variation Takaˆsi Kusano · Jelena Manojlovi´c
Received: 26 August 2011 / Accepted: 30 January 2012 / Published online: 18 February 2012 © Springer Science+Business Media B.V. 2012
Abstract The fourth order nonlinear differential equations x (4) = q(t)|x|γ sgn x,
0 < γ < 1,
(A)
with regularly varying coefficient q(t) are studied in the framework of regular variation. It is shown that thorough and complete information can be acquired about the existence of all possible regularly varying solutions of (A) and their accurate asymptotic behavior at infinity. Keywords Fourth order · Nonlinear differential equations · Positive solutions · Regularly varying functions · Asymptotic behavior Mathematics Subject Classification (2000) 34C11 · 26A12
1 Introduction This paper is concerned with the fourth order differential equations x (4) = q(t)|x|γ sgn x,
(A)
where 0 < γ < 1 and q(t) is a positive continuous function on [a, ∞), a > 0. Equations (A) are called sublinear or superlinear according as 0 < γ < 1 or γ > 1.
T. Kusano Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan e-mail: [email protected] J. Manojlovi´c () Faculty of Science and Mathematics, Department of Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia e-mail: [email protected]
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T. Kusano, J. Manojlovi´c
A solution x(t) of (A) existing in an infinite interval of the form [Tx , ∞) is said to be proper if sup |x(t)| : t ≥ T > 0 for any T ≥ Tx . A proper solution is called oscillatory if it has an infinite sequence of zeros clustering at infinity and nonoscillatory otherwise. Thus, a nonoscillatory solution is eventually positive or eventually negative. It is clear that if x(t) satisfies (A), then so does −x(t), and hence to examine the nonoscillatory solutions it suffices to restrict attention to positive solutions of (A). Beginning with the papers [6, 7] of Kiguradze, oscillation theory of higher order nonlinear differential equations has been the subject of intensive investigations in recent years. Kiguradze’s oscillation theorem (see also the book of Kiguradze et al. [8, Theorems 15.1 and 15.3]) specialized to (A) read as follows. Theorem A Any proper solution of (A) is either oscillatory or satisfies |x (i) (t)| ↓ 0
as t ↑ ∞ (i = 0, 1, 2, 3),
(1.1)
|x (i) (t)| ↑ ∞
as t ↑ ∞ (i = 0, 1, 2, 3)
(1.2)
or
if and only if
∞
t 3γ q(t)dt = ∞.
(1.3)
a
The questions that naturally arise from Theorem A are: (i) If (1.3) holds, does (A) really possess positive solutions satisfying (1.1) and (1.2)? If such solutions exists, what can be said about the exact asymptotic decay or growth of such solutions? (ii) If (1.3) does not hold, is it possible to characterize the existence of all possible positive solutions of (A) and moreover to determine their asymptotic behavior at infinity accurately? These questions seem to be extremely difficult to answer in general (see Kigura
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