Interval criteria for oscillation of second-order differential equations with mixed nonlinearities
Introduction
In this paper, we investigate the oscillatory behavior of solutions of the second order nonlinear differential equationwhere p(t) is positive and differentiable and f(t,x) is of the form:where q(t), qi(t) ∈ C[0,∞), and α1 > ⋯ > αm > 1 > αm+1 > ⋯ > αn > 0.
As usual, a solution x(t) of (1) is said to be oscillatory if it is defined on some ray [T,∞) with T ⩾ 0 and has unbounded set of zeros. Eq. (1) is said to be oscillatory if all solutions extendable throughout [0,∞) are oscillatory. When f(t, x) takes the form (2), it is known that if q(t), qi(t) are also positive and continuously differentiable then all solutions of (1) are extendable throughout [0,∞). However, when q(t), qi(t) change signs as t tends to infinity, it is also known that Eq. (1) can have solutions with finite escape time, i.e. it can become infinite at some finite t. See Coffman and Wong [6]. We shall for simplicity confine our discussion only to extendable solutions throughout this paper.
For the particular case when p(t) ≡ 1, q(t) ≡ 0, and n = 1 in (2), Eq. (1) reduces to the familiar Emden–Fowler equationWhen α1 > 1, Eq. (3) is known as the superlinear equation and when 0 < α1 < 1, it is known as the sublinear equation. The oscillation of Eq. (3) has been the subject of much attention during the last 50 years, see the seminal book by Agarwal, Grace and O’Regan [2]. Most of the results on oscillation of (3) are valid for the more general equationwhere xf(x) > 0 for x ≠ 0 and f(x) satisfies certain conditions of superlinearity and sublinearity (see [2]). In particular, f(x) can be finite sum of powers of x and if there exist in this sum exponents of x which are both greater than and less than 1, then Eq. (4) is known as Emden–Fowler equation of the mixed type. When f(t,x) in (1) takes the form of (2) and Eq. (1) is of a mixed type, results on oscillation are more or less the same when q(t), qi(t), i = 1, 2 …, n, are non-negative. This is however not the case when q(t), qi(t) are oscillatory. When n = 1, it is known (see Butler [4]) that Eq. (3) is oscillatory if q1(t) is periodic and of mean value zero, e.g. sint or cost. When f(x) is a finite sum of powers of x, the more general equation is also oscillatory as shown in a subsequent paper by Butler [5] (see also [12]). However, for Eq. (1) with f(t, x) as given in the form of (2) there seems to us no known oscillation criterion applicable to the simple equation of the mixed type:where α1 > 1 and 0 < α2 < 1.
On the other hand, if we consider the forced equationwhere e(t) is itself an oscillatory function, we also wish to establish oscillation theorems when n > 1. When n = 1, this approach was initiated by Agarwal and Grace [1, pp. 244–249] for higher order equations and subsequently developed in papers of Ou and Wong [14], Sun and Agarwal [22], [23], Sun and Saker [24], Yang [29], and Yang [30].
For the simple case when n = 1, p(t) ≡ 1, q(t) ≡ 0 and q1(t) is non-negative in (5), one can use a technique first introduced by Kartsatos [8], [9] by assuming that e(t) is the second derivative of an oscillatory function to obtain oscillation criteria for Eq. (5) as described in the earlier work [27]. See also previous results in Keener [10], Rankin [16], Skidmore and Leighton [17], Skidmore and Bowers [18] and Teufel [26] for special cases of the forced Eq. (5).
When the assumption that e(t) is the second derivative of an oscillatory function is not imposed, one can develop interval oscillation criteria for special cases of (5), in case qi(t) ≡ 0 for all i = 1, 2, …, n, see El-Sayed [7], Sun, Ou and Wong [21] and Wong [28], in case q(t) ≡ 0 and n = 1 see Nazr [13] when α1 > 1 and Sun [19], Sun and Wong [20] when 0 < α1 < 1.
In this paper, we will employ the method in Kong [11] and the arithmetic-geometric mean inequality (see [3]) to establish several interval oscillation criteria for the unforced Eq. (1) and the forced Eq. (5), respectively. In the former case our results are generalizations of the main results in [11]. Examples are given to illustrate our results when compared with known results on Emden–Fowler equations. We hope to kindle reader’s interest in further research on the oscillation of equations of mixed type, which arise for example in the growth of bacteria population with competitive species.
Section snippets
Main results
We will need the following lemmas that have been proved in our recent paper [25]: Lemma 1 Let {αi}, i = 1, 2, …, n, be the n-tuple satisfying α1 > ⋯ > αm > 1 > αm+1 > ⋯ αn > 0. Then there exists an n-tuple (η1, η2, …, ηn) satisfyingwhich also satisfies either Lemma 2 Let u, A, B, C and D be positive real numbers. Then Auγ + B ⩾ γ(γ − 1)1/γ−1 A1/γB1−1/γu, γ > 1; Cu − Duγ ⩾ (γ − 1)γγ/(1−γ) Cγ/(γ−1)D1/(1−γ), 0 < γ < 1.
Remark 1
For a given set or exponents {αi} satisfying α1 > ⋯ > αm > 1 > αm+1 > ⋯ αn > 0, Lemma 1 ensures the
Examples
In this section, we will work out two numerical examples to illustrate our main results. Example 1 Consider the following equation:where k, l, m are positive constants, α1 > 1 and 0 < α2 < 1. According to the direct computation, we have , where , η0 can be any positive number satisfying 0 < η0 < (α1 − 1)/α1, η1 and η2 satisfy condition (6). For any T ⩾ 0, we can choose a1 = 2iπ, b1 = a2 = 2iπ + π/4, b2 = 2iπ + π/2, c1 = 2iπ + π
Acknowledgements
This work was supported by the National Natural Science Foundations of China (10771118 and 10671105) and the Natural Science Foundations of Shandong Province of China (Q2006A01 and Y2005A06).
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