Oscillation theorem for superlinear second order damped differential equations

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Abstract

In this paper, we are concerned with the oscillation of second order superlinear differential equations of the form(a(t)y(t))+p(t)y(t)+q(t)f(y(t))=0.A new oscillation criterion is established under quite general assumptions. Our results are based on the information on a sequence of subintervals of [t0, ∞) only, rather than on the whole half-line. Our methodology is somewhat different from that of previous authors [2], [3], [6], [7]. Our results generalize and extend some earlier results of Yu [6] and Lu and Meng [7]. An example is also given to illustrate the results.

Introduction

We consider the oscillatory behavior of the second order superlinear differential equation with damping(a(t)y(t))+p(t)y(t)+q(t)f(y(t))=0,where a, p, q: [t0, ∞)  R = (−∞, ∞) and f: R  R are continuous functions, and a(t) > 0 in t  [t0, ∞).

We recall that a function y: [t0, t1)  R, t1 > t0 is called a solution of Eq. (1.1) if y(t) satisfies Eq. (1.1) for all t  [t0, t1). In the sequel, it will always be assumed that solutions of Eq. (1.1) exist on some half-line [T, ∞) (T  t0). A solution y(t) of Eq. (1.1) is called oscillatory if it has arbitrarily large zeros, otherwise it is called nonoscillatory. Eq. (1.1) is called oscillatory if all its solutions are oscillatory.

Recently, Philos and co workers [2], [3], Meng [4], Li and Yan [5], Yu [6] and Lu and Meng [7] have studied the oscillatory behavior of superlinear differential equations. The following oscillation criteria are their study results.

For the special case of Eq. (1.1), i.e., for the Emden–Fowler equationy(t)+q(t)|y(t)|αsgny(t)=0,α>0.Wong [1] established the following oscillation criterion.

Theorem A

Iflimtinft0tq(s)ds>0andlimtsup1tn-1t0t(t-s)n-1q(s)ds=,for some integern>2.Then, Eq. (1.2) is oscillatory for every α > 0.

Recently, Wong’s result has been extended in Philos [2] to more general equations of the formy(t)+q(t)f(y(t))=0,where q: [t0, ∞)  R and f:R  R are continuous functions.

More precisely, Philos [2] presented the following oscillation criteria for differential equation (1.5).

Theorem B

Let the following conditions hold:

  • (i)

    yf(y) > 0, f(y)  0, when y  0;

  • (ii)

    duf(u)<,-duf(u)< ;

  • (iii)

    f(u)f(u)du<,-f(u)f(u)du<;

  • (iv)

    min{infu>0f(u)uf(s)f(s)ds,infu<0f(u)u-f(s)f(s)ds}>0

and suppose that there exists a continuously differentiable function φ: [t0, )  (0, ), which leads that φis nonnegative and decreasing function, and we havelimtinft0tφ(s)q(s)ds>-,limtsup1t2t0tφ(s)t0sduφ(u)ds<andlimtsup1tn-1t0t(t-s)n-1φ(s)q(s)ds=,for some integern>2,Then Eq. (1.5) is oscillatory.

Several years ago, Yu [6] obtained a similar oscillation criterion to Theorem B for the Eq. (1.1). Yu [6] extended the main results of Philos [2] to Eq. (1.1), and obtained the more general result.

In this paper, we shall continue in this direction the study of oscillatory properties of Eq. (1.1). The purpose of this paper is to continue to improve and extend the above-mentioned results. We shall further the investigation and offer some new criteria for the oscillation of Eq. (1.1). Our methodology (Tiryaki et al. [8]) is somewhat different from that of previous authors [2], [4]. We believe that our approach is simpler and more general than a recent result of Yu [6]. Our results improve the main results of [2], [6]. Example is given to illustrate the superiority of our results at the end of this paper.

There is no doubt that the Riccati substitution and its generalized forms play a very important role in the oscillatory theory of superlinear differential equations. In this paper, we shall employ a integral operator Aba (Çkmak and Tiryaki [9], [10]) to derive several oscillation criteria for Eq. (1.1), which are still new even in some particular cases.

Section snippets

Main results

In order to discuss our main results, we introduce the general mean and some properties that will be used in the proof of our results.

Let D(a, b) = {u  C1[a, b]: u(t)  0, u(a) = u(b) = 0}, where interval [a, b] is an arbitrary subinterval on [t0, ∞), and let ρ  C1([t0, ∞)), and ρ(t) > 0 on [t0, ∞). We take the integral operator Aab in terms of H  D(a, b) and ρ(t) asAab(h;t)=abH2(t)h(t)ρ(t)dt,atb,where h  C([t0, ∞)). It is easily seen that Aab is linear and positive, and in fact satisfies the following:Aab(α1h1+α2

An example

In order to show the application of our results obtained in this paper, let us consider the following second order superlinear differential equation with damping(tλ-1y(t))-tλ-2y(t)+Ktλ|y(t)|αsgny(t)=0,t1,where α > 1, 0 < λ  1, and K = (2 + λ)2/C are constants. Let H(t) = sint, φ(t) = t, and ρ(t) = t−(λ+1). For any T  1, choose n sufficiently large so that  = 2  T and set a = 2, b = (2k + 1)π. It is easy to verify thatAab(φq;t)=abH2(t)φ(t)q(t)ρ(t)dt=K2kπ(2k+1)πsin2tdt=12Kπand14CAab1v2HH+ρρ+ξη2;t14C2kπ(2k+

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