1. INTRODUCTION

We consider the second order nonlinear differential equation

\[u'' + f(t, u) = 0, t \ge 0\] (1.1)

uhere the function /satisfies certain conditions to be given later.

Definition 1.1

• (a) A solutionu(t) of () I) is called oscillatory in [0,cc) if for every rp e l0,oc) there exists a t, ) lo such that u(tr) = 0.
• (b) A solution u(t) of (l) is called non oscillatory in [0,o) if there exists a l, > 0such that u(t)*0 for t > tr.
• (c) The differential equation (l.l) is called oscillatory [O,cc) if every solution of (l l) is oscillatory in [0,co).

Oscillation criteria for the differential eouation

\[u'' + a(t)f(u) = 0, t \ge 0\] (1.2)

includrng the Emden-Fou'ler equation, where/(D -- | ul, ,g, u, has been developed b1'man1'authors, in particular, the papers of [2,3,6.7,8,9j. Oscillation criteria for second order nonlinear inequalities which contain Eqn (l.l) has been developed in [4]. However, the conditions for function/(r, u) in this paper are quite different to conditions in [4] and also the oscillation critena In [,5] the oscillation criteria for Eqn (l.l) has been discussed. Some conditions in this paper are less weaker than those in [] and our results are anothertype of oscillatorl'criteria for Eqn (l.l).

Our main results will contain and extend some previous results in [2,5].Oscillation criteria will be established b1' modif ing the methods that have been used previously in [] and [5]

2. OSCILLATION THEOREI\{S

The following assumptions on function/willbe retained in the sequel.

Assumption A There exist a function a e C [0,co) and a function Q. Cl (-*,*) such that

\[\frac{f(t,u)}{\phi(u)} \ge a(t), \ t \ge 0,\] with /'>O,uQ@) > 0 for u +0 and

\[G(u) = \int_{0}^{u} \frac{d\tau}{\phi(\tau)} < \infty.\]

We now have the following theorems.

Theorem 2.1 Assume that Assumption A holds and there exists a function \(\rho \in C^1\) \([0,\infty)\) with \(\rho \geq 0\) and \(\rho' \leq 0\) such that

\[\lim_{\tau \to \infty} \sup \frac{1}{T} \int_{0}^{\tau} \rho^{\lambda}(t) \int_{0}^{t} a(\tau) d\tau dt = \infty\] (2.1)

for some \(\lambda \ge 0\). Then Eqn (1.1) is oscillatory in \([0,\infty)\).

Proof Suppose that Eqn (1.1) has a solution u(t) which is non oscillatory in \([0,\infty)\). Assume that u(t) > 0 on \([t_0,\infty)\) for some \(t_0 \ge 0\). Let

\[v(t) = \frac{\rho^{\lambda}(t)u'(t)}{\phi(u(t))}, \ t \ge t_0\]

Then

\[v' = \frac{\rho^{\lambda} u'' + \lambda \rho^{\lambda - 1} \rho' u'}{\phi(u)} - \rho^{\lambda} \frac{u'^{2} \phi'(u)}{\phi^{2}(u)}\]

\[= -\rho^{\lambda} \frac{f(t,u)}{\phi(u)} + \frac{\lambda \rho' v}{\rho} - \frac{v^2}{\rho^{\lambda}} \phi'(u)\]

By Assumption A and \(\rho > 0\), we obtain

\[v' \leq -\rho^{\lambda}(t)a(t) + \lambda \frac{\rho'}{\rho} v, \ t \geq t_{0}\] \[\frac{\rho v' - \lambda' v}{\rho} \leq -\rho^{\lambda}(t)a(t)\] \[\frac{\rho v' - \lambda' v}{\rho^{\lambda+1}} \leq -a(t)\] \[\frac{d}{dt} \left(\frac{v}{\rho^{\lambda}}\right) \leq -a(t), \ t \geq t_{0}\] (2.2)

Integrating on \([t_0,t]\), we have

\[\frac{v(t)}{\rho^{\lambda}(t)} \le \frac{v(t_0)}{\rho^{\lambda}(t_0)} - \int_{t_0}^{t} a(\tau) d\tau\]

Let \(v(t_0) > 0\), in case \(v(t_0) \le 0\), we omit \(\frac{v(t_0)}{\rho^{\lambda}(t_0)}\) from above equation.

Multiply by \(\rho^{\lambda}(t) > 0\), we obtain

\[v(t) \leq \frac{v(t_0)}{\rho^{\lambda}(t_0)} \rho^{\lambda}(t) - \rho^{\lambda}(t) \int_{t_0}^{t} a(\tau) d\tau\] \[\frac{\rho^{\lambda}(t)u'(t)}{\phi(u(t))} \leq \frac{v(t_0)}{\rho^{\lambda}(t_0)} \rho^{\lambda}(t) - \rho^{\lambda}(t) \int_{t_0}^{t} a(\tau) d\tau\]

Integrating on [t₀,T], we obtain

\[\int_{t_0}^{T} \frac{\rho^{\lambda}(t)u'(t)}{\phi(u(t))} dt \leq \frac{v(t_0)}{\rho^{\lambda}(t_0)} \int_{t_0}^{T} \rho^{\lambda}(t) dt - \int_{t_0}^{T} \rho^{\lambda}(t) \int_{t_0}^{t} a(\tau) d\tau dt\]

Since \(\rho(t)\) decreasing on \([0,\infty)\), \(\rho > 0\) and \(\lambda \ge 0\), we get

\[\int_{t_{0}}^{T} \rho^{\lambda}(t) dG(u(t)) \leq \frac{v(t_{0})}{\rho^{\lambda}(t_{0})} \rho^{\lambda}(t_{0}) (T - t_{0}) - \int_{t_{0}}^{T} \rho^{\lambda}(t) \int_{t_{0}}^{t} a(\tau) d\tau dt\] \[\rho^{\lambda}(t) G(u(t)) - \rho^{\lambda}(t_{0}) G(u(t_{0})) - \lambda \int_{t_{0}}^{T} G(u(t)) \rho^{\lambda - 1}(t) \rho^{1}(t) dt \leq\] \[\leq v(t_{0}) (T - t_{0}) - \int_{t_{0}}^{T} \rho^{\lambda}(t) \int_{t_{0}}^{t} a(\tau) d\tau dt\]

Since \(\rho^{\lambda}(T) > 0\), \(G(u(T) \ge 0\), and \(\rho'(t) \le 0\), \(\lambda \ge 0\), we obtain

\[\int_{0}^{T} \rho^{\lambda}(t) \int_{0}^{t} a(\tau) d\tau dt \le v(t_0)(T - t_0) + C, \quad \text{where} \quad C = \rho^{\lambda}(t_0) G(u(t_0)).\]

Dividing by T, we have

\[\frac{1}{T}\int_{t_0}^{T} \rho^{\lambda}(t)\int_{t_0}^{t} a(\tau)d\tau dt \leq v(t_0)\left(1-\frac{t_0}{T}\right) + \frac{C}{T}.\]

Condition (2.1) gives a contradiction. Similar condition will be obtained when u(t) < 0 on \([0,\infty)\).

Remark In case \(\rho = 1\), or \(\lambda = 0\), we obtain the oscillation criteria in [2]. When \(\lambda = 1\), we obtain the oscillation criteria in [6]. The condition for \(\rho\) is less weaker than in [1] and condition (2.1) is another type of oscillation criteria for Eqn (1.1).

Theorem 2.2 Assume that Assumption A holds and there exists a function \(\rho \in \mathbb{C}^2\) \([0,\infty)\) with \(\rho > 0\), \(\rho' \le 0\) and \(\rho'' \ge 0\) such that

\[\lim_{T \to \infty} \sup \frac{1}{T} \int_{0}^{T} \int_{0}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau dt = \infty, \tag{2.3}\] for some \(\lambda \ge 1\). Then Eqn (1.1) is oscillatory in \([0,\infty)\).

Proof Suppose that u (t) is a non oscillatory solution of Eqn (1.1) in \([0,\infty)\). Let u(t) > 0 on \([t_0,\infty)\) for some \(t_0 \ge 0\). Let

\[v(t) = \frac{\rho^{\lambda}(t)u'(t)}{\phi(u(t))}, \ t \ge t_0\]

Arguing as in the proof of theorem 2.1, we obtain the eqn (2.2), i.e

\[\frac{d}{dt} \left( \frac{v}{\rho^{\lambda}} \right) \le -a(t)\] \[\rho^{\lambda} \frac{d}{dt} \left( \frac{v}{\rho^{\lambda}} \right) \le -\rho^{\lambda}(t)a(t) \tag{2.4}\]

Integrating (2.4) on [t₀, t], we get

\[\int_{t_0}^{t} \rho^{\lambda}(\tau) d\left(\frac{v}{\rho^{\lambda}}\right) \leq -\int_{t_0}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau\] \[v(t) - v(t_0) - \lambda \int_{t_0}^{t} \frac{v(\tau) \rho'(\tau)}{\rho} d\tau \leq -\int_{t_0}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau\] \[v(t) \leq v(t_0) + \lambda \int_{t_0}^{t} \rho^{\lambda-1}(\tau) \rho'(\rho) dG(u(\tau)) - \int_{t_0}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau\] \[v(t) \leq v(t_0) + \lambda [\rho^{\lambda-1}(t) \rho'(t) G(u(t)) - \rho^{\lambda-1}(t_0) \rho'(t_0)) G(u(t_0))] - -\lambda \int_{t_0}^{t} G(u(\tau)) [\rho^{\lambda-1}(\tau) \rho''(\tau) + (\lambda - 1) \rho^{\lambda-2}(\tau) (\rho'(\tau))^2] d\tau - \int_{t_0}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau.\]

Since \(\rho > 0\), \(\rho' \le 0\), \(\rho'' \ge 0\), \(G(u(t)) \ge 0\) and \(\lambda \ge 1\), we have

\[v(t) \leq C_1 - \int_{t_0}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau\] \[C_1 = v(t_0) - \lambda \rho^{\lambda-1}(t_0) \rho'(t_0) G(u(t_0))\] \[\frac{\rho^{\lambda}(t) u'(t)}{\phi(u(t))} \leq C_1 - \int_{t}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau \quad (2.5)\] where

Integrating on [to, T] to obtain

\[\int_{t_0}^{T} \rho^{\lambda}(t) dG(u(t)) \leq C_1(T - t_0) - \int_{t_0}^{T} \int_{t_0}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau dt\] \[\rho^{\lambda}(T) G(u(T)) - \rho^{\lambda}(t_0) G(u(t_0)) - \lambda \int_{t_0}^{T} G(u(t)) \rho^{\lambda - 1}(t) \rho^{1}(t) dt \leq\] \[\leq C_1(T - t_0) - \int_{t_0}^{T} \int_{t_0}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau dt\]

Since \(\rho > 0\), \(\rho' \le 0\) and \(G(u(T)) \ge 0\), we obtain

\[\int_{t_0}^T \int_{t_0}^t \rho^{\lambda}(\tau) a(\tau) d\tau dt \le C_2 + C_1(T - t_0),\] where \(C_2 = \rho^{\lambda}(t_0)G(u(t_0))\).

Dividing by T, we have

\[\frac{1}{T} \int_{t_0}^T \int_{t_0}^t \rho^{\lambda}(\tau) a(\tau) d\tau dt \le \frac{C_2}{T} + C_1 \left(1 - \frac{t_0}{T}\right)\]

By taking limit sup and \(T\to\infty\), from (2.3) we arrive at a contradiction. Similar result will be obtained when u(t) < 0, on \([t_0, \infty)\), for some \(t_0 \ge 0\).

Remark Condition (2.3) improves and extends the results in [2] and [5]. Condition (2.3) is another type oscillation for Eqn (1.1).

The following theorem weakens and extends the oscillation criterion in [1].

Theorem 2.3 Assume that Assumption A holds and there exists a function \(\rho \in \mathbb{C}^2[0,\infty]\) with \(\rho > 0\), \(\rho' \le 0\) and \(\rho'' \ge 0\) such that

\[\int_{a}^{\infty} \rho^{\lambda}(t)a(t)dt = \infty, \tag{2.6}\] for some \(\lambda \ge 1\) and \(\alpha \ge 0\). Then Eqn (1.1) is oscillatory in \([0,\infty]\).

Proof Suppose that u(t) is a non oscillatory solution of Eqn (1.1) in \([0, \infty]\). Let u(t) > 0 on \([t_0, \infty)\) for some \(t_0 \ge 0\). Let

\[v(t) = \frac{\rho^{\lambda}(t)u'(t)}{\phi(u(t))}, \ t \ge t_0\]

Arguing similar to proof of theorem 2.2, we obtain eqn (2.5)

\[\frac{\rho^{\lambda}(t)u'(t)}{\phi(u(t))} \le C_1 - \int_{t_0}^t \rho^{\lambda}(\tau)a(\tau)d\tau, \tag{2.7}\] where \(C_1 = v(t_0) - \lambda \rho^{\lambda - 1}(t_0) \rho'(t_0) G(u(t_0))\)

From condition (2.6), it follows that there exists a \(t_1 \ge t_0\) such that

\[\int_{t_0}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau \ge 2 |C_1|, \ t \ge t_1\]

Hence (2.7) becomes the inequality

\[\frac{\rho^{\lambda}(t)u'(t)}{\phi(u(t))} \leq -\frac{1}{2}\int_{t_0}^t \rho^{\lambda}(\tau)a(\tau)d\tau, \ t \geq t_1.\]

Integrating on [t₁,T), we have

\[\rho^{\lambda}(T)G(u(T)) - \rho^{\lambda}(t_1)G(u(t_1) - \lambda \int_{t_1}^{\tau} G(u(t))\rho^{\lambda-1}(t)\rho'(t)dt \le\] \[-\frac{1}{2}\int_{t_1}^{\tau} \int_{t_0}^{t} \rho^{\lambda}(\tau)a(\tau)d\tau dt\]

Since \(\rho(T) > 0\), G(u(T)) > 0, \(\rho'(t) \le 0\) and \(\lambda \ge 1\), we obtain

\[\int_{t_1}^{T} \int_{t_0}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau dt \le 2\rho^{\lambda}(t_1) G(u(t_1)) = K, \ T \ge t_1\] \[(2.8)\]

From condition (2.6) we can choose a \(t_2 \ge t_1\) such that

\[\int_{t_o}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau \ge K, \ t \ge t_2\]

It follows that

\[\int_{t_1}^{T} \int_{t_2}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau dt \ge \int_{t_2}^{T} \int_{t_2}^{t} \rho^{\lambda}(\tau) a(\tau) d\tau dt \ge K(T - t_2).\]

Letting \(T \to \infty\) we have a contradiction to (2.8). Similar conclusion will be obtained when u(t) < 0 in \([t_0, \infty]\). Hence Eqn (1.1) is oscillatory in \([0, \infty]\).

3. CONCLUSIONS

Oscillation criteria have been proved by using assumption on f which is less weaker than used before, i.e the function a(t) is not necessarily nonnegative on \([0, \infty)\).

The results in this paper will improve and extend the results in [2] and [5]. By using less weaker conditions in [1], we obtained another type of oscillation criteria for the equation.

The method in this paper can be used to establish oscillation criteria for nonlinear differential inequalities.

Acknowledgment The author wishes to express his gratitude to the referees for their remarks and valuable comments

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