Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 25, 1-11; http://www.math.u-szeged.hu/ejqtde/

OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR PERTURBED DIFFERENTIAL EQUATIONS

MOUSSADEK REMILI

Abstract. Sufficient conditions for the oscillation of the nonlinear second order differential equation (a(t)x′)′ + Q(t, x′) = P (t, x, x′) are established where the coefficients are continuous and a(t) is nonnegative.

1. INTRODUCTION

We are concerned here with the oscillatory behavior of solutions of the following second order nonlinear differential equation:

(1.1)  (a(t)x′)′ + Q(t, x) = P (t, x, x′),

where a : [T_{0}, ∞) → ℝ, Q : [T_{0}, ∞) × ℝ → ℝ, and P : [T_{0}, ∞) × ℝ × ℝ → ℝ are continuous and a(t) > 0. Throughout the paper, we shall restrict our attention only to the solutions of the differential equation (1.1) which exist on some ray of the form [T_{0}, ∞).

In this paper we give more general integral criteria to the oscillation of (1.1), which contain the results in [8] as particular cases.

A solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros, and otherwise it is said to be nonoscillatory. If all solutions of (1.1) are oscillatory, (1.1) is called oscillatory. The oscillatory behavior of solutions of second order ordinary differential equation including the existence of oscillatory and nonoscillatory solutions has been the subject of intensive investigations. This problem has received the attention of many authors. Many criteria have been found which involve the average behavior of the integral of the alternating coefficient. Among numerous papers dealing with this subject we refer in particular to [1, 3, to 16 and 19, 20].

2. MAIN RESULTS

Assume that there exist continuous functions p, q : [T_{0}, ∞) → ℝ and f : ℝ → ℝ, such that

(2.1)  xf(x) > 0 for x ≠ 0,

(2.2)  f′(x) ≥ k > 0 for x ≠ 0,

EJQTDE, 2010 No. 25, p. 1

1991 Mathematics Subject Classification. 34C10,34C15.

Key words and phrases. Oscillation, second order nonlinear differential equation.

(2.3)  \frac{Q(t, x)}{f(x)} ≥ q(t) and \frac{P (t, x, x′)}{f(x)} ≤ p(t) for x ≠ 0.

Theorem 1. Suppose that conditions (2.1),(2.2), and (2.3) hold and let ρ be a positive continuously differentiable function on the interval [T, ∞) such that ρ′ ≥ 0 on [T_{0}, ∞). Equation (1.1) is oscillatory if

(2.4)  lim_{t→∞} ∫_{T_{0}}^{t} \frac{1}{ρ(s)a(s)}ds = ∞,

(2.5)  ∫_{T_{0}}^{∞} R(s)ds = ∞,

where

R(t) = ρ(t)[q(t) − p(t)] − \frac{1}{4k} \frac{ρ′^{2}(t)}{ρ(t)}a(t).

Proof. Let x be a nonoscillatory solution on an interval [T, ∞), T ≥ T_{0} of the differential equation (1.1). Without loss of generality, this solution can be supposed such that x(t) ≠ 0. We assume that x(t) is positive on [T, ∞) (the case x(t) < 0 can be treated similarly and will be omitted).

Then

(2.6)  [\frac{a(t)x′(t)}{f[x(t)]}]′ = \frac{P [t, x′(t), x(t)]}{f[x(t)]} − \frac{Q[t, x(t)]}{f[x(t)]} − \frac{a(t)f′(x(t)[x′(t)]^{2}}{f^{2}[x(t)]} .

Multiplying (2.6) by ρ(t) and integrating from T to t , we obtain (2.7)

\frac{ρ(t)a(t)x′(t)}{f[x(t)]} ≤ C_{T} −∫_{T}^{t} ρ(s)[q(s)−p(s)]ds+∫_{T}^{t} ρ′(s)\frac{a(s)x′(s)}{f[x(s)]}ds−∫_{T}^{t} ρ(s)\frac{a(s)f′(x(s)[x′(s)^{2}]}{f^{2}[x(s)]}ds.

Where C_{T} = \frac{ρ(T)a(T)x′(T)}{f[x(T)]}. We use the following notation

ω(t) = \frac{a(t)x′(t)}{f[x(t)]} and W (t) = ω(t) − \frac{ρ′(t)a(t)}{2kρ(t)}.

Then we have by condition (2.2)

\frac{ρ(t)a(t)x′(t)}{f[x(t)]} ≤ C_{T} − ∫_{T}^{t} ρ(s)[q(s) − p(s)]ds + ∫_{T}^{t} [ρ′(s)ω(s) − k \frac{ρ(s)}{a(s)}ω^{2}(s)] ds

≤ C_{T} −∫_{T}^{t} ρ(s)[q(s)−p(s)]ds−∫_{T}^{t} \frac{kρ(s)}{a(s)} [W^{2}(s) − (\frac{ρ′(s)a(s)}{2kρ(s)})^{2}] ds

(2.8)  ≤ C_{T} − ∫_{T}^{t} R(s)ds,

EJQTDE, 2010 No. 25, p. 2 we see from (2.5) that

lim_{t→∞} \frac{ρ(t)a(t)x′(t)}{f[x(t)]} = −∞,

hence, there exist T_{1} ≥ T such that

x′(t) < 0 for t ≥ T_{1}.

Condition (2.5) also implies ∫_{T}^{∞} ρ(s)[q(s)−p(s)]ds = ∞ and there exists T_{2} ≥ T_{1} such that

∫_{T_{1}}^{T_{2}} ρ(s)[q(s) − p(s)]ds = 0 and ∫_{T_{2}}^{t} ρ(s)[q(s) − p(s)]ds ≥ 0 for t ≥ T_{2}. Now multiplying (1.1) by ρ(t) and integrating by parts we obtain

ρ(t)a(t)x′(t) ≤ C_{T_{2}} + ∫_{T_{2}}^{t} ρ′(s)a(s)x′(s)ds − ∫_{T_{2}}^{t} f[x(s)]ρ(s)[q(s) − p(s)]ds
≤ C_{T_{2}} − f [x(t)] ∫_{T_{2}}^{t} ρ(s)[q(s) − p(s)]ds
+ ∫_{T_{2}}^{t} x′(s)f′ [x(s)] ∫_{T_{2}}^{s} ρ(u)[q(u) − p(u)]duds
≤ C_{T_{2}} for every t ≥ T_{1},

where C_{T_{2}} = ρ(T_{2})a(T_{2})x′(T_{2}) < 0. Thus

x(t) ≤ C_{T_{2}} ∫_{T_{2}}^{t} \frac{1}{ρ(s)a(s)}ds.

from (2.4) it follows that x(t) → −∞ as t → ∞ which is a contradiction.

Example 1. Consider the equation

[a(t)x′]′ + [\frac{1}{2}t^{− \frac{3}{2}} (2 + cos(t)) + te^{x}] x = xt^{− \frac{1}{2}} sin(t) + \frac{1}{t^{3}} \frac{x^{3} cos(x′)}{x^{2} + 1} for t ≥ \frac{π}{2} .

If we choose f(x) = x, a(t) = Log(t) and ρ(t) = t, then

\frac{Q(t, x)}{f(x)} ≥ \frac{1}{2}t^{− \frac{3}{2}} (2 + cos(t)) = q(t); \frac{P (t, x, x′)}{f(x)} ≤ t^{− \frac{1}{2}} sin(t) + \frac{1}{t^{3}} = p(t).

For every t ≥ T_{0} = \frac{π}{2} we obtain

∫_{T_{0}}^{t} R(s)ds = ∫_{T_{0}}^{t} s(\frac{1}{2}s^{− \frac{3}{2}} (2 + cos(s)) − s^{− \frac{1}{2}} sin(s) − \frac{1}{s^{3}} ) − \frac{1}{4} \frac{Log(s)}{s})ds

= ∫_{T_{0}}^{t} s(\frac{1}{2}s^{− \frac{3}{2}} (2 + cos(s)) − s^{− \frac{1}{2}} sin(s))ds − ∫_{T_{0}}^{t} \frac{1}{s^{2}} ds − ∫_{T_{0}}^{t} \frac{1}{4} \frac{Log(s)}{s}ds

= ∫_{T_{0}}^{t} d(s^{\frac{1}{2}} (2 + cos(s)) + \frac{1}{t} − \frac{2}{π} − \frac{1}{8}Log^{2}(t) + \frac{1}{8}Log^{2}(\frac{π}{2} )

= t^{\frac{1}{2}} (2 + cos t) − 2(\frac{π}{2} )^{\frac{1}{2}} + \frac{1}{t} − \frac{2}{π} − \frac{1}{8}Log^{2}(t) + \frac{1}{8}Log^{2}(\frac{π}{2} )

EJQTDE, 2010 No. 25, p. 3

≥ t^{\frac{1}{2}} − 2(\frac{π}{2} )^{\frac{1}{2}} − \frac{2}{π} − \frac{1}{8}Log^{2}(t).

Thus we have

∫_{T_{0}}^{∞} R(s) = ∞ and lim_{t→∞} ∫_{T_{0}}^{t} \frac{1}{ρ(s)a(s)}ds = ∫_{T_{0}}^{∞} \frac{1}{sLog(s)}ds = ∞,

i.e. (2.1),(2.2),(2.3),(2.4) and (2.5) are satisfied. Hence the differential equation is oscillatory.

Theorem 2. If the conditions (2.1),(2.2),(2.3) ,(2.4) hold, and let ρ be a positive continuously differentiable function on the interval [T, ∞) such that ρ′ ≥ 0 on [T_{0}, ∞) with

(2.9)  ∫_{T_{0}}^{∞} ρ(s)[q(s) − p(s)]ds < ∞,

(2.10)  lim inf_{t→∞} [∫_{T}^{t} R(s)ds] ≥ 0 for all large T,

(2.11)  lim_{t→∞} ∫_{T_{0}}^{t} \frac{1}{ρ(s)a(s)} ∫_{s}^{∞} R(u)duds = ∞,

and

(2.12)  ∫_{ϵ}^{∞} \frac{dy}{f(y)} < ∞ and ∫_{−ϵ}^{−∞} \frac{dy}{f(y)} < ∞ for every ϵ > 0.

Then all solutions of (1.1) are oscillatory.

Remark 1. Condition (2.9) implies that

∫_{T}^{∞} R(s)ds < ∞ and lim inf_{t→∞} [∫_{T}^{t} R(s)ds] = ∫_{T}^{∞} R(s)ds,

hence (2.10) takes the form

∫_{T}^{∞} R(s)ds ≥ 0 for all large T,

Proof. Let x be a nonoscillatory solution on an interval [T,∞) of the differential equation (1.1). We suppose, as in Theorem 1, that x is positive on [T, ∞). We consider the following three cases for the behavior of x′(t).

Case 1: x′(t) > 0 for t ≥ T_{1} for some T_{1} ≥ T, then from (2.8) we have

∫_{T_{1}}^{t} R(s)ds ≤ \frac{ρ(T_{1})a(T_{1})x′(T_{1})}{f[x(T_{1})]} − \frac{ρ(t)a(t)x′(t)}{f[x(t)]}.

Hence, for all t ≥ T_{1}

∫_{t}^{∞} R(s)ds ≤ ρ(t)\frac{a(t)x′(t)}{f[x(t)]}.

EJQTDE, 2010 No. 25, p. 4 Using (2.12), we obtain

∫_{T_{1}}^{t} \frac{1}{ρ(s)a(s)} ∫_{s}^{∞} R(u)duds ≤ ∫_{T_{1}}^{t} \frac{x′(s)}{f[x(s)]}ds
≤ ∫_{x(T_{1})}^{∞} \frac{dy}{f(y)} < ∞.

This contradicts condition (2.11).

Case 2: x′(t) changes signs, then there exists a sequence ( α_{n}) → ∞ in [T, ∞) such that x′(α_{n}) < 0 . Choose N large enough so that

∫_{α_{N}}^{∞} R(s)ds ≥ 0

Then from (2.8) we have

\frac{ρ(t)a(t)x′(t)}{f[x(t)]} ≤ C_{α_{N}} − ∫_{α_{N}}^{t} R(s)ds.

So

lim sup_{t→∞} \frac{ρ(t)a(t)x′(t)}{f[x(t)]} ≤ C_{α_{N}} + lim sup_{t→∞} [− ∫_{α_{N}}^{t} R(s)ds]
= C_{α_{N}} − lim inf_{t→∞} [∫_{α_{N}}^{t} R(s)ds]
< 0.

Which contradicts the fact that x′(t) oscillates. Case 3: x′(t) < 0. for t ≥ T_{1} for some T_{1} ≥ T, Wong[16] showed that (2.10)

implies that for any t_{0} ≥ T_{0} there exists t_{1} ≥ t_{0} such that ∫_{t_{1}}^{∞} ρ(s)[q(s)−p(s)]ds ≥ 0 for all t ≥ t_{1}. Choosing t_{1} ≥ T_{1} and then integrating (1.1) we have

ρ(t)a(t)x′(t) ≤ C_{t_{1}} + ∫_{t_{1}}^{t} ρ′(s)a(s)x′(s)ds − ∫_{t_{1}}^{t} f[x(s)]ρ(s)[q(s) − p(s)]ds
≤ C_{t_{1}} − f [x(t)] ∫_{t_{1}}^{t} ρ(s)[q(s) − p(s)]ds
+ ∫_{t_{1}}^{t} x′(s)f′ [x(s)] ∫_{t_{1}}^{t} ρ(u)[q(u) − p(u)]duds
≤ C_{t_{1}} for every t ≥ t_{1},

where C_{t_{1}} = ρ(t_{1})a(t_{1})x′(t_{1}) < 0. Thus

x(t) ≤ C_{t_{1}} ∫_{t_{1}}^{t} \frac{1}{ρ(s)a(s)}ds,

from (2.4) it follows that x(t) → −∞ as t → ∞ which is a contradiction. EJQTDE, 2010 No. 25, p. 5 Theorem 3. Suppose (2.1),(2.2),(2.3) hold and assume that there exists a constant A > 0 such that

(2.13)  \frac{a(t)}{ρ(t)} ≤ A,

(2.14)  lim_{t→∞} [∫_{T}^{t} \frac{1}{ρ(s)}ds]^{−1} ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} R(u)duds = ∞,

(2.15)  lim_{t→∞} ∫_{T}^{t} \frac{1}{sρ(s)}ds = ∞.

Then (1) is oscillatory.

Proof. Let x be a nonoscillatory solution on an interval [T, ∞), of the differential equation (1). Without loss of generality, this solution can be supposed such that x(t) > 0 for all t ≥ T (the case x(t) < 0 can be treated similarly and will be omitted).

defining for every t ≥ T

g(t) = \{∫_{T}^{t} \frac{ds}{ρ(s)}\}^{−1} .

From (2.6) we have

(2.16)  ρ(t)ω(t) + ∫_{T}^{t} R(s)ds + ∫_{T}^{t} \frac{kρ(s)}{a(s)}W^{2}(s)ds ≤ C_{T} .

Therefore, for every t ≥ T we have

(2.17)  g(t) ∫_{T}^{t} ω(s)ds + g(t) ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} \frac{kρ(s)}{a(s)}W^{2}(u)duds

≤ C_{T} − g(t) ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} R(u)duds.

Now, by condition (2.14)

lim_{t→∞} \{g(t) ∫_{T}^{t} ω(s)ds + g(t) ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} \frac{kρ(s)}{a(s)}W^{2}(u)duds\} = −∞.

Hence, there exist T_{1} ≥ T such that

(2.18)  ∫_{T}^{t} ω(s)ds + ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} \frac{kρ(s)}{a(s)}W^{2}(u)duds < 0 for t ≥ T_{1},

Defining

H(t) = |∫_{T}^{t} \frac{a(s)}{kρ(s)}W (s)ds|
Ψ(t) = ∫_{T}^{t} \frac{H^{2}(s)}{sρ(s)}ds for all t ≥ T,

EJQTDE, 2010 No. 25, p. 6 we may use the Schwart inequality to obtain

H^{2}(t) ≤ ∫_{T}^{t} [\frac{a(s)}{kρ(s)}]^{2} ds ∫_{T}^{t} W^{2}(s)ds,

from (2.13) we have

H ^{2}(t) ≤ Ct ∫_{T}^{t} W ^{2}(s)ds,

where C = \frac{A^{2}}{k^{2}} .Thus, by condition (2.18) for t ≥ T_{1}

−H(t)g(t) + g(t)\frac{1}{C} ∫_{T}^{t} \frac{H ^{2}(s)}{sρ(s)}ds ≤ g(t) ∫_{T}^{t} \frac{a(s)}{kρ(s)}W (s)ds + g(t) ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} W^{2}(u)duds
≤ 0,

then

H^{2}(t) ≥ \frac{1}{C^{2}} [∫_{T}^{t} \frac{H^{2}(s)}{sρ(s)}ds]^{2} for all t ≥ T_{1},

and

\frac{1}{C^{2}} \frac{1}{tρ(t)} ≤ \frac{Ψ′(t)}{Ψ^{2}(t)} for all t ≥ T_{1}.

So for any t ≥ T_{1} ≥ T

\frac{1}{C^{2}} ∫_{T_{1}}^{t} \frac{1}{sρ(s)}ds ≤ ∫_{T_{1}}^{t} \frac{Ψ′(s)}{Ψ^{2}(s)}ds = \frac{1}{Ψ(T_{1})} − \frac{1}{Ψ(t)} ≤ \frac{1}{Ψ(T_{1})} < ∞.

This contradicts condition (2.15). The proof of the theorem is now complete.

Remark 2. Theorem 3 generalizes Theorem 4 in [8].

Theorem 4. Suppose (2.1), (2.2), (2.3), hold and assume that there exist a constant λ > 0 such that

(2.19)  lim_{t→∞} inf ∫_{T}^{t} R(s)ds > −∞ for all large T,

(2.20)  lim sup_{t→∞} \frac{1}{t} ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} R(u)duds = ∞ for all large T,

(2.21)  \frac{a(t)}{ρ(t)} ≤ λt.

Then all solutions of (1) are oscillatory.

Proof. Let x be a nonoscillatory solution on an interval [T, ∞), of the differential equation (1). Without loss of generality, this solution can be supposed such that x(t) > 0. for all t ≥ T. We consider the following three cases for the behavior of x′.

Case 1: x′ is oscillatory. Then there exists a sequence (t_{n}) in [T, ∞) with lim_{n→∞} t_{n} = ∞ and such that x′(t_{n}) = 0.(n ≥ 1). Thus (2.8) gives

∫_{T}^{t_{n}} \frac{kρ(s)}{a(s)}W^{2}(s)ds ≤ C_{T} − ∫_{T}^{t_{n}} R(s)ds,

EJQTDE, 2010 No. 25, p. 7 and hence, by taking into account condition (2.19), we conclude that

∫_{T}^{∞} \frac{kρ(s)}{a(s)}W^{2}(s)ds < ∞.

So, for some constant M we have

(2.22)  ∫_{T}^{t} \frac{kρ(s)}{a(s)}W^{2}(s)ds ≤ M for every t ≥ T.

By the Schwarz’s inequality, we have

|− ∫_{T}^{t} W (s)ds|^{2} = ∫_{T}^{t} \frac{kρ(s)}{a(s)}W^{2}(s)ds ∫_{T}^{t} \frac{a(s)}{kρ(s)}ds ≤ M ∫_{T}^{t} \frac{a(s)}{kρ(s)}ds
≤ \frac{1}{2k} Mλt^{2}.

and hence for every t ≥ T

− ∫_{T}^{t} W (s)ds = − ∫_{T}^{t} ω(s) − \frac{ρ′(s)a(s)}{2kρ(s)}ds ≤ \sqrt{\frac{1}{2k} Mλ}t.

Furthermore, (2.16) gives

\frac{1}{ρ(t)} ∫_{T}^{t} R(s)ds ≤ C_{T} − ω(t),

and therefore for all t ≥ T

\frac{1}{t} ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} R(u)duds ≤ \frac{C_{T}}{t} ∫_{T}^{t} \frac{1}{ρ(s)}ds + \sqrt{\frac{1}{2k} Mλ}

≤ \frac{C_{T}}{t} \frac{(t − T)}{ρ(T)} + \sqrt{\frac{1}{2k} Mλ},

and

lim sup_{t→∞} \frac{1}{t} ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} R(u)duds ≤ \frac{C_{T}}{ρ(T)} + \sqrt{\frac{1}{2k} Mλ} < ∞.

This contradicts condition (2.20). Case 2: x′ > 0 on [T_{1}, ∞), T_{1} ≥ T . Using (2.8) we get

∫_{T}^{t} R(s)ds ≤ C_{T} ,

and consequently

lim sup_{t→∞} \frac{1}{t} ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} R(u)duds ≤ 0.

Which again contradicts (2.20). Case 3: x′(t) < 0. From (2.7), and (2.19) it follows that

(2.23) \frac{ρ(t)a(t)x′(t)}{f[x(t)]} ≤ C_{T} −∫_{T}^{t} ρ(s)[q(s)−p(s)]ds−∫_{T}^{t} ρ(s)\frac{a(s)[x′(s)]^{2}}{(f[x(s)])^{2}} f′(x(s))ds.

We distinguish two mutually exclusive cases where − ∫_{T}^{∞} ρ(s)\frac{a(s)[x′(s)]^{2}}{(f[x(s)])^{2}} f′(x(s))ds is finite or infinite.

EJQTDE, 2010 No. 25, p. 8

i) If − ∫_{T}^{∞} ρ(s)\frac{a(s)[x′(s)]^{2}}{(f[x(s)])^{2}} f′(x(s))ds is finite. In this case, it follows that (2.22) holds for t ≥ T. Once again, we can complete the proof by the procedure of the proof of Case 1.

ii) If − ∫_{T}^{∞} Rρ(s)\frac{a(s)[x′(s)]^{2}}{(f[x(s)])^{2}} f′(x(s))ds is infinite. By Condition (2.19), and from (2.22) it follows that there exists a constant μ such that

−\frac{ρ(t)a(t)x′(t)}{f[x(t)]} ≥ μ + ∫_{T}^{t} [\frac{x′(s)f′(x(s))}{f[x(s)]}] \frac{ρ(s)a(s)x′(s)}{f[x(s)]}ds for all t ≥ T.

Put

G(t) = \frac{x′(t)f′(x(t))}{f[x(t)]} ≤ 0.

Furthermore, we choose a T_{1} ≥ T so that

μ + ∫_{T}^{T_{1}} G(s)\frac{ρ(s)a(s)x′(s)}{f[x(s)]}ds = μ_{1} > 0,

and then for every t ≥ T_{1} we have

\frac{ρ(t)a(t)x′(t)}{f[x(t)]}G(t) [μ + ∫_{T}^{t} G(s)\frac{ρ(s)a(s)x′(s)}{f[x(s)]}ds]^{−1} ≥ −G(t),

and integrating from T_{1}to t, we obtain

Log \frac{[μ + ∫_{T}^{t} G(s)\frac{ρ(s)a(s)x′(s)}{f[x(s)]}ds]}{μ_{1}} ≥ Log \frac{ρ(t)f(x(T))}{ρ(T)f(x(t))}.

Thus

μ + ∫_{T}^{t} G(s) (\frac{ρ(s)a(s)x′(s)}{f[x(s)]}) ds ≥ μ_{1} \frac{ρ(t)f(x(T))}{ρ(T)f(x(t))}.

The last inequality implies for t ≥ T_{1}

x′(t) ≤ −\frac{η}{a(t)},

where η = \frac{μ_{1}+f(x(T))}{ρ(T)} > 0. And consequently for t ≥ T_{1}

x(t) ≤ x(T_{1}) − η ∫_{T_{1}}^{t} \frac{1}{a(s)}ds ≤ −\frac{η}{b} (t − T_{1}).

Therefore, we conclude that lim_{t→∞} x(t) = −∞ . This contradicts the assumption that x(t) > 0. This completes the proof of the theorem.

Example 2. Consider [a(t)x′]′+[\frac{1}{2}t^{− \frac{5}{6}} (2 + cos(t) + tx^{2}] x = xt^{− \frac{1}{6}} sin(t)+\frac{1}{t^{3}} \frac{x^{3} cos^{2}(x′)}{x^{2}+1}

for t ≥ \frac{π}{2} , with f(x) = x a(t) = t^{2/3}, ρ(t) = t^{1/3} then EJQTDE, 2010 No. 25, p. 9

\frac{Q(t, x)}{f(x)} ≥ \frac{1}{2}t^{− \frac{5}{6}} (2 + cos(t) = q(t); \frac{P (t, x, x′)}{f(x)} ≤ t^{− \frac{1}{6}} sin(t) + \frac{1}{t^{3}} = p(t).

For every t ≥ T_{0} = \frac{π}{2} , we obtain

∫_{T_{0}}^{t} R(s)ds = ∫_{T_{0}}^{t} s(\frac{1}{2}s^{− \frac{5}{6}} (2 + cos(s) − s^{− \frac{1}{6}} sin(s) − \frac{1}{s^{3}} ) − \frac{1}{36} \frac{1}{s})ds

= ∫_{T_{0}}^{t} s(\frac{1}{2}s^{− \frac{3}{2}} (2 + cos(s) − s^{− \frac{1}{2}} sin(s))ds − ∫_{T_{0}}^{t} \frac{1}{s^{2}} ds − ∫_{T_{0}}^{t} \frac{1}{36} \frac{1}{s}ds

= ∫_{T_{0}}^{t} d(s^{\frac{1}{2}} (2 + cos(s)) + \frac{1}{t} − \frac{2}{π} − \frac{1}{36}Log(t) + \frac{1}{36}Log(\frac{π}{2} )

≥ t^{\frac{1}{2}} − 2(\frac{π}{2} )^{\frac{1}{2}} − \frac{2}{π} − \frac{1}{36}Log(t).

Thus we have

\frac{1}{t} ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} R(u)duds ≥ \frac{1}{t} ∫_{T}^{t} s^{\frac{−1}{3}} [s^{\frac{1}{2}} − 2(\frac{π}{2} )^{\frac{1}{2}} − \frac{2}{π} − \frac{1}{36}Log(s)] ds

≥ \frac{1}{t} ∫_{T}^{t} s^{\frac{−1}{3}} [s^{\frac{1}{2}} − 2(\frac{π}{2} )^{\frac{1}{2}} − \frac{2}{π} − \frac{1}{36}s^{\frac{1}{3}} ] ds

≥ \frac{6}{7}t^{\frac{1}{6}} − [2(\frac{π}{2} )^{\frac{1}{2}} + \frac{2}{π} ] t^{\frac{−1}{3}} − \frac{1}{36} − \frac{6}{7}(\frac{π}{2} )^{\frac{7}{6}} ,

and consequently,

lim_{t→∞} inf ∫_{T}^{t} R(s)ds > −∞ ; lim sup_{t→∞} \frac{1}{t} ∫_{T}^{t} \frac{1}{ρ(s)} ∫_{T}^{s} R(u)duds = ∞; and \frac{a(t)}{ρ(t)} ≤

t^{1/3} ≤ t.

This means that (2.19), (2.20) hold. Thus, from Theorem 4 it follows that, when (2.21) is satisfied, our differential equation is oscillatory.

Acknowledgement

The author wishes to thank the referee for all his/her suggestions and remarks.

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(Received June 23, 2009)

Department of Mathematics Faculty of Sciences University of Oran BP 1524 Algeria

E-mail address: m.remili@yahoo.fr

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