Oscillation Theorems for Nonlinear Differential Equations

of Second Order

Jelena V. Manojlović

University of Niš, Faculty of Science, Ćirila i Metodija 2, 18000 Niš, Yugoslavia e-mail: jelenam@bankerinter.net

Abstract We establish new oscillation theorems for the nonlinear dif-

ferential equation

[a(t)ψ(x(t))|x′(t)|^{α−1}x′(t)]′ + q(t)f(x(t)) = 0, α > 0

where a, q : [t_{0}, ∞) → R, ψ, f : R → R are continuous, a(t) > 0 and ψ(x) > 0, xf(x) > 0 for x ≠ 0. These criteria involve the use of averaging functions.

1. Introduction

In this paper we are interested in obtaining results on the oscillatory behaviour of solutions of second order nonlinear differential equation

(E)  [a(t)ψ(x(t))|x′(t)|^{α−1}x′(t)]′ + q(t)f(x(t)) = 0

1991 Mathematics Subject Classification. 34C10, 34C15

Keywords. Oscillation, Nonlinear differential equations, Integral averages

Supported by Grant 04M03E of RFNS through Math.Inst. SANU

where a, q : [t_{0}, ∞) → R, ψ, f : R → R are continuous, α > 0 is a constant, a(t) > 0 and ψ(x) > 0, xf(x) > 0 for x ≠ 0.

This nonlinear equation can be considered as a natural generalization of the half-linear equation

(HL)  [a(t)|x′(t)|^{α−1}x′(t)]′ + q(t)|x(t)|^{α−1}x(t) = 0,

which has been the object of intensive studies in recent years.

By a solution of (E) we mean a function x ∈ C^{1}[T_{x}, ∞), T_{x} ≥ t_{0}, which has the property |x′(t)|^{α−1}x′(t) ∈ C^{1}[T_{x}, ∞) and satisfies (E). A solutions is said to be global if it exists on the whole interval [t_{0}, ∞). The existence and uniqueness of solutions of (HL) subject to the initial condition x(T) = x_{0}, x′(T) = x_{1} has been investigated by Kusano and Kitano [12]. They have shown that the initial value problem has a unique global solution for any given values x_{0}, x_{1} provided q(t) is positive and locally of bounded variation on [t_{0}, ∞).

The solution x of (E) which exists on some interval (T_{1}, +∞) ⊂ [t_{0}, ∞) is singular solution of the first kind, x ∈ S_{1}, if there exists t^{∗} ∈ (T_{1}, ∞) such that max{(x(s)| : t ≤ s ≤ t^{∗}} > 0 for t_{0} < t < t^{∗} and x(t) = 0 for all t ≥ t^{∗}. The solution x of (E) which exists on some interval (T_{1}, T_{2}) ⊂ [t_{0}, ∞) is singular solution of the second kind, x ∈ S_{2}, if lim sup_{t→T_{2}} x(t) = +∞. On the other hand, the solution x of (E) which exists on some interval (T_{x}, +∞), T_{x} ≥ t_{0} is called proper if

sup{ |x(t)| : t ≥ T } > 0 for all T ≥ T_{x}.

The existence of proper and singular solutions for the semilinear equations was investigated by Mirzov [24] and for the nonlinear second order equation by Kiguradze and Chanturia [11]. They established sufficient conditions that nonlinear and semilinear differential equation of the second order does not have singular solutions as well as that it has a proper solution and sufficient conditions for all global solutions to be proper. So, we shall suppose that the equation (E) has the proper solutions and our attention will be restricted only to those solutions.

A nontrivial solution of (E) is called oscillatory if it has arbitrarily large zeroes, otherwise it is said to be nonoscillatory. Equation (E) is called oscillatory if all its solution are oscillatory.

During the last two decades there has been a great deal of work on the oscillatory behavior of solutions of the equation (HL) (see Hsu, Yeh [10], Kusano, Naito [13], Kusano, Yoshida [14], Li, Yeh [16], [17], [18], [19], [20], Lian, Yeh, Li [22]). Wang in [27], [28] established oscillation criteria for the more general equation [a(t)|x′(t)|^{α−1}x′(t)]′ + Φ(t, x(t)) = 0. Wong, Agarwal [30] considered a special case of this equation for Φ(t, x(t)) = q(t) f(x(t)). We refer to that equation as to the equation (A). Afterward, in 1998. Hong in [9] generalized criteria of oscillation of half-linear differential equation due to Hsu, Yeh [10] to the nonlinear differential equation (E). Thereafter, our purpose here is to develop oscillation theory for a general case of the equation (E) in which f(x) is not necessarily of the form | x |^{α−1}x, α > 0 and ψ(x) ≠ 1, without any restriction on the sign of q(t), which is of particular interest.

Some of the very important oscillation theorems for second order linear and nonlinear differential equations involve the use of averaging functions. As recent contribution to this study we refer to the papers of Grace, Lalli and Yeh [2], [3], Grace and Lalli [5], Grace [4], [6], [7], Li and Yeh [21], Philos [26], Wong and Yeh [29] and Yeh [31]. Using a general class of continuous functions

H : D = { (t, s) | t ≥ s ≥ t_{0} } → R,

which is such that

H(t, t) = 0 for t ≥ t_{0}, H(t, s) > 0 for all (t, s) ∈ D

and has a continuous and nonpositive partial derivative on D with respect to the second variable, Philos [26] presented oscillation theorems for linear differential equations of second order

x′′(t) + q(t)x(t) = 0.

His results has been extended by Grace [7] and Li and Yeh [21] to the nonlinear differential equation

[a(t)ψ(x(t))x′(t)]′ + q(t)f(x(t)) = 0.

In this paper, we are interested in extending the results of Grace to a broad class of second order nonlinear differential equations of type (E) by using a well-known inequality stated in Lemma 2.1.

2. Main results

Throughout this paper we assume that

(C_{1})  \frac{f′(x)}{(ψ(x)|f(x)|^{α−1})^{\frac{1}{α}}} ≥ K > 0, x ≠ 0,

and in order to simplify notation we denote by

β = \frac{1}{αK^{α}} (\frac{α}{α + 1})^{α+1} .

Notice that in the special case of the equation (HL), for ψ(x) ≡ 1 and f(x) = |x|^{α−1}x, the condition (C_{1}) is satisfied.

We also need the following well-known inequality which is due to Hardy, Little and Polya [8].

Lemma 2.1 If X and Y are nonnegative, then

X^{q} + (q − 1)Y^{q} − qXY^{q−1} ≥ 0, q > 1,

where equality holds if and only if X = Y.

Theorem 2.1 Let condition (C_{1}) holds. Suppose that there exists a continuous function

H : D = { (t, s) | t ≥ s ≥ t_{0} } → R

such that

(H_{1})  H(t, t) = 0, t ≥ t_{0}, H(t, s) > 0, (t, s) ∈ D

(H_{2}) h(t, s) = −\frac{∂H(t, s)}{∂s} is nonnegative continuous function on D.

If

(C_{2}) lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} [q(s)H(t, s) − β a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)}] ds = ∞,

then the equation (E) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of the equation (E). Without loss of generality, we assume that x(t) ≠ 0 for t ≥ t_{0}. We define

w(t) = \frac{a(t)ψ(x(t))|x′(t)|^{α−1}x′(t)}{f(x(t))} for t ≥ t_{0}.

Then, by taking into account (C_{1}), for every s ≥ t_{0}, we obtain

(1)  w′(s) = −q(s) − \frac{f′(x(s))|w(s)|^{\frac{α+1}{α}}}{(a(s)ψ(x(s))|f(x(s))|^{α−1})^{\frac{1}{α}}}

≤ −q(s) − K \frac{|w(s)|^{\frac{α+1}{α}}}{a^{\frac{1}{α}} (s)} .

Multiplying (1) by H(t, s) for t ≥ s ≥ t_{0} and integrating from t_{0} to t, we get

∫_{t_{0}}^{t} w′(s)H(t, s) ds ≤ − ∫_{t_{0}}^{t} q(s)H(t, s) ds − K ∫_{t_{0}}^{t} H(t, s)\frac{|w(s)|^{\frac{α+1}{α}}}{a^{\frac{1}{α}} (s)} ds.

Since,

∫_{t_{0}}^{t} w′(s)H(t, s) ds = −w(t_{0})H(t, t_{0}) − ∫_{t_{0}}^{t} w(s)\frac{∂H(t, s)}{∂s} ds, (2)

we have

∫_{t_{0}}^{t} q(s)H(t, s) ds ≤ w(t_{0})H(t, t_{0}) + ∫_{t_{0}}^{t} |w(s)|h(t, s) ds (3)

− K ∫_{t_{0}}^{t} H(t, s)\frac{|w(s)|^{\frac{α+1}{α}}}{a^{\frac{1}{α}} (s)} ds.

Taking

X = (K H(t, s))^{\frac{α}{α+1}} \frac{|w(s)|}{a^{\frac{1}{α+1}} (s)}, q = \frac{α + 1}{α}

Y = (\frac{α}{α + 1})^{α} \frac{a^{\frac{α}{α+1}} (s)h^{α}(t, s)}{[K H(t, s)]^{\frac{α^{2}}{α+1}}},

according to Lemma 2.1, we obtain for t > s ≥ t_{0}

|w(s)|h(t, s) − K H(t, s)\frac{|w(s)|^{\frac{α+1}{α}}}{a^{\frac{1}{α}} (s)} ≤ β a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)}.

Hence, (3) implies

\frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} q(s)H(t, s) ds ≤ w(t_{0}) (4)

+ \frac{β}{H(t, t_{0})} ∫_{t_{0}}^{t} a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)} ds,

for all t ≥ t_{0}. Consequently,

\frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} [q(s)H(t, s) − β a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)}] ds ≤ w(t_{0}), t ≥ t_{0}.

Taking the upper limit as t → ∞, we obtain a contradiction, which completes the proof.

Corollary 2.1 Let condition (C_{2}) in Theorem 2.1 be replaced by

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)} ds < ∞,

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} q(s)H(t, s) ds = ∞

then the conclusion of Theorem 2.1 holds.

Remark 2.1 For a(t) ≡ 1, ψ(x) ≡ 1, H(t, s) = t − s from Theorem 2.1. we derive Corollary 3.2. in [28]. Taking H(t − s)^{λ} for some constant λ > 1, which obviously satisfies the conditions (H_{1}), (H_{2}), in the case of the equation (HL) as a special case of (E), Theorem 2.1. reduces to the oscillation criterion of Li and Yeh [16].

For illustration we consider the following example.

Example 2.1 Consider the differential equation

(E_{1}) (\frac{|x(t)|^{3−α}}{t^{ν}} |x′(t)|^{α−1}x′(t))′ + t^{λ} (λ \frac{2 − cos t}{t} + sin t) x^{3}(t) = 0,

for t ≥ t_{0}, where ν, λ, α are arbitrary positive constants and α ≠ 2. Then,

\frac{f′(x)}{(ψ(x)|f(x)|^{α−1})^{\frac{1}{α}}} = 3 for x ≠ 0.

On the other hand, for any t ≥ t_{0}, we have

∫_{t_{0}}^{t} q(s) ds = ∫_{t_{0}}^{t} d[s^{λ}(2 − cos s)] = t^{λ}(2 − cos t) − t_{0}^{λ}(2 + cos t_{0})
= t^{λ}(2 − cos t) − k_{0} ≥ t^{λ} − k_{0}.

Taking H(t, s) = (t − s)^{2}, for t ≥ s ≥ t_{0}, we have

\frac{1}{t^{2}} ∫_{t_{0}}^{t}[(t − s)^{2}q(s) − β 2^{α+1} \frac{(t − s)^{1−α}}{s^{ν}} ] ds

= \frac{1}{t^{2}} ∫_{t_{0}}^{t}[2 (t − s) (∫_{t_{0}}^{s} q(u) du) − β 2^{α+1} \frac{(t − s)^{1−α}}{s^{ν}} ] ds

≥ \frac{2}{t^{2}} ∫_{t_{0}}^{t}(t − s) (s^{λ} − k_{0}) ds − \frac{β 2^{α+2}}{t_{0}^{ν} t^{2}} ∫_{t_{0}}^{t}(t − s)^{1−α} ds

= \frac{2 t^{λ}}{(λ + 1)(λ + 2)} + \frac{k_{1}}{t^{2}} + \frac{k_{2}}{t} − k_{0} − \frac{k_{3}}{t^{α}} (1 − \frac{t_{0}}{t} )^{2−α} ,

where

k_{1} = \frac{2 t_{0}^{λ+2}}{λ + 2} − k_{0}t_{0}^{2}, k_{2} = 2k_{0}t_{0} − \frac{2 t_{0}^{λ+1}}{λ + 1} , k_{3} = \frac{β 2^{α+2}}{t_{0}^{ν}(2 − α)}.

Consequently, condition (C_{2}) is satisfied. Hence, the equation (E_{1}) is oscillatory by Theorem 2.1.

Remark 2.2 We note that since ∫_{0}^{∞} q(s) ds is not convergent the oscillation criteria in [9] fail to apply to the equation (E_{1}).

In the case of the half–linear differential equation we have the following corollary:

Corollary 2.2 The equation (HL) is oscillatory if the condition (C_{2}) is satisfied for some continuous function H(t, s) on D which satisfies (H_{1}) and (H_{2}).

Remark 2.3 As in the previous example, we conclude that (HL) for q(s) = t^{λ} (λ \frac{2−cos t}{t} + sin t), a(s) = s^{−ν} is oscillatory for λ and ν positive and α ≠ 0. On the other hand, criteria in [10], [13] and [18] (Section 2) can not be applied, since q(t) is not positive function (assumed in [13]) and ∫_{t}^{∞} q(s) ds < ∞.

Theorem 2.2 Let condition (C_{1}) holds and let the functions H and h be defined as in Theorem 2.1 such that conditions (H_{1}), (H_{2}),

(H_{3})  0 < inf_{s≥t_{0}} [lim inf_{t→∞} \frac{H(t, s)}{H(t, t_{0})}] ≤ ∞,

and

(C_{3})  lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)} ds < ∞

are satisfied. If there exists a continuous function φ on [t_{0}, ∞) such that for every T ≥ t_{0}

(C_{4}) lim sup_{t→∞} \frac{1}{H(t, T)} ∫_{T}^{t}[q(s)H(t, s) − β a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)}] ds ≥ φ(T),

and

(C_{5})  ∫_{t_{0}}^{∞} \frac{φ+^{\frac{α}{α+1}} (s)}{a^{\frac{1}{α}} (s)} ds = ∞,

where φ_{+}(s) = max{φ(s), 0}, then the equation (E) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of the equation (E), say x(t) ≠ 0 for t ≥ t_{0}. Next, we define the function w as in the proof of Theorem 2.1, so that we have (3) and (4). Then, for t > T ≥ t_{0} we have

lim sup_{t→∞} \frac{1}{H(t, T)} ∫_{T}^{t}[q(s)H(t, s) − β a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)}] ds ≤ w(T).

Therefore, by conditions (C_{4}), we have

(5) φ(T) ≤ w(T) for every T ≥ t_{0} and

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} q(s)H(t, s) ds ≥ φ(t_{0}). (6)

We define functions

F (t) = \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} |w(s)|h(t, s) ds,
G(t) = \frac{K}{H(t, t_{0})} ∫_{t_{0}}^{t} H(t, s)\frac{|w(s)|^{\frac{α+1}{α}}}{a^{\frac{1}{α}} (t)} ds.

From (3), we get for t ≥ t_{0}

G(t) − F (t) ≤ w(t_{0}) − \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} q(s)H(t, s) ds, (7)

so that (6) implies that

(8) lim inf_{t→∞} [G(t) − F (t)] ≤ w(t_{0}) − lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} q(s)H(t, s) ds

≤ w(t_{0}) − φ(t_{0}) < ∞.

Now, consider a sequence {T_{n}}_{n=1}^{∞} in (t_{0}, ∞) with lim_{n→∞} T_{n} = ∞ and such that

lim_{n→∞} [G(T_{n}) − F (T_{n})] = lim inf_{t→∞} [G(t) − F (t)].

Because of (8), there exists a constant M such that G(T_{n}) − F (T_{n}) ≤ M, n = 1, 2, … (9) We shall next prove that

∫_{t_{0}}^{∞} \frac{|w(s)|^{\frac{α}{α+1}} (s)}{a^{\frac{1}{α}} (s)} ds < ∞. (10)

EJQTDE, 2000 No. 1, p. 9 If we suppose that (10) fails, there exists a t_{1} > t_{0} such that

∫_{t_{0}}^{t} \frac{|w(s)|^{\frac{α}{α+1}} (s)}{a^{\frac{1}{α}} (s)} ds ≥ \frac{μ}{K ξ} , for t ≥ t_{1},

where μ is an arbitrary positive number and ξ is a positive constant such that

(11)  inf_{s≥t_{0}} [lim inf_{t→∞} \frac{H(t, s)}{H(t, t_{0})}] > ξ > 0.

Therefore, for all t ≥ t_{1}

G(t) = \frac{K}{H(t, t_{0})} ∫_{t_{0}}^{t} H(t, s)d (∫_{t_{0}}^{s} \frac{|w(τ)|^{\frac{α+1}{α}}}{a^{\frac{1}{α}} (τ)} dτ)

= − \frac{K}{H(t, t_{0})} ∫_{t_{0}}^{t} \frac{∂H}{∂s} (t, s) (∫_{t_{0}}^{s} \frac{|w(τ)|^{\frac{α+1}{α}}}{a^{\frac{1}{α}} (τ)} dτ) ds

≥ − \frac{K}{H(t, t_{0})} ∫_{t_{1}}^{t} \frac{∂H}{∂s} (t, s) (∫_{t_{0}}^{s} \frac{|w(τ)|^{\frac{α+1}{α}}}{a^{\frac{1}{α}} (τ)} dτ) ds
≥ − \frac{μ}{ξH(t, t_{0})} ∫_{t_{1}}^{t} \frac{∂H}{∂s} (t, s) ds = \frac{μH(t, t_{1})}{ξH(t, t_{0})}

By (11), there is a t_{2} ≥ t_{1} such that \frac{H(t,t_{1})}{H(t,t_{0})} ≥ ξ for all t ≥ t_{2}, and accordingly G(t) ≥ μ for all t ≥ t_{2}. Since μ is arbitrary,

lim_{t→∞} G(t) = ∞,

which ensures that

lim_{n→∞} G(T_{n}) = ∞. (12)

Hence, (9) gives

lim_{n→∞} F (T_{n}) = ∞. (13)

From (9) we derive for n sufficiently large

\frac{F (T_{n})}{G(T_{n})} − 1 ≥ −\frac{M}{G(T_{n})} > −\frac{1}{2}.

Therefore,

\frac{F (T_{n})}{G(T_{n})} > \frac{1}{2} for all large n,

which by (13) ensures that

lim_{n→∞} \frac{F^{α+1}(T_{n})}{G^{α}(T_{n})} = ∞. (14)

On the other hand, by Hölder’s inequality, we have for all n ∈ N

F (T_{n}) = \frac{1}{H(T_{n}, t_{0})} ∫_{t_{0}}^{T_{n}} |w(s)|h(T_{n}, s) ds,
= ∫_{t_{0}}^{T_{n}} (\frac{K^{\frac{α}{α+1}}}{H^{\frac{α}{α+1}} (T_{n}, t_{0})} \frac{|w(s)|H^{\frac{α}{α+1}} (T_{n}, s)}{a^{\frac{1}{α+1}} (s)})

× (\frac{K^{− \frac{α}{α+1}}}{H^{\frac{1}{α+1}} (T_{n}, t_{0})} \frac{h(T_{n}, s)a^{\frac{1}{α+1}} (s)}{H^{\frac{α}{α+1}} (T_{n}, s)}) ds

≤ (\frac{K}{H(T_{n}, t_{0})} ∫_{t_{0}}^{T_{n}} \frac{|w(s)|^{\frac{α+1}{α}} H(T_{n}, s)}{a^{\frac{1}{α}} (s)} ds)^{\frac{α}{α+1}}

× (\frac{K^{−α}}{H(T_{n}, t_{0})} ∫_{t_{0}}^{T_{n}} a(s)\frac{h^{α+1}(T_{n}, s)}{H^{α}(T_{n}, s)} ds)^{\frac{1}{α+1}}

and accordingly

\frac{F^{α+1}(T_{n})}{G^{α}(T_{n})} ≤ \frac{K^{−α}}{H(T_{n}, t_{0})} ∫_{t_{0}}^{T_{n}} a(s)\frac{h^{α+1}(T_{n}, s)}{H^{α}(T_{n}, s)} ds.

So, because of (14), we have

lim_{n→∞} \frac{1}{H(T_{n}, t_{0})} ∫_{t_{0}}^{T_{n}} a(s)\frac{h^{α+1}(T_{n}, s)}{H^{α}(T_{n}, s)} ds = ∞,

which gives

lim_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)} ds = ∞,

EJQTDE, 2000 No. 1, p. 11 contradicting the condition (C_{3}). So, (10) holds. Now, from (5), we obtain

∫_{t_{0}}^{∞} \frac{φ+^{\frac{α}{α+1}} (s)}{a^{\frac{1}{α}} (s)} ds ≤ ∫_{t_{0}}^{∞} \frac{|w(s)|^{\frac{α}{α+1}} (s)}{a^{\frac{1}{α}} (s)} ds < ∞,

which contradicts (C_{5}). This completes the proof.

Theorem 2.3 Let condition (C_{1}) holds and let the functions H and h be defined as in Theorem 2.1 such that conditions (H_{1}), (H_{2}), (H_{3}) and

(C_{6})  lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} |q(s)|H(t, s) ds < ∞

are satisfied. If there exists a continuous function φ on [t_{0}, ∞) such that for every T ≥ t_{0}

(C_{7}) lim inf_{t→∞} \frac{1}{H(t, T)} ∫_{T}^{t}[q(s)H(t, s) − β a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)}] ds ≥ φ(T),

and condition (C_{5}) holds, then the equation (E) is oscillatory.

Proof. For the nonoscillatory solution x(t) of the equation (E), as in the proof of Theorem 2.1, (3) and (4) are fulfilled. Thus, for t > T ≥ t_{0}, we have

lim inf_{t→∞} \frac{1}{H(t, T)} ∫_{T}^{t}[q(s)H(t, s) − β a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)}] ds ≤ w(T),

so that, according to condition (C_{7}), (5) is satisfied. By conditions (C_{7}) is

φ(t_{0}) ≤ lim inf_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} q(s)H(t, s) ds

− lim inf_{t→∞} \frac{β}{H(t, t_{0})} ∫_{t_{0}}^{t} a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)} ds,

so that (C_{6}) implies

lim inf_{t→∞} \frac{β}{H(t, t_{0})} ∫_{t_{0}}^{t} a(s)\frac{h^{α+1}(t, s)}{H^{α}(t, s)} ds < ∞.

Condition (C_{6}) together with (7) implies

lim sup_{t→∞} [G(t) − F (t)] ≤ w(t_{0}) − lim inf_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} q(s)H(t, s) ds < ∞.

This shows that there exists a sequence {T_{n}}_{n=1}^{∞} in (t_{0}, ∞) with lim_{n→∞} T_{n} = ∞, such that

lim_{n→∞} [G(T_{n}) − F (T_{n})] = lim sup_{t→∞} [G(t) − F (t)].

Following the procedure of the proof of Theorem 2.2, we conclude that (10) is satisfied. Then, we come to the contradiction as in the proof of Theorem 2.2.

We observe that Theorem 2.2 can be applied in some cases in which Theorem 2.1 is not applicable. Such a case is described in the following example.

Example 2.2 Consider the differential equation

(E_{2})  (t^{ν}|x(t)|^{3−α}|x′(t)|^{α−1}x′(t))′ + t^{λ} cos t x^{3}(t) = 0,

for t ≥ t_{0}, where ν, λ, α are constants such that λ < 0, α > 0, α ≠ 2 and ν < α. Then, condition (C_{1}) is satisfied. Moreover, taking H(t, s) = (t − s)^{2}, for t > s ≥ t_{0}, we have

\frac{1}{t^{2}} ∫_{t_{0}}^{t} s^{ν}(t − s)^{1−α} ds ≤ \{ \begin{matrix}\frac{t^{ν}}{t^{2}} \frac{(t − t_{0})^{2−α}}{2 − α} , & ν > 0 \\ \frac{t_{0}^{ν}}{t^{2}} \frac{(t − t_{0})^{2−α}}{2 − α} , & ν < 0 \end{matrix}

= \{ \begin{matrix}\frac{t^{ν−α}}{2 − α} (1 − \frac{t_{0}}{t} )^{2−α} , & ν > 0 \\ \frac{t_{0}^{ν}}{2 − α} \frac{1}{t^{α}} (1 − \frac{t_{0}}{t} )^{2−α} , & ν < 0 \end{matrix}

Therefore, condition (C_{3}) is satisfied and for arbitrary small constant ε > 0, there exists a t_{1} ≥ t_{0} such that for T ≥ t_{1}

lim sup_{t→∞} \frac{1}{t^{2}} ∫_{T}^{t}[(t − s)^{2}s^{λ} cos s − β s^{ν} (t − s)^{1−α}] ds ≥ −T^{λ} sin T − ε.

Now, set φ(T) = −T^{λ} sin T − ε and consider an integer N such that 2Nπ + 5π/4 ≥ max\{t_{1}, (1 + \sqrt{2}ε)^{1/λ}\}. Then, for all integers n ≥ N, we have

φ(T) ≥ \frac{1}{\sqrt{2}} for every T ∈ [2nπ + \frac{5π}{4} , 2nπ + \frac{7π}{4} ] .

Taking into account that ν < α, we obtain

∫_{t_{0}}^{∞} \frac{φ+^{\frac{α}{α+1}} (s)}{a^{\frac{1}{α}} (s)} ds ≥ ∑_{n=N}^{∞}(\sqrt{2})^{− \frac{α}{α+1}} ∫_{2nπ+5π/4}^{2nπ+7π/4} s^{\frac{ν}{α}} ds

≥ (\sqrt{2})^{− \frac{α}{α+1}} ∑_{n=N}^{∞} ∫_{2nπ+5π/4}^{2nπ+7π/4} \frac{ds}{s}

= (\sqrt{2})^{− \frac{α}{α+1}} ∑_{n=N}^{∞} ln (1 + \frac{\frac{π}{2}}{2nπ + \frac{5π}{4}} ) = ∞.

Accordingly, all conditions of Theorem 2.2 are satisfied and hence the equation (E_{2}) is oscillatory.

On the other hand, the condition (C_{2}) is not satisfied for λ < −1, so that by Theorem 2.1 we conclude that (E_{2}) is oscillatory only for −1 ≤ λ < 0.

Remark 2.4 It is interesting to note that by Corollary 3.1. in [28] we have that (E_{2}), where ψ(x) ≡ 1, is oscillatory for λ ≥ 0 and ν < α. Therefore, by the previous deduction, we have that such equation is oscillatory for ν < α and all λ.

Theorem 2.4 Suppose that condition (C_{1}) holds and let the functions H and h be defined as in Theorem 2.1, such that conditions (H_{1}) and (H_{2}) hold. If there exists a differentiable function ρ : [t_{0}, ∞) → (0, ∞) such that ρ′(t) ≥ 0 for all t ≥ t_{0} and (C_{8})

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} ρ(s) [q(s)H(t, s) − \frac{β a(s)}{H^{α}(t, s)}G^{α+1}(t, s)] ds = ∞,

where G(t, s) = h(t, s) + \frac{ρ′(s)}{ρ(s)}H(t, s), then the equation (E) is oscil-

latory.

Proof. Let x be a solution on [t_{0}, ∞) of the differential equation (E) with x(t) ≠ 0 for all t ≥ t_{0}. Now, we define

W (t) = ρ(t)\frac{a(t)ψ(x(t))|x′(t)|^{α−1}x′(t)}{f(x(t))} for t ≥ t_{0}.

Then, for every t ≥ t_{0}, we obtain

W′(t) = −q(t)ρ(t) + \frac{ρ′(t)}{ρ(t)}W (t) − \frac{f′(x(t))|W (t)|^{\frac{α+1}{α}}}{(a(t)ρ(t)ψ(x(t))|f(x(t))|^{α−1})^{\frac{1}{α}}} .

Therefore,

∫_{t_{0}}^{t} W′(s)H(t, s) ds ≤ − ∫_{t_{0}}^{t} q(s)ρ(s)H(t, s) ds

+ ∫_{t_{0}}^{t} \frac{ρ′(s)}{ρ(s)}W (s)H(t, s) ds − K ∫_{t_{0}}^{t} H(t, s)\frac{|W (s)|^{\frac{α+1}{α}}}{(a(s)ρ(s))^{\frac{1}{α}}} ds.

Using (2), we have

(15)  \begin{matrix}∫_{t_{0}}^{t} q(s)ρ(s)H(t, s) ds ≤ W (t_{0})H(t, t_{0}) \\ + ∫_{t_{0}}^{t} G(t, s)|W (s)| ds − K ∫_{t_{0}}^{t} H(t, s)\frac{|W (s)|^{\frac{α+1}{α}}}{(a(s)ρ(s))^{\frac{1}{α}}} ds. \end{matrix}

If we take

X = (K H(t, s))^{\frac{α}{α+1}} \frac{|W (s)|}{(a(s)ρ(s))^{\frac{1}{α+1}}} , q = \frac{α + 1}{α}

Y = (\frac{α}{α + 1})^{α} \frac{[a(s)ρ(s)]^{\frac{α}{α+1}}}{[K H(t, s)]^{\frac{α^{2}}{α+1}}} (h(t, s) + \frac{ρ′(s)}{ρ(s)}H(t, s))^{α} ,

according to Lemma 2.1, we get

(16)  \begin{matrix}|W (s)|(h(t, s) + \frac{ρ′(s)}{ρ(s)}H(t, s)) − K H(t, s)\frac{|W (s)|^{\frac{α+1}{α}}}{[a(s)ρ(s)]^{\frac{1}{α}}} \\ ≤ β \frac{a(s)ρ(s)}{H^{α}(t, s)}(h(t, s) + \frac{ρ′(s)}{ρ(s)}H(t, s))^{α+1}. \end{matrix}

From (15) and (16) we obtain

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t}[q(s)ρ(s)H(t, s) − β \frac{a(s)ρ(s)}{H^{α}(t, s)}

× (h(t, s) + \frac{ρ′(s)}{ρ(s)}H(t, s))^{α+1}] ds ≤ W (t_{0}),

which contradicts (C_{8}). Remark 2.5 For α = 1 Theorem 2.4 reduces to Theorem 1 in Grace [7]. Remark 2.6 If α = 1 and H(t, s) = (t−s)^{γ} for some constant γ > 1, Theorem 2.4 include as a special case Theorem 2 in Grace [4]. Corollary 2.3 Let condition (C_{8}) in Theorem 2.4 be replaced by

(C_{9}) lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} \frac{a(s)ρ(s)}{H^{α}(t, s)}(h(t, s) + \frac{ρ′(s)}{ρ(s)}H(t, s))^{α+1} ds < ∞,
(C_{10})  lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} q(s)ρ(s)H(t, s) ds = ∞,

then the conclusion of Theorem 2.4 holds. Example 2.3 Consider the differential equation (E_{3}) (t^{ν}|x(t)|^{3−α}|x′(t)|^{α−1}x′(t))′ + [λ t^{λ−3}(2 − cos t) + t^{λ−2} sin t]x^{3}(t) = 0, for t ≥ t_{0} > 0, where λ is arbitrary positive constant and ν, α are constants such that ν < α − 2, α < 1. Here, we choose ρ(t) = t^{2} and

EJQTDE, 2000 No. 1, p. 16 H(t, s) = (t−s)^{2} for t ≥ s ≥ t_{0}. Then, since ρ(t)q(t) = \frac{d}{ds}[s^{λ}(2−cos s)], as in Example 2.1, we get

∫_{t_{0}}^{t} ρ(s)q(s) ds ≥ t^{λ} − k_{0}

and therefore,

\frac{1}{t^{2}} ∫_{t_{0}}^{t}(t − s)^{2}ρ(s)q(s) ds ≥ \frac{2t^{λ}}{(λ + 1)(λ + 2)} + \frac{k_{1}}{t^{2}} + \frac{k_{2}}{t} − k_{0},

where

k_{1} = \frac{2 t_{0}^{λ+2}}{λ + 2} − k_{0}t_{0}^{2}, k_{2} = 2k_{0}t_{0} − \frac{2 t_{0}^{λ+1}}{λ + 1} .

Hence, condition (C_{10}) is satisfied. On the other hand,

\frac{1}{t^{2}} ∫_{t_{0}}^{t} \frac{s^{ν+2}}{(t − s)^{2α}} (2(t − s) + \frac{2}{s}(t − s)^{2})^{α+1} ds

= t^{α−1}2^{α+1} ∫_{t_{0}}^{t} s^{ν−α+1}(t − s)^{1−α} ds

≤ 2^{α+1} (1 − \frac{t_{0}}{t} )^{1−α} \frac{t^{ν−α+2} − t_{0}^{ν−α+2}}{ν − α + 2} ,

so that condition (C_{9}) is also satisfied. Consequently, by Corollary 2.3, the equation (E_{3}) is oscillatory.

Using Theorem 2.4 and the same technique as in the proof of Theorem 2.2 and 2.3, we have the following two theorems which extend two Grace’s theorems [7, Theorem 3 and 4].

Theorem 2.5 Let condition (C_{1}) holds and let the functions H and h be defined as in Theorem 2.1 such that conditions (H_{1}), (H_{2}), (H_{3}) are satisfied. If there exists a nonnegative, differentiable, increasing function ρ(t) such that

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} \frac{a(s)ρ(s)}{H^{α}(t, s)}(h(t, s) + \frac{ρ′(s)}{ρ(s)}H(t, s))^{α+1} ds < ∞,

and there exists a continuous function φ on [t_{0}, ∞) such that for every T ≥ t_{0}

lim sup_{t→∞} \frac{1}{H(t, T)} ∫_{T}^{t}[q(s)ρ(s)H(t, s) − β \frac{a(s)ρ(s)}{H^{α}(t, s)}

× (h(t, s) + \frac{ρ′(s)}{ρ(s)}H(t, s))^{α+1}] ds ≥ φ(T),

and condition (C_{5}) is satisfied, then the equation (E) is oscillatory.

Theorem 2.6 Let condition (C_{5}) holds and let the functions H and h be defined as in Theorem 2.1 such that conditions (H_{1}), (H_{2}), (H_{3}) are satisfied. If there exists a nonnegative, differentiable, increasing function ρ(t) such that

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} |q(s)|ρ(s)H(t, s) ds < ∞

and there exists a continuous function φ on [t_{0}, ∞) such that for every T ≥ t_{0}

lim inf_{t→∞} \frac{1}{H(t, T)} ∫_{T}^{t}[q(s)H(t, s) − β \frac{a(s)ρ(s)}{H^{α}(t, s)}

× (h(t, s) + \frac{ρ′(s)}{ρ(s)}H(t, s))^{α+1}] ds ≥ φ(T),

and condition (C_{5}) holds, then the equation (E) is oscillatory.

References

[1] A. Elbert, T. Kusano, T. Tanigawa, An oscillatory half-linear differential equations, Acta Math. (Brno), 33 (1997), 355-361

[2] S. R. Grace, B. S. Lalli, C. C. Yeh, Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term, SIAM J. Math. Anal. 15 (1984) 1082–1093

[3] S. R.Grace, B. S.Lalli, C. C. Yeh, Addendum: Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term, SIAM J. Math. Anal. 19 (1988) 1252–1253 [4] S. R. Grace, Oscillation theorems for second order nonlinear differ-

ential equations with damping, Math. Nachr. 141 (1989) 117–127

[5] S. R. Grace, B. S. Lalli, Integral averaging techniques for the oscillation of second order nonlinear differential equations, J. Math. Anal. and Appl. 149 (1990) 277–311

[6] S. R. Grace, Oscillation criteria for second order differential equations with damping, J. Austral. Math. Soc. (Series A) 49 (1990) 43–54

[7] S. R. Grace, Oscillation theorems for nonlinear differential equations of second order, J. Math. Anal. and Appl. 171 (1992) 220–241

[8] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities 2nd edition, Cambridge University Press, Cambridge (1)988

[9] H. L. Hong, On the oscillatory behaviour of solutions of second order nonlinear differential equations, Publ. Math. Debrecen 52 (1998) 55– 68

[10] H. B. Hsu, C. C. Yeh, Oscillation theorems for second order halflinear differential equations, Appl. Math. Lett. 9 (1996) 71–77

[11] I. T. Kiguradze, A. T. Chanturia, Asymptotic properties of solutions of the nonautonomous ordinary differential equations, (in Russian), ”Nauka” Moskva, (1990)

[12] T. Kusano, M. Kitano, On a class pof second order quasilinear ordinary differential equations, Hiroshima Math. Jour. 25 (1995), 321– 355

[13] T. Kusano, Y. Naito, Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar., 76 (1-2) (1997), 81–99

[14] T. Kusano, N. Yoshida, Nonoscillation theorems for a class of quasilinear differential equations of second order, J. Math. Anal. and Appl. 189 (1995) 115–127

[15] H. J. Li, Oscillation criteria for second order linear differential equations, J. Math. Anal. and Appl. 194 (1995) 217–234 [16] H. J. Li, C. C. Yeh, An integral criterion for oscillation of nonlinear

differential equations, Math. Japonica 41 (1995) 185–188

[17] H. J. Li, C. C. Yeh, Oscillation criteria for nonlinear differential equations, Houston Jour. Math. 21 (1995) 801–811

[18] H. J. Li, C. C. Yeh, Nonoscillation criteria for second order halflinear differential equations, Appl. Math. Lett. 8 (1995) 63–70

[19] H. J. Li, C. C. Yeh, Nonoscillation theorems for second order quasilinear differential equations, Publ. Math. Debrecen 47/3-4 (1995) 271–279

[20] H. J. Li, C. C. Yeh, Oscillations of half-linear second order differential equations, Hiroshima Math. Jour. 25 (1995) 585–594

[21] H. J. Li, C. C. Yeh, Oscillation of second order sublinear differential equations, Dynamic Systems and Applic. 6 (1997) 529–534

[22] W. C. Lian, C. C. Yeh, H. J. Li, The distance between zeros of an oscillatory solution to a half-linear differential equations, Computers Math. Applic. 29 (1995) 39–43

[23] D. D. Mirzov, On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Applic., 53 (1976), 418–425

[24] D. D. Mirzov, Asymptotic properties of solutions of the systems of nonlinear nonautonomous ordinary differential equations, (in Russian), Maikop, Adygeja Publ. (1993)

[25] S. Pena, Conjugacy criteria for half-linear differential equations, Acta Math. (Brno), 34 (1998), 1-11

[26] Ch. G. Philos, Oscillation theorems for linear differential equations of second order, Arch. Math. (Basel) 53 (1989) 482–492

[27] J. Wang, Oscillation and nonoscillation theorems for a class of second order quasilinear functional differential equations, Hiroshima Math. Jour., 27 (1997), 449–466

[28] J. Wang, On second order quasilinear oscillations, Funkcialaj Ekvacioj, 41 (1998), 25–54 [29] J. S. W. Wong, C. C. Yeh, An oscillation criterion for second order

sublinear differential equations, J. Math. Anal. and Appl. 171 (1992) 346–351

[30] P. J. Y. Wong, R. P. Agarwal, Oscillatory behaviour of solutions of certain second order nonlinear differential equations, J. Math. Anal. and Appl. 198 (1996) 337–354

[31] C. C. Yeh, Oscillation criteria for second order nonlinear perturbed differential equations, J. Math. Anal. and Appl. 138 (1989) 157–165