数理解析研究所講究録1083 巻 1999 年 219-230

219

Comparison Theorems for Neutral Differential Equations
田中敏 (Satoshi Tanaka)

愛媛大理工

1. INTRODUCTION

We shall be concerned with the oscillatory behavior of solutions of the even orderneutral differential equation(1.1)$\frac{d^{n}}{dt^{n}}[x(t)+h(t)X(\mathcal{T}(i))]+f(t, x(g(t)))=0$,

are assumed to holds:
where $n\geq 2$ is even and the following conditions(H1) $h\in C[t0, \infty)$ and $h(t)\geq 0$ for)is strictly increasing and satisfies $\tau(t)<t$ for(H2)$\lim_{tarrow\infty}\tau(t)/t=1$
;
and
;
) and
(H3)
$f(t,
u)$
for each fixed
is nondecreasing in(H4).
and satisfies $uf(t, u)\geq 0$ for
holds:
Moreover, we assume that one of the following cases
for some constants and ;(R1) $0\leq\mu\leq h(t)\leq\lambda<1$ on
for some constants and ;(R2) $1<\mu\leq h(t)\leq\lambda<\infty$ on$\lim_{tarrow\infty}h(t)=\infty$.
(R3)
By a solution of (1.1), we mean a function $x(t)$ which is continuous and satisfies. Therefore, if $x(t)$ is a solution of (1.1), thenfor some(1.1) on$x(t)+h(t)X(\tau(t))$ is -times continuously differentiable on. Note that, in$x(t)$
itself is not -times continuously differentiable.
general,A solution of (1.1) is called oscillatory if it has arbitrarily large zeros; otherwise
it is called nonoscillatory. This means that a solution $x(t)$ is oscillatory if andsuch that
as $iarrow\infty$ and $x(t_{i})=0$
only if there is a sequence$(i=1,2, \ldots)$ , and a solution $x(t)$ is nonoscillatory if and only if $x(t)\cdot\neq 0$ for all
large . Equation (1.1) is said to be oscillatory if every solution of (1.1) oscillatory,
and nonoscillatory if at least one solution of (1.1) is nonoscillatory.
The purpose of this paper is to present sufficient conditions for (1.1) to be oscillatory or nonoscillatory.
. .
has
been
an
years
there
increasing
interest in oscillation theory for even
In recentorder neutral differential equations, and a number of results have been obtained.
For typical results we refer to the papers |1, 2, 4-9, 12, 13, 16, 18, 19, 21, 24] and theequations find numerous applications
monographs [3] and [10]. Neutral
in natural science and technology. For instance, they are frequently used for the
study of distributed networks containing lossless transmission lines. See Hale [11].
Now consider the equation$(\mathrm{H}1)-(\mathrm{H}4)$$t\geq t_{0;}$$\tau\in C[t_{0}, \infty)$$t\geq t_{0},$$\lim_{tarrow\infty^{\tau(t}}$$=\infty$$g\in C[t_{0_{)}}\infty$$\lim_{tarrow\infty \mathit{9}(t)}=\infty$$t\geq t_{0}$$u\in \mathbb{R}$$f\in C([t_{0}, \infty)\cross \mathbb{R}),$$(t, u)\in[t_{0,\infty})\cross \mathbb{R}$$(\mathrm{R}1)-(\mathrm{R}3)$$[t_{0}, \infty)$$\lambda$$\mu$$[t_{0}, \infty)$$\lambda$$\mu$$t_{x}\geq t_{0}$$[t_{x}, \infty)$$[t_{x}, \infty)$

$n$

$n$

$\{t_{i}\}_{i=1}^{\infty}$$t_{i}arrow\infty$$t$$\mathrm{i}\mathrm{s}_{\vee}$$\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\dot{\mathrm{e}}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}$(1.2). sgn $x(t-\rho)=0$ ,$\frac{d^{n}}{dt^{n}}[x(t)+\lambda x(t-\tau)]+p(t)|X(t-\rho)|\gamma$

220$\gamma>0,$ $p\in C[t_{0,\infty),(t)}p\geq 0$ . We note$\tau>0,$where $n\geq 2$ is even,here that if $\gamma=1$ , then (1.2) becomes the linear equation$\lambda>0,$$\rho\in \mathbb{R},$$.$$\frac{d^{n}}{dt^{n}}[x(t)+\lambda x(t-\tau)]+p(t)X(t-\rho)=0$.

The following result was obtained by Jaro\v{s} and Kusano [12, Theorems 3.1 and
4.1].

Theorem A. Let $\gamma=1$ and $\lambda\in(0,1)$ . If
(1.3)$\int^{\infty}t^{n-1-}\mathrm{g}p(t)dt=\infty$

for some$\epsilon>0$,

then (1.2) is oscillatory. If(1.4)$\int^{\infty}t^{n-1}p(t)dt<\infty$,(1.2) is nonoscillatory.

However, very little is known about the oscillation of (1.2) with $\gamma=1$ andin the case where both conditions (1.3) and (1.4) fail, such as $p(t)=ct^{-n}$$\lambda\in(0,1)$$(C>0)$ .

The following result, which is a characterization of the oscillation of (1.2) withand $\lambda\in(0,1)$ , has been established by Y\={u}ki Naito [19, Theorems 5.3 and5.4].

$\gamma\neq 1$

Theorem B. Let

and $\lambda\in(0,1)$ . Then (1.2) is oscillatory if and only if$\gamma\neq 1$(1.5)$\int^{\infty}t^{\min\{\}}\gamma,1(n-1)p(t)dt=\infty$.

, the following sufficient condition for (1.2) to be nonoscillatory
For the casehas been obtained in [22].
$\lambda>1$

Theorem C. Let$\lambda>1$. If$\int^{\infty}t^{\min\{\gamma,1\}}(n-1)p(t)dt<\infty$,

then (1.2) is nonoscillatory.
to be oscillatory were obtained in [5],Sufficient conditions for (1.2) with[6], [8] and [9]. All of them, however, assume that$\lambda>1$$\int^{\infty}p(t)dl=\infty$.

In this paper we have the following results.

Theorem 1.1. Let(1.5) holds.

$\gamma\neq 1$

and$\lambda\neq 1$. Then (1.2) is oscillatory if and only if.
Theorem 1.2. Let $\gamma=1$ and(i) Equation (1.2) is oscillatory if (1.3) holds.
$\lambda\neq 1$

221(ii) Suppose that
(1.6)$\int^{\infty}t^{n-2}p(t)dt<\infty$.

Equation (1.2) is oscillatory if$\lim_{tarrow}\sup_{\infty}t\int_{t}^{\infty}s^{n-2}p(_{S)s>(+}d1\lambda)(n-1)!$

,

or if
$\lim_{tarrow\infty}\inf t\int_{t}^{\infty}s-2p(n)SdS>(1+\lambda)(n-1)!/4$.

Equation (1.2) is nonoscillatory if$\lim_{tarrow}\sup_{\infty}t\int_{t}^{\infty}s^{n-2}p(S)ds<(1+\lambda)(n-2)!/4$.

Remark 1.1. If (1.4) holds, then (1.6) is satisfied and$0 \leq\lim ttarrow\infty\int_{t}^{\infty}s^{n-2}p(S)d_{S}\leq\lim_{\infty tarrow}\int_{t}^{\infty}s-1p(_{S}n)dS=0$

Hence, Theorem 1.2 implies that (1.2) with $\gamma=1$ and(1.4) holds.

$\lambda\neq 1$.

is nonoscillatory if

We give an example illustrating Theorem 1.2.
Example 1.1. We consider the second order linear neutral differential equation(1.7)$\frac{d^{2}}{dt^{2}}[x(t)+\lambda X(t-\mathcal{T})]+Ct\alpha x(t-\rho)=0$,. Applying Theorem 1.2 to (1.7),$\tau>0,$$c>0,$
where$\alpha>-2$
or $\alpha=-2$ and $c>(1+\lambda)/4$ ;we conclude that: (1.7) is oscillatory if either(1.7) is nonoscillatory if either $\alpha<-2$ or $\alpha=-2$ and $c<(1+\lambda)/4$ .
Our approach is to compare the oscillation of neutral differential equations of theform (1.1) with that of non-neutral differential equations of the form$\lambda>0,$(1.8)$\lambda\neq 1,$$\rho\in \mathbb{R},$$\alpha\in \mathbb{R}$$x^{(n)}(t)+f(t, X(g(t)))=0$ .

Such an approach as this has been conducted by Tang and Shen [23], and Zhang
and Yang [25]. The oscillatory behavior of solutions of (1.8) has been intensively
studied in the last three decades. See, for example, Kitamura [15], Manabu Naito[17] and the references cited therein.
2. COMPARISON THEOREMS
The main results of this paper are as follows.

Theorem 2.1. Suppose that (R1) hold8. If, for some 6 with $0<\epsilon<(1-\lambda)/(1-$, the differential equation$\mu^{2})$$x^{(n)}(t)+( \frac{1-\lambda}{1-\mu^{2}}-\epsilon)f(t, x(g(t)))=0$

222

is

$\mathit{0}\mathit{8}cillatory$, then (1.1) is oscillatory. If, for some$\epsilon>0$, the differential equation$x^{(n)}(t)+( \frac{1-\mu}{1-\lambda^{2}}+\epsilon)f(t, x(g(t)))=0$

is $nono\mathit{8}Cillatory$ , then (1.1) is nonoscillatory.
Theorem 2.2. Suppose that (R2) holds. If, for some, the differential equation$\epsilon$with $0<\epsilon<(\mu-$$1)/(\lambda^{2}-1)$$x^{(n)}(t)+( \frac{\mu-1}{\lambda^{2}-1}-\epsilon)f(t, X(g(t)))=0$

is$\mathit{0}\mathit{8}cillatory$, then (1.1) is oscillatory. If, for some, the differential equation$\epsilon>0$$x^{(n)}(t)+( \frac{\lambda-1}{\mu^{2}-1}+\epsilon)f(t, X(g(t)))=0$

is $nono\mathit{8}Cillatory$ , then (1.1) is nonoscillatory.
Theorem 2.3. Suppose that (R3) holds. If, for someequation$\epsilon\in(0,1)$, the differential$x^{(n)}(t)+(1-\epsilon)f(t, [h(\tau^{-1}(g(t)))]^{-1}x(g(t)))=0$

is oscillatory, then (1.1) is oscillatory. If, for some$\epsilon>0$, the differential equation$x^{(n)}(t)+(1+\epsilon)f(t, [h(\tau^{-1}(g(t)))]^{-1_{X}}(g(t)))=0$$i\mathit{8}$nonoscillatory, then (1.1) is nonoscillatory. Here,of$\tau(t)$$\tau^{-1}(t)$.

is the inverse function

From Theorems 2.1 and 2.2, we have the following results.
Corollary 2.1. Suppose thatfor some $\lambda>0$ withfor some with $0<\epsilon<1/(1+\lambda)$ , the differential equation$\lim_{tarrow\infty}h(t)=\lambda$$\lambda\neq 1$. If,$\epsilon$(2.1)$x^{(n)}(t)+( \frac{1}{1+\lambda}-\epsilon \mathrm{I}^{f}(t, X(g(t)))=0$

is oscillatory, then (1.1) is(2.2)$i\mathit{8}$$\mathit{0}\mathit{8}cillatory$. If, for some$\epsilon>0$, the differential equation$x^{(n)}(t)+( \frac{1}{1+\lambda}+\epsilon)f(t, x(g(t)))=0$

nonoscillatory, then (1.1) is nonoscillatory.

Corollary 2.2.
ferential equation$Suppo\mathit{8}e$that $\lim_{tarrow\infty}h(t)=0$ . If, for some $\epsilon\in(0,1)$ , the dif-$x^{(n)}(t)+(1-\epsilon)f(t, X(g(t)))=0$

is oscillatory, then (1.1) is oscillatory. If, for some$\epsilon>0$$x^{(n)}(t)+(1+\epsilon)f(t, x(g(t)))=0$$i\mathit{8}$nonoscillatory, then (1.1) is nonoscillatory.

, the differential equation

223
for the caseand Corollary. Likewise, we can be prove Corollary 2.1 for the case
2.2. First suppose that (2.1) is oscillatory for some 6 with $0<\epsilon<1/(1’+\lambda)$ . Thereexists a number
such that

Proof of Corollaries 2.1 and 2.2. We give only the proofof$\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{r}\dot{\mathrm{y}}2.1$$0<\lambda<1$$\lambda>1$$\delta>0$and$0<\lambda-\delta<\lambda+\delta<1$.

$| \frac{1-(\lambda+\delta)}{1+(\lambda-\delta)^{2}}-\frac{1}{1+\lambda}|<\frac{\epsilon}{2}$

Put
,

and

$\overline{\lambda}=\lambda+\delta$$\overline{\mu}=\lambda-\delta$$\overline{\epsilon}--\frac{1-\overline{\lambda}}{1-\overline{\mu}^{2}}-\frac{1}{1+\lambda}+\epsilon$.

Then$0<\overline{\mu}<\lambda<\overline{\lambda}<1$,$\frac{1}{1+\lambda}-\epsilon=\frac{1-\overline{\lambda}}{1-\overline{\mu}^{2}}-\overline{\epsilon}$and$\overline{\epsilon}>\frac{\epsilon}{2}>0$.

. Hence,onfor some(R1) withand holds. Theorem 2.1 implies that (1.1)
andis oscillatory. In the same way, we conclude that if (2.2) is nonoscillatory for some
Since$\lim_{\iotaarrow\infty}h(t)=\lambda$$\lambda$$\mu,$$\epsilon>0$$t_{0}$, we see that
replaced by$\overline{\mu}\underline{\leq}h(\mathrm{t})\leq\overline{\lambda}$$\overline{\mu},$$\lambda$$[t_{1}, \infty)$$t_{1}\geq t_{0}$$t_{1}$, then (1.1) is nonoscillatory.

Now we are concerned with the oscillatory behavior of solutions of (1.2). It is
possible to discuss more general neutral differential equations of the form (1.1).
But, for simplicity, we restrict our attention to neutral differential equations of theform (1.2).
Consider the equation$x^{(n)}(t)+p(t)|x(t-\rho)|\gamma$ sgn $x(t-\rho)=0$ ,(2.3)
where $n\geq 2$ is even,$\rho\in \mathbb{R},$$\gamma>0,$$p\in C[t_{0,\infty),(t)}p\geq 0$.

Lemma 2.1. Let $\gamma=1$ . Then (2.3) is oscillatory if (1.3) holds.
Lemma 2.2. Let$\gamma\neq 1$. Then (2.3) is oscillatory if and only if (1.5) hold8.

For the proof of Lemmas 2.1 and 2.2, see Kitamura [15, Corollaries 3.1 and 5.1].
The following results was obtained by Manabu Naito [17, Theorems 2 and 4].

Lemma 2.3. Let $\gamma=1$ . Suppose that (1.6) holds. Then (2.3) is oscillatory if$\lim_{tarrow}\sup_{\infty}t\int_{t}^{\infty}s-2p(n)SdS>(n-1)!$

,

or if
$\lim_{tarrow\infty}\mathrm{i}\mathrm{n}\mathrm{f}t\int_{t}^{\infty}s^{n-2}p(S)dS>(n-1)!/4$.

Lemma 2.4. Let $\gamma=1$ . Suppose that (1.6) hold8. Then (2.3) is nonoscillatory

if$\lim_{tarrow}\sup_{\infty}t\int_{t}^{\infty}s^{n-2}p(s)ds<(n-2)!/4$.

Combining Corollary 2.1 with Lemmas 2.1-2.4, we obtain Theorems 1.1 and 1.2.

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3. PREPARATION FOR THE PROOPs OF THEOREMS 2.1-2.3
In this section we prepare for the proofs of Theorems 2.1-2.3.
We make use of the following well-known lemma of K. iguradze [14].

Lemma 3.1. Let $n\geq 2$ be even and let$u(t)\neq 0$

and$u\in C^{n}[t0, \infty)\mathit{8}atisfy$

for$u(t)u^{()}(nt)\leq 0$$t\geq t_{0}$.

Then there exist an integer $k\in\{1,3, \ldots, n-1\}$ and a number(3.1)$u(t)u^{(i)}(t)>0$ ,$0\leq i\leq k-1$ ,$T\geq t_{0}$$t\geq T$,$t\geq T$.

such that$\{$$(-1)^{i-k}u(t)u^{(})(it)\geq 0$,$k\leq i\leq n$,

A function $u(t)$ satisfying (3.1) is said to be a function of Kiguradze degree .
$k$

$k$

Lenma 3.2. Suppose that (H2) holds. Let $u(t)$ be a function of Kiguradze degreewith $k\geq 1$ . Then

(3.2)$\lim_{tarrow\infty}\frac{u(\tau(t))}{u(t)}=1$.

Proof. We may assume that $u(t)>0$ for all large , since $-u(t)$ is a function of$t$

Kiguradze degree . It is easy to see that one of the following three cases holds:
$k$

(3.3)
(3.4)
(3.5)

;

$\lim_{tarrow\infty}u(t)/t^{k}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}>0$$\lim_{tarrow\infty}u(t)/t^{k}=0$

and$\lim_{tarrow\infty}u(t)/t^{k-}1=+\infty$;.

$\lim_{tarrow\infty}u(t)/t^{k-}1=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}>0$

In the case (3.3), we find that$\lim_{tarrow\infty}\frac{u(\tau(t))}{u(t)}=\lim_{arrow t\infty}\frac{u(\tau(t))}{[\tau(t)]^{k}}[\frac{u(t)}{t^{k}}]^{-1}[\frac{\tau(t)}{t}]^{k}=1$.

In exactly the same way, (3.2) holds for the case (3.5). Assume that (3.4) holds. We
can take a number $T>0$ so large that $u(t)$ satisfies (3.1) and $\tau(t)>0$ for $t\geq T$ .
There exists a function
such that $0<\rho(t)\leq\tau(t)<t$ for $t\geq T$ and$\lim_{tarrow\infty}\rho(t)/t=\lim_{tarrow\infty}\rho’(t)=1$ . In fact,$\rho\in C^{1}[\tau, \infty)$$\rho(t)=\int_{T}^{t}r\geq \mathrm{i}\mathrm{n}\mathrm{f}S\frac{\tau(r)}{r}d_{S}$is such a function, since$\rho(t)\leq\int_{T}^{t}\inf_{r\geq l}\frac{\tau(r)}{r}d_{S}\leq\frac{\tau(t)}{t}(t-T)\leq\tau(\mathrm{t})<t$

and
(3.6)

,

$t\geq T$. Now we claim that$\lim_{tarrow\infty^{\rho’}}(t)=\mathrm{l}\mathrm{i}\mathrm{m}tarrow\infty \mathrm{i}\mathrm{n}\mathrm{f}r\geq t^{\mathcal{T}}(r)/r=1$$\lim_{tarrow\infty}\frac{u^{(k-1)}(\rho(t))}{u^{(k-1)}(t)}=1$

.

,

225
To see this, it is sufficient to show that$\lim_{tarrow\infty}\frac{1}{u^{(k-1)}(t)}\int_{\rho(t)}^{t}u^{(k}(_{S})ds)=0$

,

since
$\frac{1}{u^{(k-1)}(t)}\int_{\rho()}^{t}\iota\frac{u^{(k-1)}(\rho(t))}{u^{(k-1)}(t)}u^{(k})(S)ds=1-$

Notice that $u^{(k-1)}(t)$ is nondecreasing and positive, and. We conclude that
nonnegative on$u^{(k)}(t)$.
is nonincreasing and$[T, \infty)$$u^{(k-1)}(t) \geq u^{(k-1)}(\rho(t))=\int_{T}^{\rho(t)}u^{(}(s)dk)s+u^{(k-1)}(T)\geq(\rho(t)-\tau)u^{(}(k)(\rho t))$for all large $t>T$ , so that$0 \leq\frac{u^{(k)}(\rho(t))}{u^{(k-1)}(t)}\leq\frac{1}{\rho(t)-T}$

for all large $t>T$ . We obtain$0 \leq\frac{1}{u^{(k-1)}(t)}\int_{\rho(t)}^{t}u^{()}(k)Sd_{S\leq}\frac{u^{(k)}(\rho(t))}{u^{(k-1)}(t)}(t-\rho(t))$$\leq\frac{t-\rho(t)}{\rho(t)-T}=\frac{1-\rho(t)/t}{\rho(t)/t-T/t}arrow 0$$(tarrow\infty)$.

Consequently, (3.6) holds as claimed. From (3.6) and the fact that $\lim_{tarrow\infty}\rho’(t)=1$ ,it follows that$\lim_{tarrow\infty}\frac{u(\rho(t))}{u(t)}=tarrow\lim\frac{u’(\rho(t))\rho’(t)}{u(t)}\infty’=\lim_{tarrow\infty}\frac{u’(\rho(t))}{u(t)},\cdot\lim_{arrow t\infty}\rho’(t)$$= \lim_{tarrow\infty}\frac{u’(\rho(t))}{u(t)},=\cdots=\lim\iotaarrow\infty\frac{u^{(k-1)}(\rho(t))}{u^{(k-1)}(t)}=1$

Since $u(t)$ is nondecreasing on$[T, \infty)$.

, we have$\frac{u(\rho(t))}{u(t)}\leq\frac{u(\tau(t))}{u(t)}\leq 1$

for all large $t>T$ . This implies that (3.2) holds.

A close look at the proof of the result of Onose [20] enable us to obtain the next
result.
Lemma 3.3. Let $n\geq 2$ be even. Suppose that (H3) and (H4) holds. Then the
differential equation$x^{(n)}(t)+f(t, x(g(t)))=0$

is oscillatory if and only if the differential inequality$\{x^{(n)}(t)+f(t, X(g(t)))\}$

has no nonoscillatory$\mathit{8}olution$.

sgn $x(t)\leq 0$

226
Now consider the functional equation(3.7)$\omega(t)+h(t)\omega(_{\mathcal{T}(t)})=1$.

The following results have been established in [22].
Lemma 3.4. Suppose that (H1), (H2) and (R1) holds. Iffunction $sati\mathit{8}fying(3.7)$ for all large , thena continuous$\omega(t)i\mathit{8}$$t$$0< \frac{1-\lambda}{1-\mu^{2}}\leq\lim_{\infty tarrow}\inf\omega(t)\leq\lim_{tarrow}\sup_{\infty}\omega(t)\leq\frac{1-\mu}{1-\lambda^{2}}$If

Lemma 3.5. Suppose that (H1), (H2) and (R2) holds.
continuous function satisfying (3.7) for all large , then.
is a bounded$\omega(t)$$t$$0< \frac{\mu-1}{\lambda^{2}-1}\leq\lim_{tarrow\infty}\inf\omega(t)\leq\lim_{tarrow}\sup_{\infty}\omega(t)\leq\frac{\lambda-1}{\mu^{2}-1}$

If

Lemma 3.6. Suppose that (H1), (H2) and (R3) holds.
continuous function satisfying (3.7) for all large , then.
is a bounded$\omega(t)$$\lim_{tarrow\infty}\omega(t)h(\mathcal{T}^{-1}(t))=1$$t$

We regard $C[T, \infty)$ as the Fr\’echet space of all continuous functions onwith the topology of uniform convergence on every compact subinterval of
We introduce the operator : $C[T, \infty)arrow C[T, \infty)$ such that.

$[T, \infty)$$[T, \infty)$.

$\Phi$$\Phi[u](t)+h(t)\Phi[u](\mathcal{T}(t))=u(t)$,$u\in C[T, \infty)$.

This operator is useful to discuss the existence of solutions of the neutral differential equation (1.1). The following propositions concerning this operator havebeen obtained in [22].
$\Phi$

and be numbers
Proposition 3.1. Suppose that (H1) and (H2) holds. Let.
such that $t_{0}\leq T_{*}\leq\tau(T)$ and let $r\in C[T_{*}, \infty)$ with $r(t)>0$ for$0\leqh(t)[r(\tau(t))/r(t)]\leq\lambda<1$onmoreover thatfor some . Then therea mapping : $C[\tau_{*}, \infty)arrow C[T_{*}, \infty)$ which satisfies the following properties:-topology;(i) the mapping is continuous in the(ii) for each $u\in C[T_{*}, \infty),$satisfies $\Phi[u](t)+h(t)\Phi[u](\mathcal{T}(t))=u(t)$ for $t\geq T$ .
$T$

$T_{*}$$t\geq T_{*}$$[T, \infty)$$exi_{\mathit{8}}t_{S}$$A_{\mathit{8}}sume$$\lambda$$\Phi$$C[T_{*}, \infty)$$\Phi$$\Phi$

and be numbers
Proposition 3.2. Suppose that (H1) and (H2) holds. Let$r\in
C[T_{*},\infty)$$t_{0}\leq
T_{*}\leq\tau(T)$. Assume
and letwith $r(t)>0$ for
such that.
somemoreover that $h(t)[r(\tau(t))/r(t)]\geq\mu>1$ on
Definefor$T$$T_{*}$$t\geq T_{*}$$[T, \infty)$$U=\{u\in C[T_{*}, \infty) :$\mu$|u(t)|\leq r(t), t\geq T\}$.

the following: $Uarrow C[T_{*}, \infty)$ which
Then there exist8 a mapping
properties:-topology;(i) the mapping is continuous in the$\Psi[u](t)+h(t)\Psi[u](\mathcal{T}(t))=u(t)$ for $t\geq T$ ;(ii) for each $u\in U,$
satisfies(iii) if $u\in U,$ $u(t)>0$ forand$\Psi$$saii\mathit{8}fies$$C[T_{*}, \infty)$$\Psi$$\Psi$$t\geq T_{*}$$\lim_{tarrow}\sup_{\infty}\frac{u(t)r(\tau(\mathrm{t}))}{u(\tau(t))r(t)}\leq 1$then $\Psi[u](t)/u(t)$ is bounded on$[T_{*}, \infty)$

.

,

227
4. PROOFS OF THEOREMS 2.1-2.3
In this section we give the proofs of Theorems 2.1-2.3.

Proofs of Theorems 2.1-2.3. Suppose that one of conditions

fine the constants$c_{*}\mathrm{a}\mathrm{n}\mathrm{d}_{C^{*}}$$(\mathrm{R}1)-(\mathrm{R}3)$

holds. De-, and the function $H(t)$ by$c_{*}=(1-\lambda)/(1-\mu^{2})$,$c^{*}=(1-\mu)/(1-\lambda^{2})$,$H(t)=1$

if (R1) holds,$c_{*}=(\mu-1)/(\lambda^{2}-1)$,$c^{*}=(\lambda-1)/(\mu^{2}-1)$,$H(t)=1$

if (R2) holds,$c_{*}=1$,$c^{*}=1$,

if (R3) holds,$H(t)=[h(\tau^{-1}(t))]^{-1}$First we show the first halves of Theorems 2.1-2.3. Suppose that
(4.1)$v^{(n)}(t)+(c_{*}-\epsilon)f(t, H(g(t))v(g(t)))=0$. Assume to the contrary that (1.1) has a nonoscilis oscillatory for somelatory solution $x(t)$ . We may suppose without loss of generality that $x(t)>0$
for all large , since the case $x(t)<0$ can be treated similarly. Then we easily
see that $y(t)\equiv x(t)+h(t)X(\tau(t))$ is a function of Kiguradze degree for some$k\in\{1,3, \ldots , n-1\}$ and $y(t)>0$ for all large. Observe that$\epsilon\in(0, c_{*})$

$t$

$k$

$t\geq t_{0}$$\frac{x(t)}{y(t)}+h(t)\frac{y(\tau(t))}{y(t)}\frac{x(\tau(t))}{y(\tau(t))}=1$

for all large$t\geq t_{0}$. Put $\omega(t)=x(t)/y(t)$ and $\overline{h}(t)=h(t)y(\tau(t))/y(t)$ . Then$\omega(t)+\overline{h}(t)\omega(_{\mathcal{T}(t)})=1$.

is bounded.
Since $0<\omega(t)=1-\overline{h}(t)\omega(\tau(t))\leq 1$ for all large , we find thatandNow we assume that (R1) holds. By Lemma 3.2, there are numbers$\delta\in(0,1)$ such that $(1+\delta)\lambda<1$ ,$\omega(t)$$t$$t_{1}$

and$\frac{1-\lambda(1+\delta)}{1-[\mu(1-\delta)]^{2}}\geq\frac{1-\lambda}{1-\mu^{2}}-\frac{\epsilon}{2}$. ThenPutandConsequently, form Lemma 3.4 it follows that$\overline{\mu}=\mu(1-\delta)$$\overline{\lambda}=\lambda(1+\delta)$,$1- \delta<\frac{y(\tau(t))}{y(t)}<1+\delta$$t\geq t_{1}$$0\leq\overline{\mu}\leq\overline{h}(t)\leq\overline{\lambda}<1$,$\frac{x(t)}{y(t)}=\omega(t)\geq\frac{1-\overline{\lambda}}{1-\overline{\mu}^{2}}-\frac{\epsilon}{2}\geq\frac{1-\lambda}{1-\mu^{2}}-\epsilon=(_{C_{*}}-\epsilon)H(t)$for.

$t\geq t_{1}$.

$t\geq t_{2}$. Likewise, using Lemmas 3.5 and 3.6, we can prove that $x(t)\geq$
in the case where (R2) or (R3) holds.
onfor some
By virtue of (1.1) and the monotonicity of , we see thatfor some$t_{2}\geq t_{1}$$(c_{*}-\epsilon)H(t)y(t)$$[t_{2}, \infty)$$t_{2}\geq t_{1}$$f$$y^{(n)}(t)+f(t, (c_{*}-\epsilon)H(g(t))y(g(t)))\leq 0$

for all large. Put $z(t)=(c_{*}-\epsilon)y(t)$ and $F(t, u)=(c_{*}-\epsilon)f(t, H(g(t))u)$ .
Then $z(t)$ is a nonoscillatory solution of the differential inequality$t\geq t_{0}$$z^{(n)}(t)+F(t, z(g(t)))\leq 0$

From Lemma 3.3 it follows that$z^{(n)}(t)+F(t, z(g(t)))=0$.

228
has a nonoscillatory solution, which implies that (4.1) is nonoscillatory. This is acontradiction. The proof of the first halves of Theorem 2.1-2.3 is complete.
Now let us show the second halves of Theorems 2.1-2.3.
Assume that, for some, the differential equation$\epsilon>0$$v^{(n)}(t)+(C^{*}+\epsilon)f(t, H(g(t))v(g(t)))=0$

has a nonoscillatory solution $v(t)$ . Set $w(t)=(c^{*}+\epsilon)^{-1}v(t)$ . Then $w(t)$ is anonoscillatory solution of(4.2)$w^{(n)}(t)+f(t, (c^{*}+\epsilon)H(g(t))w(g(t)))=0$ .

We may assume that $w(t)>0$ and $w(g(t))>0$ for all large , say. It is easy
to verify that $w(t)$ is a function of Kiguradze degree for some $k\in\{1,3,$$n-$)$\lim_{tarrow\infty^{w^{(i}(t)}})=0(i=k+1, \ldots , n-1)$ and$(kt)$
exists in.
By Lemma 3.2, we have $\lim_{tarrow\infty}w(\tau(t))/w(t)=1$ . Thus we can take a number$T\geq T_{1}$ so large that $T_{*} \equiv\min\{\tau(\tau), \inf\{g(t) : t\geq T\}\}\geq T_{1},$$w^{(i)}(T)>0$ for$k-1$$i=0,1,2,$, and if (R1) holds, then$t\geq T_{1}$

$t$

$k$

$\ldots,$$1\},$$\lim_{tarrow\infty}w^{(}$$[0, \infty)$$\ldots,$$0\leq h(t)[w(\mathcal{T}(t))/w(t)]\leq\overline{\lambda}<1$,

for some ,$t\geq T$$\overline{\lambda}$

and if (R2) or (R3) holds, then$h(t)[w(\tau(t))/w(t)]\geq\overline{\mu}>1$,

for some .

$t\geq T$$\overline{\mu}$Integration of (4.2) yields(4.3)$w(t)=P(t)+ \int_{T}^{t}\frac{(t-S)k-1}{(k-1)!}\int_{s}^{\infty}\frac{(r-s)n-k-1}{(n-k-1)!}f(r, (c*+\epsilon)H(g(r))w(g(r)))drds$

for $t\geq T$ , where$P(t)= \sum_{i0}k=-1\frac{(t-T)^{i}}{i!}w((i)T)+\frac{(t-T)^{k}}{k!}w((k)\infty)$.

Notice that $P(t)\geq P(T)=w(T)>0$ for $t\geq T$ . We define the set$y\in C[T_{*}, \infty)$ satisfying$P(t)\leq y(t)\leq w(t)$ ,$t\geq T$

and $y(t)=y(T)$ ,$\mathrm{Y}$of functions$t\in[T_{*}, T]$.

We use Proposition 3.1 or 3.2 with $r(t)=w(t)$ . Then there exists a continuous
mapping A :such that$\mathrm{Y}arrow C[T_{*}, \infty)$(4.4)$\Lambda[y](t)+h(t)\Lambda[y](\tau(t))=y(t)$,$t\geq T$,$y\in \mathrm{Y}$and if (R2) or (R3) holds andsatisfies $\lim_{tarrow}y\infty(\mathcal{T}(t))/y(t)=1$ , then$\Lambda[y](t)/y(t)$ is bounded on. We define the mappingas$y\in \mathrm{Y}$$[T_{*}, \infty)$$\mathcal{F}:\mathrm{Y}arrow C[T_{*}, \infty)$

follows:$P(t)+ \int_{T}^{t}\frac{(t-S)k-1}{(k-1)!}\int_{s}^{\infty}\frac{(r-s)^{n-k}-1}{(n-k-1)!}\overline{f}(r, \Lambda[y](g(r)))drds$$(\mathcal{F}y)(t)=\{$$(\mathcal{F}y)(T)$,$t\in[T_{*}, T]$,,$t\geq T$

,

229

where

$\overline{f}(t, u)=$We note that
(4.5)

$0\leq\overline{f}(t, u)\leq f(t, (c^{*}+\epsilon)H(g(t))w(g(t)))$for all$(t, u)\in[T, \infty)\cross \mathbb{R}$.

Then it is easy to see that is well defined on and maps into itself, by (4.3).
Since A is continuous on , the Lebesgue dominated convergence theorem showsthat is continuous on Y.
Now we claim thatis relatively compact. In view of, wefind thatis uniformly bounded on every compact subinterval of.
By the Ascoli-Arzel\‘a theorem, it suffices to verify thatis equicontinuous on
every compact subinterval of. By (4.5), we easily see that there exists afunction $G\in C[T, \infty)$ which is independent ofand satisfies, for each,$|(\mathcal{F}y)’(t)|\leq G(t)$ on. Let be an arbitrary compact subinterval of.
Then we see that $\{(\mathcal{F}y)’(t) : y\in Y\}$ is uniformly bounded on . The mean value
theorem implies thatis equicontinuous on . Since$t_{2}\in[T_{*}, T]$ , we conclude thatforis equicontinuous on ever.y compact
subinterval of.
By applying the Schauder-Tychonoff fixed point theorem to the operator , there
exists asuch that. It is easy to verify thatsatisfies$t\geqT_{*}$
forand is a function of Kiguradze degree . Lemma 3.2 implies that
. From (4.4), we see that$\mathcal{F}$$Y$$\mathrm{Y}$$Y$$\mathcal{F}$$\mathcal{F}(Y)$$\mathcal{F}(\mathrm{Y})\subset \mathrm{Y}$$\mathcal{F}(\mathrm{Y})$$[T_{*}, \infty)$$\mathcal{F}(Y)$$[T_{*}, \infty)$$y\in \mathrm{Y}$$[T, \infty)$$y\in \mathrm{Y}$$I$$[T, \infty)$

$I$

$I$

$\mathcal{F}(Y)$$t_{1},$$|(\mathcal{F}y)(t_{1})-(\mathcal{F}y)(t_{2})|=0$$\mathcal{F}(\mathrm{Y})$$[T_{*}, \infty)$$\mathcal{F}$$\overline{y}\in \mathrm{Y}$$\overline{y}=\mathcal{F}\overline{y}$$\overline{y}(t)$$0$$\overline{y}(t)>$$k$$\lim_{tarrow}\overline{y}\infty(\mathcal{T}(t))/y(\sim t)=1$$\frac{\Lambda[y\neg(t)}{\overline{y}(t)}+h(t)\frac{y(\sim\tau(t))}{\overline{y}(t)}\frac{\Lambda[y\neg(\mathcal{T}(t))}{\overline{y}(\tau(t))}=1$,$t\geq T$,andis bounded on
if (R2) or (R3) holds. By the same arguments
as in the proofs of the first halves of Theorems 2.1-2.3 and using Lemmas 3.4-3.6,
we obtain$\Lambda[y\neg(t)/\overline{y}(t)$$[T_{*}, \infty)$$0< \frac{\Lambda[y\neg(t)}{\overline{y}(t)}\leq(c^{*}+\epsilon)H(t)$Since $0<\overline{y}(t)\leq w(t)$ for $t\geq T$ , we havefor all large , say. Hence,and (4.4) imply that$t$$t\geq\overline{T}$for all large $t\geq T$ .

$0<\Lambda[y\neg(g(t))\leq(c^{*}+\epsilon)H(g(t))\underline{w}(\mathit{9}(t))$$\overline{f}(t, \Lambda[y\neg(g(t)))=f(t, \Lambda[y\neg(g(t)))$for $t\geq T$ . This$\frac{d^{n}}{dt^{n}}[\Lambda[y\neg(t)+h(t)\Lambda[y\neg(\mathcal{T}(t))]=\overline{y}(\langle n)t)=-\overline{f}(t, \Lambda[y\neg(g(t)))=-f(t, \Lambda[y\gamma(g(t)))$for, which means thatcompletes the proof.
$t\geq\overline{T}$$\Lambda[y\neg(t)$is a nonoscillatory solution of (1.1). This

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