数理解析研究所講究録
第 1702 巻 2010 年 98-107

98

Uniform stability and attractivity for linear systems
with periodic coefficients
杉江 実郎 (Jitsuro Sugie)

Department of Mathematics and Computer Science, Shimane University

1 Introduction
In this paper, we consider the linear system$x’=A(t)x=(\begin{array}{ll}-r(t) p(t)-p(t) -q(t)\end{array})x$,(1)where the prime denotes $d/dt$ ; the coefficients $p(t),$ $q(t)$ and $r(t)$ are continuous for $t\geq 0$ , and$p(t)$ is a periodic fimction with period $\omega>0$ . System (1) has the zero solution.
We say that the zero solution of (1) is attractive if every solution $x(t)$ of (1) tends to as time
$x(t)\equiv 0\in \mathbb{R}^{2}$$0$

increases.
If $q(t)$ and $r(t)$ are also periodic fimctions with period , Floquet’s theorem is available. Let
isidentity matrix. Thenbe the fimdamental matrix of (1) with $\Phi(0)=E$ , thecalled the monodromy matrix of(l). Let and be the eigenvalues ofthe monodromy matrix. The eigenvalues and are often called the Floquet multipliers of (1). By Abel’s$t$$\omega$$2\cross 2$$\Phi(t)$$\mu_{1}$$\Phi(\omega)$$\Phi(\omega)$$\mu_{2}$$\mu_{2}$$\mu_{1}$formula,$\det\Phi(\omega)=\det\Phi(0)\exp(-\int_{0}^{\omega}(q(s)+r(s))ds)=\exp(-\int_{0}^{\omega}(q(s)+r(s))ds)$

Thus, the Floquet multipliers$\mu_{1}$and$\mu_{2}$.

are the roots of the equation$\mu^{2}-tr\Phi(\omega)\mu+\exp(-\int_{0}^{\omega}(q(s)+r(s))ds)=0$.

It is well-known that the zero solution of (1) is attractive if and only if the Floquet multipliers
and have magnitudes strictly less than 1. Hence, in the case where $p(t),$ $q(t)$ and $r(t)$ areperiodic, necessary and sufficient conditions for the zero solution of (1) to be attractive are that$\mu_{1}$$\mu_{2}$$| tr\Phi(\omega)|<1+\exp(-\int_{0}^{\omega}(q(s)+r(s))ds)$

and$\exp(-\int_{0}^{\omega}(q(s)+r(s))ds)<1$.

For example, we can find Floquet’s theorem in the books [2, 3, 5, 8, 16].

99
Although the above conditions are necessary and sufficient for the zero solution of (1) to
be attractive, it is difficult to estimate the absolute value of the trace of, because it is
impossible to find a hndamental matrix of (1) in general. Of course, Floquet’s theorem isuseless when $q(t)$ or $r(t)$ is not periodic. Then, without knowledge of a ffindamental matrix of(1), can we decide whether the zero solution is attractive? What kind of condition on $A(t)$ will
guarantee the attractivity ofthe zero solution of (1)?
$\Phi(\omega)$

2 The main theorem
To give an answer to the above question, we prepare some notations. Let$R(t)= \int_{0}^{t}r(s)ds$

and$\psi(t)=2(q(t)-r(t))$

for $t\geq 0$ and denote a positive part and a negative part of$\psi_{+}(t)=\max\{0, \psi(t)\}$and$\psi(t)$by$\psi_{-}(t)=\max\{0, -\psi(t)\}$,respectively. Note that $\psi(t)=\psi_{+}(t)-\psi_{-}(t)$ and $|\psi(t)|=\psi_{+}(t)+\psi_{-}(t)$ .
We introduce an important concept here. A nonnegative ffinction
is said to be weakly$\phi(t)$integrally positive if$l\phi(t)dt=\infty$

for every set$\triangle>0$$I= \bigcup_{n=1}^{\infty}[\tau_{n}, \sigma_{n}]$such that $\tau_{n}+\delta<\sigma_{n}<\tau_{n+1}<\sigma_{n}+\Delta$ for some $\delta>0$ and. For example, $1/(1+t)$ and $\sin^{2}t/(1+t)$ are weakly integrally positive Rnctions (see[6, 7, 13-15] .
$)$Our main result is stated as follows:

Theorem 1. Suppose that $q(t)$ and $R(t)$ are boundedfor $t\geq 0$ . Suppose also that
(i)
(ii)

$\psi_{+}(t)$is weakly integrally positive;$\int_{0}^{\infty}\psi_{-}(t)dt<\infty$.

Then the zero solution of (1) is attractive.
To prove Theorem 1, we need some lemmas. We present the lemmas without the proofs.

Lemma 2. Suppose that assumption (ii) in Theorem 1 holds. Let $v(t)$ be nonnegative andcontinuously differentiable on
for some $t_{0}>0$ . If$[t_{0}, \infty)$$v’(t)\leq\psi_{-}(t)v(t)$for$t\geq t_{0}$

,

(2)

100
then $v’(t)$ is absolutely integrable, and therefore $v(t)$ has a nonnegative limiting value.

Lemma 3. Suppose that $R(t)$ is boundedfor $t\geq 0$ . If assumption (ii) in Theorem 1 holds, thenall solutions of (1) are unifomly stable and uniformly bounded.
Recall that $p(t)$ is a periodic fimction with period $\omega>0$ . Let$\overline{p}=\max_{t\in[0,\omega]}p(t)$and$\underline{p}=\min_{t\in[0,\omega]}p(t)$.

; if, then, then. Since
into account, we see that if
Taking$p(t)$ is continuous for $t\geq 0$ , we see that $p(t)$ has the following property (we omit the proof).
$\overline{p}+\underline{p}\geq 0$$\overline{p}\geq\underline{p}$Lemma 4. Let $m$ be any integer. If
such that$\underline{p}+\overline{p}\geq 0$$\overline{p}>0$$\underline{p}<0$$\overline{p}+\underline{p}<0$, then there exist numbers$a$and with $0\leq a<$$b$$b\leq\omega$for $m\omega+a\leq t\leq m\omega+b$ .

$p(t) \geq\frac{1}{2}\overline{p}>0$

If$\underline{p}+\overline{p}<0$, then there exist numbers$a$and with $0\leq a<b\leq\omega$ such that$b$for $m\omega+a\leq t\leq m\omega+b$ .

$p(t) \leq\frac{1}{2}\underline{p}<0$

3 Proof of the main theorem
We are now ready to prove Theorem 1.

.
Proof of Theorem 1. Let $x(t;t_{0}, x_{0})$ be a solution of(l) passing throughIt follows ffom Lemma 3 that for any $\alpha>0$, there exists a $\beta(\alpha)>0$ such that $t_{0}\geq 0$ andimply that
(3)
.
for
$(t_{0}, x_{0})\in[0, \infty)\cross \mathbb{R}^{2}$$\Vert x_{0}\Vert<\alpha$$\Vert x(t;t_{0}, x_{0})\Vert<\beta$$t\geq t_{0}$For the sake of brevity, we write $(x(t), y(t))=x(t;t_{0}, x_{0})$ and$v(t)=V(t, x(t), y(t))$ .

Then, we have$v(t)= \frac{1}{2}e^{2R(t)}(x^{2}(t)+y^{2}(t))$(4)$v^{f}(t)=-(q(t)-r(t))e^{2R(t)}y^{2}(t)\leq\psi_{-}(t)v(t)$

(5)

and

. If $v_{0}=0$ , then. Hence, ffom Lemma 2, we see that $v(t)$ has a limiting valuefor. This completes the proof. Thus, we needby (4) the solution $(x(t), y(t))$ tends to as
consider only the case in which $v_{0}>0$ . We will show that this case does not occur.
. Hence, $|y(t)|$ has an inferior limit
Because of (3), we see that $|y(t)|$ is bounded forand a superior limit. First, we will show that the inferior limit of $|y(t)|$ is zero, and we will thenshow that the superior limit of $|y(t)|$ is also zero.
$v_{0}\geq 0$$t\geq t_{0}$$0$$tarrow\infty$$t\geq t_{0}$101
Suppose that $hm\inf_{tarrow\infty}|y(t)|>0$ . Then, there exist a $\gamma>0$ and a$|y(t)|>\gamma$ for. It follows from (5) and Lemma 2 thatsuch that$T_{1}\geq t_{0}$$t\geq T_{1}$$\infty>\int_{t_{0}}^{\infty}|v’(s)|ds=\frac{1}{2}\int_{t_{0}}^{\infty}|\psi(s)|e^{2R(s)}y^{2}(s)ds$$\geq\frac{1}{2}\gamma^{2}\int_{T_{1}}^{\infty}\psi_{+}(s)e^{2R(s)}ds\geq\frac{1}{2}\gamma^{2}e^{-2L}\int_{T_{1}}^{\infty}\psi_{+}(s)ds$,

is the number given in the proof of Lemma 3. This contradicts assumption (i). Thus,
we see that $\lim\inf_{tarrow\infty}|y(t)|=0$.
Suppose that $\lim\sup_{tarrow\infty}|y(t)|>0$ . Let $\nu=\lim\sup_{tarrow\infty}|y(t)|$ . Since $q(t)$ is bounded, we
can find asuch that
where

$L$

$\overline{q}>0$$|q(t)|\leq\overline{q}$Since $v(t)$ tends to a positive valueas$v_{0}$$tarrow\infty$

for $t\geq 0$ .

(6)

, there exists a$0< \frac{1}{2}v_{0}<v(t)<\frac{3}{2}v_{0}$for$T_{2}\geq t_{0}$$t\geq T_{2}$such that.

(7)

Let be so small that$\epsilon$$0< \epsilon<\min\{\frac{1}{2}\nu,$$\sqrt{\frac{\overline{p}^{2}e^{-2L}v_{0}}{4(\overline{q}+2/(b-a))^{2}+\overline{p}^{2}}},$$\sqrt{\frac{\underline{p}^{2}e^{-2L}v_{0}}{4(\overline{q}+2/(b-a))^{2}+\underline{p}^{2}}}\}$,(8)

where and are the numbers given in Lemma 4. Then, since $\lim\inf_{tarrow\infty}|y(t)|=0$ , we canselect two intervals
and
with
and
as
$|y(t_{n})|=\nu/2,$ $|y(s_{n})|=3\nu/4$ andsuch that$b$$a$$[t_{n}, s_{n}]$$[\tau_{n}, \sigma_{n}]$$[t_{n}, s_{n}]\subset[\tau_{n}, \sigma_{n}],$ $T_{2}<\tau_{n}$$\tau_{n}arrow\infty$$narrow\infty$$|y(\tau_{n})|=|y(\sigma_{n})|=\epsilon,$

for

,

(9)

for $\sigma_{n}<t<\tau_{n+1}$ ,(10)$|y(t)|\geq\epsilon$$|y(t)|\leq\epsilon$$\frac{1}{2}\nu<|y(t)|<\frac{3}{4}\nu$$\tau_{n}<t<\sigma_{n}$for $t_{n}<t<s_{n}$ .

(11)

By (4), (7) and (10), we have(12)$|x(t)|=\sqrt{2e^{-2R(t)}v(t)-y^{2}(t)}\geq\sqrt{e^{-2L}v_{0}-\epsilon^{2}}$for$\sigma_{n}\leq t\leq\tau_{n+1}$.

Claim. The sequences$\{\tau_{n}\}$and$\{\sigma_{n}\}$satisfy$\tau_{n+1}-\sigma_{n}\leq 2\omega$

Suppose that there exists an $n_{0}\in N$ such thatsuch that $(m-1)\omega<\sigma_{n_{0}}\leq m\omega$ . Hence, we havefor any integer$\tau_{no+1}-\sigma_{n0}>2\omega$$[rr\iota\omega, (m+1)\omega]\subset[\sigma_{n_{0}}, \tau_{n_{0}+1}]$$\overline{p}+\underline{p}<0$.

. We can choose an $m\in N$$\tau_{n_{0}+1}>\sigma_{n_{0}}+2\omega>(m-1)\omega+2\omega=(m+1)\omega$

and thereforeand (b)$n$,. There are two cases to consider: (a)$\overline{p}+\underline{p}\geq 0$. In case (a), by Lemma 4, $p(t)\geq\overline{p}/2>0$ for $t\in[a+m\omega, b+m\omega]\subset$

102$[m\omega, (m+1)\omega]$. Hence, using the second equation in system (1) with (6), (10) and (12), we

have
(13)

$|y’(t)| \geq|p(t)||x(t)|-|q(t)||y(t)|\geq\frac{1}{2}\overline{p}\sqrt{e^{-2L}v_{0}-\epsilon^{2}}-\overline{q}\epsilon$

for $a+m\omega<t<b+m\omega$ . It follows $\theta om(8)$ that$\frac{1}{2}\overline{p}\sqrt{e^{-2L}v_{0}-\epsilon^{2}}-\overline{q}\epsilon>\frac{2}{b-a}\epsilon$

.

(14)

From (10) and (13), we can estimate that$2 \epsilon\geq|y(b+m\omega)|+|y(a+m\omega)|\geq|\int_{a+\pi\omega}^{b+m\omega}y^{f}(s)ds|$$= \int_{a+\pi w}^{b+\pi\omega}|y’(s)|ds\geq(b-a)(\frac{1}{2}\overline{p}\sqrt{e^{-2L}v_{0}-\epsilon^{2}}-\overline{q}\epsilon)$.

This contradicts (14). In case (b), by Lemma 4, $p(t)\leq\underline{p}/2<0$ for $t\in[a+m\omega, b+m\omega]\subset$$[m\omega, (m+1)\omega]$ . Hence, combining this with (6), (10) and (12), we obtain(15)$|y^{f}(t)| \geq|p(t)||x(t)|-|q(t)||y(t)|\geq-\frac{1}{2}\underline{p}\sqrt{e^{-2L}v_{0}-\epsilon^{2}}-\overline{q}\epsilon$

for $a+m\omega<t<b+m\omega$ . It follows Rom (8) that$- \frac{1}{2}\underline{p}\sqrt{e^{-2L}v_{0}-\epsilon^{2}}-\overline{q}\epsilon>\frac{2}{b-a}\epsilon$

.

(16)

From (10) and (15), we can estimate that$2 \epsilon\geq|y(b+m\omega)|+|y(a+m\omega)|\geq|\int_{a+\pi w}^{b+m\omega}y’(s)ds|$$= \int_{a+mtd}^{b+\pi\omega}|y^{f}(s)|ds\geq(b-a)(-\frac{1}{2}\underline{p}\sqrt{e^{-2L}v_{0}-\epsilon^{2}}-\overline{q}\epsilon)$.

This contradicts (16). Thus, the claim is proved.
Let$I= \bigcup_{n=1}^{\infty}[\tau_{n}, \sigma_{n}]$. Then, by means of Lemma 2 with (5) and (9), we get$\infty>\int_{t_{0}}^{\infty}|v’(s)|ds=\frac{1}{2}\int_{t_{0}}^{\infty}|\psi(s)|e^{2R(s)}y^{2}(s)ds$$\geq\frac{1}{2}e^{-2L}\int_{t_{0}}^{\infty}\psi_{+}(s)y^{2}(s)ds\geq\frac{1}{2}\epsilon^{2}e^{-2L}\int_{I}\psi_{+}(s)ds$.

Hence, it follows from assumption (i) and the claim that $\lim\inf_{narrow\infty}(\sigma_{n}-\tau_{n})=0$ . Since$[t_{n}, s_{n}]\subset[\tau_{n}, \sigma_{n}]$, it follows that$\lim_{narrow}\inf_{\infty}(s_{n}-t_{n})=0$

.

(17)

103
By (4), (7) and (11), we have$|x(t)|=\sqrt{2e^{-2R(t)}v(t)-y^{2}(t)}\leq\sqrt{3e^{2L}v_{0}-\frac{\nu^{2}}{4}}$for$t_{n}\leq t\leq s_{n}$. Let$K= \max\{|\overline{p}|, |\underline{p}|\}$. Then, from (6) and (11), we see that$|y’(t)| \leq|p(t)||x(t)|+|q(t)||y(t)|<K\sqrt{3e^{2L}v_{0}-\frac{\nu^{2}}{4}}+\frac{3}{4}\overline{q}\nu$for$t_{n}$$t_{n}\leq t\leq s_{n}$to$s_{n}$. Lettingand integrating this inequality ffom

$N=K\sqrt{3e^{2L}v_{0}-\nu^{2}}/4+3\overline{q}\nu/4$, we obtain$\frac{1}{4}\nu=|y(s_{n})|-|y(t_{n})|\leq|y(s_{n})-y(t_{n})|$$=| \int_{t_{n}}^{s_{\hslash}}y^{f}(s)ds|\leq\int_{t_{n}}^{s_{n}}|y’(s)|ds\leq N(s_{n}-t_{n})$.

This contradicts (17). We therefore conclude that $\lim\sup_{tarrow\infty}|y(t)|=\nu=0$.
In summary, $y(t)$ tends to zero as. Hence, there exists asuch that$tarrow\infty$$|y(t)|<\epsilon$$T_{3}\geq T_{2}$for$t\geq T_{3}$

.

(18)

be an integer satisfying $l\omega>T_{3}$ . Using (18) instead of (10) and following the sameprocess as in the proof of the claim, we see that if, thenLet$l$$\overline{p}+\underline{p}\geq 0$$2 \epsilon\geq|y(b+l\omega)|+|y(a+l\omega)|\geq|\int_{a+l\omega}^{b+l\omega}y’(s)ds|$$= \int_{a+l\omega}^{b+l\omega}|y’(s)|ds\geq(b-a)(\frac{1}{2}\overline{p}\sqrt{e^{-2L}v_{0}-\epsilon^{2}}-\overline{q}\epsilon)>2\epsilon$

which is a contradiction; if$\overline{p}+\underline{p}<0$,, then$2 \epsilon\geq|y(b+l\omega)|+|y(a+l\omega)|\geq|\int_{a+l\omega}^{b+l\omega}y’(s)ds|$,$= \int_{a+l\omega}^{b+l\omega}|y’(s)|ds\geq(b-a)(-\frac{1}{2}\underline{p}\sqrt{e^{-2L}v_{0}-\epsilon^{2}}-\overline{q}\epsilon)>2\epsilon$

which is again a contradiction. Thus, the case of $v_{0}>0$ cannot happen.
The proof of Theorem 1 is thus complete.

4$\square$Examples

We illustrate our main result with simple examples in which $p(t),$ $q(t)$ and $r(t)$ are periodic. It is

well-known that ifthe zero solution of a linear periodic system is attractive, then it is uniformly
asymptotically stable (for example, see [5, 18]).

104
Example 1. Let$\lambda>0$. Consider system (1) with$p(t)=\cos t$ ,and $r(t)=0$ .

$q(t)= \frac{\lambda}{2-nt}$(19)

Then the zero solution is attractive.
Since
and $R(t)\equiv 0$ , it is clear that $q(t)$ and $R(t)$ are bounded for $t\geq 0$ .
Also, assumptions (i) and (ii) are satisfied. In fact, we have$\lambda/3\leq q(t)\leq\lambda$$\psi(t)=2(q(t)-r(t))=\frac{2\lambda}{2-\sin t}$,

and therefore$\psi_{+}(t)=\frac{2\lambda}{2-\sin t}$

for $t\geq 0$ . Hence,$\psi_{+}(t)$and$\psi_{-}(t)=0$

is weakly integrally positive and$\int_{0}^{\infty}\psi_{-}(t)dt=0$.

Thus, by means of Theorem 1, we conclude that the zero solution is attractive.

Figure l(a) shows a positive orbit of (1) with (19) and $\lambda=0.1$ . The starting pointis$(-1,0)$ and the initial timeis . The positive orbit moves around the origin in a clockwise
and a counter-clockwise direction altemately, because $p(t)$ changes its sign. The positive orbit
approaches the origin as it goes up and down.
$x_{O}$$0$$t_{0}$

$0$

$0$

(a)

(b)

Figure 1: (a) A positive orbit of (1) with (20); (b) a positive orbit of (1) with (21)
Example 2. Let$\lambda\geq 1$. Consider system (1) with$p(t)=\cos\lambda t$,

Then the zero solution is attractive.

$q(t)=\cos^{2}t+\sin t$

and$r(t)=\sin$ t.

(20)

105
It is easy to check that $q(t)$ and $R(t)$ are bounded for $t\geq 0$ and that assumptions (i) and (ii)
are satisfied. We omit the details.
In Figure 1 (b), we show a positive orbit of (1) with (20) and. The positive orbit starts$(-1,0)$
Rom the pointat the initial time . The positive orbit goes to the right and then goes tothe left, and it repeats such a movement regularly. Although the positive orbit displays intricate
behavior, it approaches the origin ultimately.
In Examples 1 and 2, all coefficients of (1) are periodic hncbons with period . However,
we camot find the monodromy matrix. It is particularly hard to estimate the absolutevalue of the trace of. For this reason, we cannot apply Floquet’s theorem to Examples 1
and 2 directly. Theorem 1 has the advantage ofbeing applicable to cases where the monodromy
matrix of (1) camot be found and cases where $q(t)$ or $r(t)$ is not periodic.
Fortunately, in Examples 1 and 2 the Floquet multipliers
andcan be calculated by anumerical scheme. As shown in Tables 1 and 2, $|\mu_{1}|<1$ and $|\mu_{2}|<1$ . Hence, we see that thezero solution of (1) is attractive.
$\lambda=4$

$0$

$0$

$2\pi$$\Phi(2\pi)$$\Phi(2\pi)$$\mu_{1}$$\mu_{2}$Table 1: Floquet multipliers of (1) with (20)

Table 2: Floquet multipliers of (1) with (21)Remark. The zero solution of system (1) with (19) is attractive if and only if, then$\lambda>0$$\lambda\leq 0$.

$\exp(-\int_{0}^{\omega}(q(s)+r(s))ds)=\exp(-\int_{0}^{\omega}\frac{\lambda}{2-\sin s}ds)\geq 1$

Hence, as mentioned in Section 1, the zero solution is not attractive in this case.

. In fact, if

106References[1] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, 2nd ed., Spninger-

Verlag, Berlin-Heidelberg-New York, 2005.

[2] F. Brauer and J. Nohel, The Qualitative Theory of Ordinary Differential Equations, W. A. Benjamin, New York and Amsterdam, 1969; (revised) Dover, New York, 1989.

[3] J. Cronin, Differential Equations: Introduction and Qualitative Theory, 2nd ed., Monographs andTextbooks in Pure and Applied Mathematics 180, Marcel Dekker, New York-Basel-Hong Kong,
1994.

[4] A. Halanay, Differential Equations: Stability, Oscillations, fime Lags, Academic Press, New York–

London, 1966.

[5] J. K. Hale, Odinary Differential Equations, Wiley-Interscience, New York-London-Sydney, 1969;(revised) Knieger, Malabar, 1980.
[6] L. Hatvani, On the asymptotic stability by nondecrescent Ljapunovfunction, Nonlinear Anal., 8(1984), 67-77.
[7] L. Hatvani, On the asymptotic stability for a two-dimensional linear nonautonomous differential
system, Nonlinear Anal., 25 (1995), 991-1002.
[8] D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An intmduction to Dynamical Systems, 3rd ed., Oxford Texts in Applied and Engineering Mathematics 2, Oxford University
Press, Oxford, 1999.

[9] J. P. LaSalle and S. Lefschetz, Stability by LiapunovS Direct Method with Applications, Mathematics in Science and Engineering 4, Academic Press, New York-London, 1961.
[10] D. R. Merkin, Introduction to the Theory of Stability, Texts in Applied Mathematics 24, Springer-

Verlag, New York-Berlin-Heidelberg, 1997.

[11] A. N. Michel, L. Hou and D. Liu, Stobility Dynamical Systems: Continuous, Discontinuous, andDiscrete Systems, Birkh\"auser, Boston-Basel-Berlin, 2008.
[12] N. Rouche, P. Habets and M. Laloy, Stability Theo y by Liapunov ’s Direct Method, Applied Mathematical Sciences 22, Springer-Verlag, New York-Heidelberg-Berlin, 1977.
[13] J. Sugie, Convergence of solutions of time-varying linear system with integrable forcing term,
Bull. Austral. Math. Soc., 78 (2008), 445-462.

[14] J. Sugie, Influence of anti-diagonals on the asymptotic stability for linear differential systems,
Monatsh. Math., 157 (2009), 163-176.
[15] J. Sugie and Y. Ogami, Asymptotic stability for three-dimensional linear differential systems with

time-varying coefficients, to appear in Quart. Appl. Math.

[16] F. Verhulst, Nonlinear Differential Equations andDynamical Systems, Springer-Verlag, New York-

Berlin-Heidelberg, 1990.

[17] T. Yoshizawa, Note on the boundedness and the ultimate boundedness ofsolutions of $x’=F(t,$ x),
Mem. Coll. Sci. Univ. Kyoto, Ser. A. Math., 29 (1955), 275-291.

107[18] T. Yoshizawa, Stability Theory by Liapunov ’s Second Method, Math. Soc. Japan, Tokyo, 1966.
[19] T. Yoshizawa, Stability Theo,y and the Existence of Periodic Solutions andAlmost Periodic Solu-

tions, Applied Mathematical Sciences 14, Springer-Verlag, New York-Heidelberg-Berlin, 1975.