Funkcialaj Ekvacioj, 22 (1979), 67-76 Stability Theory for Delay Equations By T. A. BURTON (Southern Illinois University, U.S.A.) We consider a system of delay differential equations together with a Liapunov functional and present conditions under which solutions will approach certain sets. In particular, we obtain new results on asymptotic stability when the delay is un- bounded-and when the right hand side of the delay equation is unbounded. The me- thods from these results are then applied to the equation $x^{ prime/}(t)+ psi(t, x(t), x^{ prime}(t))x^{ prime}(t)+g(x(t-r(t)))=0$ . §1. Introduction We consider a system of delay-differential equations (1) $x^{ prime}(t)=f(t, x(t), x(t-r(t)))$ with $f:[0,$ $ infty$ ) $ times R^{n} times R^{n} rightarrow R^{n}$ , $r:[0,$ $ infty$ ) $ rightarrow[0$ , $ infty$ ), $f$ and $r^{ prime}$ continuous, and $r(t) leq t$ . We usually suppose that $f(t, 0, 0)=0$ so that the zero function is a solution. Our results can easily be translated into results for systems with several delays and, in fact, will usually apply to a system of functional differential equations with arbitra- rily large time lags. However, when dealing with such general systems one must either assume that bounded solutions are continuable or make severe restrictions on /. Under the conditions stated with (1), continuation of bounded solutions is no problem. In this paper we assume that there is a Liapunov functional for (1) and give conditions which insure various properties of solutions of (1). Most of the classical results call for the functional to be positive definite, its derivative to be negative de- finite, $r(t)$ to be bounded, and $f$ to be bounded for $x$ bounded in order to conclude that solutions tend to zero. Our main result shows that if the functional satisfies a certain Lipschitz property in $t$ , then the bounds on $r$ and $f$ may be dropped. §2. Notation and definitions The following notation and definitions will be used. Given $t_{0} geq 0$ we define $E_{t mathrm{o}}= {t_{0} } cup {s:s=t-r(t) leq t_{0} }$ which we call the initial in-
Transcript
Funkcialaj Ekvacioj, 22 (1979), 67-76
Stability Theory for Delay Equations
By
T. A. BURTON
(Southern Illinois University, U.S.A.)
We consider a system of delay differential equations together with a Liapunovfunctional and present conditions under which solutions will approach certain sets.In particular, we obtain new results on asymptotic stability when the delay is un-bounded-and when the right hand side of the delay equation is unbounded. The me-thods from these results are then applied to the equation
We consider a system of delay-differential equations
(1) $x^{¥prime}(t)=f(t, x(t), x(t-r(t)))$
with $f:[0,$ $¥infty$ ) $¥times R^{n}¥times R^{n}¥rightarrow R^{n}$ , $r:[0,$ $¥infty$ ) $¥rightarrow[0$ , $¥infty$ ), $f$ and $r^{¥prime}$ continuous, and $r(t)¥leq t$.We usually suppose that $f(t, 0, 0)=0$ so that the zero function is a solution. Ourresults can easily be translated into results for systems with several delays and, infact, will usually apply to a system of functional differential equations with arbitra-rily large time lags. However, when dealing with such general systems one musteither assume that bounded solutions are continuable or make severe restrictions on/. Under the conditions stated with (1), continuation of bounded solutions is noproblem.
In this paper we assume that there is a Liapunov functional for (1) and giveconditions which insure various properties of solutions of (1). Most of the classicalresults call for the functional to be positive definite, its derivative to be negative de-finite, $r(t)$ to be bounded, and $f$ to be bounded for $x$ bounded in order to concludethat solutions tend to zero. Our main result shows that if the functional satisfies acertain Lipschitz property in $t$, then the bounds on $r$ and $f$ may be dropped.
§2. Notation and definitions
The following notation and definitions will be used.Given $t_{0}¥geq 0$ we define $E_{t¥mathrm{o}}=¥{t_{0}¥}¥cup¥{s:s=t-r(t)¥leq t_{0}¥}$ which we call the initial in-
68 T. A. BURTON
terval. For a given $t_{0}$ and a continuous function $¥phi:E_{t¥mathrm{o}}¥rightarrow R^{n}$ , there is a solution$x(t, ¥phi)$ on $ t_{0}¥leq t<¥alpha$ for some $¥alpha>0$ and $ x(t, ¥phi)=¥phi$ if $t$ $¥in E_{t¥mathrm{o}}$ . If $x(t, ¥phi)$ is bounded,then $¥alpha$ may be chosen as $+¥infty$ . Basic facts about delay and functional differentialequations may be found in Driver [1], EPsgol’ts [2], Hale [3], and Yoshizawa [5; pp.183- ].
If $x$ $¥in R^{n}$ , then $|x|$ is the euclidean length. If $¥phi:E_{t¥mathrm{o}}¥rightarrow R^{n}$ , then$||¥phi||=¥sup_{t¥in E_{t_{0}}}|¥phi(t)|$.
The reader is referred to Driver [1; p. 410] for definitions of stability, uniformstability, asymptotic stability, and uniform asymptotic stability.
We shall employ certain continuous functions $W_{i}$ : [0, $¥infty$ ) $¥rightarrow[0$, $¥infty$ ) with $W_{i}(0)$
$=0$, $W_{i}(s)>0$ if $s>0$, and $W_{i}$ strictly increasing. The $¥mathrm{n}_{i}^{¥gamma}$ are called wedges.
where $A$ and $B$ are $n¥times n$ matrices. If we express the second term of $V$ as $Z(t, x(¥cdot))$,we notice that $(d/dt)Z(t, x)=x^{T}(t)Bx(t)-x^{T}(t-r(t))Bx(t-r(t))(1-r^{¥prime}(t))$ so that ifboth $B$ and $r^{¥prime}$ are bounded, then $Z$ is Lipschitz in $t$ for any bounded function $¥mathrm{x}$ .That property can be used to good advantage.
Theorem 1. Let $V$ be a functional satisfying (4) and the conditions with (2).Suppose there is a functional $Z(t, x(¥cdot))$ and wedges $W$ and $W_{1}$ such that $W(|x(t)|)+$
$Z(t, x(¥cdot))¥leq¥nabla(t, x(¥cdot))¥leq W_{1}(|x(t)|)+Z(t, x(¥cdot))$ . If for each $t_{0}¥geq 0$ and each boundedcontinuous function $x:E_{t¥mathrm{o}}¥cup[t_{0},$ $¥infty$ ) $¥rightarrow R^{n}$ , there exists $K>0$ such that $t_{0}¥leq t_{1}<t_{2}$ implies
Stability Theory for Delay Equations 69
$Z(t_{2}, x(¥cdot))-Z(t_{1}, x(¥cdot))¥leq K(t_{2}-t_{1})$ , then each bounded solution of (I) converges to zero.
Proof. If the theorem is false, then there is a bounded solution $x(t)$ on $[t_{0},$ $¥infty)$ ,an $¥epsilon>0$ , and a sequence $¥{¥overline{t}_{n}¥}$ with $|x(¥overline{t}_{n})|¥geq¥epsilon$ . As $V_{(1)}^{¥prime}(t, x(¥cdot))¥leq-W_{4}(|x(t)|)$ and$V¥geq 0$, an integration from $t_{0}$ to $t$ shows that there is a sequence $¥{t_{n}^{*}¥}$ with $x(t_{n}^{*})¥rightarrow 0$ .Now $V^{¥prime}(t, x(¥cdot))¥leq 0$ and so $V(t, x(¥cdot))¥rightarrow L$.
Choose $¥gamma>0$ so that $W_{1}(¥gamma)<W(¥epsilon)/4$ . Then choose sequences $¥{t_{n}¥}$ and $¥{T_{n}¥}$
with $|x(t_{n})|=¥epsilon$ , $|x(T_{n})|=¥gamma$ , and $¥gamma¥leq|x(t)|¥leq¥epsilon$ for $t_{n}¥leq t¥leq T_{n}$ , and $ t_{n}¥rightarrow¥infty$ as $ n¥rightarrow¥infty$ .We have
Thus, on intervals $[t_{n}, T_{n}]$ we have $V_{(1)}^{¥prime}(t, x(¥cdot))¥leq-W_{4}(¥gamma)$ . As $T_{n}-t_{n}¥geq T$, anintegration yields $ V(t, x(¥cdot))¥rightarrow-¥infty$ as $ t-¥rightarrow¥infty$ , a contradiction. This completes theproof.
Remark 1. The right-hand Lipschitz condition can be replaced by a left-handone
for $k>0$ and to be determined. We have $W(s)=W_{1}(s)=¥frac{1}{2}s^{2}$ and $Z(t, x(¥cdot))=$
$k¥int_{t-r(t)}^{t}x^{2}(s)ds$. Clearly, $Z$ is Lipschitz in $t$ for bounded $x(¥cdot)$ . Note that $r(t)$ need
not be bounded. We find $V^{¥prime}(t, x(¥cdot))¥leq-ax^{2}(t)+bx(t)x(t-r(t))+k[x^{2}(t)-X^{2}(t-$
$r(t))(1-r^{¥prime}(t))]¥leq-k(1-¥alpha)x^{2}(t-r(t))+bx(t-r(t))x(t)-(a-k)x^{2}(t)$. We completethe square, select $k=a/2$, and ask that $a^{2}(1-¥alpha)>b^{2}$ to insure that $ V^{¥prime}(t, x(¥cdot))¥leq$
$-¥mu x^{2}(t)$ for some $¥mu>0$ . As $V^{¥prime}¥leq 0$ , $W(|x(t)|)¥leq V(t, x(¥cdot))$, and $ W(s)¥rightarrow¥infty$ as $ s¥rightarrow¥infty$ ,all solutions are bounded.
The reader may compare Theorem 1 with Theorem 6 of Driver [1; pp. 419-20].Driver allows $r(t)$ unbounded, but requires the right side of (1) bounded for $x$ bound-ed. Driver’s Theorem 4 [1; p. 415] allows the right side of (1) unbounded and $r(t)$
unbounded, but asks $V_{(1)}^{J}(t, x(¥cdot))¥leq¥omega(t, V(t, x(¥cdot)))$ with $h^{¥prime}=¥omega(t, h)$ having an asym-ptotically stable zero solution. We have been unsuccessful in the search for $¥omega$ innumerical examples such as Example 1.
As our Theorem 1 concerns the behavior of bounded solutions we give condi-tions yielding bounded solutions.
Corollary. Let the conditions of Theorem 1 hold. If $ W(|x|)¥rightarrow¥infty$ as $|x|¥rightarrow¥infty$
and if $Z(t, x(¥cdot))$ is bounded from below, then $aff$ solutions are bounded. If. $Z(t, x(¥cdot))$
$¥geq 0$ and if $V(t_{0}, ¥phi)¥rightarrow 0$ as $||¥phi||¥rightarrow 0$ for each $t_{0}¥geq 0$, then $x=0$ is asymptotically stable.
Proof. If $Z(t, x(¥cdot))¥geq-M$ for some $M>0$ and if $W$ is unbounded, then$W(|x(t)|)-M¥leq V(t, x(¥cdot))¥leq V(t_{0}, ¥phi)$ yields every solution $x(t_{0}, ¥phi)$ bounded.
Suppose $Z(t, x(¥cdot))¥geq 0$ and $V(t_{0}, ¥phi)¥rightarrow 0$ as $||¥phi||¥rightarrow 0$ . Let $¥epsilon>0$ and $t_{0}¥geq 0$ be given.Then there exists $¥delta>0$ such that $||¥phi||<¥delta$ implies $V(t_{0}, ¥phi)<W(¥epsilon)$ . Thus, if $x(t)=$
$x(t, ¥phi)$ , then $t¥geq t_{0}$ implies $W(|x(t)|)¥leq V(t, x(¥cdot))¥leq V(t_{0}, ¥phi)<W(¥epsilon)$ or $|x(t)|<¥epsilon$ . Thisyields Liapunov stability. By Theorem 1 these bounded solutions tend to zero,yielding asymptotic stability.
Our next result asks $r(t)$ bounded and, hence, is better presented in a more gen-eral form.
For a given $h>0$, $C$ denotes the space of continuous functions mapping [?h,$ $0]into $R^{n}$ . If $¥phi¥in C$, then $||¥phi||=¥sup_{-h¥leq¥theta¥leq 0}|¥phi(¥theta)|$ . $C_{¥mathrm{j}1f}$ denotes the set of $¥phi¥in C$ with
Stability Theory for Delay Equations 71
$||¥phi||¥leq M$. If $x$ : [ $-h$, $A)¥rightarrow R^{n}$ for $A>0$, and if $t$ $¥in[0,$ $A$), then $x_{t}$ is the restriction of$x(u)$ to the interval $[t-h, t]$ shifted back to the domain [?h,$ $0].
Consider the system
(5) $x^{¥prime}(t)=G(t, x_{t})$
where $G$ is continuous, $G:[0,$ $¥infty$ ) $¥times C_{M}¥rightarrow R^{n}$ , and $G$ takes bounded sets into boundedsets.
Theorem 2. Let $V$ and $Z:[0,$ $¥infty$ ) $¥chi C_{M}¥rightarrow[0,$ $¥infty$ ) be continuous and let $V$ be locallyLipschitz in the second argument. Suppose that $Z(t_{2}, ¥phi)-Z(t_{1}, ¥phi)¥leq K(t_{2}-t_{1})$ for some$K>0$, all and $t_{2}$ satisfving $ 0<t_{1}<t_{2}<¥infty$ , and $aff$ $¥phi¥in C_{M}$ . If $ W(|¥phi(0)|)+Z(t, ¥phi)¥leq$
$V(t, ¥phi)¥leq W_{1}(|¥phi(0)|)+Z(t, ¥phi)$ , $Z(t, ¥phi)¥leq W_{2}(||¥phi||)$ , and $V_{(4)}^{J}(t, x_{t})¥leq-W_{4}(|x(t)|)$ , then $x=$
$0$ is uniformly asymptotically $stabfe$ ]$¥dot{o}r$ $(5)$ .
Proof. We first show uniform stability. Let $¥overline{¥epsilon}>0$ be given. We will find $¥overline{¥delta}>0$
such that [ $t_{0}¥geq 0$, $||¥phi||<¥delta$, and $t¥geq t_{0}$] imply $|x(t, ¥phi)|<¥overline{¥epsilon}$. There exists $¥delta>0$ such that$W_{1}(¥delta)+W_{2}(¥delta)<W(¥epsilon)$ . Thus, if $||¥phi||<¥delta$ and $x(t)=x(t, ¥phi)$ , then $ W(|x(t)|)¥leq V(t, x_{t})¥leq$
$V(t_{0}, ¥phi)¥leq W_{1}(|¥phi(0)|)+W_{2}(||¥phi||)<W(¥epsilon)$ so that $|x(t)|<¥epsilon$ if $t¥geq t_{0}$ , yielding uniformstability.
To complete the proof we must find $¥eta>0$ such that for any $¥epsilon>0$ there exists $T$
for which [ $t_{0}¥geq 0$ , $||¥phi||<¥eta$ , and $t¥geq t_{0}+T$] imply $|x(t, ¥phi)|<¥epsilon$ . Pick the $¥overline{¥delta}$ of uniformstability when $¥overline{¥epsilon}=M$. Select $¥eta=¥overline{¥delta}$ . Now, let $¥epsilon>0$ be given and let $t_{0}¥geq 0$ , $||¥phi||<¥eta$,
and let $x(t)=x(t, ¥phi)$ .For the given $¥epsilon>0$ , find the $¥delta$ of uniform stability as in the above proof. By
that proof we see that on each interval of length $h$ either $|x(t)|¥geq¥delta$ for some $t$ in theinterval, or $|x(t)|<¥epsilon$ for all subsequent $t$ . Thus, let $t_{0}¥geq 0$ be arbitrary and supposethere is a sequence $¥{t_{n}¥}$ with $ t_{0}¥leq t_{1}¥leq t_{0}+h¥leq t_{2}¥leq t_{0}+2h¥leq t_{8}¥leq$ ? $¥leq t_{0}+(n-1)h¥leq t_{n}¥leq$
$t_{n}+nh$ ? with $|x(t_{i})|¥geq¥delta$. We will show that $n$ may not exceed a fixed integer. Asin the proof of Theorem 1 we use the right-Lipschitz condition on $Z$ to find $¥gamma>0$
and $P>0$ with $P¥leq h$ and $|x(t)|¥geq¥gamma$ if $t_{i}-P¥leq t¥leq t_{i}$ . We select only the $t_{i}$ with evenindices so that the intervals over which we now integrate do not overlap. For $t¥geq t_{2n}$
we have $V(t, x_{t})¥leq V(t_{0}, ¥phi)-¥sum_{i=1}^{n}¥int_{t_{2i}-P}^{t_{2i}}W_{4}(¥gamma)dt¥leq W_{1}(¥eta)+W_{2}(¥eta)-nPW_{4}(¥gamma)$. Choose $N$
$>[W_{1}(¥eta)+W_{2}(¥eta)]/PW_{4}(¥gamma)$ and select $T=2Nh$ . Then for $n>N$ we would have $V<0$ .
Thus, $t_{2N}$ does not exist and so $|x(t)|<¥epsilon$ if $t¥geq t_{0}+2Nh$ . This yields uniform asym-
ptotic stability.An example of Theorem 2 may be obtained by letting $r(t)¥leq h$ in Example 1.
§3. Limit sets
One usually asks that $V(t, x(¥cdot))¥leq W_{8}(|||x(¥cdot)|||)$ where $|||¥cdot|||$ is some suitablenorm ($¥sup$ , $L^{2}$ , or a combination). One then tries to drive $x(t)$ to zero by driving
72 T. A. BURTON
$W_{8}(|||x(¥cdot)|||)$ to zero. However, when $ r(t)¥rightarrow¥infty$ it usually appears that $W_{3}$ is increas-ing without bound, even when $x$ is bounded. To clarify this, look at $V$ in Example1. Even when $x(t)$ is bounded, if $r(t)$ is unbounded one intuitively feels that
$¥int_{t-r(t)}^{t}x^{2}(s)ds$ may be unbounded. When properly viewed, it is easy to see that this
is wrong. In fact, such an apparently unbounded functional is actually boundedand can be used to locate limit sets even when $V_{(1)}^{¥prime}(t, x(¥cdot))$ is merely non-positive.
Definition. A functional $Z(t, x(¥cdot))¥geq 0$ expands relative to a closed set $E¥subset R^{n}$
if for each $¥epsilon>0$ and each $M>0$, if $x:E_{0}¥cup[0,$ $¥infty$ ) $¥rightarrow R^{n}$ and if there exists $T>0$ with$¥epsilon¥leq d(x(t), E)¥leq M$ for $t¥geq T$, then $ Z(t, x(¥cdot))¥rightarrow¥infty$ as $ t¥rightarrow¥infty$ .
Here, $d(x(t), E)$ is the distance from $x(t)$ to $E$.
Example 2. The functional $x^{2}(t)+¥int_{t-r(t)}^{t}x^{2}(s)ds$, where $ r(t)¥rightarrow¥infty$ as $ t¥rightarrow¥infty$ , ex-
pands relative to {0}.
Theorem 3. Let $V$ be a functional satisfying the conditions with (2), let$V_{(1)}^{¥prime}(t, x(¥cdot))¥leq 0$, and suppose that $V$ expands relative to a closed set E. If $x(t)$ is $a$
solution of {1) on [ $t_{0}$ , $¥infty)$ with $d(x(t), E)$ bounded, then there is a sequence $¥{t_{n}¥}$ tendingto infinity with $x(i_{n}^{¥}})¥rightarrow E$.
Proof. If the theorem is false, then there is a solution $x(t)$ on [ $t_{0}$ , $¥infty)$ and con-stants $M$ and $¥epsilon$ with $0<¥epsilon<d(x(t), E)¥leq M$ for all large $t$ . Thus, $ V(t, x(¥cdot))¥rightarrow¥infty$ as$ t¥rightarrow¥infty$ ; however, $V_{(1)}^{¥prime}(t, x(¥cdot))¥leq 0$ and so $V$ is bounded. This contradiction completesthe proof.
Remark 2. PropertiesA) $V(t, x(¥cdot))¥geq 0$ and $V_{(1)}^{¥prime}(t, x(¥cdot))¥leq-W_{4}(|x(t)|)$ andB) $V_{(1)}^{¥prime}(t, x(¥cdot))¥leq 0$ , $V(t, x(¥cdot))¥geq 0$ , and $V$ expands relative to {0}
are related. Both yield the same conclusion; that is, if A) or B) hold and if $x(t)$ isa bounded solution of (1), then there is a sequence $¥{t_{n}¥}$ with $x(t_{n})¥rightarrow 0$ . Without fur-ther assumptions one can show that A) implies nothing more. To see this, construct
a differentiable function $g:[0, ¥infty)¥rightarrow(0,1]$ with $g(n)=1$ and $¥int_{0}^{¥infty}g(s)ds<¥infty$ . Consider
the differential equation $x^{¥prime}(t)=(g^{¥prime}(t)/g(t))x(t)$ having solution $g(t)$ . Consider a Li-apunov function of the form $V(t, x)=a(t)x^{2}$ and construct $a(t)$ so that $V^{¥prime}(t, x)=-x^{2}$.A simple computation will yield an $a(t)¥geq 1$ . It would be interesting to determine ifA) and B) are, in fact, equivalent (or equivalent with other mild conditions). Forexample, if A) holds and if $f$ is bounded for $x$ bounded, then one may conclude thatbounded solutions of (1) tend to zero. One would like to conclude the same for B)when $f$ is bounded since it is often difficult to make $V^{¥prime}$ negative definite.
Stability Theory for Delay Equations 73
§4. A special case
One may observe from the general theory developed by Krasovskii [4; pp. 161-175] that for $V(t, x(¥cdot))¥leq W_{1}(|x(t)|)+Z(t, x(¥cdot))$ there is a special relationship between$Z(t, x(¥cdot))$ and $V_{(1)}^{¥prime}(t, x(¥cdot))$ . When such a relation holds, then a surprisingly simpleresult may be obtained.
Theorem 4. Let $V$ be a functional satisfying (4) and the conditions with (2).Suppose there is a functional $Z(t, x(¥cdot))$ with $W(|x(t)|)¥leq V(t, x(¥cdot))¥leq W_{1}(|x(t)|)+$
$Z(t, x(¥cdot))$ and $Z(t, x(¥cdot))¥leq¥int_{t-r(t)}^{t}p(x(s))ds$ where $p:R^{n}¥rightarrow[0,$ $¥infty$ ) is continuous and $p(x)$
$¥leq KW_{4}(|x|)$ for some $K>0$. If t?r(t)-∞ as $ t¥rightarrow¥infty$ , then each bounded solution of(1) tends to zero as $ t¥rightarrow¥infty$ .
Proof. If the theorem is false then there is a bounded solution $x(t)$ of (1) whichdoes not tend to zero. As $V_{(1)}^{J}(t, x(¥cdot))¥leq-W_{4}(|x(t)|)$ we have $0¥leq V(t, x(¥cdot))¥leq V(t_{0}, ¥phi)$
$¥int_{t-r(t)}^{t}p((x))ds¥leq K¥int_{t-r(t)}^{t}W_{4}(|x(s)|)ds¥rightarrow 0$ as $ t¥rightarrow¥infty$ by the Cauchy criteria since $t-$
$ r(t)¥rightarrow¥infty$ as $ t¥rightarrow¥infty$ . Now there is a sequence $¥{t_{n}¥}$ tending to infinity with $x(t_{n})¥rightarrow 0$.Thus, for $t¥geq t_{n}$ we have $W(|x(t)|)¥leq V(t, x(¥cdot))¥leq W_{1}(|x(t_{n})|)+Z(t_{n}, x(¥cdot))¥rightarrow 0$ as $ n¥rightarrow¥infty$ .This completes the proof.
Example 3. Consider the scalar equation of Example 1 again
$x^{¥prime}(t)=-[a+(t¥sin t)^{2}]x(t)+bx(t-r(t))$
with $a>0$ and constant, $b$ constant, and $ r^{¥prime}(t)¥leq¥alpha$ for $0<¥alpha<1$ . To apply Theorem4 it is not necessary to ask that $r^{¥prime}(t)$ be bounded from below as we no longer need aLipschitz condition on $Z$. As $r^{¥prime}(t)¥leq¥alpha<1$ , t?r(t)→∞ as $ t¥rightarrow¥infty$ . The remainderof Example 1 is as before. Then
and $V^{¥prime}¥leq-¥mu x^{2}(t)$ . The conditions of Theorem 4 are satisfied and as all solutionsare bounded, we conclude that all solutions tend to zero.
The result applies equally well to $¥mathrm{n}$-dimensional systems
$x^{¥prime}(t)=Ax(t)+g(t, x(t-r(t)))$
where $A$ is a negative definite constant matrix and $g$. is bounded near $x=0$ by a li-
with $A^{T}B+BA=-L$ $B=B^{T}$ positive definite, and $C$ is chosen to make the derivativenegative near zero. This yields $V^{¥prime}¥leq-k|x(t)|^{2}$ for $k>0$ so long as $|x|$ is small. Theconclusion follows as in Example 3. Our theorems are stated globally for conveni-ence, but they apply equally well for local results.
§5. An example
Let $x$ and $y$ be scalars and consider the system
where $0¥leq r(t)¥leq¥alpha(t)¥leq M$ with $¥alpha^{¥prime}(t)¥leq¥alpha<1$ , $¥alpha(t)¥geq m>0$, $¥psi(t, x, y)/¥alpha(t)¥geq b>0$ , and$|g^{¥prime}(x)|¥leq L$.
This example has been discussed extensively in the litrature and is treated inseveral monographs ([4; p. 173], [5; p. 208], and [3; p. 134]), usually under severerestrictions.
Here, we develop more fully some of the ideas in the foregoing sections to provethe following result.
Theorem 5. Suppose that the conditions with (6) hold and that $b^{2}(1-¥alpha)>L^{2}$ .
If $G(x)=¥int_{0}^{x}g(s)ds$, then $ G(x(t))¥rightarrow$constant and $y(t)¥rightarrow 0$ as $ t¥rightarrow¥infty$ .
Note that $V^{¥prime}¥leq 0$ and $V¥geq y^{2}$ yields $y$ bounded. As $x^{¥prime}(t)=y$ , $x(t)$ is bounded bya linear function and, hence, each solution can be continued for all future time. Wecan then conclude from $0¥leq V$ and $V^{¥prime}¥leq-¥mu my^{2}(t)$ that if $(x(t), y(t))$ is any solution,
then $¥int_{t¥mathrm{o}}^{¥infty}y^{2}(t)dt<¥infty$ .
We apply this result to the integral in $V$ and obtain (for an arbitrary solution$(x(t), y(t)))$
$¥leq M¥int_{t-a(t)}^{t}y^{2}(u)du¥rightarrow 0$ as $ t¥rightarrow¥infty$
by the Cauchy criterion. As $V^{¥prime}¥leq 0$ it then follows that $2¥mathrm{G}(¥mathrm{x}(¥mathrm{t}))+¥mathrm{y}^{2}(¥mathrm{t})¥rightarrow ¥mathrm{c}¥mathrm{o}¥mathrm{n}¥mathrm{s}¥mathrm{t}¥mathrm{a}¥mathrm{n}¥mathrm{t}$
as $ t¥rightarrow¥infty$ .If the theorem is false, then there is a solution $(x(t), y(t))$ such that $y(t)¥neq 0$ or
$G(x(t))$ has no limit.If $y(t)¥neq 0$ , then there exists $¥gamma>0$ and a sequence $¥{t_{n}¥}¥rightarrow¥infty$ with $ y^{2}(t_{n})¥geq¥gamma$ . As
$¥int_{t¥mathrm{o}}^{¥infty}y^{2}(t)dt<¥infty$ , there is a sequence $¥{T_{n}¥}¥rightarrow¥infty$ with $y(T_{n})¥rightarrow 0$ . But $ 2G(x(t)+y^{2}(t)¥rightarrow$
constant and so there is an $¥epsilon>0$ with $|G(x(t_{n}))-G(x(T_{n}))|¥geq¥epsilon$ for all large $n$ , say$n¥geq 1$ .
Now $|x^{¥prime}(t)|=y(t)|$ is bounded and so there exists $T>0$ with $|G(x(t))-$
$G(x(t_{n}))|¥leq¥epsilon/2$ for $t_{n}¥leq t¥leq t_{n}+T$. Thus, as $2¥mathrm{G}(¥mathrm{x}(¥mathrm{t}))+¥mathrm{y}^{2}(¥mathrm{t})¥rightarrow ¥mathrm{c}¥mathrm{o}¥mathrm{n}¥mathrm{s}¥mathrm{t}¥mathrm{a}¥mathrm{n}¥mathrm{t}$ and $y^{2}(t_{n})$
$¥geq¥gamma$, there exists $K>0$ with $y^{2}(t)¥geq¥gamma/K$ for $t_{n}¥leq t¥leq t_{n}+T$. But this contradicts
$¥int_{t¥mathrm{o}}^{¥infty}y^{2}(t)dt<¥infty$ . This shows that the assumption $y(t)¥neq 0$ is false. A review of the
above two paragraphs will show that the assumption of $G(x(t))$ having no limit leadsto the same contradiction. This completes the proof.
Remark 3. A commonly used technique is to utilize the fact that $x^{¥prime}=y$ isbounded to conclude that $|x(t_{1})-x(t_{2})|¥geq¥epsilon>0$ implies $|t_{1}-t_{2}|¥geq¥delta$ for some $¥delta>0$ .The above proof goes a step farther. When $2G(x(t))+y^{2}(t)$ approaches a constant,then $|y(t_{1})-y(t_{2})|¥geq¥epsilon>0$ implies $|t_{1}-t_{2}|¥geq¥delta$ for some $¥delta>0$ and this holds evenwhen $y^{¥prime}$ is unbounded.
76 T. A. BURTON
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[1] Driver, Rodney D., Existence and stability of solutions of a delay-differential system,
Arch. Rational Mech. Anal., 10 (1962), 401-426.[2] EFsgol’ts, L. E., Introduction to the theory of differential equations with deviating argu-
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Vol. 3, 2nd ed.), Springer-Verlag, New York, 1977.[4] Krasovskii, N. N., Stability of motion, Stanford Univ. Press, Stanford, 1963.[5] Yoshizawa, T., Stability of Liapunov’s second method, Math. Soc. Japan, 1966.
nuna adreso:Department of MathematicsSouthern Illinois UniversityCarbondale, 111. 62901U.S.A.
(Ricevita la 4-an de januaro, 1978)(Reviziita la 3-an de marto, 1978)
