Funkcialaj Ekvacioj, 16 (1973), 79-88

Behavioral Relationships between Ordinary and

Functional Differential Equations

By David Lowell LOVELADY

(Florida State University)

I. Introduction.

Let $Y$ be a Banach space with norm $|$ $|$ , let $R$ be the set of all realnumbers, and let $R^{+}$ be the set of all nonnegative real numbers. Let $F[Y]$ bethe Frechet space of all continuous functions from $R^{+}$ to $Y$, with the topologyof uniform convergence on compact intervals, and define $F[R]$ analogously. Wepropose to study the boundedness and convergence relationships between theordinary differential equation

(1) $v^{¥prime}(t)=A(t, v(t));v(0)=z$

and the functional differential equation

(2) $u^{¥prime}(t)=A(t, u(t))+B[t, u];u(0)=z$ ,

where $A$ is a continuous function from $R^{+}¥times Y$ to $Y$ and $B$ is a continuous nonan-ticipative function from $R^{+}¥times F[Y]$ to $Y$.

The additional hypotheses which we shall place on $A$ and $B$ will be suchthat global existence and uniqueness for (1) follows from [19] and such thatglobal existence and uniqueness for (2) follows from [16]. Herein we shallobtain conditions to ensure that the solution $u$ of (2) is bounded relative to aprescribed function $¥phi$ if and only if the solution $v$ of (1) is bounded relative tothe same function $¥phi$ . We shall obtain an additional condition which ensuresthat $u$ and $v$ are asymptotically equivalent relative to $¥phi$ .

The problem of determining when solutions are in a particular space ofrelatively bounded functions appears to have been initiated by A. Bielecki [1],and has been studied by C. Corduneanu [4], T. G. Hallam [8], P. Talpalaru [22],

and this author [14]. The problem of asymptotic behavior for systems relatedto ours has also been studied by several authors (see J. K. Hale [5], Hale andPerello [6], Hallam, G. Ladas, and V. Lakshmikantham [9], Y. Hino [10], [11],

J. Kurzweil [12], T. Yoshizawa [23], Hallam [8], and Talpalaru [22] $)$ . Theprimary tool of the present work will be the circle of ideas involving theHolder inequality developed by R. Conti [2] $¥mathrm{a}¥dot{¥mathrm{n}}¥mathrm{d}$ used by V. A. Staikos [21],Talpalaru [22], and the present author [18]. The primary advantage of the

80 D. L. LoVELADY

present work, in comparison with previous works, is that we do not requireequation (1) to be linear.

II. Behavioral Relationships.Let $Y$, $R$, $R^{+}$ , $F[Y]$ , and $F[R]$ be as in our introduction. Let $¥Delta$ be that

function from $F[Y]$ to $F[R]$ having the property that if $(t,f)$ is in $R^{+}¥times F[Y]$

then $¥Delta[f](t)$ is the least number $c$ such that $|f(s)|¥leqq c$ whenever $s$ is in $[0, t]$ .Let $¥phi$ be a nondecreasing function from $R^{+}$ to $R^{+}$ with $¥phi(0)$ positive. Weshall say that a member $f$ of $F[Y]$ is $¥phi$-bounded only in case there is a positivenumber $b$ such that $|f(t)|¥leqq b¥phi(t)$ whenever $t$ is in $R^{+}$ . Define $¥phi$-boundednessfor members of $F[R]$ analogously. Let $A$ and $B$ be continuous functions as inthe introduction, let a and $¥beta$ be members of $F[R]$ , and require that $¥beta$ haveonly nonnegative values (note that a is permitted to have negative values).Let $p$ and $q$ be members of $(1, ¥infty)$ , and suppose that $p+q=pq$ . We will findconditions (C1), (C2), $(¥mathrm{C}3)$ , (C1), $(¥mathrm{C}5)$ , and $(¥mathrm{C}6)$ to be useful.

$(¥mathrm{C} 1)$ : If $(t, x, y)$ is in $R^{+}¥times Y¥times Y$ and $c$ is a positive number then

$|[x-y]-c[A(t, x)-A(t, y)]|¥geqq[l-ca(t)]|x-y|$ .

$(¥mathrm{C} 2)$ : If $(¥mathrm{t} , g)$ is in $R^{+}¥times F[Y]¥times F[Y]$ then

$|B[t,f]-B[t, g]|¥leqq¥beta(t)¥Delta[f-g](t)$ .

$(¥mathrm{C}3)$ : There is a positive number $K$ such that

$¥mathrm{o}¥int^{t}$($¥exp[_{s}¥int^{t}$a(r)$dr]$) it

whenever $t$ is in $R^{+}$ .

$(¥mathrm{C} 4):0¥int^{¥infty}¥beta(s)^{p}ds$ is finite.

$(¥mathrm{C} 5)$ : The member $¥omega$ of $F[R]$ described by

$¥omega(t)=(_{0}¥int^{t}|B[s, 0]|^{p}ds)^{1/p}$

is $¥phi$-bounded.

$(¥mathrm{C} 6)$ : If $¥omega$ is as in $(¥mathrm{C}5)$ then $¥lim_{i¥rightarrow¥infty}¥frac{¥omega(t)}{¥phi(t)}=0$ .

It follows from [19] that if (C1) holds and $z$ is in $Y$ then there is exactlyone member $v$ of $F[Y]$ such that (1) holds whenever $t$ is a positive number,and it follows from [16] that if both $(¥mathrm{C}1)$ and (C2) hold and $z$ is in $Y$ thenthere is exactly one member $u$ of $F[Y]$ such that (2) holds whenever $t$ is apositive number.

Behavioral Relationships between Ordii ary and function al Differential Equations 81

Theorem 1. Suppose that each $(¥mathrm{C}1)$ , $(¥mathrm{C}2)$ , $(¥mathrm{C}3)$ , $(¥mathrm{C}4)$ and $(¥mathrm{C}5)$ is true.

Let $z$ be in 7, let $v$ in $F[Y]$ solve (1), and $fet$ $u$ in $F[Y]$ solve (2). Then$u$ is $¥phi-$bounded if and only if $v$ is $¥phi$-bounded.

Theorem 2. Suppose that each of $(¥mathrm{C}1)$ , $(¥mathrm{C}2)$ , $(¥mathrm{C}3)$ , $(¥mathrm{C}4)$ and $(¥mathrm{C}6)$ istrue. Let $z$ , $v$ , and $u$ be as in Theorem 1, and suppose that $v$ is $¥phi-$bounded.Then $u$ and $v$ are asymptotically equivalent relative to $¥phi$ in the sense thatin $¥rightarrow¥infty|u(t)-v(t)|/¥phi(t)=0$ .

If $m_{¥_}$ is that function from $Y¥times Y$ to $R$ given by

$m_{¥_}[x, y]=¥lim_{¥delta¥rightarrow 0-}¥frac{1}{¥delta}(|x+¥delta y|-|x|)$,

then condition $(¥mathrm{C}1)$ is equivalent to requiring that$m_{¥_}[x-y, A(t, x)-A(t, y)]¥leqq ¥mathrm{a}(t)|x-y|$

whenever $(t, x, y)$ is in $R^{+}¥times Y¥times Y$ (compare [3, p. 3]). Furthermore, if $f$ is in$F[Y]$ , if $c$ is a positive number, if $f_{-}^{¥prime}(c)$ (the left derivative of $f$ at $c$) exists,and if $P$ in $F[R]$ is given by $P(t)=|f(t)|$ , then $P_{-}^{¥prime}(c)$ exists, and $P_{¥_}^{¥prime}(c)=m_{¥_}$

$[f(c), f_{¥_}^{¥prime}(c)]$ (compare [3, p. 3]). Note also that if $(x, y, z)$ is in $Y¥times Y¥times Y$

then $m_{¥_}[x, y+z]¥leqq m_{¥_}[x, y]+|z|$ (see [15, Lemma 6]).Note that conditions $(¥mathrm{C}5)$ and $(¥mathrm{C}6)$ are both gratis if $B$ [ , 0] is the zero

member of $F[Y]$ , and that conditions $(¥mathrm{C}1)$ , (C2), $(¥mathrm{C}3)$ , and $(¥mathrm{C}4)$ are indepen-dent of $¥phi$ . Thus Theorem 1 tells us that if $(¥mathrm{C}1)$ , (C2), $(¥mathrm{C}3)$ , and $(¥mathrm{C}4)$ aretrue and $B[ , 0]=0$ then $u$ is $(¥Delta[v]+1)$-bounded, and, under these same circum-

stances, Theorem 2 tells us that $¥mathrm{l}¥mathrm{i}¥mathrm{m}t¥rightarrow¥infty|u(t)-v(t)|/(¥Delta[v](t)+1)=0$. Specializingeven more we see that if $v$ is bounded then so also is $u$ and $¥mathrm{l}¥mathrm{i}¥mathrm{m}t¥rightarrow¥infty|u(t)-$

$v(t)|=0$ ; in particular, if $x_{0}$ is in $Y$ and $¥mathrm{l}¥mathrm{i}¥mathrm{m}t¥rightarrow¥infty|v(t)-x_{0}|=0$ , then $¥lim t¥rightarrow¥infty|u(t)-$

$x_{0}|=0$ ; all of these statements being subject to the assumptions that $(¥mathrm{C}1)$ ,(C2), $(¥mathrm{C}3)$ , and $(¥mathrm{C}4)$ are true, and $B[, 0]=0$ . Note that in [17] conditionshave been given which ensure that there is a member $x_{0}$ of $Y$ such that $¥lim_{t¥rightarrow¥infty}$

$|v(t)-x_{0}|=0$ .

Before proving Theorems 1 and 2, we shall prove two lemmas which willsimplify the proofs of the theorems. In the proofs of the lemmas we shall usewithout further comment the Holder inequality, for which we refer the readerto [20, Theorem 3. 5, p. 62].

Lemma 1. Suppose that $(¥mathrm{C}3)$ and $(¥mathrm{C}4)$ are true. Then there is a number$b_{1}$ having the property that if $c$ is in $R^{+}$ and $t$ is in $¥mathrm{c}$ , $¥infty$ ) then

$C¥int^{t}¥beta(s)¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds¥leqq b_{1}$ .

Furthermore,

32 D. L. LoVELADY

$¥lim t¥rightarrow¥infty ¥mathrm{o}¥int^{t}¥beta(s)¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds=0$ .

Proof. Let $b_{1}=K[_{0}¥int^{¥infty}¥beta(s)^{p}ds]^{1/p}$ Now, if $c$ is in $R^{+}$ and $t$ is in $[c,$ $¥infty)$ ,

then

$¥mathrm{C}¥int^{t}¥beta(s)^{p}¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds$

$¥leqq[_{c}¥int^{t}¥beta(s)^{p}ds]^{1/p}[_{c}¥int^{t}(¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr])^{q}ds]^{1/q}$

$¥leqq[_{0}¥int^{¥infty}¥beta(s)^{p}ds]^{1/p}K=b_{1}$ ,

and the first part of the proof is complete.

Now recall that W. A. Coppel [3, Lemma 1, p. 68] has shown that $(¥mathrm{C}3)$

implies

$¥lim_{t¥rightarrow¥infty}¥exp[_{0}¥int^{t}q¥mathrm{a}(r)dr]=0$

and hence

$¥mathrm{l}¥mathrm{i}¥mathrm{m}¥mathrm{t}¥rightarrow¥infty$ $¥exp[_{0}¥int^{t}¥mathrm{a}(r)dr]=0$ .

Let $¥epsilon$ be a positive number. Let $c_{1}$ be a positive number such that if $t$ is in[$c_{1}$ , $¥infty)$ then

$[_{t}¥int^{¥infty}¥beta(s)^{p}ds]^{1/p}¥leqq¥frac{¥epsilon}{2K}$.

Let

$d=_{0}¥int^{c_{1}}¥beta(¥mathrm{s})¥exp[-_{0}¥int^{s}¥mathrm{a}(r)dr]ds$ ,

and let $c_{2}$ a positive number such that $c_{2}¥geqq c_{1}$ and such that

$¥exp[_{0}¥int^{t}¥mathrm{a}(r)dr]<¥frac{¥epsilon}{2(d+1)}$

whenever $t$ is in [$c_{2}$ , $¥infty)$ . Suppose that $t$ is in [$c_{2}$ , $¥infty)$ . Now

$¥mathrm{o}¥int^{t}¥beta(s)¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds$

$=0¥int^{c_{1}}¥beta(s)¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds+_{c_{1}}¥int^{t}¥beta(s)¥exp[_{s}¥oint^{t}¥mathrm{a}(r)dr]ds$

Behavioral Relationships between Ordinary and Functional Differential Equations 83

$¥leqq(d)¥exp[_{0}¥int^{t}¥mathrm{a}(r)dr]$

$+[_{c_{1}}¥int^{t}¥beta(s)^{p}ds]^{1/p}[_{c_{1}}¥int^{t}(¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr])^{q}ds]^{1/q}$

$<¥frac{¥epsilon}{2}+[_{c¥iota}¥int^{¥infty}¥beta(s)^{p}ds]^{1/p}K<¥frac{¥epsilon}{2}+¥frac{¥epsilon}{2}=¥epsilon$ .

This completes the proof of the lemma.Lemma 2. If $(¥mathrm{C}5)$ is true then there is a number $b_{2}$ such that if $c$ is in

$R^{+}$ and $t$ is in [$c$ , $¥infty)$ then

$c¥int^{t}|B[s, ¥mathrm{O}]|¥exp[_{S}¥int^{t}¥mathrm{a}(r)dr]ds¥leqq b_{2}¥phi(t)$ .

Furthermore, if $(¥mathrm{C}6)$ is true, we have

$¥lim_{t¥rightarrow¥infty}(_{0}¥int^{t}|B[s, ¥mathrm{O}]|¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds)/¥phi(t)=0$ .

Proof. Suppose that $(¥mathrm{C}5)$ is true, and let $¥omega$ be as in $(¥mathrm{C} 5)$ . Choose $b_{2}$

such that $¥omega(t)¥leqq(b_{2}/K)¥phi(t)$ whenever $t$ is in $R^{+}$ . Now, if $c$ is in $R^{+}$ and $t$ isin [$c$ , $¥infty)$ , then

$c¥int^{t}|B[s, ¥mathrm{O}]|¥exp[_{S}¥int^{t}¥mathrm{a}(r)dr]ds$

$¥leqq_{0}¥int^{t}|B[s, ¥mathrm{O}]|¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds$

$¥leqq¥omega(t)[_{0}¥int^{t}(¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr])^{q}ds]^{1/Q}$

$¥leqq K¥omega(t)¥leqq b_{2}¥phi(t)$ .

It is also clear that if $t$ is in $R^{+}$ then

$(_{0}¥int^{t}|B[s, ¥mathrm{O}]|¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds)/¥phi(t)¥leqq¥frac{K¥omega(t)}{¥phi(t)}$

so the proof is complete.Before proving Theorem 1, let us recall that Corduneanu [4, Lemme, p. 121]

has shown that if $f$ is in $F[Y]$ then $f$ is $¥phi$-bounded if and only if $¥Delta[f]$ is $¥phi-$

bounded; furthermore, if $b$ is a number and $|f(t)|¥leqq b¥phi(t)$ whenever $t$ is in $R^{+}$

then $¥Delta[f](t)¥leqq b¥phi(t)$ whenever $t$ is in $R^{+}$ .Proof of Theorem 1. Suppose that $u$ is $¥phi$-bounded. Let $b_{1}$ and $b_{2}$ be as

in Lemmas 1 and 2, let $b_{3}$ be a number such that $|u(t)|¥leqq b_{3}¥phi(t)$ whenever $t$ isin $R^{+}$ . Let $Q$ in $F[R]$ be given by $Q(t)=|u(t)-v(t)|$ . Now $Q(0)=0$, and, if$t$ is a positive number,

84 D.L.LoVELADY

$Q_{-}^{¥prime}(t)=m_{¥_}[u(t)-v(t), u^{¥prime}(t)-v^{¥prime}(t)]$

$=m_{¥_}[u(t)-v(t), A(t, u(t))-A(t, v(t))+B[t, u]]$

$¥leqq ¥mathrm{a}(t)Q(t)+|B[t, u]|$

$¥leqq ¥mathrm{a}(t)Q(t)+|B[t, u]-B[t, 0]|+|B[t, 0]|$

$¥leqq ¥mathrm{a}(t)Q(t)+¥beta(t)¥Delta[u](t)+|B[t, 0]|$

$¥leqq ¥mathrm{a}(t)Q(t)+b_{3}¥beta(t)¥phi(t)+|B[t, 0]|$ .

Hence [13, Theorem 1. 4. 1, p. 15],

$Q(t)¥leqq_{0}¥int^{t}b_{3}¥beta(s)¥phi(s)¥exp[_{s}¥int^{t}¥mathrm{a}(r)drds]+_{0}¥int^{t}|B[s, 0]|¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds$

$¥leqq b_{3}¥phi(t)¥mathrm{o}¥int^{t}¥beta(s)¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds+b_{2}¥phi(t)$

$¥leqq b_{3}b_{1}¥phi(t)+b_{2}¥phi(t)=(b_{3}b_{1}+b_{2})¥phi(t)$

whenever $t$ is in $R^{+}$ . Thus, if $t$ is in $R^{+}$ ,

$|v(t)|¥leqq|u(t)|+|u(t)-v(t)|¥leqq(b_{3}+b_{1}b_{3}+b_{2})¥phi(t)$,

so $v$ is $¥phi$-bounded, and the proof of the first part of the theorem is complete.Now suppose that $v$ is $¥phi$-bounded. Let $b_{3}$ be a number such that $|v(t)|¥leqq$

$b_{3}¥phi(t)$ whenever $t$ is in $R^{+}$ . Let $b_{1}$ and $b_{2}$ be as in Lemmas 1 and 2. Let $c$

be a positive number such that $(_{c}¥int^{¥infty}¥beta(s)^{p}ds)^{1/p}K¥leqq¥frac{1}{2}$. Let $B(c, ¥phi)$ be the linearspace to which $f$ belongs only in case $f$ is a continuous function from $[c,$ $¥infty)$

to $Y$ and $f$ is $¥phi$-bounded on [$c,$ $¥infty)$ . If $f$ is in $B(c, ¥phi)$ let $N[f]$ be the leastnumber $b$ such that $|f(t)|¥leqq c¥phi(t)$ whenever $t$ is in [$c$ , $¥infty)$ . Now $N$ is a normon $B(c, ¥phi)$ and $B(c, ¥phi)$ is a Banach space with respect to $N$. If $f$ is in $B(c, ¥phi)$

let $¥Lambda[f]$ from [$c,$ $¥infty)$ to $R^{+}$ be given by $¥Lambda[f](t)=¥max$ { $|f(s)|:s$ is in $[c,$ $t]$ },and note again [4, Lemme, p. 121] that $¥Lambda[f](t)¥leqq N[f]¥phi(t)$ whenever $t$ is in[$c$ , $¥infty)$ . If $f$ is in $B(c, ¥phi)$ let $f^{*}$ be that member of $F[Y]$ such that $f^{*}(t)=$

$u(t)+f(c)-u(c)$ if $t$ is in $[0, c]$ and $f^{*}(t)=f(t)$ if $t$ is in [$c$ , $¥infty)$ . Note that if$f$ is in $B(c, ¥phi)$ then $f^{*}$ is $¥phi$-bounded. Let $B^{*}$ be the function from $[c,$ $¥infty)$

$¥times B(c, ¥phi)$ to $Y$ given by $B^{*}[t,f]=B[t,f^{*}]$ . Now if $(t,f, g)$ is in $[c,$ $¥infty)$

$¥times B(c, ¥phi)¥times B(c, ¥phi)$ then $¥Lambda[f-g](t)=¥Delta[f^{*}-g^{*}](t)$ , so $|B^{*}[t,f]-B^{*}[t, g]|¥leqq¥beta(t)$

$¥Lambda[f-g](t)$ . It follows from [19] that if $f$ is in $B(c, ¥phi)$ then there is exactlyone continuously differentiate function $g$ from [$c,$ $¥infty)$ to $Y$ such that $g(c)=u(c)$

and such that $g^{¥prime}(t)=A(t, g(t))+B^{*}[t,f]$ whenever $t$ is in $B(c, ¥phi)$ . If $f$ is in$B(c, ¥phi)$ , let $T[f]$ be the function $g$ mentioned in the previous sentence. Firstwe shall show that $T$ has its range in $B(c, ¥phi)$ and then that $T$ is a (strict)contraction on $¥beta(c, ¥phi)$ .

Let $f$ be in $B(c, ¥phi)$ , and let $b_{4}$ be a number such that $|f^{*}(t)|¥leqq b_{4}¥phi(t)$

whenever $t$ is in $R^{+}$ . It was observed above that

Behavioral Relationships between Ordinary and Functional Differential Equations 85

$¥lim_{t¥rightarrow¥infty}¥exp[_{0}¥int^{i}¥mathrm{a}(r)dr]=0$ ,

and thus

$¥lim_{t¥rightarrow¥infty}¥exp[_{c}¥int^{t}¥mathrm{a}(r)dr]=0$ .

Consequently, since $¥phi$ is nondecreasing, there is a number $b_{5}$ such that

$|u(c)-v(c)|¥exp[_{c}¥int^{t}¥mathrm{a}(r)dr]¥leqq b_{5}¥phi(t)$

whenever $t$ is in [$c,$ $¥infty)$ . Let $g=T[f]$ , and let $Q$ from [$c,$ $¥infty)$ to $R^{+}$ be givenby $Q(t)=|g(t)-v(t)|$ . Now $Q(c)=|u(c)-v(c)|$ , and, if $t$ is in $(c, ¥infty)$ ,

$Q_{-}^{¥prime}(t)=m_{¥_}[g(t)-v(t), g^{¥prime}(t)-v^{¥prime}(t)]$

$=m_{¥_}[g(t)-v(t), A(t, g(t))-A(t, v(t))+B[t, f^{*}]]$

$¥leqq ¥mathrm{a}(t)Q(t)+|B[t,f^{*}]|$

$¥leqq ¥mathrm{a}(t)Q(t)+|B[t,f^{*}]-B[t, 0]|+|B[t, 0]|$

$¥leqq ¥mathrm{a}(t)Q(t)+¥beta(t)¥Delta[f^{*}](t)+|B[t, 0]|$

$¥leqq ¥mathrm{a}(t)Q(t)+b_{4}¥beta(t)¥phi(t)+|B[t, 0]|$ .

Hence [13, Theorem 1. 4. 1, p. 15]

$Q(t)¥leqq|u(c)-v(c)|¥exp[_{c}¥int^{t}¥mathrm{a}(r)dr]$

$+_{c}¥int^{t}b_{4}¥beta(s)¥phi(s)¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds$

$+_{c}¥int^{t}|B[s, ¥mathrm{O}]|¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds$

$¥leqq(b_{5}+b_{2})¥phi(t)+b_{4}¥phi(t)c¥int^{t}¥beta(s)¥exp[_{S}¥int^{t}¥mathrm{a}(r)dr]ds$

$¥leqq(b_{2}+b_{5}+b_{1}b_{4})¥phi(t)$

whenever $t$ is in [$c,$ $¥infty)$ . Thus, if $t$ is in $[c,$ $¥infty)$ .

$|g(t)|¥leqq|g(t)-v(t)|+|v(t)|¥leqq(b_{2}+b_{3}+b_{5}+b_{1}b_{4})¥phi(t)$,

and $g$ is in $B(c, ¥phi)$ .Now suppose that $(/, g)$ is in $B(c, ¥phi)¥times B(c, ¥phi)$ and $x=T[f]$ and $y=T[g]$ .

Let $Q$ from [$c,$ $¥infty)$ to $R^{+}$ be given by $Q(t)=|x(t)-y(t)|$ . Now $Q(0)=0$, and,if $t$ is in $(c, ¥infty)$ ,

$Q_{-}^{¥prime}(t)=m_{¥_}[x(t)-y(t), x^{r}(t)-y^{¥prime}(t)]$

$=m_{¥_}[x(t)-y(t), A(t, x(t))-A(t, y(t))+B^{*}[t,f]-B^{*}[t, g]]$

$=¥mathrm{a}(t)Q(t)+|B^{*}[t,f]-B^{*}[t, g]|$

86 D. L. LoVELADY

$=¥mathrm{a}(t)Q(t)+¥beta(t)¥Lambda[f-g](t)$

$=¥mathrm{a}(t)Q(t)+N[f-g]¥beta(t)¥phi(t)$ .

Hence, if $t$ is in $[c,$ $¥infty)$ ,

$Q(t)¥leqq N[f-g]c¥int^{t}¥beta(s)¥phi(t)¥exp[_{S}¥int^{t}¥mathrm{a}(r)dr]ds$

$¥leqq N[f-g]¥phi(t)c¥int^{t}¥beta(s)¥exp[_{S}¥int^{t}¥mathrm{a}(r)dr]^{-}ds$

$¥leqq N[f-g]¥phi(t)[_{C}¥int^{t}¥beta(s)^{p}ds]^{1/p}[_{¥mathrm{C}}¥int^{t}(¥exp[_{S}¥int^{t}¥mathrm{a}(r)dr])^{q}ds]^{1/q}$

$¥leqq¥frac{1}{2}¥phi(t)N[f-g]$ .

Thus

$N[T[f]-T[g]]=N[x-y]¥leqq¥frac{1}{2}N[f-g]$,

and $T$ is a strict contraction from $B(c, ¥phi)$ to $B(c, ¥phi)$ .

Let $w$ be that member of $B(c, ¥phi)$ such that $w=T[w]$ . Now $w(c)=u(c)$ ,

so there is a $¥phi$-bounded member $¥tilde{u}$ of $F[Y]$ such that $¥tilde{u}(t)=u(t)$ if $t$ is in $[0, c]$

and such that $¥tilde{u}(t)=w(t)$ if $t$ is in [$c$ , $¥infty)$ . It is now easy to see that if $t$ is in$R^{+}$ then

$¥tilde{u}(t)=z+_{0}¥int^{t}A(s,¥tilde{u}(s))ds¥dagger_{0}¥int^{t}B[s,¥tilde{u}]ds$ ,

so (2) holds whenever $t$ is a positive number. By uniqueness this says that$¥tilde{u}=u$ , and hence $u$ is $¥phi$-bounded. This completes the proof of Theorem 1.

Proof of Theorem 2. Since $(¥mathrm{C}6)$ implies $(¥mathrm{C}5)$ , the supposition that $v$ is$¥phi$-bounded, in conjunction with Theorem 1, tells us that $u$ is $¥phi$-bounded. Let$b$ be a number such that $|u(t)|¥leqq b¥phi(t)$ whenever $t$ is in $R^{+}$ . Let $¥lambda$ and $¥mu$ bemembers of $F[R]$ given by

$¥lambda(t)=_{0}¥int^{t}¥beta(s)¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds$

and

$¥mu(t)=(_{0}¥int^{t}|B[s, ¥mathrm{O}]|¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds)/¥phi(t)$ .

Let $Q$ in $F[R]$ be given by $Q(t)=|u(t)-v(t)|$ . Now $Q(0)=0$ and, if $t$ is apositive number,

$Q_{-}^{¥prime}(t)=m_{¥_}[u(t)-v(t), u^{¥prime}(t)-v^{¥prime}(t)]$

Behavioral Relationships between Ordinary and Functional Differential Equations $8¥mathcal{T}$

$=m_{¥_}[u(t)-v(t), A(t, u(t))-A(f, v(f))+B[t, u]]$

$¥leqq ¥mathrm{a}(t)Q(t)+|B[t, u]|$

$¥leqq ¥mathrm{a}(t)Q(t)+|B[t, u]-B[t,¥mathrm{O}]|+|B[t, 0]|$

$¥leqq¥alpha(t)Q(t)+b¥beta(t)¥phi(t)+|B[t, 0]|$ .

Thus, if $t$ is in $R^{+}$ ,

$|u(t)-v(t)|=Q(t)¥leqq_{0}b¥int^{t}¥beta(s)¥phi(s)¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr¥underline{]}ds$

$+_{0}¥int^{t}|B[s, ¥mathrm{O}]|¥exp[_{s}¥int^{t}¥mathrm{a}(r)dr]ds$

$¥leqq b¥phi(t)¥lambda(t)+¥phi(t)¥mu(t)$ ,

and

$¥frac{|u(t)-v(t)|}{¥phi(t)}¥leqq b¥lambda(t)+¥mu(t)$ .

But Lemmas 1 and 2 tell us that $¥lim_{t¥rightarrow¥infty}¥lambda(t)=0$ and that $¥lim_{t¥rightarrow¥infty}¥mu(t)=0$, so theproof of Theorem 2 is complete.

Remark. Care must be exercised if one wishes to extend our results to

the “extreme” cases $p=1$ , $ q=¥infty$ and $ p=¥infty$ , $q=1$ . In particular, if we take thecase $p=1$ , $ q=¥infty$ , our proofs will break down unless, in addition to the hypo-

thesis that there is a number $K$ such that

$¥exp[_{s}¥int^{t}¥mathrm{a}(r)$ $dr]¥leqq K$

whenever $(s, t)$ is in $R^{+}¥times R^{+}$ and $s¥leqq t$ , we also assume that

$¥lim_{t¥rightarrow¥infty}¥exp[_{0}¥int^{i}¥mathrm{a}(r)dr]=0$.

On the other hand, in the case $ p=¥infty$ , $q=1$ , we must add the hypothesis $¥lim_{l¥rightarrow¥infty}$

$¥beta(t)=0$ in order to use our proofs.

References

[1] A. Bielecki, Une remarque sur la methode de Banach Cacciopoli-Tikhonov dansla theorie des equations differentielles ordinaires, Bull. Acad. Polon. Sci. Cl. III,4 (1956), 261-264.

[2] R. Conti, On the boundedness of solutions of ordinary diffenential equations,Funkcialaj Ekvacioj, 9 (1966), 23-26.

[3] W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C.Heath and Co., Boston, 1965.

[4] C. Corduneanu, Sur certaines equations fonctionelles de Volterra, FunkcialajlEkvacioj, 9 (1966), 119-127.

[5] J. K. Hale, Asymptotic behavior of the solutions of differential-difference equa-

$¥mathfrak{Z}8$ D. L. LovELADY

tions, Proc. Int. Symp. Nonlinear Vibrations, II (1961), 409-426. Izdat. Akad.Math. Ukrain. S. S. R., Kiev, 1963.

[6] J. K. Hale and P. Perello, The neighborhood of a singular point of functionaldifferential equations, Contr. Diff. Eqns., 3 (1965), 351-375.

[7] T. G. Hallam, On the asymptotic growth of the solutions of a system of nonho-mogeneous linear differential equations, J. Math. Anal. Appl., 25 (1969), 254-265.

[8] T. G. Hallam, On nonlinear functional perturbation problems for ordinary dif-ferential equations, J. Diff. Eqns., 121 (1972), 63-80.

[9] T. G. Hallam, G. Ladas an-d V. Lakshmikantham, On the asymptotic bahavior offunctional differential equations, SIAM J. Math. Anal., 3 (1972), 58-64.

[10] Y. Hino, Asymptotic behavior of solutions of some functional differential equa-tions, Tohoku J. Math., 22 (1970), 98-108.

[11] Y. Hino, On the stability of the solutions of some functional differential equa-tions, Funkcialaj Ekvacioj, 14 (1971), 47-60.

[12] J. Kurzweil, Global solutions of functional differential equations, Univ. MarylandInst. Fluid Dynamics Appl. Math. Tech. Note # BN-629, College Park, 1969.

[13] V. Lakshmikantham and S. Leela, Differential and integral inequalities, Vol. 1,Academic Press, New York, 1969.

[14] D. L. Lovelady, Bounded solutions of Stieltjes integral equations, Proc. Amer.Math. Soc., 28 (1971), 127-133.

[15] D. L. Lovelady, A funtional differential equation in a Banach space, FunkcialajEkvacioj, 14 (1971), 111-122.

[16] D. L. Lovelady, A global existence theorem for a functional differential equation,An. st. Univ. “Al. I. Cuza”, Iasi, Matematica 18 (1972), 343-349.

[17] D. L. Lovelady, Global attraction and asymptotic equilibrium for nonlinearordinary equations, Proc. Conf. Theory of Ordinary and Partial DifferentialEquations, Springer-Verlag, Lecture Notes Math., #280, pp. 303-307.

[18] D. L. Loveladv, Asymptotic equivalence for two nonlinear systems, Math. SystemsTheory, 7 (1973), 170-176.

[19] D. L. Loveladv and R. H. Martin, Jr., A global existence theorem for a nonauto-nomous differential equation in a Banach space, Proc. Amer. Math. Soc., S5(1972), 445-449.

[20] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966.[21] v.A.Staikos, A note on the boundedness of solutions of ordinary differential

equations, Boll. Un. Mat. Ital. (4), 1 (1968), 256-261.[22] P. Talpalaru, Quelques problemes concernant l’equivalence asymptotique des

systemes differentiels, Boll. Un. Mat. Ital. (4), 4 (1971), 164-186.[23] T. Yoshizawa, Stability theory by Liapunov’s second method, The Mathematical

Society of Japan, 1966.

(Ricevita la 11-an de decembro, 1972)