ON FUNCTIONAL DIFFERENTIAL EQUATIONS WITH · FUNCTIONAL DIFFERENTIAL EQUATIONS 113 Theorem 2.2. If f is Lebesgue integrable on the interval [a, 6], then it is H-K integrable on this

NEW ZEALAND JOURNAL OF MATHEMATICS Volume 28 (1999), 111-123

ON FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY AND HENSTOCK-KURZWEIL

INTEGRALS

T i m - L a m T o h a n d T u a n - S e n g C h e w

(Received January 1997)

Abstract. In this note we establish an Existence Theorem for Functional Differential Equations with unbounded delays under Henstock-Kurzweil integral setting. W e also establish a theorem concerning the continuous dependence of solutions on a parameter

1. Introduction

The Henstock-Kurzweil integration has proved useful in the study of Ordinary Differential Equations. See [1], [2], [9] and [13]. A further step of generalisation was done in [3] which applies the Henstock-Kurzweil integrals to the study of Retarded Functional Differential Equations with finite delays, i.e. equations of the form

x'{t) = f( t ,x t) (1)

where xt{9) = x(t + 0) and 9 takes values from [—r, 0] for some finite positive number r, subject to some initial function 0 at r, where (j) is some Henstock-Kurzweil integrable function over [—r, 0]. It is well-known that Henstock-Kurzweil integrals encompasses Newton, Riemann and Lebesgue integrals (See [8]). A particular feature of this integral is that integrals of highly oscillating functions such as F'{t), where F(t) = t2sin£-2 on (0,6], for any positive value of b, and F(0) = 0, can be defined. The integral was introduced independently by Henstock and Kurzweil independently in 1957 - 58.

The theory of Retarded Functional Differential Equations of (1) has been well- understood when (f> and / are continuous functions, hence Riemann integrable. Hale in [6] notes that the results still hold true when continuity of / is weakened to satisfaction of a Caratheodory condition. M.C. Delfom and S.K. Mitter in [4] further generalises the theory to the case where the initial function (j) and/or / is (are) Lebesgue integrable. The further step of generalisation which was made in[3] is such that / and/or <f> are only assumed to be Henstock-Kurzweil integrable. In this note we shall further generalise the result of [3] to Retarded Functional Differential Equations with unbounded delays under Henstock-Kurzweil integral settings.

1991 AMS Mathematics Subject Classification: 34K15. Masters Thesis, National University of Singapore, 1997.

112 TIM-LAM TOH AND TUAN-SENG CHEW

As a motivation, let us consider a numerical example of an Retarded Functional Differential Equation

subject to the prescription that x(Q = when r — y/r < C < r for someprescribed initial function ^ at initial value r > 1. Here the initial function is defined over [r — y/r, r]. An immediate observation is that t — y/t is a continuous and strictly increasing function for t > 1 . Also, t — y/t —> oo when t —► oo, hence there exists exactly one value £o > 1 such that to — y/to = r.

Clearly equation (2) cannot be expressed directly in the form of (1). The delay term, y/t, is unbounded while t — y/t < t — y /r for all t > t . This type of equations, in which the delay term is unbounded but the past history is not really so, is called “retarded functional differential equations with unbounded delay but finite memory” . From the fact that as t —> oo we have t — y/t —> oo, we can also say that the equation (2) “forgets its past history” .

Finding a solution to (2) is equivalent to solving the integral equation

the integral in this last equation is meaningful for a more general class of integrands if)(s — y/s) - not only restricted to Lebesgue integrable functions. If the function r](t) = ip(t — y/t) is Henstock-Kurzweil integrable over [r, to], then solutions to (2) exist. The existence of solutions of this type of Retarded Functional Differential Equations (which may not be Lebesgue or Riemann integrable) cannot be directly established from any of the Existence Theorems established in the classical results.

2. Preliminaries

In this section we first mention some definitions and basic results on Henstock- Kurzweil integration (see [8] and [12]) and then we shall state the Ascoli’s Lemma and Schauder’s Fixed Point Theorem, which will be required to establish our results in the later sections.

Definition 2.1. A real-valued function / is said to be Henstock-Kurzweil integrable to a value A on [a, b] if for any e > 0, there is a positive function <5(£) such that whenever a division D given by

a = xq < xi < x2 < ■ ■ - xn = b and {£1,62, • • • ,£n}

satisfying & - <$(&) < Xi-i < & < £ » < & + £(&) for i = 1 , 2, . . . , n, we have

x'(t) = x(t — y/t) (2)

For t < t < to (recall that to — y/to — t), we have

x(t) = ?/>(t) + f ^(s - y/s)ds,J T

n

- X i - i ) - A < £.

For succintness we shall simply call such functions H - K integrable.

FUNCTIONAL DIFFERENTIAL EQUATIONS 113

Theorem 2.2. If f is Lebesgue integrable on the interval [a, 6], then it is H -K integrable on this interval. If f is H -K integrable on [a, b] and nonnegative, then it is Lebesgue integrable there.

Definition 2.3. A function F is said to be absolutely continuous in the restricted sense on X, or in short AC*(X) if for every e > 0 there exists a ij > 0 such that for every finite or infinite sequence of non-overlapping intervals {[a*, 6*]} with ai,bi € X and satisfying 1 - a*| < rj, we have ^ w(F; [a*, &*]) < e, where uj(F; [ai,bi]) denotes the oscillation of F over [a*, 6*]. A function F is said to be generalised absolutely continuous in the restricted sense on [a, b] or ACG* if [a, b} is the union of a sequence closed sets Xi such that on each set the function F is AC*(Xi).

Theorem 2.4. A function f is H -K integrable on [a, b] if and only if there exists a continuous function F which is ACG * on [a, b] such that F'(x) = f(x) almost everywhere.

Theorem 2.5 (Controlled Convergence Theorem). Let { / n} be a sequence of H -K integrable functions on [a, b] satisfying the following conditions:

(i) fn{x) —*■ f(x) almost everywhere in [a, 6] as n —*■ oo;(ii) the set of primitives of fn, {Fn(x)}, where Fn(x) = f * fn(s)ds, is uniformly

ACG* in n;

(iii) the primitives Fn are equicontinuous on [a, b], then f is H -K integrable on [a, b] and

n X p X

I fn * I f J a J a

uniformly on [a, b] as n —* oo.Conditions (ii) and (iii) can be replaced by the following condition :

(iv) there exists functions g and h which are H -K integrable on [a, b] such that g(x) < fn (x ) < h{x) almost everywhere on [a, b\.

In this case the above theorem corresponds to the well-known Dominated Convergence Theorem.Theorem 2.6 (Ascoli’s Lemma). Let F = { / Q} ,a £ I (where the index set I may either be countable or uncountable) be an uniformly bounded and equicontinuous set of functions defined on [a, b]. Then F contains a sequence { / n} which is uniformly convergent on [a, b].

Theorem 2.7 (Schauder’s Fixed-point Theorem). [3, Lemma 3.3]. Let U be a closed bounded convex subset of a Banach space X and suppose that the mapping T : U —> U is completely continuous. Then T has a fixed point in U.

3. Existence Theorem

Let M denote the set of all real numbers and C(X) denote the class of all continuous real-valued functions defined on the domain X. In particular, C(R) denotes the class of all continuous real-valued functions defined on the set of all real numbers. Throughout this section, let n, r, k > 0 be fixed. Let / G C(R) with f(t) > n > 0 for all t € [r, r + «].


Definition 3.1. p(t,9) = t + 9 f (t ) G C([t , t + k] x [—1,0]) is called a p-function if p(t, —1) is nondecreasing in t whenever t > t , i.e. t\ — f ( t i ) < t2 — f ( t 2) for any t < h < t2\

Remark 3.2. In [11], p-functions of a more general form are considered. However in this note, we only consider p-functions of the form p(t,9) = t + 9f(t). This is due to the restriction of H - K integral setting. See Theorem 3.3 below.

Theorem 3.3 ([13, p. 19]). Let 7? : [c, d] —* R be a strictly monotone linear function on [c,d\ (i.e. r}(t) — a + /3t for all t G [c, d] where a is a real number and (3 nonnegative real number). Let f be a function defined on [77(c), r)(d)] and H - K integrable on [r](c),r)(d)]. Then

(i) <3>(0) = f(ri(0)) is H - K integrable on [c,d];

f 77(d)

'ri(c)

rv(d) rd

(ii) / f(6)de = / f(r,(6))r]'(e)de.Jri(c) Jc

Fix a function </> which is H - K integrable over [—1,0] and a p-function p over [r, r + k] x [—1,0]. For any H - K integrable function x defined on [p(r, — 1), t + a] and continuous on [r, r + a], where a > 0, we define the translated function for any t G [r, r + a] as xt(0) = x(p(t, 9)) which is defined for 9 G [—1,0].

Clearly p(r, — 1) < p(t, 9) < r = p(r, 0). For each t G [r, t + k ] , p(t, 9) is monotone increasing and continuous. Hence there exists exactly one (q G [—1,0] such that p(r, (0) —p(t,9), i.e. (e = p~1p(t,9), where pT(9) =p(r,9) . The auxiliary function Xt is defined as

xt(9) = x(p(t,0j) ='x(p(t, 9)) if p(t, 9) > t

>(Pr e)) if - 1) < 0) < T-

Note that xT(9) = (f)(9) if —1 < 9 < 0 in the above definition.We will consider the RFDE

x'(t) - f ( t ,x t) (3)

with initial function (p which is H - K integrable on [—1,0], where xt(9) = x(p(t, 9)). We shall call (3) the general p-type Retarded Functional Differential Equation.

The equation x'(t) = x(t — y/t) can be expressed as (3) by choosing p(t, 9) = .t + 9\ t, and /(t, ip) = ip(—l).

Notation.1. For any u, v G R with u < v let H[u,v] denote the equivalence class of all

H - K integrable functions equal almost everywhere on [u, v]. H[u,v\ is a normed linear space with norm || • ||/f such that for any P G H[—1,0]

1 r||F||// = sup / (f)(u)du u£[—1,0] \ J- 1

where (f) G P. As the H -K integrable functions are given as part of the problem, the equivalence classes to which they belong to play a minor role.


We shall refer to an H -K integrable function 0 on [u,v] as fixed in H[u,v], meaning that a specific function 0 belonging to some equivalence class P G H[u,v] has been selected. In this case we write 0 G H[u,v] and the symbol ||0 | |h is defined as ||0 | |h — \\P\\h for 0 G P.

2. For any (3 > 0 we shall denote the interval [r, r + /3\ by Ip and for any b > 0, {ijj G H[— 1,0] such that ||-0 — (j>\\h < b} is denoted by Qj,. For any a > 0 and 0 G H[—1,0], let A(<j>,a) denote the set

a) = ( iG C(Ia) : x(r) = 0(0), sup |x(i)| (u)du : 0 G J-j is a compact subset of C[—1,0]. Then there exists an a > 0 such that A(4>, a) is nonempty for all 0 G T .

Proof. Tj: is compact, hence equicontinuous. Let /ii > 0 such that

|$(u)-$(u)| < ^for all 0 G T whenever |u — v| < n\. Here pi is independent of the choice of 0. We next construct a function x G A(0, a), where a is to be determined later. Let x(t) = 0(0) over t G Ia. Then

xt{6) -’0 (0) if p { t , 6 ) e l a

> (P r V ( i , 0)) if p(t, 6) G [p(i, - 1 ) ,

Clearly x G C (/a),x(r) = 0(0) and supxe/a \x(t)\ = |0(O)| < 6 + |0(O)|. For fixed t, p~1p(t, 0) is linear and strictly increasing in 0, hence (j)(p~lp(t,6)) is H -K integrable over [—l,p^ (T)] by Theorem 3.3. So \\xt — 0||jj is bounded by

fS fssup / <!>\Pt lp{t,0)\ - (0) dd + sup / 0(0) dd

«e[-l,pt_1(r)] J-1 sebt-1(T)’°]

+ IPt (T)l 10(0)1- (4)

Choose cti > 0 such that |r — t\ < ai implies Ip^C7")! < Hi, so the second integral in (4) is bounded as

supsebr1 )-0]

[ 0(0) d0 J p T H t )

(pt x(r))| < se[pt rJ.O]

b_10

Choose o;2 > 0 such that \t — r| < a2 implies

mf(r)

- 1 M l H < 20 •

Consider the first integral of (4). Let

A = sups6[-l,pt_1(r)]

J 0 (pT V(*>0)) - (f>(O)d0

(5)


By changing variables, we have

rPT1p(t>s)A =

<

sups 6 [ - l ,p r 1(r)]

sups e [- l ,p t_1(r)]

rpT P ( Z , S ) rs

/ m m do - mJpT p(t,—i ) J \T) J — 1

dO- i )

r p ~ 1p(t,s)

hr i)0(0) d6 - J (f)(9) d9

mf(r) - 1

c

pT V (M )0(0) d0

Pr p(t,~ 1)

 0 such that 0 < £ — r < <23 implies \pT 1p(t, s) — s| = |p~1p(t,s) - p~1p(r,s)I < for all s € [—1,0]. Hence

A < sup {|$(pTV (* ,s ))-$ (s )| + |$(pT1p ( i , - l ) ) - $ ( - l ) | }sG [-l,pt_1(r)]

b b b b 36H------< ------- 1------- 1-----= —

10 “ 10 10 10 10 (6)

Combining (4), (5) and (6),

P « - * l l » < f j j + ^ + b ,“ ‘ M l |0(o)|.

Let 0:4 > 0 be such that for any t > r with 0 < t — r < 0:4, we have

\PT\r)\ < 110 |0(O)| + 1

(7)

Let a < min(Q1, a 2, 0:3, 0 :4 ), then ||xf — <P\\h < | < 6, thereby completing the proof. □

Subsequently we assume a > 0 has been chosen for A((p,a) to be nonempty.

Lemma 3.5. Let {xn} be a subset from C (/M) for some a > 0, and x* an element from C(Ifj,) such that xn tends to x* uniformly on I

Then, for any t G 1^, x™ converges to x£ uniformly on [—1,0], and in addition II%t ~ xI\\h tends to zero.

Proof. The proof is easy hence omitted. □

Theorem 3.6. A((f),a) is bounded, convex and closed for any fixed 0 € i f [—1,0].

Proof. That 4(0, cu) is bounded follows from its definition, as swpteIa |rc(t)| ,a) follows from the Triangle Inequality for the norms || • || and || • ||i/. We just need to show that A(0, a) is closed. Let {xn} C A(0, a) be a sequence such that xn —> x* uniformly, i.e. limn_+OG ||a;n — x*|| = 0.


Completeness of C(Ia) implies that x* G C(Ia). Also, x * ( t ) = <j>(0), and

sup |x*(t)| \\h + \\x? ~ ^Wh < b+\\x*t -x?\\H-

Lemma 3.5 implies \\x —x \\h —> 0, hence x* G A(<j), a ), thus showing that A(<f>, a)

Definition 3.7. A function f(t, x) defined on Rab = Iax fib is said to be Caratheodory there if for each x G fit, f(t, x) is measurable in t and for almost all t G Ia, f(t, x) is continuous with respect to x. It is continuous if for any sequence {xn : n = 1,2, . . . } from fib which converges uniformly to some x* G fib, then f( t ,xn) converges to f(t,x*) for almost all t G Ia-

Theorem 3.8. Let T C # [ —1,0] be such that = {$(£) = <j>(u)du : (j) G J-} is a compact subset of C[—1,0]. For each (f> G T , suppose f^(t,x) is Caratheodory on Ia x flb and / ^ ( t ,x t) is H -K integrable on Ia for any x G A(<p,a).

Define

Suppose the set {F^x : x G A(<j>,ct),(j) G F ] is equicontinuous on Ia. Define T^ as

for t G Ia- Then there exists /i > 0 independent of (j), where 0 < /z < a, such that for all <\> e T , is a mapping from A((j>, /i) into A((f),n).

Proof. {F^x : x G A(cf),a),(f) G T } is equicontinuous. Choose /ii > 0 such that0 < t — r < fix implies

for all x G A(<f>, a) and all 0 G T . As the function T^x is equicontinuous, let > 0 also satisfy the condition that whenever 0 , a ) . Prom (8), for any t G [r, t + / i i ] ,

|(T**)(t)| < |0(O)| + | < |0 (O)| + 6.


Clearly T^x(r) = 0(0) and T^x G (7(7^). We just need to show that ||T ^xt — 0||h < 6 for all f G 7M for some p > 0. We have

IIT * x t - 0 Hi/ = sups6[—1,0]

{Prlp(t,Q)) ~(j>{0)d6

[ Tx(p(t,O))-<t>{9)d0J p T 1 ( t )

From the proof of Theorem 3.4, choose fi2 > 0 be such that 0 < t — r < p2 implies

J ^ { P r 1 Pit, 0)) ~ 0(0) d6A = sup®G[-i.pr1(r)]

b< -

- 2

We also assume /i2 > 0 satisfies the condition that |$(w) — ‘JK'u)! < 6/4 for all $ G Tjr whenever \u — v\ < p2, where u,v G [-1,0]. Choose /u3 > 0 such that \Pt\r)\ < l/nmin(//i,/x2) whenever 0 < t — r < p3. Hence

B = supsebt_1(r),o]

[ ° r*x(p(t,e)) - m Jpt X(T)

dO

x(p{t,9))d9 + sup [ (f)(9) d9sebt_1(T)>0] sebr1(r)-°] Jp7\r)

s ~Pt 1(r )l 1/(01 < min(/ii,/x2) < Pi for any e [pt_1(^)?0]- Consequently

r8 1m

rt+sf{t)sup / T<t>x(p{t,9))d9 = sup / T*x{u) du

sebt^M.o] JpT\r) s€bt_1(r)>°] ^+pr1(r)/(t)

1 bn b ~ n 4 4

Therefore B < b/4 + b/4 = 6/2, and hence ||T a;t — 0||h < 6/4 + 6/2 + 6/4 < 6 for all t G where p < min(^i, p2, ^3). Thus T^ is a mapping from A(0,/i) to A((f>,fi) for p > 0 thus chosen as above. As pi,p2 and ps are independent of 0, so is p. .. □

Theorem 3.9 (Existence Theorem). Let 0 G # [ —1,0]. Consider the equation (3). Suppose that f(t ,x) is Caratheodory for some a, 6 > 0 on Rab such that f(t ,x t) is H - K integrable over Ia for all x G A((/>,a) and {Fx : x G A((f>, a )} is uniformly ACG* and equicontinuous on Ia, where

(.Fx){t) = J f ( s ,x s)ds, t e l a.

Then there exists a solution to (3) with initial function 0 at t .


Proof. Take T = {</>}. From Theorem 3.4, for any fixed 0 G H[—1,0], A(<f>,a) is nonempty for some a > 0. From Theorem 3.6, A((f>,a) is closed, bounded and

convex. Define a mapping T^ as

(T*x)(t) = 0(0) + J f c/){s,xs)ds.

As there is only one element 0 in T, we shall simply write T& as T. Theorem 3.8

assures there exists a fi > 0 with 0 < fi < a such that T is a mapping from A(<f>, fi)

to A(<j>, fi).We shall next show that T is completely continuous, i.e. T is continuous (we

shall prove later) and that for any subset W C A{(j>,fi), T(W) = {Tx : x G W} is compact. {Tx : x G A(<f>, fi)} is equicontinuous because {Fx : x G A(4>,fi)} is

equicontinuous. Thus T(W) is equicontinuous.

We next show that T(W) is uniformly bounded. For fixed m > 0, there exists

8 > 0 such that for any x G A((j>, fi), \Tx{t\) — Tx(t2)\ < rn whenever \ti — t2\ < 8. Consider a partition of [r, r + fj] with partition points: r = t0 < ti < t2 < ... <

tn-1 < tn = r + fi, such that 0 ti],

\Tx(t) | < |Tx{t) - Tx(t) | + \Tx{r) \

< m+|0(O)|.

For t G [ti,t2], \Tx(t)\ < |Tx(t) — Tx(ti)\ + \Tx(t\)\ < 2m+ |0(O)|. Inductively,

\Tx(t)\ < nm + |0(O)| for all t G [r,r + /x] and x G A(<f>,fi) showing that T(W) is uniformly bounded. By Ascoli’s Lemma, T(W) is sequentially compact, hence

compact. It remains to show that T is continuous. Let {£n} C A((f),fi) such that

||a;n— £c° || —> 0 as n —* oo. From Lemma 3.5, xf converges uniformly to x® on [—1,0],

hence f(s,x” ) converges to f(s,x^). The set {Tx71} is uniformly ACG* on and

equicontinuous there. By Controlled Convergence Theorem, \\Txn — Tx°|| —» 0 as

n —> oo. Thus T is continuous. By Schauder’s Fixed Point Theorem, the mapping

T : A(4>, fi) —> A(cj), fi) has a fixed point, implying that (3) has a solution on /M. □

A consequence of Theorem 3.9 with the Dominated Convergence Theorem yields:

Corollary 3.10 (Existence Theorem). Let 0, r and f be as defined in the above

Existence Theorem. Suppose that there exist two functions g and h which are H-K integrable over Ia such that

g(t) < f(t,x t) < h(t)

for almost all t G I a and all x G A(cj),a). Then there exists a solution to (3) with

initial function 0 at r.

4. Some Examples

Example 4.1. Consider the equation (3). Let to > r such that p{to,— 1) = r,

and 0 G # [ —1,0] such that ip(t) = 0 (p“ 1p(t, —1)) is H-K integrable over [r,to}. Here f{t,ij>) = iJj(—1) over Rab = I to x flb for some b > 0. Now (Fx)(t) =

fr f{s,xs)ds = f*xs(-l)ds = f l (j){p~1p{s,-l)) ds. If 0(p“ 1p ( i,- 1)) is H-K

integrable over [r, to], then {Fx : x G A((j),a)} is ACG* and equicontinuous. By


Theorem 3.9, (3) has a solution. We may also deduce the existence of solution from

Corollary 3.10. Clearly

which is H-K integrable. Thus f(t, xt) is H-K integrable, so there definitely exist

two H-K integrable functions g and h such that

for almost all t E I t o- Corollary 3.10 thus asserts that the RFDE has a solution.

The solution to (3) by direct integration is

Example 4.2. Let pit, 0), to and </> be as in Example 4.1. Consider the RFDE (3)

with f(t,ip) = c(t)i>(-1) + d(t) on Rab = I a x Clb, where a > to. The function

c(t) is of bounded variation over Ia, and d(t) is H-K integrable over Ia. Suppose

a > to such that A(<f>, a) is nonempty. Let g and h be defined as follows:

Clearly g(t) < f(t,x t) < h(t) for almost all t € Ia, and both functions are H-K integrable. By Corollary 3.10, (3) has a solution. This example illustrates the

importance of the bound ||a;|| (0)| in Theorems 3.9.

We remark that in Example 4.1 if we take p(t, 9) = t + Q\ft, then

is a special case of Example 4.1.

5. Continuous Dependence

In this section we generalise the result that was established in [3] to the gener

alised p-type RFDEs. Let I and ft, be open subsets of E and H[— 1,0] respectively

and let D = I x Q,. We fix a ^-function throughout the entire section here. We

shall consider the one parameter family of p-type FDEs given by

For any c > 0, let Kc denote the set {v £ E : \u — vq\ < c} where uo is a fixed

number. Suppose that for each v 6 Kc (for some c > 0) f v is Caratheodory on

D with ( t , ^ ) G D. Let E = {(f>u G i/[—1,0] : v € Kc} be a subset of i f [—1,0]

such that I# " : (j>u € E} is a sequentially compact subset of C[—1, 0], and that as

v —> uq we have ^ (0 ) —► 0I/°(O).

f ( t , x t ) = xt(- 1) = x(p(t,- l)) = <f>(pTlp(t,~ 1))

g{t) < (j)(pTlp it,- 1)) < hit)

cj)(pT 1p(t, - 1 ))c(t) + d(t), if t e [t , t 0]

-(b + |0(O)|)|c(t)| + d(t), if t e [t0, r + a]

and

<t>(pT 1P(t > ~ 1))CW + if 1 e K o]

(6 + |0(O)|)|c(t)| + d(t), if t G [t0, r + a].

rl>(t) = 0(pT1(t ,- l)) = (f){t-Vt- 1),

hence the existence of solution of equation

x'(t) — x(t — Vt)

x'{t) = f uit,xt), xT = 4r (10)


Theorem 5.1. Suppose the above conditions hold, and that for each u G Kc andIa C I, and for all x G C(Ia) with xt G fi, we have

for almost all t G I a . Also assume that when v — v0, the equation (10) has a unique solution xu° on Ia . Then for u sufficiently close uq the equation (10) has a solution, say xu. This solution is valid over Ia and as v —> v0, xu —> xu° uniformly

on Ia .

Proof. Prom Corollary 3.10, for each u G Kc, (10) admits a solution. Here /j, is

independent of u, since g and h are independent of u G Kc. The solutions to (10)

can be expressed as

Choose M > 10"° (0)| and c\ > 0 be such that |0"(O)| < M whenever v € KCl. M exists because <^(0) —* </>"°(0). We claim that Q — {xv : v G Kc} is equicontinuous

on I^ and uniformly bounded there. Equicontinuity follows from the equations (10)

and (11). The proof of uniform boundedness is similar to that in Theorem 3.9.

sequence {xUk} which converges uniformly to some function 7 (t) on /M. For each

t (z I^ and k as defined above,

showing that 7 (2) is a solution to (10) with u = vq. Uniqueness of solution of (10)

with v = vq implies xu°(t) = 7 (t) on /M. Therefore xUk(t) —> xv°(t) uniformly on

By reductio ad absurdum, xu converges to xu° uniformly on /M. We extend

our result to Ia. Suppose there exists to e Ia = [r, r + a) such that xv converges

uniformly to xu° on [r, to] but not on [r, to + h\ for any h > 0. Choose 8 > 0 such

that [to, to + 6] C Ia C I. As x\° G Q, and that Q, is open, there exists a q > 0 such

that Oq — {ijj G H[—1,0] : ||ip — x °||h < q} C fi. Now xu converges uniformly

to xVo on [r,to]- Hence x”o converges uniformly to x on [—1,0]. We next show

that for v sufficiently close to vq, any function close to xv is also close to Xq. Let

0 < C2 < c be such that whenever v G KC2, we have \\xQ — x°\\h < q/2. So

whenever ||£ — ||h < q/2 for u G KC2, we have

Now f is defined on [to, to + 5] x Oq. Let Q = {xv(t) : v G KC3}. It is uniformly

bounded and equicontinuous (the proof of uniform boundedness is similar to Theo-

g(t) < r ( t ,x t) < h(t)

(11)

Ascoli’s Lemma implies that every sequence {xv} with v —> vq contains a sub-

and from Controlled Convergence Theorem

{C : lie - $roll// < |} C O q C n .

rem 3.9), hence Q is compact. Clearly 71 = < f _ 1 x o(s)ds : u G KC2 \ is compact


Define the translated function xt as

\xV{P&9)) if P M )> * o

l^ o {VtoPW)) if PfaO)<to

and the set A(v, 8) to be

{y e C[to,to+p] : y{t0) = xv(tQ), sup \y(t)\ < q+\xv(t0)\, \\yt-xvto\\H < 9/ 2}.t€[to,to+p]

From Theorem 3.4, there exists pi > 0 such that A(u,pi) is nonempty for all

v € Kc2, and p\ is independent of u. Thus Corollary 3.10 implies that there exists a

fi2 > 0 such that the equation x'(t) = f v(t,Xt), where v G KCa with initial function

x% at to has a solution which exists over [to, Iq + p2] for some fi2 independent

of v. Repeat the above argument of the uniform convergence over [r, to] to the

interval [to, to + <5]. So we conclude that xu converges uniformly to xu° uniformly

over [r,to + 8 + p,2]. This contradicts our assumption that the convergence is over

[r, to] only, therefore the result is proven. □

6. Further Remark

We remark that the results of classical comparison theorems and stability of

solutions can easily be carried over to generalised p-type Functional Differential

Equations under Henstock-Kurzweil integral setting, see [15].

References

1. T.S. Chew and Flordeliza Francisco, On x' = f[t,x) and Henstock-Kurzweil integrals, Differential Integral Equations, 4 (1991), 861-868.

2. T.S. Chew and P.Y. Lee , Comparison theorems and Perron integrals, Nonlinear

Times Digest, 2 (1995), 125-139.

3. T.S. Chew, B. Van-Brunt, and G.C. Wake, On retarded functional differential

equations and Henstock-Kurzweil integrals, Differential and Integral Equations,

9 (1996), 569-580.4. M.C. Delfour and S.K. Mitter, Hereditary differential systems with constant

delays, I. General case, J. Differential Equations, 9 (1972), 213-235.

5. J. Hale, Delay Differential Equations and Dynamical Systems, Springer-Verlag,

1990.

6. J. Hale, Functional Differential Equations, Springer-Verlag, 1971.

7. V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, 1992.

8. P.Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, Singa

pore, 1989.9. T.K. Lee, Existence of Solutions of x' — f(t,x), NUS Honours Project, 1988.

10. V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Aca

demic Press, 1969.11. V. Lakshmikantham, Lizhi Wen and Binggen Zhang , Theory of Differential

Equations with Unbounded Delay, Kluwer Academic Publishers, 1995.

12. W.F. Pfeffer, The Riemann Approach to Integration, Cambridge University

Press, 1993.


13. S. Schwabik, Generalised Ordinary Differential Equations, World Scientific,

Singapore, 1992.14. T.L. Toh and T.S. Chew, On p-Type Functional Differential Equations, Na

tional University of Singapore, Maths Dept, Research Report No. 683, June

1996.15. T.L. Toh, Functional Differential Equations and Henstock-Kurzweil Integra

tion, Masters Thesis, National University of Singapore, 1997.

Tim-Lam Toh and Tuan-Seng Chew

Department of Mathematics

The National University of Singapore

SINGAPORE 119260

matcts@nus. edu. sg

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ON FUNCTIONAL DIFFERENTIAL EQUATIONS WITH · FUNCTIONAL DIFFERENTIAL EQUATIONS 113 Theorem 2.2. If f is Lebesgue integrable on the interval [a, 6], then it is H-K integrable on this