Dynamic Systems and Applications 11 ( 2002 ) 545-556
An Interval Analytic Method in Constructive 
Existence Theorems for Initial Value Problems
Peter 0 . Arawomo1 and Olusola Akinyelo2
1 Department of Mathematics, University of Ibadan,
Ibadan, Nigeria.
department of Mathematics, Bowie State University,
Bowie, MD 20716-9405.
Abstract
The method of interval analysis is employed to show that the solution, if it exists, 
of a first order initial value problem is majorised l>y an interval function whose end- 
functions satisfy some prescribed conditions. An interval operator is constructed and 
shown to be a contraction on the majorising interval function. Using this operator, 
the existence and uniqueness of the solution is established.
AMS (M OS) Subject Classification: 34A12
1. INTRODUCTION
Consider the scalar initial value problem
u/ =  /(£ ,u(0), u (0 )= u o (11)
where /  6 C ( / x 1R, JR) and 1 =  {t : 0 < t < T  < oo}. Let us assume that tliere 
exist functions a and 0  € C l(/, 3Uch that
a(t) < 0{t), t e l  and o(0) < u0 < 0(  0) (1.2)
Assume further that the function /  has continuous first order partial derivative with 
respect to its second argument and that
a'{t) <  /(<, u(0) +  /•(*, u(t))(n{t) -  «(£)) 
and (1.3)
F (t )  > / ( t yu{t)) +  / . ( * . - u( O)
for any function u € C ! ( /, Ml) satisfying a(t) < u(t) < 0(t) on /.
In this paper we shall use an interval analytic: method to establish existence of 
a solution and subsequently its uniqueness. Some authors (Clean & Vatsala, 1990; 
TAkshmikanthAm & Swansundaran, 1987) have used this method to obtain existence 
of solutions and sometimes solution sets of differential equations. However, the real 
integral operators equivalent to the problems were so constructed to ensure their
Received August 26, 2002 1056-2176 $03.50 © Dynamic Publishers, Inc.
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546 Arawomo and Akinyele
monotonicity. The techniques employed here rely solely on the inherent m onotone in­
clusion property o f interval functions and as such neither the underlying real function 
nor the equivalent real integral operator need l>e monotone.
2. B A S IC  D E F IN IT IO N S  A N D  R E S U L T S  IN  IN T E R V A L  A N A L Y S IS
We give some basic definitions and results iu interval analysis that will be needed in 
subsequent discussions. Readers who are however not familiar with this subject arc 
referred to (Moore, 1979; Rail, 1981; Caparani et al, 1981).
The basic objects of interval analysis used here are the dosed, non-empty and 
bounded real intervals
X  =  [i , x) =  {x|x <  x  <  £ }  (2.1)
and the real number x is identified with the degenerate interval
x =  [x, x] (2.2)
D efin ition  2.1: The width w (X ) ,  midpoint tn (X ) and modulus \X\ o f  the interval X  
are respectively defined as
w (X )  = x - £  (2.3)
m (X ) =  i ( x  +  £ ) (2.4)
and
\X\ =  max{|2 |,|x|} (2.5)
D efin ition  2.2: An interval function Y  is defined as the function which assigns to 
each x  in its interval o f  definition X  =  [a:, x] the interval denoted by
Y ( x )  =  (g(*).y(*)|  (2.G)
where the real functions y  and y are call<*l the ondfunctions o f Y.
D e fin it io n  2 .3 : An interval function Y  is said to  l>e an interval extension  o f  a 
real function y  if it has the property o f  inclusion o f  y
v W  =  {v (* ) l*  e  X ]  C  Y ( X )  (2.7)
for each interval X  =  fc, £] on which y  is defined.
A n  interval extension Y  is railed a natural interval extm .iion  <.r y  if it is obtained 
from  y  by  replacing the real variables with the corresponding interval variables and 
the real arithmetic operations with the corrospouding interval arithmetic, operations. 
T he interval arithmetic operations used in this work are those defined in chapter tw o 
o f  [2] and they preserve the inclusion projXTty.
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Constructive Existence Theorems for Initial Value Problems 547
I f y  is a differentiable function with derivative i / ,  then the interval extension Y  
is called an internal mean-value extension o f  y if it is given by
Y ( X )  =  y (m p O ) +  Y ' ( X ) ( X  -  ?n (X )) (2.8)
where Y* is a natural interval extension o f  the derivative y '.
D e fin it io n  2 .4 : An interval function Y  is said to be inclusion monotone if
X l Q X 2 => Y ( X x) C Y ( X 2) (2.9)
for intervals, X y , X 2 on which Y  is defined.
D e fin ition  2 .5 : Let Y  be a non-degenerate interval and X  another interval which 
may.be degenerate or not. The interval Y  is said to be an interval majorant o f X  if
X ( t ) Q Y ( t )  t e l  (2.10)
for an interval I  on which X  and Y  are defined.
\
D efin ition  2 .6 : The interval integral o f an interval function Y  over an interval 
X  — [x» x] on which it is defined is the interval
£  Y(t)dt =  Jx Y ( t )d t=  x y(t)dt^ (2.11)
where J denotes the lower Darboux integral over X  and J ̂ denotes the upper 
Darboux integral over X.
L em m a 2.1 (R a il, 1981): If X  and Y  are intervals, then
x c r «  |m(V) -  m (X )| <  i  M V )  -  .» (* )}  (2.12)
T h eorem  2.1 (M o o re , 1979): If P  is an inclusion mouolonic interval majorant o f 
a real operator p  and if
P ( X o) C  X 0 (2.13) .
for an interval X 0 in the domain of P, thou the sequence {A\.} o f intervals defined by
X k+x =  P (X k), k =  0 , 1 ,2 , . . . .  (2.14)
•*
has the following properties:
(i) X k+i C X kl k =  0 ,1 ,2 ,. . .
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548
Araw om o and Akinyele
(ii) for t in the interval /  o f definition o f  X , the limit
* ( 0 -  n  * * ( ')  (2. ia)
t-o
exists as an interval function and
X ( t ) Q X k(t), k =  0, 1,2,...
(iii) for any solution x  of the operator equation
x ( t ) = p ( x ) ( t ) ,  t e l  (2.16)
such that x(t) e  X 0(t), t e l  
we have x(t) e  X k(t) V k and t e l .
(iv) if there exists a real number c, such that 0 <  r <  1, for which
Z  C X 0
=» sut pw(P(Z)(t) )  <  csutp(w(Z(t)),
then (2.16) lias the unique solution x (/) in A' given hy (2.15).
3. IN T E R V A L  M A J O R A N T  O F S O L U T IO N
In this section we present a result which guarantees the majorisation of a solution of 
the initial value problem (1.1), if it exists, by an interval function.
T h eorem  3 .1 : Suppose that in addition to being continuous, the function /  ap­
pearing in equation (1.1) has continuous first order partial derivative with respect to 
x  and that it also satisfies conditions (1.3). Then, if u is a solution of the i.v.p. (1.1), 
it is majorised by the interval function Y  given by
n o  =  k o , /?(*)].
where a  and p  are the functions defined in (1.2).
P r o o f :  I f  u is a solution o f  (1.1), we need to show that n €  Y  and this we shall
show by  contradiction.
Suppose u(t)  Y ( t )  for some t e  J C  / .  Then 
either u(t) <  n(t)
(3-1)
or u(t) >  f/(t), t e . /
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Constructive Existence Theorems for Initial Value Problems 549
First we suppose u(t)  <  a(t), t >  0, so the interval function C  defined by
has a strictly positive width. Also since u is assumed to be a solution o f equation 
(1.1), it also solves the integral equation
u(t) =  u(0) +  jf* / ( s ,  u(.i))ds, t €  J. (3.2)
From (1.3) we also have
a(t) <  a(0) +  [*  f ( s ,a (3 ) )ds t t €  J. (3.3)
From (3.2) and (3.3) we have
w(G(t)) <  a(0) -  u(0) +  / '  {/(a , « (.,)) -  /(.» , « (a ))}  da. (3.4)
J o
Since
/(3 ,Q (s )) - / ( * , * ( * ) )  €  F .(« f C7(s))[0f iii(C(s))|f 
the integral inequality (3.4) gives
w(G(t )) <  a(0) -  u(0) +  J ‘ G(.i))| w(G(a))da, t 6 ./ 
which by the application o f Gronwall-Bellman’s Lemma in [1] yields 
w (G (t)) <  (« (0 ) -  «(0 )) exp Q f  |n(a, <7(.->))| da') , 
showing by (1.2) and earlier assumptions that
0 <  w(G(t)) <  0.
This contradicts the assumption that w(G(t)) Ls strictly positive. Hence the assump­
tion that u(t) <  ct(t), t >  0 must have been wrong. We now consider the other 
assumption, that
u(t) >  p(t )  t >  0.
In a similar manner we deduce that the interval function / /  defines 1 Ijy
tf(O = [0(O.«(OI
has a strictly positive width.
By (1.3) we equally have
Pit) >  P(0) + Jfo  f  (s. flis))d3, t C ./. (3.5)
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550 Arawomo and AJcinyele
Using this and (3.2) we obtain:
<  tt(0) -  /?(0) + £  { / ( * ,  « (* ) )  -  / ( * .  /* (* ) »
Again by the property o f tlie interval function Fu, wc have
»«(//(« )) <  “ (0) -  W )  +  f  '"(^(•, )) ,is
which by Gronwall-Bellman’s Lemma again yields
w(H(t)) <  (u (0)-/J (0))racp  '
By the assumption on w(H)  and (1.2) this implies that
0 < tu(//) <  0
which is a contradiction.
Hence our assumptions must be wrong and so wo have u €  Y  as required.
4. E X IST E N C E  RESU LTS
In this section an interval operator is constructed. It is shown that this interval oper­
ator is a contraction. With the use of this operator, the existence of a nested sequence 
of interval functions is established and shown to converge to a limit containing the 
solution of (1.1).
Theorem  4.1: Suppose that the function /  appearing in equation ( l . l )  is con­
tinuous and continuously differentiable with respect, to its second argument. Assume 
further that it satisfies conditions (1.3) with the functions ft and /? defined in (1.2). 
Then the interval integral operator P  defined by
P(C/(t)) =  «o +  J '  f(a,m(U(a)))dx +  £ |F„(s, f/(s))| « ,( { /( . ,) ) [ -1 , ljds (4.1)
contracts the interval function
=  M 0 ./> (0 h  t e l .  (4.2)
where w{U) and m(U) are respectively the width and midpoint of the interval function 
U and the interval function Fu is a natural interval extension o f the partial derivative 
fu o f the function /  chosen such that
V(i))| <  2 /.(< ,m (V (<))). (4.:t)
P roo f: To show that P  contracts Y  it suffices to show, by Lemma 2.1, tiiat
|m(T) -  m (P(y))| < 1 (,„ ( Y )  -  w ( r ( Y ) ) ) (4.4)
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Constructive Existence Theore ;ms for Initial Value Problems 551
Now from (4.1) and (4.2), wc have, for t e l
W * ) )  -  « W W ) ) >  =  \ { f ‘ \Fu{a,Y{*))\w(Y(*))<U -  [p(l) -  <*(*)}}
<  • j f  /.(s,m (y(s)))u»(y(5))d 3  -  \  m )  -  a(t)}
< jf ‘ {/?-(») -  f(s,m(Y(a)))} d a - \  {p(t) ~  “ (0 }
by assumption (1.3).
< ^{/J(t) +  a ( t ) } - u ( 0 ) - jT , /(5,m(y(a)))d5
by assumption (1.2).
That is
- \ («>(y(t)) -  « ( w }  <  {m (y (0 ) -  ( * v
We also, from (4.1) and (4.2), have
\ M V (t)) -  u ;(P (Y (t)))} = \ {p(t) -  <r(t) -  fo |F„(s, Y(.,))] w(y(s))rfa}
>  \ (fl(0  -  « ( 0  -  2 J ‘ Us ,m(Y(s)) )w(Y{s) )ds}
>  \ [IK') -  <>(0 +  2 ^ ' {.»'(•■.) -  /(.•*, m (V(*)))} «fa}
by assumption (1.3).
> \ W ) +  <*(')} - “ (») -  £  f(s,m(Y(a)))da
by assumption (1.2).
which implies that
\ {w{Y(t)) -  w{P(Y(t)))} > m(y(0) -  m (/»(r(t)) (4-0)
The combination of (4.5) and (4.6) yields the desired result (4.4) wliich by Lemma 
2.1 establishes that
P(Y)  C Y. (4.7)
Hence the operator P contracts the interval Y.
Example: Consider the initial value problem
u' =  u2 -  t, u(0) =  1, 0 < t < 1
a(t) =  1 4* £, P{t) =  1/(1 — t). These functions sat isfy conditions (1.2) and (1.3) 
since a'(t) < /(t , o r )  and P{t) >J{t,P).  So, we linw Y =  |l + f, 1/(1 —*)], m(Y) =  
(2 — t2) /2 ( l  — t), / .  =  2u. F. =  2U, /.(f ,m (Y )) =  (2 -  tJ) / ( l  -  t), Fu(t,Y) =  
2(1 +  t, 1/(1 — 01. |F«| =  2/(1 — t).
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552 Araw om o and Akinyele
In this case |Fu(t, Y)] <  2 /„(t,m (Y )),V £ e  10,1). and P (Y (t)) as <lcfine<l l>y (4.1) 
is given by:
F (Y (0 ) =  1 +  / 0‘ { ^ £  +■ 11}rfS-
When the integral is evaluated it gives:
P (Y (t)) =  l-t-(t3+3t5)/1 2+ [—5£/4—3£/4(l—£)—ln (l—f)3, 3 t/4+ 5£ /4 (l—£ )+ ln (l—£)]. 
It is clearly seen that P(Y(t) )  C Y(t)  for 0 <  t <  1.
T heorem  4.2: Let all the assumptions o f Theorem 4.1 be true. Then there exists a 
nested sequence of interval functions {!/„(£) : t €  / ,  n €  N }  with the initial interval 
Uo(t) =  (a(t),/?(£)], where a  and (3 arc the functions defined in (1.2). Furthermore 
the sequence is such that the limit
U{t) =  Mm Unit)
exists as an interval function on I  and is a majorant of the function u which solves 
the initial value problem (1.1).
P roof: Since the function /  in the i.v.p. (1.1) is continuous, the problem is <*quivalent 
to the integral equation
u(t) =  n0 +  Jfo / ( * ,  '«(*))</*, t e  /  (4.8)
the solution of which also satisfies the i.v.p. (1.1). We now prepjirc to obtain this 
solution by an interval analytic method:
Let U  with the representation
£/(£) =  [rn(t/(£)) -  |u,(f/(£)). m (l/(£ )) +  i..»(t/(£))|
be an interval majorant of the solution u o f expiation (4.8); by Theorem 3.1, this 
exists. Let Fu be a natural interval extension o f / u, the partial derivative o f the 
function /  satisfying assumption (4.3). Then an interval mean value extension o f  /  
is given by
F {t ,U { t ) )  =  /(£ ,m (f/(£ )) +  /•’.(£ , l / ( t ) ) ( l / ( 0  ~  m (( /( t ) ) )
=  / ( t ,m ( t '( 0 )  +  I]
Substituting these interval functions for u anil /  in equation (4.8) wilh the interval 
integral and interval arithmetic operations where appropriate we obtain an interval 
integral operator P, which is an interval extension o f the real operator (4.8), given by
p {U{t)) = «o +Jo f(s,m(U(s)))ds + 1 £  |F,.(s, t/(s))|u;((/(s))[-l, \]d3 (4.9)
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Constructive Existence Theorems for Initial Value Problems 553
and we then have u(t) 6 P[U(t))1 t e l .
We now define a sequence of interval functions {£/«} by
Un+l(t) =  P(Un(t)), n =  0 ,1 ,2 ,... (4.10)
with
Uo{t) = [a (0 ./*«l-
All we need to establish the nestedness of this sequence by Tlieorem 2.1, is to
show that
P(U0{t)) C U0(t).
This by Theorem 4.1 is true and so the sequence (4.10) is nested with each term 
containing the solution u of (4.8). Finally the sequence converges as well by Tlieorem 
2.1, to the limit (7, given by
oo
m  =  n  tf.W . t e  l (4.11)
and also contains the solution u of problem (4.8). This concludes t he proof.
5. UNIQUENESS OF SOLUTION
In what follows we show that the limit of the interval sequence is unique irrespective 
of the initial interval functions, ns long as the prescribed conditions are satisfi^l by 
the end functions of such an interval. We finally give condition under which the limit 
interval function coincides with the real valued solution of the i.v.p. (1.1).
Theorem 5.1: The limit (4.11) of the sequence of interval functions (4.10) of The­
orem 4.2 is unique as long as the endfunctions of the initial interval function satisfy 
conditions (1.2) and (1.3).
P roof: Suppose the limit varies with the initial interval function. Then if {X ,,} 
is another sequence of interval functions with initial interval function X 0 given by 
X 0(t) =  [p(t), r (t)l and limit X(t)  where p and r  satisfy conditions (1.2) and (1.3) 
with a  and replaced by p and r  respectively. Since p and r  satisfy conditions (1.2) 
and (1.3), it follows, by Theorem 3.1, that the solution u of the i.v.p (1.1) satisfies 
u(t) e  X 0(t), t e l  and therefore
Let
Zn(t) =  x „ ( t ) n u 0(t)
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554 Arawomo and Akinyele
then u(t) 6  U{t) C Z0(t) and u(t) e  X{t)  C Z„(t).
This implies, by the monotone inclusion property of F„, tlmt
F.(t ,U(t ) )CF.(t ,Z„(t ) )
(5.1)
and F , M ( i ) ) C F , ( U „ ( i ) )
From (4.9) we have 
U(t) =  P(U(t)) =  uo + / ( . , m(t/(s)))<fs +  J f  |F „(.r,{/W )H C /W )(-l, 1 ]ds 
J° 2 (5.2)
and since X  is also the limit of the sequence {A’,,}, we have, by (4.9),
X(t)  = P(X(t)) =  Xo +  f  /(*, m(X(t))),h |-1  f ‘ |F„(s, *  W )M * (.i) ) |- l,  1J*
(5.3)
SO
d(X(t),U{t)) =  inax{|s(0 -  u(0|, l*(0  “ « (0 I) 
which, by the use of (5.2) and (5.3), gives
= max{|x0 + jf  f ( s , m ( X ( s ) ) ) d s - \ h \ ( s % X(s))\ w{X{s))ds 
-Uo -  jT /(* , rn([/(s)))ds + ± f ‘ IF.fs, (/(*)) | w(V (.,)),Is |,
|io + _̂  f(s,m(X(s)))ds + i  |f’u(.i,.V(3))| w(X(s))ds 
-uo -  J' /(*, m(t/(s)))da -\J‘\F.(s, U{*))\ w(|/(,))*|}
=  max|ji0 -u o  + ^  { /( j ,m (A '(s ) ) ) - /(s ,m ([ /( j) ) ) }d 3
- 5  f ‘  { !* (• . * « ) l  » ( * « )  -  |e.(», m ) l  '«(£/(*))} * | .
I| xo-tio+ v/o {/(5,m (A(s)))-/(s,m (f/(s)))}(fs 
+ J lo * M ) I « W ’ )) -  |F .(*.l'«)|  «•(!/(»))} d-d}
< |io -  Uol +  max |̂jT |F.(3 , Z0(s))\ (m(X(.»)) -  r/i(C/(.s))} ,Ls
J ‘ Z , m  {w(X(,)) -  w(U(s)) -  w(Z„(s) ) } <fa|,
|jf |F.(3, Z„(s))| {«.(*(*)) -  m(£/(.,))} J,
+ \  f ‘ |F’.(s, Z o m  {'«(*(*)) -  «>(£/(.-)) H w(Z0(s))} <b|}
< 1*0 -  Uol +  \ f o |K.(.1 , ZoW)l w(z„(.,)),h
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Constructive Existence Theorems for Initial Value Problems 555
+max || j f  |F.(s, Z„(s))| {(m(X(.,)) -  i<»(*(*))) -  -  \w (U (s)))} </*|,
IjT |F.(s,Z„(s))| {(m(X(s)) +  ^ ( X ( , ) ) )  -  (m(t/(.,)) +  \ v > (V m )  * | }
< |xo -  uol + \ fa IF.(», Z«(*))| ui(2>(.i))ds
+  jf* max ||(m (X (j))  -  \vu(X(s))) -  (m((/(s)) -  ±t»(£/(*)))|,
|(m(X(s)) + i«<(X(j))) -  (m(C/(s)) +  iu,({/(-.)))|} |F.(». Z„(s))|<*>
That is,
d(X(t), U(t)) < |i0-tio|+^ j f  Zo{*))\w{Z0(s))d.i+J d(X(s),U(s))\Fu(s, Z0(s))\ds 
and this implies
d(X(t), U{t)) < \xq -  Uol exp (5  £  |F„(s, Z0(s))|<is)
which by (1.1) => d(X(t),U(t)) < 0
and therefore d(X(t), U(t)) =  0.
Hence X(t) =  U(t), and this concludes the proof.
Theorem 5.2: Suppose the natural interval extension of the function / „  consid­
ered is such that
[ l lF*(3tU(*))\ds <  1 for t e l  (5.4)
Jo
Then the sequence of interval functions (4.9) converges to a degenerate interval func­
tion which coincides with the real valued solution u of the i.v.p. (1.1).
Proof: FYom Theorem 4.2, u(t) € U(t) =  Jim Un[t). and also U(t) =  P(U(t)). 
Therefore
w(U(t)) = w(P((/(t))).
JFYom (4.8)
Mm t))) = j[ |F„(s,l/(.i))|«»({/(*))*
and so
sup w(U(t)) =  SUV f'\F.(., ,U(*))MU{'))<h  
< (atii>ui(U(t)j) £  |F.(s, f/(.,))|
setting
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556 Arawomo and Akinyele
we have
(1 — fc)sutpu;(C/(t)) <  0.
which, by (5.4), implies sup w{U{t)) <  0 aiul lienee w(U(t))  — 0.
So U(t)  =  (r(t),r(t)], a degenerate interval, for a real function r(£) defined on / ,  
and since ti{i)  e  U(t) we have u(t) =  r(t) as required.
R em ark s: (a) If conditions in (1.3) are replaced with the conditions
/ ( i , x ) - / ( t , y ) > - A f ( : r - y ) ,  Af >  0
and
• the results o f theorems 3.1, 4.1 and 4.2 would still hold. However, an interval exten­
sion for f  which does not involve / „  would be needed to establish the results.
(b) If f x(t ,x (t ) )  in (1.3) is replaced with — Af, for positive constant A f, we obtain the 
result o f theorem 2.1 in (Lakslunikantham & Swaansundaran, 1987).
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4. Chan, C .Y . and Vatsala, A.S. (1990). Methods o f Upper and Ixjwcr Solution 
and Interval Methods for Scmilinear Euler-Poission-Darboux equations. J. Math. 
Anal, and Appl. v. 150, pp. 378-384.
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differential equations. Appl. Math and Com p. v. 23, pp. 1-5.
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S IA M  J M ath. Anal. v. 12, No. 3, pp. 321-341.
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