Journal
of Applied
Mathematics and Stochastic Analysis, 12:3
(1999),
253-260.
UNIQUENESS OF SOLUTIONS TO
INTEGRODIFFERENTIAL AND FUNCTIONAL
INTEGRODIFFERENTIAL EQUATIONS
D. BAHUGUNA
of Technology
Department of Mathematics
Indian Institute
Kanpur 208 015 India
L.E. GAREY
of New Brunswick
University
Department of Mathematics, Statistics and Computer Science
P.O. Boa: 5050, Saint John, N.B. Canada E2L 4L5
(Received April, 1998; Revised August, 1998)
In this paper
we study a class of integrodifferential and functional integrodifferential equations with infinite delay. These problems are reformulated
as abstract integrodifferential and functional integrodifferential equations.
We use Nagumo type conditions to establish the uniqueness of solutions to
these abstract equations.
Key words: Integrodifferential Equation, Mild Solution, Nagumo Condition, Finite and Infinite Delays.
AMS subject classifications: 34G20, 24K15.
1. Introduction
In the present work
we are concerned with the following integrodifferential and functional integrodifferential equations considered in a real Banach space X"
du(t)
dt
I[t’ it(t),
/ ]l(S’ u(s))ds],
t
> o
(1.1)
o
(to)
du(t)
g[t’ ut’
x
/ k2(s’ us)ds ],
> to,
(1.2)
o
Printed in the U.S.A.
(C)1999 by North
Atlantic Science Publishing Company
253
D. BAHUGUNA and L.E. GAREY
254
(1.1) the nonlinear map f is defined fromJ x X x X into X, J (to, t o + T],
t
[t0, 0 + T], 0 < T < cx, the nonlinear map k I is defined from J x X into X,
x E X, and in (1.2), 9 is defined from J x C X x C x into X, C x BUC((- cx,0];X)
(BUC(I;X) denotes the space of bounded uniformly continuous functions from the
interval I into a Banach space X endowed with the supremum norm), for any u E
BUC((- cxz, t o + T]; X) and t [to, t o + T] the map u C x is defined by
where in
J-
the nonlinear map k 2 is defined from J x Cx into Cx and G Cx.
Particular cases of (1.1) and (1.2), in which k I k 2 _= 0, have been considered by
many authors, see for instance, Rogers [10], Kotta [4, 5]. For the case X R n, we
refer to Kappel and Schappacher [3].
In the present work, we shall be concerned with the uniqueness of solutions only.
Proving the existence of solutions to (1.1) and (1.2) will be our next concern. For the
existence of solutions for the particular cases mentioned above, we refer to Hale [1],
Ladas and Lakshmikantham [6], Martin and Smith [8], and Martin [9].
2. Preliminaries
We shall establish the uniqueness of mild solutions to (1.1) and (1.2), which would
also establish the uniqueness in the case of classical solutions.
By a mild solution to (1.1) we mean a continuous function u
8
to
o
o
where x E X. By a mild solution to
BUC(( oc, t o + T], X) such that
(1.2)
where
Cx.
We consider
f of[S,
us,
", ur)d7]ds,
lim
to
/ h(t)dt
we may take
h(t)Then h satisfies condition
(H).
1
(t-to)
(2.1)
t
< t <_ to,
(2.2)
t o + T,
C(J, R + satisfying the following condition
t-,to+
For, instance,
_< t _< t o + T,
-c
f 0kl(7
a function h
__
such that
we mean a continuous function
(t- to),
(0) +
C(J ,X)
teJ.
Uniqueness
Remark:
If h
C(J, R + given by
EC(J,R +)
h
of Solutions
(H),
satisfies condition
h(t) + C,
(t)
255
then the function
eJ
for any positive constant C also satisfies condition (H).
The main tool for proving the uniqueness of solutions is the following lemma due
to Kotta [5]. This result is a generalization of an analogous result of Rogers [10].
For the sake of completeness, we shall give a proof of the lemma here.
Lemma 2.1" Let u C(,R +) and suppose that
(i)
(ii)
<
u(t)- o(e ’f h(t)dt), as t--to+
C(J, R + satisfying condition (H).
where h
Proof: Let
/ h(s)u(s)ds,
F(t)Using (i) and
(2.3)
Then u
t
0 on
.
e J.
(2.3)
o
we obtain
tF(t)
h(t)u(t) < h(t)F(t).
Therefore,
-t(e- f h(t)dtF(t)) < O.
Inequality (2.4)implies that F(t)e- f h(t)dt s a nonincreasing function on J.
From (ii) it follows that for every e > 0, there exists a 5 > 0 such that for
with 0 < t-t o < 5, we have
e- f h(t)dtF(t) e- f h(t)dt
(2.4)
EJ
/ h(s)u(s)ds
o
<_ ee- f h(t)dt
(2.5)
o
Now
Ct f
e
Integrating
/ h(s)e f h(s)dSd8.
(2.6)
over the interval
e
h(t)e f h(t)dt.
h(t)dt
(to, t),
lt
f h(t)dt to
we get
f
o
Using condition (H)in
(2.7),
we
get
(2.6)
h(s)e f h(s)dSds.
(2.7)
_
D. BAHUGUNA and L.E. GAREY
256
e
/ h(8)e h(s)dsdS.
fh(t)dt-
o
Using (2.8)in
(2.5),
we
get
e- f h(t)dtF(t)
.
(2.9)
From (2.9), we have that
lira
t--to+
Hence, F(t) -0
on
e- f h(t)dtF(t)
O.
J, which in turn implies that u-0
on
J.
3. Main Result
We first state and prove the following uniqueness theorem for (1.1).
Theorem 3.1" Suppose that f’J XX---,X is a continuous function and
satisfies the
conditions
I f(t, Xl, yl)- f(t, x2, Y2)II
for
any t
J,
xi, Yi
X,
_< h(t)[ I Xl x2 I
1,2; and
I f(t, x(t), y(t)) I
o(h(t)e f h(t)dt)
any t G if, x,y @ C(ff;X), where h G
suppose that kl: J x X--,X satisfies
for
as
-
y2
II]
t---to+
C(J,R+ satisfies
I ]Cl(t, Xl)- ]1(t, x2)II
I yl
condition
(3.2)
(FI). Further,
c(t)II Xl X2 I
any t E J,x
X,
1,2; where C(t) is a nonnegative integrable
Then (1.1) has at most one solution.
Proof: Let x(t) and y(t) be solutions of (1.1). Then we have
for
(3.1)
(3.3)
function
on
J.
8
o
o
8
f[(s,y(s),
/ kl(r,y(v))dT" I
d.
(a.4)
o
Using
(3.1)
and
(3.3)in (3.4),
I1 ,(t)- y(t)II
we
get
/ h()[ I x()- y()II + /
o
o
I ’1(7,x(7))- ]1(7", Y(7"))II d]d
Uniqueness
of Solutions
257
8
f
h(s)[ I (s)- y(s)II
+
/ C()II ()- y()II d]d
o
o
<--
/ (h(s) -+- CT) I t0<r<s I x()- (,)II }
sup
d
O
t0<v<s
o
where
to+T
/
CT
C(s)ds
o
and
h(t)-h(t)+C T.
Now, (3.5) implies that for every t o _< r _< t,
to<r<s
o
t0<r<s
o
From (3.6)
we
get
0<r<t
O<r<t
0<v<s
o
Replacing the dummy variable
finally get
,
r
on the left-hand side of inequality
t O<-<s
J
_
o
Now, using (3.2) in inequality (3.4)
< t-t o < 5,
that for 0
we have
> 0, there
/ h(s)e f h(s)dsds
o
/ (h(s)
o
-t-
we
,
we have that for e
Il x(t)- y(t) l <_
(3.7) by r,
CT)eCTSe f h(s)dSds
exists 5
> 0 such
D. BAHUGUNA and L.E. GAREY
258
(3.9)
o
<_ 7- _< t,
Again, for t o
we have
7"
o
</ () f ()d.
(3.10)
o
Taking supremum in
sup
to<r<t
(3.10)
we
get
I ()- y()II
-/ (s)eJ" (s)dSds eeJh’" (t)dt.
J
(3.11)
o
We obtained the desired result using Lemma 2.1. This completes the proof of
Theorem 3.1.
Next, we state and prove below a similar uniqueness result for (1.2).
Theorem 3.2: Suppose that g:J x C x x C x---C x is continuous and satisfies the
condition
I g(t,l,l)-g(t, dp2,2)II c x < h(t)[ I
for
any t E J
i,iGC X, i-1,2; and
I[ g(t, tt, tt) I x
for any t e
(It). Suppose
-
I1 c x / I 1-2 I Cx],
o(h(t)ef h(t)dt), as t--,to+
(3.12)
(3.13)
,2 e C(-cxz, t o-t-7"];X), where h @ C(J,I?. +) satisfies condition
k2: J C x--C x is continuous and satisfies
that
[I k2(t,l) k(t,2) I c X_< D(t)II 1- 2 I C x
(3.14)
where D(t) is a nonnegative inlegrable function on J. Then (1.2) has at most one
solution.
Proof: Suppose that x and y are two solutions to (1.2). Then
implies that
Xto-Yto
Therefore,
I[xt Yt [] C X
s
_
x- y, on
sup
(-cx,O]
(- cxz, to].
[[ x(t + s)- yt + s)[[
max
I (0)- y(t)II.
0E[t0, t]
From definition (2.2) of mild solutions to (1.2) and (3.15)
we have
(3.5)
of Solutions
Uniqueness
259
8
to
Using
-
to
to
(3.12)and (3.14)in (3.16),
I
8
(3.16)
we obtain
, il x -< i h(,)[ I .-
8
i D(’r)II :.-
I
’, c x +
o
,.-il x dr]ds
o
8
<
.- Y. I c x
D(r)dv)[ I
(h(s) +
o
o
<
(h(s) 4- DT)II
,.
o
_<
h
()II
.-
o
where
.
y. I c X ds
I cxd,
(3.17)
to+T
/
DT
D(s)ds
o
and
h
Now, using (3.13) in (3.16),
h(t) + DT,
(t)
we have that for e
t
e J.
> 0, there
exists
> 0 such
that for
O<t-to<5,
I ,- a I c x <_
J h()
h(s)dSds
o
<--el
(h(s) + DT)eDTSe f h(s)dsds
o
<-
1
(") f 7 (")"d.
o
eef (t) dr.
(3.18)
Again, we apply Lemma 2.1 to get the desired result. This completes the proof of
Theorem 3.2.
D. BAHUGUNA and L.E. GAREY
260
Acknowledgement
The first author would like to acknowledge the financial support provided by the
National Board for Higher Mathematics under its research project No. 48/2/97-R&DII/2003 to carry out this work.
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[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[lO]
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