Journal
of Applied Mathematics
and Stochastic Analysis, 10:1
(1997),
89-94.
INTEGRAL INEQUALITIES OF GRONWALL TYPE
FOR PIECEWISE CONTINUOUS FUNCTIONS
DRUMI D. BAINOV
Medical University
of Sofia
P.O. Box 45
150 Sofia, Bulgaria
SNEZHANA G. HRISTOVA
Plovdiv University "Paissii Hilendarski"
Tsar Asen Str., 24, 4000 Plovdiv, Bulgaria
(Received March, 1995; Revised March, 1996)
In this paper
we generalize the integral inequality of Gronwall and study
the continuous dependence of the solution of the initial value problem for
nonlinear impulsive integro-differential equations of Volterra type on the
initial conditions.
Key words: Integral Inequalities, Piecewise Continuous Functions.
AMS subject classifications: 26D 10.
1. Introduction
In the present paper analogues of Gronwall’s inequality for piecewise continuous functions are introduced. The results obtained for these inequalities are applied to finding
sufficient conditions for continuous dependence on the initial conditions of the solutions of the initial value problem for nonlinear impulsive integro-differential equations.
The integral inequalities, established in this paper, can successfully be used in the
qualitative theory of the impulsive differential equations.
Let us note that the present paper generalizes some results obtained in [2-4].
2. Basic Notations. Auxihary Assertions
Let 0 _< t o < t I < t 2 <... and limk__.oot k cx3.
Denote by PC([to, oC),+) the set of all functions
u’[t0, oc)+, which are
piecewise continuous with discontinuity of the first kind at the points t k (k
u(t k + O) u(t k 0) < oc and u(tk) u(t k 0).
Lemma 1: (Theorem 16.4, [1]) Let for t > t o the inequality
Printed in the U.S.A.
()1997 by North
Atlantic Science Publishing Company
89
DRUMI D. BAINOV AND SNEZHANA G. HRISTOVA
90
u(t) <_ a(t) +
hold,
where
(kElP)
ilk(t)
PC([to, C),+)
ito
g(t,s)u(s)ds +
Then, for t
function for t,
(1)
k
nondecreasing functions
are
is a nondecreasing function,
continuous nonnegative
any fixed s t O.
E < k(t)u(tk )’
o<
for t>_to,
and g(t,s)
u PC([to, cX), +),
t 0 and nondecreasing with respect to t
s
a
E
is a
for
t o the following inequality is valid:
u(t)
(1 + k(t)) exp
a(t)
o
g(t,s)ds
k
(2)
to
3. Main Results
>_ t o
Theorem 1" Let for t
u(t) <_ a(t) +
the inequality
i b(s)u(s)ds,+
o
O
+
ito
ds
o
+
o<
to
k
PC([to, o), + ), a is nondecreasing, b C([to, cx), + ), k(t, s)
h(t,s, 7") are continuous and nonnegative functions for t, s, 7" >_ t o and flk >-- 0
(k [) are constants.
hold, where a, u
and
Then, the following inequality is valid:
u(t)
a(t)
0
<
k
<
(1 + ilk)
b(s)ds +
exp
to
k(s, r)dvds
to to
(4)
v
s
0 0 0
Pf;
Denote he right-hand side of lnequaliy (3) by (). The function
is nondecreasing, v(to) a(to) u(t) J v(t) for t t o and it satisfies
PC([to, ), +
the inequality
v(t)
a(t) +
/_
b(.) +
o
k(., r)dr +
o
o
+
to<tk<t
We apply Lemma 1 to inequality (5) for
o
h(., v,.)dadr v(s)d.
o
o
o
Integral Inequalities
and obtain inequality (4).
Theorem 2" Let for t
of Gronwall Type for Piecewise
Continuous Functions
>_ t o the following inequality hold
u(t) <_ a(t) +
/ b(t,s)u(s)ds +
k(t,s, 9")u(q’)d"
o
o
ds
o
k(t)u(tk )’
E
to<tk<t
+
,a
PC([to,),u+),
and nonnegative
(t) ( )
a i
(6)
oncan, (t,) a (t,,)
a coto
are nondecreasing with respect to t,
functions for t,s,v >_ t o and
a
91
odcai o t >_ to.
Then, for t >_ to, the following inequality is valid:
H
u(t) <_ a(t)
(1 + k(t)) exp
O<t k<:t
b(i, s)ds +
to
Proof: Denote the right-hand side of inequality
PC([to, cx), + is nondecreasing, u(t) <_ v(t) and
/
v(t) <_ a(t) +
5(t,) +
k(t, s, v)dq’ds
to to
(6) by v(t).
(t,,)d ()d +
The function v E
Z(/)(t).
o<
k
(7)
(8)
<
to
We apply Lemma 1 to inequality (8) and obtain inequality (7).
Theorem 3: Let for t >_ t o the following inequality hold
o
u(t) <_ a(t) +
ffto
b(s) u(s) +
k(v)u(v)dq" ds +
E < ku(tk )’
o<
to
where u, a G PC([t o,oo), + ), a is nondecreasing, b, k G
(k G N) are constants.
Then, for t >_ to, the following inequality is valid:
u(t) <_ a(t)--
k(v)a(v)dv
b(s) a(s)-t-
to
to
(9)
k
C([t o,c),N +),
E
ds -to< k<
>o
ka(tk
(10)
H < (1 +/) exp
o<
k
b(s)
1
to
+
k(v)dr
ds
to
Proof: Consider the function defined by the equality
v(t)-
]to 5()
()+
()()d
to
d+
o< k<
Z(t).
(11)
92
DRUMI D. BAINOV AND SNEZHANA G. HRISTOVA
The function v E
PC(It o, cx), +
is nondecreasing and satisfies the inequalities
u(t)<_a(t)+v(t)
(12)
and
f b(s)
v(t) <_
a(s) h-
k(7")a(r)
o
+
o
f
dvlds
+
E < ka(tk
o<
k
(13)
ds +
o
to<tk<t
o
From inequality (13) and Theorem 1
v(t) <_
b(s) a(s)-b
we obtain the inequality
k(7)a(v)dv
to
to
(1 +flk)exp
II
to<tk<t
E
ds +
o< k<
ka(tk
(14)
b(s)
1
+
o
k(r)dr
o
Thus, (10) follows from inequalities (12) and (14).
Corollary 1: Let the conditions of Theorem 3 hold for a(t)
Then, for t >_ to, the following inequality is valid:.
u(t) <_a
b(s) 1+
1+
to
H < (l+flk) exp
x
o<
k
ds
k(v)dT" ds/
const >_ O.
E<
to
o<
b(s) 1+
k(v)dv
to
a
k
ds
to
4. Application
With the aid of the established inequalities we shall analyze the continuous
dependence of the solutions of the initial value problem for impulsive integrodifferential equations on the initial data.
Consider the nonlinear impulsive integro-differential equation
ic
k(t,s,x(s))ds
f t,x,
o
for t
7 tk,
(15)
Integral Inequalities
of Gronwall Type for Piecewise
93
Continuous Functions
(16)
with initial condition
(17)
X(to)- Xo,
1.
It
t k x(tk + 0)- x(tk- 0).
Theorem 4: Let the following conditions hold:
The function f E C([t 0, c)x x ,N) and it satisfies the inequality
where Ax
If( t, Xl’ Yl)-- f(t, x2, Y2)
<- g(t)
xl
x2
+ h(t)
, h e C([t0,
),
k
+ ).
The function G C([t 0’ oo)x [to, oo)x N,N)
w
.
](t, 8, Xl)-k(t,s, x2) _< m(t,s)
and it
c([to, oo) [to, oo), + ).
Ik(x)’N--N (k e N) satisfy the
’
l,
Xl,X2
e
inequality
Ik(Xl)--Ik(X2) <_ flklXl--X21,
4.
Xl’ X2’ Yl’ Y2
satisfies the inequality
x 1-x 2
The functions
Y2
yl
Xl,X 2
eN,
,
to.
where /k const > O.
For each point x o e
the initial value problem (15), (16), (17) has a solution
(; to, o) fo t >_
Then, the solutions of equation (15), (16) depend continuously on the initial conditions, i.e., for any number e >0, there exists a number 5 >0 such that for
Xo- Yol < 5 the inequality
(t; to, o) (t; to, yo) <
[to,T], T
holds for t
Proof."
Let
e
> to, T < oo.
an arbitrary number.
const
> 0 be
x(t;to, Xo)- x(t;to, Yo) I,
Consider the function u(t)which by the condition of Theorem 4 satisfies the
inequality
(t) _< o yo
+
/
o
I
()() + h()
/s
o
+
(18)
to<tk<t
<-
Xo YO I+
/to g(s)u(s)ds /to h(s)m(s,)u()dTds
4-
E < #ku(tk )"
4o<
From inequality (18), by Theorem 2 we obtain the inequality
k
DRUMI D. BAINOV AND SNEZHANA G. HRISTOVA
94
<
Htlc < (1 +/k) exp
I :o-
g(s)ds +
h(s)m(s, v)d7ds
to
to
g(s)ds +
h(s)m(s, r)dvds
(19)
We choose 5 > 0 such that
0
< 5 <e
/
o<
Inequalities
H<
k
(19)
(1 +/k) exp
T
and
to
(20)
(20)
o
yield the assertion of Theorem 4.
Acknowledgement
The present investigation was supported by the Bulgarian Ministry of Education,
Science and Technologies under Grant MM-511.
References
[1]
[4]
Bainov, D.D. and Simeonov, P.S., Integral Inequalities and Applications,
Kluwer Academic Publishers, Dordrecht 1992.
Simeonov, P.S. and Bainov, D.D., On an integral inequality for piecewise continuous functions, J. Math. Phys. Sci. 21:4 (1987), 315-323.
Simeonov, P.S. and Bainov, D.D., Integral and differential inequalities for a
class of piecewise continuous functions, Ann. Polon. Math. XLVIII (1988), 207216 (in Russian).
Simeonov, P.S. and Bainov, D.D., Perturbation theorems for systems with impulse effect, Int. J. Systems Sci. 19:7 (1988), 1213-1223.