COMPARISON THEOREMS RELATED TO A CERTAIN

SOOCHOW JOURNAL OF MATHEMATICS
Volume 22, No. 3, pp. 383-394, July 1996
COMPARISON THEOREMS RELATED TO A CERTAIN
INEQUALITY USED IN THE THEORY
OF DIFFERENTIAL EQUATIONS
BY
B. G. PACHPATTE
Abstract. In the present paper we establish some new comparison
theorems related to a certain integral inequality arising in the theory of
di erential equations. The results that we propose here can be used as
tools in the theory of certain new classes of di erential, integral, nite
di erence and sum-di erence equations.
1. Introduction
Comparison method has become a major tool in the analysis of various
nonlinear di erential equations that occurs in the nature or are built by man.
The idea of studying a system of ordinary di erential equations by comparing
it with a single rst order equation derived naturally from an estimate on the
system has been used as early as 1930 by Kamke 6] to establish a general
uniqueness theorem. In 1956, Conti 4] proved a very general comparison
theorem which has been used with considerable success in studying many
problems in the theory of systems of di erential equations. For a good account
of earlier developments, see 1,3,5,12-14] and the references given therein.
In the literature there are many papers which make use of the following
Received August 14, 1995.
AMS Subject Classication. 26D15, 26D20.
Keywords. comparison theorems, integral inequality, theory of di erential equations,
nite di erence and sum-di erence equations, bounds on the solutions.
383
384
B. G. PACHPATTE
inequality.
Lemma 1. Let y and f be real-valued nonnegative continuous functions
dened for t 2 R+ = 0 1). If
y2 (t) c2 + 2
Zt
0
f (s)y(s)ds
for t 2 R+ , where c 0 is a real constant, then
y(t) c +
Zt
0
f (s)ds
for t 2 R+ .
The above inequality has been frequently used to study the global existence, stability, boundedness and other properties of the solutions for wide
classes of nonlinear di erential equations. For some recent results related
to this inequality and their applications, see 9-11] and the references cited
therein. The aim of the present paper is to establish some new comparison
theorems related to the inequality given in Lemma 1, which can be used as
handy tools in the analysis of certain new classes of di erential, integral, nite
di erence and sum-di erence equations. An interesting feature of our results
lies in their fruitful utilization to the situations for which the other available
comparison theorems do not apply directly. We also present some immediate
applications to convey the importance of our results to the literature.
2. Statement of Results
In this section we state our results to be proved in this paper. In what
follows we denote by R the set of real numbers, R+ = 0 1) and N0 =
f0 1 2 : : : g. For any function u(n) dened for n 2 N0 , we dene the operator
by u(n) = u(n + 1) ; u(n). For m > n, m, n 2 N0 and any function p(n)
dened for n 2 N0 , we use the usual conventions
n
X
s=m
p(s) = 0
n
Y
s=m
p(s) = 1:
COMPARISON THEOREMS RELATED TO A CERTAIN INEQUALITY
385
We need the following known comparison results in the proofs of our main
theorems (see, 12,13] and 7]).
Lemma 2. Let w(t r) be a real-valued continuous function dened for
t 2 R+ , 0 r < 1. Let u(t) be a real-valued dierentiable function dined
for t 2 R+ such that
u (t) w(t u(t))
0
for t 2 R+ . Let r(t) be a maximal solution of
r (t) = w(t r(t)) r(0) = r0
0
for t 2 R+ such that u(0) r0 . Then
u(t) r(t) t 2 R+:
Lemma 3. Let w(n r) be a real-valued function dened for n 2 N0,
0 r < 1, and monotone nondecreasing with respect to r for any xed
n 2 N0 . Let u(n) be a real-valued function dened for n 2 N0 such that
u(n) w(n u(n))
for n 2 N0 . Let r(n) be a solution of
r(n) = w(n r(n))
r(0) = r0
for n 2 N0 such that u(0) r0 . Then
u(n) r(n)
n 2 N0 :
Our main results are given in the following theorem.
Theorem 1. Let y, f , g be real-valued nonnegative continuous functions
dened on R+ and c be a nonnegative real constant. Let w(t r) be a realvalued nonnegative continuous function dened for t 2 R+ , 0 r < 1, and
monotone nondecreasing with respect to r for any xed t 2 R+ .
386
B. G. PACHPATTE
(a1 ) If
y2 (t) c2 + 2
for t 2 R+ , then
Zt
0
y(s)w(s y(s))ds
y(t) r(t)
(2:1)
t 2 R+
(2:2)
where r(t) is a maximal solution of
r (t) = w(t r(t)) r(0) = c
(2:3)
0
for t 2 R+ .
(a2 ) If
y2(t) c2 + 2
for t 2 R+ , then
Zt
0
y(s)f (s)y(s) + w(s y(s))]ds
y(t) A(t)r(t) t 2 R+
where
Zt
A(t) = exp( f (s)ds)
and r(t) is a maximal solution of
0
r (t) = w(t A(t)r(t))
0
for t 2 R+ .
(a3 ) If
y2 (t) c2 + 2
for t 2 R+ , then
Zt
0
where
B (t) = 1 +
y(s)f (s) y(s) +
Zs
0
Zt
0
(2:5)
t 2 R+
(2:6)
r(0) = c
(2:7)
g()y()d + w(s y(s))]ds
y(t) B (t)r(t)
(2:4)
t 2 R+
Zs
f (s) exp( f () + g()]d)ds
0
(2:8)
(2:9)
t 2 R+
(2:10)
and r(t) is a maximal solution of
r (t) = w(t B (t)r(t))
0
r(0) = c
(2:11)
COMPARISON THEOREMS RELATED TO A CERTAIN INEQUALITY
for t 2 R+ .
(a4 ) If
y2 (t) c2 + 2
for t 2 R+ , then
where
Zt
0
Zs
y(s)f (s)(
0
g()y()d) + w(s y(s))]ds
y(t) E (t)r(t)
Zt
Zs
0
0
t 2 R+
E (t) = exp( f (s)( g()d)ds)
and r(t) is a maximal solution of
r (t) = w(t E (t)r(t))
0
387
(2:12)
(2:13)
t 2 R+
r(0) = c
(2:14)
(2:15)
for t 2 R+ .
The discrete analogues of the results given in Theorem 1 are established
in the following theorem.
Theorem 2. Let y, f , g be real-valued nonnegative functions dened on
N0 and c be a nonnegative real constant. Let w(t r) be a real-valued nonnegative function dened for n 2 N0 , 0 r < 1, and monotone nondecreasing
with respect to r for any xed n 2 N0 .
(b1 ) If
nX1
y2(n) c2 + 2 y(s)w(s y(s))
(2:16)
;
s=0
for n 2 N0 , then
y(n) r(n)
n 2 N0
(2:17)
where r(n) is a solution of
r(n) = w(n r(n))
for n 2 N0 :
(b2 ) If
y2 (n) c2 + 2
nX1
;
s=0
r(0) = c
y(s)f (s)y(s) + w(s y(s))]
(2:18)
(2:19)
388
B. G. PACHPATTE
for n 2 N0 , then
y(n) L(n)r(n)
where
L(n) =
and r(n) is a solution of
n 2 N0
nY1
;
s=0
1 + f (s)]
(2:20)
n 2 N0
r(n) = w(n L(n)r(n))
(2:21)
r(0) = c
(2:22)
for n 2 N0 .
(b3 ) If
y2 (n) c2 + 2
nX1
;
s=0
y(s)f (s)(y(s) +
for n 2 N0 , then
where
s 1
X
;
t=0
g(t)y(t)) + w(s y(s))]
y(n) P (n)r(n)
P (n ) = 1 +
and r(n) is a solution of
nX1
n 2 N0
sY1
;
;
t=0
r(n) = w(n P (n)r(n))
for n 2 N0 .
(b4 ) If
y2(n) c2 + 2
for n 2 N0 , then
where
nX1
;
s=0
y(s)f (s)(
s 1
X
;
t=0
y(n) Q(n)r(n)
Q(n) =
nY1
s 1
X
s=0
t=0
;
(2:24)
f (s) 1 + f (t) + g(t)]
s=0
1 + f (s)
;
n 2 N0
r(0) = c
g(t)y(t)) + w(s y(s))]
n 2 N0
g(t) ]
(2:23)
(2:25)
(2:26)
(2:27)
(2:28)
n 2 N0
(2:29)
COMPARISON THEOREMS RELATED TO A CERTAIN INEQUALITY
389
and r(n) is a solution of
r(n) = w(n Q(n)r(n))
r(0) = c
(2:30)
for n 2 N0 .
3. Proofs of Theorems 1 and 2
Since the proofs resemble one another, we give the details for (a1 ), (a2 ),
(b3 ), (b4 ) only, the proofs of (a3 ), (a4 ), (b1 ), (b2 ) can be completed by following
the proofs of the above mentioned results.
(a1 ) we rst assume that c > 0 and dene a function z (t) by
z(t) = c2 + 2
Zt
0
y(s)w(s y(s))ds:
(3:1)
p
From (3.1) and using the fact that y(t) z (t), we observe that
q
q
z (t) 2 z(t)w(t z(t)):
0
(3:2)
p
Di erentiating z (t) we have
d
dt
q
z(t) = 12 pz (t) :
z(t)
0
By using (3.2) in (3.3) we get
q
d
dt
q
(3:3)
z (t) w t z (t) :
(3:4)
Now a suitable application of Lemma 1 to (3.4) and (2.3) yields
q
z (t) r(t) t 2 R+
(3:5)
where r(t) is a maximal solution of (2.3). Now by using the fact that y(t) z(t) in (3.5), we get the required inequality in (2.2).
If c is nonnegative, we can carry out the above procedure with c + instead
of c, where > 0 is an arbitrary small constant, and subsequently pass to the
limit ! 0 to obtain (2.2). This completes the proof of (a1 ).
p
390
B. G. PACHPATTE
(a2 ) By assuming that c > 0 and dening a function z (t) by the right side
of (2.4) and following the same steps as in the proof of (a1 ) we have
d
dt
q
q
q
z (t) f (t) z(t) + w t z(t) :
(3:6)
From (3.6) it is easy to observe that
Zt
q
z (t) c +
0
q
f (s) z(s)ds +
Dene a function m(t) by
m(t) = c +
Zt
0
Zt
0
q
w(s z(s))ds:
q
w(s z(s))ds:
(3:7)
(3:8)
By using (3.8), the inequality (3.7) can be written as
q
z (t) m(t) +
Zt
0
q
f (s) z (s)ds:
(3:9)
Since m(t) is positive and monotone nondecreasing for t 2 R+ , the inequality
(3.9) implies the estimate (see, 2, p.56])
q
z(t) A(t)m(t) t 2 R+
(3:10)
where A(t) is dened by (2.6). From (3.8) and using (3.10) we observe that
m (t) w(t A(t)m(t))
0
(3:11)
for t 2 R+ . Now a suitable application of Lemma 1 to (3.11) and (2.7) yields
m(t) r(t) t 2 R+
(3:12)
where r(t) is a maximal solution of (2.7). From (3.10) and (3.12) we have
q
z(t) A(t)r(t) t 2 R+:
p
(3:13)
Now by using the fact that y(t) z (t) in (3.13) we get the required inequality
in (2.5). The proof of the case when c is nonnegative can be completed as
mentioned in the proof of (a1 ). This completes the proof of (a2 ).
COMPARISON THEOREMS RELATED TO A CERTAIN INEQUALITY
391
(b3 ) We assume that c > 0 and dene a function z (n) by
z (n) = c2 + 2
nX1
;
s=0
y(s)f (s)(y(s) +
s 1
X
;
t=0
g(t)y(t)) + w(s y(s))]:
(3:14)
p
From (3.14) and using the fact that y(n) z (n) we observe that
q
h
z (n) 2 z (n) f (n)
q
z(n) +
nX1
;
t=0
q
q
i
g(t) z(t) + w(n z(n)) :
(3:15)
It is easy to observe that
q
z (n) pz(n) :
( z (n)) = pz (n + 1) ; p
z (n + 1) + z(n) 2 z (n)
p
(3:16)
p
Here in the last step of (3.16) we have used the fact that z (n) z (n + 1).
By using (3.15) in (3.16) we get
q
q
z (n)) f (n) z (n) +
nX1
;
t=0
q
q
q
g(t) z (t) + w(n z(n)):
(3:17)
From (3.17) it is easy to observe that
q
z(n) c +
nX1
;
s=0
q
f (s) z(s) +
s 1
X
;
t=0
g(t) z(t) +
nX1
;
s=0
q
w(s z (s)):
(3:18)
Dene a function m(n) by
m(n) = c +
nX1
;
s=0
q
w(s z(s)):
(3:19)
By using (3.19), the inequality (3.18) can be written as
q
z (n) m(n) +
nX1
;
s=0
q
f (s)( z(s) +
s 1
X
;
t=0
q
g(t) z (t)):
(3:20)
Since m(n) is positive and monotone nondecreasing for n 2 N0 , the inequality
(3.20) implies the estimate (see, 8, Theorem 20 ])
q
z(n) P (n)m(n) n 2 N0
(3:21)
392
B. G. PACHPATTE
where P (n) is dened by (2.25). From (3.19) and using (3.21) we observe that
m(n) w(n P (n)m(n))
(3:22)
for n 2 N0 . Now a suitable application of Lemma 2 to (3.22) and (2.26) yields
m(n) r(n) n 2 N0
(3:23)
where r(n) is a solution of (2.26). Using (3.23) in (3.21) we have
q
z(n) P (n)r(n) n 2 N0:
(3:24)
p
Now by using the fact that y(n) z (n) in (3.24) we get the required inequality in (2.24). The proof of the case when c is nonnegative can be completed
as mentioned in proof of (a1 ). The proof of (b3 ) is complete.
(b4 ) By assuming that c > 0 and dening a function z (n) by the right side
of (2.27) and following the same steps as in the proof of (b3 ) upto (3.17) we
have
nX1
q
q
q
(3:25)
z (n) f (n) g(t) z (t) + w(n z (n)):
;
t=0
p
From (3.25) and using the fact that z (n) is monotone nondecreasing for
n 2 N0 , it is easy to observe that
q
z (n) c +
c+
nX1
;
s=0
nX1
;
s=0
f (s)
s 1
X
;
t=0
q
q
g(t) z(t) +
s 1
X
f (s) z (s)(
;
t=0
g(t)) +
nX1
;
s=0
nX1
;
s=0
q
w(s z (s))
q
w(s z (s)):
(3:26)
Now by following the same steps as in the proof of (b3 ) below (3.18) with
suitable modications we get the required inequality in (2.28). This completes
the proof of (b4 ).
4. Some Applications
In this section, we indicate some applications of our results to obtain
bounds on the solutions of certain di erential and sum-di erence equations.
These applications are given as examples.
COMPARISON THEOREMS RELATED TO A CERTAIN INEQUALITY
393
Example 1. As a rst application we obtain a bound on the solution of
the di erential equation of the form
x (t) = h(t x(t)) + (t x(t)) x(0) = x0
(4:1)
where h : R+ R ! R are continuous functions. Multiplying both sides of
equation (4.1) by x(t), substituting t = s, and then integrating it from 0 to t
we have
Zt
x2 (t) = x20 + 2 x(s)h(s x(s)) + (s x(s))]ds:
(4:2)
0
We assume that
jh(t x(t))j f (t)jx(t)j
(4:3)
j(t x(t))j w(t jx(t)j)
(4:4)
where f and w are as dened in Theorem 1. From (4.2)-(4.4) we observe that
0
jx(t)j2
jx0
j2 + 2
Zt
0
jx(s)jf (s)jx(s)j + w(s jx(s)j)]ds:
(4:5)
Now an application of (a2 ) yields
jx(t)j A(t)r(t) t 2 R+
(4:6)
where A(t) is dened by (2.6), and r(t) is a maximal solution of (2.7) with
c = jx0 j. The inequality (4.6) gives the bound on the solution x(t) of equation
(4.1) in terms of the known function A(t) and the maximal solution r(t) of
(2.7). If r(t) is bounded and A(t) is nite, then (4.6) implies that the solution
x(t) of (4.1) is bounded on R+.
Example 2. As a second application, we shall obtain a bound on the
solution of the following sum-di erence equation
x2 (n) = 2x(n)F (n x(n)
nX1
;
t=0
k(n t x(t)) + G(n x(n))] x(0) = x0 (4:7)
where G k F are real-valued functions dened respectively on N0 R, N02 R,
N0 R2. We assume that
jG(n x(n))j w(n jx(n)j)
(4:8)
jk(n t x(t))j g(t)jx(t)j
(4:9)
jF (n x(n) u)j f (n)(jx(n)j + juj)
(4:10)
394
B. G. PACHPATTE
where f g w are as dened in Theorem 2. The equation (4.7) is equivalent to
the following equation
x2 (n) = x20 + 2
nX1
;
s=0
x(s)F (s x(s)
s 1
X
;
t=0
k(s t x(t))) + G(s x(s))]:
(4:11)
Now by using (4.8)-(4.10) in (4.11) and applying (b3 ) we have
jx(n)j P (n)r(n) n 2 N0
(4:12)
where P (n) is dened by (2.25) and r(n) is a solution of (2.26), with c = jx0 j.
If r(n) is bounded and P (n) is nite, then (4.12) implies that the solution
x(n) of (4.7) is bounded on N0 .
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Department of Mathematics, Marathwada University, Aurangabad 431 004, (Maharashtra)
India.

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