Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 908784, 15 pages
doi:10.1155/2008/908784
Research Article
New Retarded Integral Inequalities
with Applications
Ravi P. Agarwal,1 Young-Ho Kim,2 and S. K. Sen1
1
Department of Mathematical Sciences, Florida Institute of Technology, 150 West University Boulevard,
Melbourne, FL 32901, USA
2
Department of Applied Mathematics, College of Natural Sciences Changwon National University
Changwon, Kyeongnam 641-773, South Korea
Correspondence should be addressed to Young-Ho Kim, yhkim@changwon.ac.kr
Received 29 January 2008; Accepted 24 April 2008
Recommended by Yeol Je Cho
Some new nonlinear integral inequalities of Gronwall type for retarded functions are established,
which extend the results Lipovan 2003 and Pachpatte 2004. These inequalities can be used as
basic tools in the study of certain classes of functional differential equations as well as integral
equations. A existence and a uniqueness on the solution of the functional differential equation
involving several retarded arguments with the initial condition are also indicated.
Copyright q 2008 Ravi P. Agarwal et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
The celebrated Gronwall inequality 1 states that if u and f are nonnegative continuous
functions on the interval a, b satisfying
ut ≤ c t
fsusds,
t ∈ a, b,
1.1
a
for some constant c ≥ 0, then
ut ≤ c exp
t
fsds ,
t ∈ a, b.
1.2
a
Since the inequality 1.2 provides an explicit bound to the unknown function u and
hence furnishes a handy tool in the study of solutions of differential equations. Because of its
2
Journal of Inequalities and Applications
fundamental importance, several generalizations and analogous results of the inequality 1.2
have been established over years. Such generalizations are, in general, referred to as Gronwall
type inequalities 2–6. These inequalities provide necessary tools in the study of the theory
of differential equations, integral equations, and inequalities of various types. Many authors
2–21 have established several other very useful Gronwall-like integral inequalities. Among
these inequalities, the following one Theorem A due to Ou-Yang 15 needs specific mention.
It is useful in the study of boundedness of certain second-order differential equations.
Theorem A Ou-Yang 15. If u and f are nonnegative continuous functions on 0, ∞ such that
t
2
2
u t ≤ u0 2 fsusds
1.3
0
for all t ∈ 0, ∞, where u0 ≥ 0 is a constant, then
t
ut ≤ u0 fsds,
t ∈ 0, ∞.
1.4
0
The Ou-Yang inequality prompted researchers to devote considerable time for its
generalization and consequent applications 3, 4, 9, 11, 14. For instance, Lipovan established
the following generalization Theorem B of Ou-Yang’s inequality in the process of establishing
a connection between stability and the second law of thermodynamics 14.
Theorem B Lipovan 14. Let u, f, and g be continuous nonnegative functions on R and c be a
nonnegative constant. Also, let w ∈ CR , R be a nondecreasing function with wu > 0 on 0, ∞
and α ∈ C1 R , R be nondecreasing with αt ≤ t on R . If
αt
2
2
u t ≤ c 2
fsusw us gsus ds
1.5
0
for all t ∈ R , then for 0 ≤ t ≤ t1 ,
αt
αt
ut ≤ Ω Ω c gsds fsds ,
−1
0
r
ds
Ωr ,
ws
1
0
1.6
r > 0,
Ω−1 is the inverse function of Ω, and t1 ∈ R is chosen so that Ωc DomΩ−1 for all t ∈ R lying in the interval 0 ≤ t ≤ t1 .
αt
0
gsds αt
0
fsds ∈
More recently, Pachpatte established further generalization Theorem C of Theorem B as
follows 20, which is handy in the study of the global existence of solutions to certain integral
equations and functional differential equations.
Theorem C Pachpatte 20. Let u, ai , bi ∈ CI, R and let αi ∈ C1 I, I be nondecreasing with
αi t ≤ t on I for i 1, . . . , n. Let p > 1 and c ≥ 0 be constants and w ∈ CR , R be nondecreasing
with wu > 0 on 0, ∞. If for t ∈ I,
n αi t
up t ≤ c p
us ai sψ us bi s ds,
1.7
i1
αi t0 Ravi P. Agarwal et al.
3
then for t0 ≤ t ≤ t1 ,
n
ut ≤ G GAt p − 1
−1
αi t
i1
where
Gr r
r0 w
At c
ds
s1/p−1
p−1/p
ai σdσ
1/p−1
,
n αi t
αi t0 i1
1.9
bi σdσ,
r0 > 0 is arbitrary, G−1 is the inverse function of G and t1 ∈ I is so chosen that
n αi t
ai σdσ ∈ Dom G−1 .
GAt p − 1
αi t0 i1
1.8
r ≥ r0 > 0,
,
p − 1
αi t0 1.10
The present paper establishes some nonlinear retarded inequalities which extend the
foregoing theorems. In addition, it illustrates the use/application of these inequalities.
2. Main results
Let R denote the set of real numbers, R 0, ∞ and R1 1, ∞. Also, let I t0 , T be
the given subset of R. Denote by Ci M, N the class of all i-times continuously differentiable
functions defined on the set M to the set N for i 1, 2, . . . and C0 M, N CM, N.
Theorem 2.1. Let u, fi , gi ∈ CI, R , i 1, . . . , n, and let αi ∈ C1 I, I be nondecreasing with
αi t ≤ t, i 1, . . . , n. Suppose that c ≥ 0 and q > 0 are constants, ϕ ∈ C1 R , R is an increasing
function with ϕ∞ ∞ on I, and ψu is a nondecreasing continuous function for u ∈ R with
ψu > 0 for u > 0. If
n
ϕ ut ≤ c i1
αi t
αi t0 uq s fi sψ us gi s ds
2.1
for t ∈ I, then
−1
ut ≤ ϕ
G
−1
Ψ
−1
n
Ψ k t0 i1
αi t
αi t0 fi sds
2.2
for t ∈ t0 , t1 , where
Gr Ψr r
ds
,
r ≥ r0 > 0,
r0
q
ϕ−1 s
r0
ψϕ−1 G−1 s
r
ds
n
k t0 Gc i1
αi t
αi t0 ,
r ≥ r0 > 0,
gi sds,
2.3
4
Journal of Inequalities and Applications
G−1 and Ψ−1 denote the inverse functions of G and Ψ, respectively, for t ∈ I. t1 ∈ I is so chosen
that
n
Ψ k t0 αi t
αi t0 i1
fi sds ∈ Dom Ψ−1 .
2.4
Proof. Assume that c > 0. Define a function zt by the right-hand side of 2.1. Clearly, zt is
nondecreasing, ut ≤ ϕ−1 zt for t ∈ I and zt0 c. Differentiating zt we get
z t n
q u αi t
fi αi t ψ u αi t gi αi t αi t
i1
−1
≤ ϕ
n
q gi αi t αi t.
zt
fi αi t ψ ϕ−1 z αi t
2.5
i1
Using the monotonicity of ϕ−1 and z, we deduce
−1
q q q
ϕ zt ≥ ϕ−1 zt0 ϕ−1 c > 0.
2.6
That is
n
z t
≤
gi αi t αi t.
fi αi t ψ ϕ−1 z αi t
q
ϕ−1 zt
i1
2.7
Setting t s in the inequality 2.7, integrating it from t0 to t, using the function G in the
left-hand side, and changing variable in the right-hand side, we obtain
n
G zt ≤ Gc i1
αi t
αi t0 fi sψ ϕ−1 zs gi s ds.
2.8
From the inequality 2.8, we find
n
G zt ≤ pt αi t
i1
αi t0 fi sψ ϕ−1 zs ds,
2.9
where
pt Gc n αi t
i1
αi t0 gi sds.
2.10
From the inequality 2.9, we observe that
n
G zt ≤ p t1 i1
αi t
αi t0 fi sψ ϕ−1 zs ds,
2.11
for t ≤ t1 . Now, define a function kt by the right-hand side of 2.11. Clearly, kt is
nondecreasing, zt ≤ G−1 kt for t ∈ I and kt0 pt1 . Differentiating kt, we get
k t n
n
αi t ≤ ψ ϕ−1 G−1 kt
fi αi t ψ ϕ−1 z αi t
fi αi t αi t.
i1
i1
2.12
Ravi P. Agarwal et al.
5
Using the monotonicity of ψ, ϕ−1 , G−1 , and k, we deduce
n
k t
≤
fi αi t αi t.
−1
−1
ψ ϕ G kt
i1
2.13
Setting t s in the inequality 2.13, integrating it from t0 to t, using the function Ψ in the
left-hand side, and changing variable in the right-hand side, we obtain
n
Ψ kt ≤ Ψ k t0 αi t
αi t0 i1
fi sds.
2.14
From the inequalities 2.11 and 2.14, we conclude that
n αi t
−1
−1
zt ≤ G Ψ
fi sds
Ψ pt1 i1
αi t0 2.15
for t0 ≤ t ≤ t1 . Now a combination of ut ≤ ϕ−1 zt and the last inequality in 2.15 for t1 t
produces the required inequality.
If c 0 we carry out the above procedure with ε > 0 instead of c and subsequently let
ε→0. This completes the proof.
For the special case ϕu up p > q > 0 is a constant, Theorem 2.1 gives the following
retarded integral inequality for nonlinear functions.
Corollary 2.2. Let u, fi , gi ∈ CI, R , i 1, . . . , n, and let αi ∈ C1 I, I be nondecreasing with
αi t ≤ t, i 1, . . . , n. Suppose that c ≥ 0 and p > q > 0 are constants, and ψu is a nondecreasing
continuous function for u ∈ R with ψu > 0 for u > 0. If
up t ≤ c n αi t
αi t0 i1
uq s fi sψ us gi s ds
2.16
for t ∈ I, then
ut ≤
Ψ−1
0
n
p − q Ψ0 k1 t0 p i1
αi t
αi t0 1/p−q
fi sds
2.17
for t ∈ t0 , t, where
Ψ0 r r
ds
1/p−q ,
ψ
s
r0
r ≥ r0 > 0,
n
p − q
k1 t0 cp−q/p p i1
αi t
αi t0 2.18
gi sds,
Ψ−1
0 denotes the inverse function of Ψ0 for t ∈ I. t ∈ I is so chosen that
n
p − q Ψ0 k1 t0 p i1
αi t
αi t0 fi sds ∈ Dom Ψ−1
0 .
2.19
6
Journal of Inequalities and Applications
Proof. The proof follows by an argument similar to that in the proof of Theorem 2.1 with
suitable modification. We omit the details here.
Remark 2.3. When q 1, from Corollary 2.2, one derives Theorem C. When p 2, q 1, from
Corollary 2.2, one derives Theorem B.
Theorem 2.1 can easily be applied to generate other useful nonlinear integral inequalities
in more general situations. For example, one has the following result Theorem 2.4.
Theorem 2.4. Let u ∈ CI, R1 , fi , gi ∈ CI, R , i 1, . . . , n, and let αi ∈ C1 I, I be nondecreasing
with αi t ≤ t, i 1, . . . , n. Suppose that c ≥ 1 is a constant, ϕ ∈ C1 R , R is an increasing function
with ϕ∞ ∞ and ψj u, j 1, 2 are nondecreasing continuous functions for u ∈ R with ψj u > 0
for u > 0. If
n αi t
uq s fi sψ1 us gi sψ2 log us ds
ϕ ut ≤ c 2.20
i1
αi t0 for t ∈ I, then
i as the case ψ1 u ≥ ψ2 logu,
ut ≤ ϕ
−1
G
−1
Ψ−1
1
Ψ1 Gc n αi t
i1
αi t0 fi s gi s ds
2.21
for t ∈ t0 , t1 ,
ii as the case ψ1 u < ψ2 logu,
ut ≤ ϕ
−1
G
−1
Ψ−1
2
Ψ2 Gc n αi t
i1
for t ∈ t0 , t2 , where
Ψj r r
r0 ψj
ϕ−1
αi t0 ds
−1 ,
G s
fi s gi s ds
r ≥ r0 > 0,
2.22
2.23
G−1 , Ψ−1
j , j 1, 2, denote the inverse functions of G, Ψj , j 1, 2, respectively, the function
Gt is as defined in Theorem 2.1 for t ∈ I, and tj ∈ I, j 1, 2 are so chosen that
n αi t
Ψj Gc fi s gi s ds ∈ Dom Ψ−1
.
2.24
j
i1
αi t0 Proof. Let c > 0. Define a function zt by the right-hand side of 2.20. Clearly, zt is
nondecreasing, ut ≤ ϕ−1 zt for t ∈ I and zt0 c. Differentiating zt, we get
z t n
q αi t
u αi t fi αi t ψ1 u αi t gi αi t ψ2 log u αi t
i1
n
q gi αi t ψ2 log ϕ−1 z αi t
αi t.
≤ ϕ−1 zt
fi αi t ψ1 ϕ−1 z αi t
i1
2.25
Ravi P. Agarwal et al.
7
Using the monotonicity of ϕ−1 and z, we deduce
−1 q −1 q −1 q
≥ ϕ z t0
ϕ c > 0.
ϕ zt
2.26
That is
n
z t
≤
fi αi tψ1 ϕ−1 zαi t gi αi tψ2 logϕ−1 zαi tαi t.
q
ϕ−1 zt
i1
2.27
Setting t s in the inequality 2.27, integrating it from t0 to t, using the function G in the
left-hand side, and changing variable in the right-hand side, we obtain
n
G zt ≤ Gc αi t
αi t0 i1
ds.
fi sψ1 ϕ−1 zs gi sψ2 log ϕ−1 zs
2.28
When ψ1 u ≥ ψ2 logu, from the inequality 2.28, we find
n
G zt ≤ Gc αi t
i1
αi t0 fi s gi s ψ1 ϕ−1 zs ds.
2.29
Now, define a function kt by the right-hand side of 2.29. Clearly, kt is nondecreasing,
zt ≤ G−1 kt for t ∈ I and kt0 Gc. Differentiating kt, we get
k t n
fi αi t gi αi t ψ1 ϕ−1 z αi t αi t
i1
n
≤ ψ1 ϕ−1 G−1 kt
fi αi t gi αi t αi t.
2.30
i1
Using the monotonicity of ψ1 , ϕ−1 , G−1 , and k, we deduce
n
k t
≤
fi αi t αi t.
−1
−1
ψ1 ϕ G kt
i1
2.31
Setting t s in the inequality 2.31, integrating it from t0 to t, using the function Ψ1 in the
left-hand side, and changing variable in the right-hand side, we obtain
n
Ψ1 kt ≤ Ψ1 k t0 αi t
i1
αi t0 fi s gi s ds.
2.32
From the inequality 2.32, we conclude that
zt ≤ G
−1
Ψ−1
1
n
Ψ1 Gc i1
αi t
αi t0 fi s gi s ds
2.33
for t ∈ I. Now a combination of ut ≤ ϕ−1 zt and the last inequality produces the required
inequality in 2.21.
8
Journal of Inequalities and Applications
When ψ1 u < ψ1 logu, from the inequality 2.28, we find
n
G zt ≤ Gc αi t
i1
αi t0 fi s gi s ψ1 ϕ−1 zs ds.
2.34
Now, by a suitable application of the process from 2.29 to 2.32 in the inequality 2.34, we
conclude that
n αi t
−1
−1
2.35
zt ≤ G Ψ2 Ψ2 Gc fi s gi s ds
αi t0 i1
for t ∈ I. Now a combination of ut ≤ ϕ−1 zt and the last inequality produces the required
inequality in 2.22.
If c 0, we carry out the above procedure with ε > 0 instead of c and subsequently let
ε→0. This completes the proof.
For the special case ϕu up p > q > 0 is a constant, Theorem 2.4 gives the following
retarded integral inequality for nonlinear functions.
Corollary 2.5. Let u ∈ CI, R1 , fi , gi ∈ CI, R , i 1, . . . , n, and let αi ∈ C1 I, I be nondecreasing
with αi t ≤ t, i 1, . . . , n. Suppose that c ≥ 0 and p > q > 0 are constants, and ψj u, j 1, 2 are
nondecreasing continuous functions for u ∈ R with ψj u > 0 for u > 0. If
u t ≤ c p
n αi t
i1
αi t0 uq s fi sψ1 us gi sψ2 log us ds
2.36
for t ∈ I, then
i as the case ψ1 u ≥ ψ2 logu,
ut ≤
G−1
1
n
p − q
G1 cp−q/p p i1
αi t
αi t0 fi s gi s ds
1/p−q
2.37
for t ∈ t0 , t1 ,
ii as the case ψ1 u < ψ2 logu,
ut ≤
G−1
2
1/p−q
n αi t
p−q/p p − q G2 c
fi s gi s ds
p i1 αi t0 2.38
for t ∈ t0 , t2 , where G−1
j , j 1, 2, denote the inverse functions of Gj , j 1, 2,
Gj r r
r0 ψj
ds
s1/p−q
,
r ≥ r0 > 0,
2.39
for t ∈ I, and tj ∈ I, j 1, 2, are chosen so that
n
p − q
Gj cp−q/p p i1
αi t
αi t0 fi s gi s ds ∈ DomG−1
j .
2.40
Ravi P. Agarwal et al.
9
Proof. The proof follows by an argument similar to that in the proof of Theorem 2.4 with
suitable modification. We omit the details here.
Theorem 2.1 can easily be applied to generate another useful nonlinear integral inequalities in more general situations. For example, we have the following result Theorem 2.6.
Theorem 2.6. Let u, fi , gi ∈ CI, R , i 1, . . . , n, and let αi ∈ C1 I, I be nondecreasing with
αi t ≤ t, i 1, . . . , n. Suppose that c ≥ 0 and q > 0 are constants, ϕ ∈ C1 R , R is an increasing
function with ϕ∞ ∞ on I, and L, M ∈ CR2 , R satisfy
0 ≤ Lt, v − Lt, w ≤ Mt, wv − w
2.41
for t, v, w ∈ R with v ≥ w ≥ 0. If
n
ϕ ut ≤ c i1
αi t
αi t0 uq s fi sL s, us gi sus ds
2.42
for t ∈ I, then
ut ≤ ϕ
−1
G
−1
−1
Ω
n
Ω k2 t0 i1
αi t
αi t0 fi sMs gi s ds
2.43
for t ∈ t0 , t1 , where
r
ds
, r ≥ r0 > 0,
−1 G−1 s
ϕ
r0
n αi t
k2 t0 Gc fi sLsds,
Ωr 2.44
αi t0 i1
G−1 and Ω−1 denote the inverse function of G and Ω, respectively, the function G is as defined in
Theorem 2.1 for t ∈ I and t1 ∈ I is so chosen that
n
Ω k2 t0 i1
αi t
αi t0 fi sds ∈ Dom Ω−1 .
2.45
Proof. Let c > 0. Define a function zt by the right-hand side of 2.42. Clearly, zt is
nondecreasing, ut ≤ ϕ−1 zt for t ∈ I and zt0 c. Differentiating zt, we get
z t n
q u αi t
fi αi t L αi t, u αi t gi αi t u αi t αi t
i1
n
q ≤ ϕ−1 zt
fi αi t L αi t, ϕ−1 z αi t gi αi t ϕ−1 z αi t αi t.
2.46
i1
Using the monotonicity of ϕ−1 and z, we deduce
n
z t
≤
gi αi t ϕ−1 z αi t αi t.
fi αi t L αi t, ϕ−1 z αi t
q
ϕ−1 zt
i1
2.47
10
Journal of Inequalities and Applications
Setting t s in the inequality 2.47, integrating it from t0 to t, using the function G in the
left-hand side, and changing variable in the right-hand side, we obtain
n
G zt ≤ Gc i1
αi t
αi t0 fi sL s, ϕ−1 zs gi sϕ−1 zs ds.
2.48
From the inequalities 2.41, 2.48, we find
n
G zt ≤ Gc αi t1 αi t0 i1
fi sL s ds n αi t
i1
αi t0 fi sMs gi s ϕ−1 zs ds,
2.49
for t ≤ t1 . Now, define a function k2 t by the right-hand side of 2.49. Clearly, k2 t is
nondecreasing, zt ≤ G−1 k2 t for t ∈ I. Differentiating k2 t, we get
k2 t n
fi αi tMαi t gi αi tϕ−1 zsαi t
i1
−1
−1
≤ ϕ G k2 t
n
2.50
fi αi tMαi t gi αi tαi t.
i1
Using the monotonicity of ϕ−1 , G−1 , and k2 , we deduce
n
k2 t
≤
fi αi t M αi t gi αi t αi t.
ψ ϕ−1 G−1 k2 t
i1
2.51
Setting t s in the inequality 2.51, integrating it from t0 to t, using the function Ω in the
left-hand side, and changing variable in the right-hand side, we obtain
n
Ω k2 t ≤ Ω k2 t0 αi t
i1
αi t0 fi sMs gi s ds.
2.52
From the inequalities 2.49 and 2.52, we conclude that
−1
zt ≤ G
−1
Ω
n
Ω k2 t0 i1
αi t
αi t0 fi sMs gi s ds
2.53
for t0 ≤ t ≤ t1 . Now a combination of ut ≤ ϕ−1 zt and the last inequality produces the
required inequality in 2.43 for t1 t.
If c 0, we carry out the above procedure with ε > 0 instead of c and subsequently let
ε→0. This completes the proof.
For the special case ϕu up p > q > 0 is a constant, Theorem 2.6 gives the following
retarded integral inequality for nonlinear functions.
Corollary 2.7. Let u, fi , gi , and αi be as defined in Theorem 2.6. Suppose that c ≥ 0 and p > q > 0 are
constants, and L, M ∈ CR2 , R satisfy
0 ≤ Lt, v − Lt, w ≤ Mt, wv − w
2.54
Ravi P. Agarwal et al.
11
for t, v, w ∈ R with v ≥ w ≥ 0. If
up t ≤ c n αi t
i1
αi t0 uq s fi sL s, us gi sus ds
2.55
for t ∈ I, then
ut ≤
Ω−1
1
n
p − q Ω1 k3 t0 p i1
αi t
αi t0 fi sMs gi s ds
1/p−q
2.56
for t ∈ t0 , t1 , where
Ω1 r r
ds
,
1/p−q
s
r0
r ≥ r0 > 0,
n
p − q
k3 t0 cp−q/p p i1
αi t
αi t0 2.57
fi sLsds,
Ω−1
1 denotes the inverse function of Ω1 for t ∈ I and t1 ∈ I is so chosen that
n
p − q Ω1 k3 t0 p i1
αi t
αi t0 fi sMs gi s ds ∈ Dom Ω−1
1 .
2.58
Proof. The proof follows by an argument similar to that in the proof of Theorem 2.6 with
suitable modification. We omit the details here.
3. Applications
We will show that our results are useful in proving the global existence of solutions to certain
differential equations with time delay. Consider the functional differential equation involving
several retarded arguments with the initial condition:
ϕ xt x t F t, x t − h1 t , . . . , x t − hn t ,
x t0 x0 ,
t ∈ I,
3.1
where x0 is a constant, F ∈ CI × Rn , R, hi ∈ CI, R , i 1, . . . , n be nonincreasing such that
t − hi t ≥ 0, t − hi t ∈ C1 I, I, hi t < 1, and ϕ ∈ C1 R, R is an increasing function with
ϕ|x| ≤ |ϕx|. The following theorem deals with a bound on the solution of the problem 3.1.
Theorem 3.1. Assume that F : I × Rn →R is a continuous function for which there exist continuous
nonnegative functions fi t, gi t, i 1, . . . , n for t ∈ I such that
n F t, u1 , . . . , un ≤
ui q fi tψ ui gi t ,
i1
3.2
12
Journal of Inequalities and Applications
where q > 0 is a constant and ψ is as in Theorem 2.1. Let
Qi max
t∈I
1
,
1 − hi t
i 1, . . . , n.
3.3
If xt is any solution of the problem 3.1, then
n t−hi t
xt ≤ ϕ−1 G−1 Ψ−1 Ψ k t0 fi σdσ
i1
t0 −hi t0 3.4
for t ∈ I, where G, Ψ are as in Theorem 2.1 and
n
k t0 G ϕ x0 i1
t−hi t
t0 −hi t0 gi σdσ,
3.5
fi σ Qi fi σ hi s, gi σ Qi gi σ hi s for s, σ ∈ I.
Proof. It is easy to see that the solution xt of the problem 3.1 satisfies the equivalent integral
equation:
ϕ xt ϕ x t0 t
F s, x s − h1 s , . . . , x s − hn s ds.
3.6
t0
From 3.2 and 3.6, we have
ϕ xt ≤ ϕ x t0 t
F s, x s − h1 s , . . . , x s − hn s ds
t0
≤ ϕ x0 t
n
x s − hi s q fi tψ x s − hi s gi t ds
3.7
t0 i1
for t, s ∈ I. By making the change of variables on the right side of the inequality of 3.7 and
rewriting, we have
n
ϕ xt ≤ ϕ x0 i1
t−hi t
t0 −hi t0 xσq fi σψ xσ gi σ dσ,
3.8
where fi σ Qi fi σ hi s, gi σ Qi gi σ hi s for s, σ ∈ I. Now an immediate application
of the inequality established in Theorem 2.1 to the inequality 3.8 yields the result.
Ravi P. Agarwal et al.
13
Remark 3.2. Consider the functional differential equation involving several retarded arguments
with the initial condition:
pxp−1 tx t F t, x t − h1 t , . . . , x t − hn t ,
x t0 x0 ,
t ∈ I,
3.9
where p > 0 and x0 are constants, F ∈ CI × Rn , R, hi ∈ CI, R , i 1, . . . , n be nonincreasing
such that t − hi t ≥ 0, t − hi t ∈ C1 I, I, hi t < 1.
Assume that F : I × Rn →R is a continuous function for which there exist continuous
nonnegative functions fi t, gi t, i 1, . . . , n for t ∈ I such that
n ui q fi tψ ui gi t ,
F t, u1 , . . . , un ≤
3.10
i1
where q > 0 p > q is a constant and ψ is as in Theorem 2.1. If xt is any solution of the
problem 3.9, then it satisfies the equivalent integral equation:
xp t xp t0 t
F s, x s − h1 s , . . . , x s − hn s ds.
3.11
t0
From 3.10 and 3.11, one has
xtp ≤ x0 p t
F s, x s − h1 s , . . . , x s − hn s ds
t0
p
≤ x0 t
n
x s − hi s q fi tψ x s − hi s gi t ds
3.12
t0 i1
for t, s ∈ I. By making the change of variables on the right side of 3.12 and rewriting, one has
n
xtp ≤ x0 p i1
t−hi t
t0 −hi t0 xσq fi σψ xσ gi σ dσ,
3.13
where fi σ Qi fi σ hi s, gi σ Qi gi σ hi s for s, σ ∈ I. Now an immediate application
of the inequality established in Corollary 2.2 to the inequality 3.13 yields
1/p−q
n t−hi t
p − q −1
xt ≤ Ψ
fi σdσ
Ψ0 k1 t0 0
p i1 t0 −hi t0 3.14
for t ∈ I, where Ψ0 is as in Corollary 2.2,
p−q
k1 t0 x0
n
p − q
p i1
t−hi t
t0 −hi t0 gi σdσ,
fi σ Qi fi σ hi s, and gi σ Qi gi σ hi s for s, σ ∈ I.
3.15
14
Journal of Inequalities and Applications
The following theorem provides a uniqueness on the solution of the problem 3.9.
Theorem 3.3. Assume that F : I × R3 →R is a continuous function for which there exist continuous
nonnegative functions fi t, i 1, . . . , n for t ∈ I such that
n
p
p
F t, u1 , . . . , un − F t, v1 , . . . , vn ≤
fi tui − vi ,
3.16
i1
where p > 1 is a constant, then the problem 3.9 has at most one solution on I.
Proof. Let xt and xt be two distinct solutions of the problem 3.9, one has
xp t − x p t t
F s, x s−h1 s , . . . , x s − hn s −F s, x s − h1 s , . . . , x s−hn s ds.
t0
3.17
From 3.16 and 3.17, one finds
p
x t − xp t ≤
t n
t0
p
p
fi sx s − hi s − x s − hi s ds
3.18
i1
for t, s ∈ I. By making the change of variables on the right side of 3.18 and rewriting, one has
n
p p
x t − xp t1/p ≤
βi t
i1
βi t0 p
x σ − xp σ1/p p−1 fi σ xp σ − xp σ1/p dσ,
3.19
where βi t t − hi t, fi σ Qi fi σ hi s for s, σ ∈ I. Now when ψu u, q p − 1, a
suitable application of the inequality in Corollary 2.2 to the function |xp t − xp t|1/p and the
inequality 3.19, one concludes that
p
x t − xp t1/p ≤ 0
3.20
for all t ∈ I. Hence xt xt.
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