Journal of Inequalities in Pure and
Applied Mathematics
ON SOME NEW RETARD INTEGRAL INEQUALITIES IN n
INDEPENDENT VARIABLES AND THEIR APPLICATIONS
volume 7, issue 3, article 111,
2006.
XUEQIN ZHAO AND FANWEI MENG
Department of Mathematics
Qufu Normal University
Qufu 273165
People’s Republic of China.
Received 24 October, 2005;
accepted 24 May, 2006.
Communicated by: S.S. Dragomir
EMail: xqzhao1972@126.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
318-05
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Abstract
In this paper, we established some retard integral inequalities in n independent
variables and by means of examples we show the usefulness of our results.
2000 Mathematics Subject Classification: 26D15, 26D10.
Key words: Retard integral inequalities; integral equation; Partial differential equation.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminaries and Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4
Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
References
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
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1.
Introduction
The study of integral inequalities involving functions of one or more independent variables is an important tool in the study of existence, uniqueness, bounds,
stability, invariant manifolds and other qualitative properties of solutions of differential equations and integral equations. During the past few years, many new
inequalities have been discovered (see [1, 3, 4, 7, 8]). In the qualitative analysis of some classes of partial differential equations, the bounds provided by the
earlier inequalities are inadequate and it is necessary to seek some new inequalities in order to achieve a diversity of desired goals. Our aim in this paper is to
establish some new inequalities in n independent variables, meanwhile, some
applications of our results are also given.
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
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2.
Preliminaries and Lemmas
In this paper, we suppose R+ = [0, ∞), is subset of real numbers R, e
0 =
n
n
(0, . . . , 0), α
e(t) = (α1 (t1 ), . . . , αn (tn )) ∈ R+ , t = (t1 , . . . , tn ) ∈ R+ , s =
(s1 , . . . , sn ) ∈ Rn+ , re = (r1 , . . . , rn ), re0 = (r10 , . . . , rn0 ), ze = (z1 , . . . , zn ),
ze0 = (z10 , . . . , zn0 ), T = (T1 , . . . , Tn ) ∈ Rn+ .
If f : Rn+ → R+ , we suppose
1. s ≤ t ⇔ si ≤ ti (i = 1, 2, . . . , n);
R αe(t)
R α (t )
R α (t )
2. e0 f (s)ds = 0 1 1 · · · 0 n n f (s1 , . . . , sn )dsn . . . ds1 ;
3. Di =
d
,
dti
i = 1, 2,. . . , n.
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
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3.
Main Results
In this part, we obtain our main results as follows:
Theorem 3.1. Let ψ ∈ C(R+ , R+ ) be a nondecreasing function with ψ(u) >
0 on (0, ∞), and let c be a nonnegative constant. Let αi ∈ C 1 (R+ , R+ ) be
nondecreasing with αi (ti ) ≤ ti on R+ (i = 1, . . . , n). If u, f ∈ C(Rn+ , R+ ) and
Z
(3.1)
u(t) ≤ c +
α
e(t)
f (s)ψ(u(s))ds,
e
0
for e
0 ≤ t < T , then
"
(3.2)
u(t) ≤ G
−1
Z
G(c) +
#
α
e(t)
Z
ze
G(e
z) =
ze0
ds
,
ψ(s)
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ze ≥ ze0 > 0.
G−1 is the inverse of G, T ∈ Rn+ is chosen so that
Z αe(t)
(3.3)
G(c) +
f (s)ds ∈ Dom(G−1 ),
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e
0 ≤ t < T.
e
0
Define the nondecreasing positive function z(t) and make
Z αe(t)
e
(3.4)
z(t) = c + ε +
f (s)ψ(u(s))ds,
0 ≤ t < T,
e
0
Xueqin Zhao and Fanwei Meng
f (s)ds ,
e
0
where
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
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where ε is an arbitrary small positive number. We know that
u(t) ≤ z(t),
(3.5)
D1 D2 · · · Dn z(t) = f (e
α)ψ(u(e
α))α10 α20 · · · αn0 .
Using (3.5), we have
D1 D2 · · · Dn z(t)
≤ f (e
α)α10 α20 · · · αn0 .
ψ(z(t))
(3.6)
For
(3.7) Dn
D1 D2 · · · Dn−1 z(t)
ψ(z(t))
D1 D2 · · · Dn z(t)ψ(z(t)) − D1 D2 · · · Dn−1 z(t)ψ 0 Dn z(t)
=
,
ψ 2 (z(t))
0
using D1 D2 · · · Dn−1 z(t) ≥ 0, ψ ≥ 0, Dn z(t) ≥ 0 in (3.7), we get
D1 D2 · · · Dn−1 z(t)
D1 D2 · · · Dn z(t)
(3.8) Dn
≤
≤ f (e
α)α10 α20 · · · αn0 .
ψ(z(t))
ψ(z(t))
Fixing t1 , . . . , tn−1 , setting tn = sn , integrating from tn to ∞, yields
(3.9)
D1 D2 · · · Dn−1 z(t)
ψ(z(t))
Z αn (tn )
0
≤
f (α1 (t1 ), . . . , αn−1 (tn−1 ), sn )α10 α20 · · · αn−1
dsn .
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
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Using the same method, we deduce that
(3.10)
D1 z(t)
≤
ψ(z(t))
Z
α2 (t2 )
Z
αn (tn )
···
0
f (t1 , s2 , . . . , sn )α10 dsn . . . ds2 ,
0
and integration on [t1 , ∞) yields
(3.11) G(z(t))
Z
≤ G(c + ε) +
α1 (t1 )
αn (tn )
Z
···
0
f (s1 , s2 , . . . , sn )dsn . . . ds1 ,
t ∈ Rn+ .
0
From the definition of G and letting ε → 0, we can obtain inequality (3.2).
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
Remark 1. If we let G(z) → ∞, z → ∞, then condition (3.3) can be omitted.
Corollary 3.2. If we let ψ = sr , 0 < r ≤ 1 in Theorem 3.1, then for t ∈ Rn+ ,
we have
 h
1
i 1−r
R αe(t)

1−r

+ (1 − r) e0 f (s)ds
, 0 < r < 1;
 c
(3.12)
u(t) ≤


 c exp R αe(t) f (s)ds ,
r = 1.
e
0
Remark 2. If we let n = 1, r = 1, α
e(t) = t, in Corollary 3.2, we obtain the
Mate-Nevai inequality.
Theorem 3.3. Let ϕ ∈ C(R+ , R+ ) be an increasing function with ϕ(∞) = ∞.
Let ψ ∈ C(R+ , R+ ) be a nondecreasing function and let c be a nonnegative
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constant. Let αi ∈ C 1 (R+ , R+ ) be nondecreasing with αi (ti ) ≤ ti on R+ (i =
1, . . . , n). If u, f ∈ C(Rn+ , R+ ) and
Z
ϕ(u(t)) ≤ c +
(3.13)
α
e(t)
f (s)ψ(u(s))ds,
e
0
for e
0 ≤ t < T , then
(
(3.14)
u(t) ≤ ϕ−1
G−1 [G(c) +
Z
)
α
e(t)
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
f (s)ds] ,
e
0
R ze
ds
, ze ≥ ze0 > 0, ϕ−1 , G−1 are respectively the inverse
where G(e
z ) = ze0 ψ[ϕ−1
(s)]
of ϕ and G, T ∈ Rn+ is chosen so that
Z
(3.15)
G(c) +
α
e(t)
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f (s)ds ∈ Dom(G−1 ),
e
0 ≤ t < T.
e
0
Proof. From the definition of the ϕ, we know (3.13) can be restated as
Z
(3.16)
ϕ(u(t)) ≤ c +
α
e(t)
f (s)ψ[ϕ−1 (ϕ(u(s)))]ds,
t ∈ Rn+ .
e
0
Now an application of Theorem 3.1 gives
"
Z
(3.17)
ϕ(u(t)) ≤ G−1 G(c) +
Xueqin Zhao and Fanwei Meng
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α
e(t)
#
f (s)ds ,
e
0 ≤ t < T.
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e
0
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So,
(
(3.18)
u(t) ≤ ϕ−1
G−1 [G(c) +
Z
)
α
e(t)
e
0 ≤ t < T.
f (s)ds] ,
e
0
Corollary 3.4. If we let ϕ = sp , ψ = sq , p, q are constants, and p ≥ q > 0 in
Theorem 3.3, for e
0 ≤ t < T , then
 h
1
R
i p−q
α
e(t)
1− pq
q


, when p > q;
c
+
1
−
f
(s)ds

e
p
0
(3.19)
u(t) ≤


 c p1 exp 1 R αe(t) f (s)ds ,
when p = q.
p
e
0
Theorem 3.5. Let u, f and g be nonnegative continuous functions defined on
Rn+ , let c be a nonnegative constant. Moreover, let w1 , w2 ∈ C(R+ , R+ ) be
nondecreasing functions with wi (u) > 0 (i = 1, 2) on (0, ∞). Let αi ∈
C 1 (R+ , R+ ) be nondecreasing with αi (ti ) ≤ ti on R+ (i = 1, . . . , n). If
Z αe(t)
Z t
(3.20)
u(t) ≤ c +
f (s)w1 (u(s))ds +
g(s)w2 (u(s))ds,
e
0
e
0
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(i) for the case w2 (u) ≤ w1 (u),
(
u(t) ≤ G−1
1
Xueqin Zhao and Fanwei Meng
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for e
0 ≤ t < T , then
(3.21)
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
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Z
G1 (c) +
α
e(t)
Z
f (s)ds +
e
0
t
)
g(s)ds ,
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(ii) for the case w1 (u) ≤ w2 (u),
(
(3.22)
u(t) ≤
G−1
2
Z
α
e(t)
G2 (c) +
Z
f (s)ds +
e
0
)
t
g(s)ds ,
e
0
where
Z
(3.23)
ze
Gi (e
z) =
ze0
ds
,
wi (s)
ze ≥ ze0 > 0,
(i = 1, 2)
n
and G−1
i (i = 1, 2) is the inverse of Gi , T ∈ R+ is chosen so that
Z
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
α
e(t)
(3.24) Gi (c) +
f (s)ds
Xueqin Zhao and Fanwei Meng
e
0
Z
+
t
g(s)ds ∈ Dom(G−1
i ),
(i = 1, 2),
e
0 ≤ t < T.
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e
0
Contents
Proof. Define the nonincreasing positive function z(t) and make
Z αe(t)
Z t
(3.25)
z(t) = c + ε +
f (s)w1 (u(s))ds +
g(s)w2 (u(s))ds,
e
0
e
0
where ε is an arbitrary small positive number. From inequality (3.20), we know
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(3.26)
u(t) ≤ z(t)
and
(3.27)
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D1 D2 · · · Dn z(t) = [f (e
α)w1 (u(e
α))α10 α20 · · · αn0 + g(t)w2 (u(t))].
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The rest of the proof can be completed by following the proof of Theorem 3.1
with suitable modifications.
Theorem 3.6. Let u, f and g be nonnegative continuous functions defined on
Rn+ , and let ϕ ∈ C(R+ , R+ ) be an increasing function with ϕ(∞) = ∞ and
let c be a nonnegative constant. Moreover, let w1 , w2 ∈ C(R+ , R+ ) be nondecreasing functions with wi (u) > 0 (i = 1, 2) on (0, ∞), αi ∈ C 1 (R+ , R+ ) be
nondecreasing with αi (ti ) ≤ ti on R+ (i = 1, . . . , n). If
α
e(t)
Z
(3.28)
ϕ(u(t)) ≤ c +
Z
f (s)w1 (u(s))ds +
e
0
t
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
g(s)w2 (u(s))ds,
e
0
for e
0 ≤ t < T , then
Xueqin Zhao and Fanwei Meng
(i) for the case w2 (u) ≤ w1 (u),
(
"
(3.29)
u(t) ≤ ϕ−1
Title Page
G1−1 G1 (c) +
Z
α
e(t)
Z
f (s)ds +
e
0
(ii) for the case w1 (u) ≤ w2 (u),
(
"
(3.30)
u(t) ≤ ϕ
−1
G2−1
Z
G2 (c) +
#)
t
g(s)ds
;
e
0
α
e(t)
Z
f (s)ds +
e
0
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t
g(s)ds
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0
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where
Z
ze
Gi (e
z) =
ze0
ds
,
wi (ϕ−1 (s))
Page 11 of 27
ze > ze0 ,
(i = 1, 2),
J. Ineq. Pure and Appl. Math. 7(3) Art. 111, 2006
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n
and ϕ−1 , G−1
i (i = 1, 2) are respectively the inverse of Gi , ϕ, T ∈ R+ is
chosen so that
Z
α
e(t)
(3.31) Gi (c) +
f (s)ds
e
0
Z
t
g(s)ds ∈ Dom(G−1
i ),
+
(i = 1, 2),
e
0 ≤ t < T.
e
0
Proof. From the definition of ϕ, we know (3.28) can be restated as
α
e(t)
Z
(3.32)
ϕ(u(t)) ≤ c +
f (s)w1 [ϕ−1 (ϕ(u(s)))]ds
Xueqin Zhao and Fanwei Meng
e
0
t
Z
g(s)w2 [ϕ−1 (ϕ(u(s)))]ds,
+
t ∈ Rn+ .
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e
0
Now an application of Theorem 3.5 gives
(
Z
ϕ(u(t)) ≤
G−1
i
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
α
e(t)
Gi (c) +
Contents
Z
f (s)ds +
e
0
t
)
g(s)ds ,
e
0 ≤ t < T,
e
0
where T satisfies (3.31). We can obtain the desired inequalities (3.29) and
(3.30).
Theorem 3.7. Let u, f and g be nonnegative continuous functions defined on
Rn+ and let c be a nonnegative constant. Moreover, let ϕ ∈ C(R+ , R+ ) be
an increasing function with ϕ(∞) = ∞, ψ ∈ C(R+ , R+ ) be a nondecreasing
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function with ψ(u) > 0 on (0, ∞) and αi ∈ C 1 (R+ , R+ ) be nondecreasing with
αi (ti ) ≤ ti on R+ (i = 1, . . . , n). If
Z αe(t)
(3.33)
ϕ(u(t)) ≤ c +
[f (s)u(s)ψ(u(s)) + g(s)u(s)]ds,
e
0
for e
0 ≤ t < T , then
(3.34) u(t)
(
≤ϕ
−1
"
Ω
−1
G
Z
−1
G[Ω(c) +
α
e(t)
Z
g(s)ds] +
e
0
!#)
α
e(t)
f (s)ds
,
e
0
where
Z
re
Ω(e
r) =
re0
Z ze
G(e
z) =
ze0
Xueqin Zhao and Fanwei Meng
ds
,
−1
ϕ (s)
re ≥ re0 > 0 ,
Title Page
ds
,
−1
ψ{ϕ [Ω−1 (s)]}
ze ≥ ze0 > 0,
Contents
Ω−1 , ϕ−1 , G−1 are respectively the inverse of Ω, ϕ, G. And T ∈ R+ is chosen so
that
"
# Z
Z αe(t)
α
e(t)
G Ω(c) +
g(s)ds +
f (s)ds ∈ Dom(G−1 ), e
0 ≤ t < T,
e
0
e
0
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( "
G−1
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Z
G Ω(c) +
α
e(t)
#
Z
g(s)ds +
e
0
α
e(t)
)
f (s)ds
∈ Dom(Ω−1 ),
e
0 ≤ t < T.
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e
0
J. Ineq. Pure and Appl. Math. 7(3) Art. 111, 2006
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Proof. Let us first assume that c > 0. Defining the nondecreasing positive
function z(t) by the right-hand side of (3.33)
Z αe(t)
z(t) = c +
[f (s)u(s)ψ(u(s)) + g(s)u(s)]ds,
e
0
we know
u(t) ≤ ϕ−1 [z(t)]
(3.35)
and
(3.36)
D1 D2 · · · Dn z(t) = [f (e
α)u(e
α)ψ(u(e
α)) + g(e
α)u(e
α)]α10 α20 · · · αn0 .
Using (3.35), we have
(3.37)
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
D1 D2 · · · Dn z(t)
≤ [f (e
α)ψ(ϕ−1 (z(α))) + g(e
α)]α10 α20 · · · αn0 .
ϕ−1 (z(t))
For
Title Page
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D1 D2 · · · Dn−1 z(t)
ϕ−1 (z(t))
D1 D2 · · · Dn z(t)ϕ−1 (z(t)) − D1 D2 · · · Dn−1 z(t)(ϕ−1 (z(t)))0 Dn z(t)
=
,
(ϕ−1 (z(t)))2
(3.38) Dn
using D1 D2 · · · Dn−1 z(t) ≥ 0, Dn z(t) ≥ 0, (ϕ−1 (z(t)))0 ≥ 0 in (3.38), we get
D1 D2 · · · Dn−1 z(t)
D1 D2 · · · Dn z(t)
(3.39) Dn
≤
−1
ϕ (z(t))
ϕ−1 (z(t))
≤ [f (e
α)ψ(ϕ−1 z(e
α)) + g(e
α)]α10 α20 · · · αn0 .
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Fixing t1 , . . . , tn−1 , setting tn = sn , integrating from 0 to tn with respect to sn
yields
(3.40)
Z
D1 D2 · · · Dn−1 z(t)
ϕ−1 (z(t))
αn (tn )
[f (α1 (t1 ), . . . , αn−1 (tn−1 ), sn )ψ(ϕ−1 (z(α1 (t1 ), . . . , αn−1 (tn−1 ), sn )))
≤
0
0
+ g(α1 (t1 ), . . . , αn−1 (tn−1 ), sn )]α10 α20 · · · αn−1
dsn .
Using the same method, we deduce that
D1 z(t)
ϕ−1 (z(t))
Z α2 (t2 )
Z
≤
···
(3.41)
0
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
αn (tn )
[f (α1 (t1 ), s2 , . . . , sn )ψ(ϕ(z(α1 (t1 ), s2 , . . . , sn )))
Title Page
0
+ g(α1 (t1 ), s2 , . . . , sn )]α10 dsn . . . ds2 .
Setting t1 = s1 , and integrating it from 0 to t1 with respect to s1 yields
Z αe(t)
Z αe(t)
−1
f (s)ψ(ϕ (z(s))ds +
g(s)ds,
(3.42)
Ω(z(t)) ≤ Ω(c) +
e
0
Let T1 ≤ T be arbitrary, we denote p(T1 ) = Ω(c) +
we deduce that
Z αe(t)
Ω(z(t)) ≤ p(T1 ) +
f (s)ψ[ϕ−1 z(s)]ds,
0
e
0
R αe(T1 )
0
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e
0 ≤ t ≤ T1 ≤ T.
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Now an application of Theorem 3.3 gives
(
"
#)
Z αe(t)
z(t) ≤ Ω−1 G−1 G(p(T1 )) +
f (s)ds
,
e
0 ≤ t ≤ T1 ≤ T,
e
0
so
(
u(t) ≤ ϕ−1
"
Ω−1 G−1
Z
!#)
α
e(t)
f (s)ds
G(p(T1 )) +
,
e
0 ≤ t ≤ T1 ≤ T.
e
0
Taking t = T1 in the above inequality, since T1 is arbitrary, we can prove the
desired inequality (3.34).
If c = 0 we carry out the above procedure with ε > 0 instead of c and
subsequently let ε → 0.
Setting f (t) = 0, n = 1, we can obtain a retarded Ou-Iang inequality.
Let u, f and g be nonnegative continuous functions defined on Rn+ and let c
be a nonnegative constant. Moreover, let ψ ∈ C(R+ , R+ ) be a nondecreasing
function with ψ(u) > 0 on (0, ∞) and αi ∈ C 1 (R+ , R+ ) be nondecreasing with
αi (ti ) ≤ ti on R+ (i = 1, . . . , n). If
Z αe(t)
2
2
u (t) ≤ c +
[f (s)u(s)ψ(u(s)) + g(s)u(s)]ds,
e
0
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"
u(t) ≤ Ω
Xueqin Zhao and Fanwei Meng
Close
for e
0 ≤ t < T , then
−1
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
1
Ω c+
2
Z
α
e(t)
!
g(s)ds
e
0
1
+
2
Z
α
e(t)
#
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f (s)ds ,
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where
Z
ze
Ω(e
z) =
ze0
ds
ψ(s)
ze > ze0 ,
Ω−1 is the inverse of Ω, and T ∈ Rn+ is chosen so that
!
Z
Z
1 αe(t)
1 αe(t)
Ω c+
g(s)ds +
f (s)ds ∈ Dom(Ω−1 ),
2 e0
2 e0
e
0 ≤ t < T.
Corollary 3.8. Let u, f and g be nonnegative continuous functions defined on
Rn+ and let c be a nonnegative constant. Moreover, let p, q be positive constants
with p ≥ q, p 6= 1. Let αi ∈ C 1 (R+ , R+ ) be nondecreasing with αi (ti ) ≤ ti on
R+ (i = 1, . . . , n). If
p
Z
u (t) ≤ c +
α
e(t)
[f (s)uq (s) + g(s)u(s)]ds,
t≥0
e
0
for e
0 ≤ t < T then
 p
p−1
h R
i
R e(t)
e(t)

(1− p1 )
p−1 α
1 α

c
+
g(s)ds
exp
f
(s)ds
,
when p = q;

e
p
p e
0
0


u(t) ≤
1
p−q
p−q

p−1

R e(t)
R e(t)

(1− p1 )
p−1 α
p−q α

c
+ p e0 g(s)ds
+ p e0 f (s)ds
, when p > q.

Theorem 3.9. Let u, f and g be nonnegative continuous functions defined on
Rn+ , and let ϕ ∈ C(R+ , R+ ) be an increasing function with ϕ(∞) = ∞ and
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
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let c be a nonnegative constant. Moreover, let w1 , w2 ∈ C(R+ , R+ ) be nondecreasing functions with wi (u) > 0 (i = 1, 2) on (0, ∞), and αi ∈ C 1 (R+ , R+ )
be nondecreasing with αi (ti ) ≤ ti (i = 1, . . . , n). If
Z t
Z αe(t)
(3.43) ϕ(u(t)) ≤ c +
f (s)u(s)w1 (u(s))ds +
g(s)u(s)w2 (u(s))ds,
e
0
e
0
for e
0 ≤ t < T , then
(i) for the case w2 (u) ≤ w1 (u),
(3.44) u(t)
(
≤ ϕ−1
"
Ω−1 G−1
1
Z
α
e(t)
G1 (Ω(c)) +
Z
!#)
t
f (s)ds +
e
0
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
g(s)ds
,
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(ii) for the case w1 (u) ≤ w2 (u),
Contents
(3.45) u(t)
(
≤ϕ
−1
Ω
Xueqin Zhao and Fanwei Meng
e
0
"
−1
G−1
2
Z
α
e(t)
G2 (Ω(c)) +
Z
f (s)ds +
e
0
!#)
t
g(s)ds
e
0
,
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where
Z
re
Ω(e
r) =
re0
Z ze
Gi (e
z) =
ze0
ds
,
ϕ−1 (s)
Close
re ≥ re0 > 0 ,
ds
wi
{ϕ−1 [Ω−1 (s)]}
,
ze ≥ ze0 > 0
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(i = 1, 2)
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Ω−1 , ϕ−1 , G−1 are respectively the inverse of Ω, ϕ, G, and T ∈ R+ is
chosen so that
!
Z αe(t)
Z t
e
Gi Ω(c) +
f (s)ds +
g(s)ds ∈ Dom(G−1
0 ≤ t ≤ T,
i ),
e
0
e
0
and
"
G−1
Gi Ω(c) +
i
Z
α
e(t)
Z
f (s)ds +
e
0
!#
t
∈ Dom(Ω−1 ), e
0 ≤ t ≤ T.
g(s)ds
e
0
Proof. Let c > 0 and define the nonincreasing positive function z(t) and make
Z αe(t)
Z t
(3.46) z(t) = c +
f (s)u(s)w1 (u(s))ds +
g(s)u(s)w2 (u(s))ds.
e
0
e
0
From inequality (3.43), we know
Xueqin Zhao and Fanwei Meng
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−1
u(t) ≤ ϕ [z(t)],
(3.47)
and
(3.48) D1 D2 · · · Dn z(t)
= [f (e
α)u(e
α)w1 (u(e
α))α10 α20 · · · αn0 + g(t)u(t)w2 (u(t))].
Using (3.47), we have
(3.49)
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
D1 D2 · · · Dn z(t)
≤ f (e
α)w1 (u(e
α))α10 α20 · · · αn0 + g(t)w2 (u(t)).
−1
ϕ (z(t))
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For
D1 D2 · · · Dn−1 z(t)
(3.50) Dn
ϕ−1 (z(t))
D1 D2 · · · Dn z(t)ϕ−1 (z(t)) − D1 D2 · · · Dn−1 z(t)(ϕ−1 (z(t)))0 Dn z(t)
,
=
(ϕ−1 (z(t)))2
using D1 D2 · · · Dn−1 z(t) ≥ 0, (ϕ−1 (z(t)))0 ≥ 0, Dn z(t) ≥ 0 in (3.50), we get
D1 D2 · · · Dn−1 z(t)
D1 D2 · · · Dn z(t)
Dn
≤
−1
ϕ (z(t))
ϕ−1 (z(t))
≤ f (e
α)α10 α20 · · · αn0 w1 (ϕ−1 (e
α(t))) + g(t)w2 (ϕ−1 (t)).
Fixing t1 , . . . , tn−1 , setting tn = sn , integrating from tn to ∞, yields
Z αn (tn )
D1 D2 · · · Dn−1 z(t)
≤
f (α1 (t1 ), . . . , αn−1 (tn−1 ), sn )
ϕ−1 (z(t))
0
0
× w1 (ϕ−1 (α1 (t1 ), . . . , αn−1 (tn−1 ), sn ))α10 α20 · · · αn−1
dsn
Z tn
+
g(t1 , . . . , tn−1 , sn )w2 (ϕ−1 (t1 , . . . , tn−1 , sn ))dsn .
0
Deductively
Z α2 (t2 )
Z αn (tn )
D1 z(t)
(3.51)
≤
···
f (α1 (t1 ), s2 , . . . , sn )
ϕ−1 (z(t))
0
0
× w1 (ϕ−1 (α1 (t1 ), s2 , . . . , sn ))α10 dsn . . . ds2
Z t2
Z tn
+
···
g(t1 , s2 , . . . , sn )w2 (ϕ−1 (t1 , s2 , . . . , sn ))dsn . . . ds2 .
0
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
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Fixing t2 , . . . , tn , setting t1 = s1 , integrating from 0 to t1 with respect to s1
yields
Z
(3.52) Ω(z(t)) ≤ Ω(c) +
α
e(t)
e
0
f (s1 , . . . , sn )w1 (ϕ−1 (z(s))
Z t
+
g(s)w2 (ϕ−1 (z(s))ds,
t ∈ Rn+ .
e
0
From Theorem 3.6, we obtain
"
z(t) ≤ Ω−1 G−1
1
Z
G1 (Ω(c)) +
α
e(t)
Z
f (s)ds +
e
0
!#
t
g(s)ds
,
e
0
using (3.47), we get the inequality (3.44).
If c = 0 we carry out the above procedure with ε > 0 instead of c and
subsequently let ε → 0.
(ii) when w1 (u) ≤ w2 (u).
The proof can be completed with suitable changes.
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
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4.
Some Applications
Example 4.1. Consider the integral equation:
(4.1) up (t1 , . . . , tn )
Z αe(t)
= f (t1 , . . . , tn )+
K(s1 , . . . , sn )g(s1 , . . . , sn , u(s1 , . . . , sn ))ds1 . . . dsn ,
e
0
where f, K : Rn+ → R, g : Rn+ × R → R are continuous functions and p > 0
and p 6= 1is constant, α
ei (t) ∈ C 1 (R+ , R+ ) is nondecreasing with αi (t) ≤ ti
on R+ (i = 1, . . . , n). In [8] B.G. Pachpatte studied the problem when α(t) =
t, n = 1. Here we assume that every solution under discussion exists on an
interval Rn+ . We suppose that the functions f , K, g in (4.1) satisfy the following
conditions
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
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|f (t1 , . . . , tn )| ≤ c1 ,
|K(t1 , . . . , tn )| ≤ c2 ,
|g(t1 , . . . , tn , u)| ≤ r(t1 , . . . , tn )|u|q + h(t1 , . . . , tn )|u|,
(4.2)
Contents
Rn+
where c1 , c2 , are nonnegative constants, and p ≥ q > 0, and r :
→ R+ ,
n
h : R+ → R+ are continuous functions. From (4.1) and using (4.2), we get
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p
(4.3) |u (t1 , . . . , tn )|
Z αe(t)
≤ c1 +
[c2 r(s1 , . . . , sn )|u|q + c2 h(s1 , . . . , sn )|u|ds1 . . . dsn .
e
0
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Now an application of Corollary 3.8 yields
 p−1
p

R αe(t)
(1− p1 )

c
(p−1)
2

+ p
h(s)ds
.
.
.
ds
c1

1
n
e

0


i
h R

α
e(t)

c
2

× exp p e0 r(s)ds1 . . . dsn





when p = q,




"
p−q
p−1
|u(t)| ≤
(1− p1 )

e(t)
c2 (p−1) R α


+ p
h(s)ds1 . . . dsn
c1
e

0




1
# p−q



R

α
e(t)


+ c2 (p−q)
r(s)ds1 . . . dsn

e
p
0




when p > q.
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
Title Page
If the integrals of r(s), h(s) are bounded, then we can have the bound of the
solution u(t) of (4.1). Similarly, we can obtain many other kinds of estimates.
Example 4.2. Consider the partial delay differential equation:
(4.4)
∂ 2 up (x, y)
= f (x, y, u(x, y), u(x − h1 (x), y − h2 (y))),
∂x1 ∂x2
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(4.5)
(4.6)
up (x, 0) = a1 (x)
a1 (0) = a2 (0) = 0,
up (0, y) = a2 (y)
|a1 (x) + a2 (y)| ≤ c,
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where f ∈ C(R+ × R+ × R2 , R), a1 ∈ C 1 (R+ , R), a2 ∈ C 1 (R+ , R), c and p
are nonnegative constants. h1 ∈ C 1 (R+ , R+ ), h2 ∈ C 1 (R+ , R+ ), such that
x − h1 (x) ≥ 0,
h01 (x) < 1,
y − h2 (y) ≥ 0,
h02 (y) < 1.
Suppose that
|f (x, y, u, v)| ≤ a(x, y)|v|q + b(x, y)|v|,
(4.7)
where a, b ∈ C(R+ × R+ , R) and let
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
(4.8)
1
,
M1 = max
x∈R+ 1 − h01 (x)
1
M2 = max
.
y∈R+ 1 − h02 (y)
If u(x, y) is any solution of (4.4) – (4.7), then
(4.9) |u(x, y)|
≤
c
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(i) if p = q, we have
(1− p1 )
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p
! p−1
Z
Z
M1 M2 (p − 1) φ1 (x) φ1 (y) e
b(σ, τ )dσdτ
+
p
0
0
"
#
Z
Z
M1 M2 φ1 (x) φ1 (y)
e
a(σ, τ )dσdτ
× exp
p
0
0
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(ii) if p > q, we have
(4.10) |u(x, y)|

≤ c
(1− p1 )
+
M1 M2 (p − 1)
p
Z
φ1 (x)
Z
p−q
! p−1
φ1 (y)
eb(σ, τ )dσdτ
0
M1 M2 (p − q)
+
p
0
Z
0
φ1 (x)
Z
0
1
# p−q
φ1 (y)
e
a(σ, τ )dσdτ
.
In which φ1 (x) = x − h1 (x), x ∈ Rn+ , φ2 (y) = y − h2 (y), y ∈ Rn+ and
eb(σ, τ ) = b(σ + h1 (s), τ + h2 (t)), e
a(σ, τ ) = a(σ + h1 (s), τ + h2 (t)),
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
Rn+ .
for σ, s, τ, t ∈
In fact, if u(x, y) is a solution of (4.4) – (4.7), then it satisfies the equivalent
integral equation:
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p
(4.11) [u(x, y)]
Z
x
y
Z
f (s, t, u(s, t), u(s − h1 (s), t − h2 (t)))dtds.
= a1 (x) + a2 (y) +
0
0
for x, y ∈ (Rn+ × Rn+ , R).
Using (4.5), (4.7) in (4.11) and making the change of variables, we have
(4.12) |u(x, y)|p
Z
≤ c + M1 M2
0
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φ1 (x) Z
0
φ1 (y)
e
a(σ, τ )|u(σ, τ )|q + eb(σ, τ )|u(σ, τ )|dσdτ.
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Now a suitable application of the inequality in Corollary 3.8 to (4.12)
yields (4.9) and (4.10).
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
Xueqin Zhao and Fanwei Meng
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References
[1] B.G. PACHPATTE, On some new inequalities related to certain inequalities
in the theory of differential equations, J. Math. Anal. Appl., 189 (1995),
128–144.
[2] B.G. PACHPATTE, Explicit bounds on certain integral inequalities, J. Math.
Anal. Appl., 267 (2002), 48–61.
[3] B.G. PACHPATTE, On some new inequalities related to a certain inequality
arising in the theory of differential equations, J. Math. Anal. Appl., 251
(2000), 736–751.
On Some New Retard Integral
Inequalities in n Independent
Variables and Their
Applications
[4] B.G. PACHPATTE, On a certain inequality arising in the theory of differential equations, J. Math. Anal. Appl., 182 (1994), 143–157.
Xueqin Zhao and Fanwei Meng
[5] I. BIHARI, A generalization of a lemma of Bellman and its application
to uniqueness problems of differential equations, Acta. Math. Acad. Sci.
Hungar., 7 (1956), 71–94.
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[6] M. MEDVED, Nonlinear singular integral inequalities for functions in two
and n independent variables, J. Inequalities and Appl., 5 (2000), 287–308.
Contents
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[7] O. LIPOVAN, A retarded Gronwall-like inequality and its applications, J.
Math. Anal. Appl., 252 (2000), 389–401.
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[8] O. LIPOVAN, A retarded integral inequality and its applications, J. Math.
Anal. Appl., 285 (2003), 436–443.
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