Journal of Inequalities in Pure and
Applied Mathematics
ON SOME ADVANCED INTEGRAL INEQUALITIES AND
THEIR APPLICATIONS
volume 6, issue 3, article 60,
2005.
XUEQIN ZHAO AND FANWEI MENG
Department of Mathematics
Qufu Normal University
Qufu 273165
People’s Republic of China.
Received 08 November, 2004;
accepted 03 April, 2005.
Communicated by: S.S. Dragomir
EMail: xqzhao1972@126.com
EMail: fwmeng@qfnu.edu.cn
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
214-04
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Abstract
In this paper, we obtain a generalization of advanced integral inequality and by
means of examples we show the usefulness of our results.
2000 Mathematics Subject Classification: 26D15, 26D10.
Key words: Advanced integral inequality; Integral equation.
Project supported by a grant from NECC and NSF of Shandong Province, China
(Y2001A03).
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
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Contents
Xueqin Zhao and Fanwei Meng
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminaries and Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4
An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References
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1.
Introduction
Integral inequalities play an important role in the qualitative analysis of the solutions to differential and integral equations. Many retarded inequalities have
been discovered (see [2], [3], [5], [7]). However, we almost neglect the importance of advanced inequalities. After all, it does great benefit to solve the bound
of certain integral equations, which help us to fulfill a diversity of desired goals.
In this paper we establish two advanced integral inequalities and an application
of our results is also given.
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
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Xueqin Zhao and Fanwei Meng
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2.
Preliminaries and Lemmas
In this paper, we assume throughout that R+ = [0, ∞), is a subset of the set of
real numbers R. The following lemmas play an important role in this paper.
Lemma 2.1. 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
constant. Let α ∈ C 1 (R+ , R+ ) be nondecreasing with α(t) ≥ t on R+ . If
u, f ∈ C(R+ , R+ ) and
Z ∞
(2.1)
ϕ(u(t)) ≤ c +
f (s)ψ(u(s))ds,
t ∈ R+ ,
α(t)
then for 0 ≤ T ≤ t < ∞,
Xueqin Zhao and Fanwei Meng
Z
−1
−1
u(t) ≤ ϕ
G [G(c) +
(2.2)
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
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∞
f (s)ds] ,
α(t)
Rz
ds
, z ≥ z0 > 0, ϕ−1 , G−1 are respectively the inverse
where G(z) = z0 ψ[ϕ−1
(s)]
of ϕ and G, T ∈ R+ is chosen so that
Z ∞
(2.3a)
G(c) +
f (s)ds ∈ Dom(G−1 ),
t ∈ [T, ∞).
α(t)
Z ∞
−1
(2.3b)
G
G(c) +
f (s)ds ∈ Dom(ϕ−1 ),
t ∈ [T, ∞).
α(t)
Proof. Define the nonincreasing positive function z(t) and make
Z ∞
(2.4)
z(t) = c + ε +
f (s)ψ(u(s))ds,
t ∈ R+ ,
α(t)
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where ε is an arbitrary small positive number. From inequality (2.1), we have
(2.5)
u(t) ≤ ϕ−1 [z(t)].
Differentiating (2.4) and using (2.5) and the monotonicity of ϕ−1 , ψ, we deduce
that
z 0 (t) = −f α(t) ψ u α(t) α0 (t)
≥ −f α(t) ψ ϕ−1 z(α(t)) α0 (t)
≥ −f α(t) ψ ϕ−1 z(t) α0 (t).
For
ψ[ϕ−1 (z(t))] ≥ ψ[ϕ−1 (z(∞))] = ψ[ϕ−1 (c + ε)] > 0,
from the definition of G, the above relation gives
0
d
z (t)
G(z(t)) =
≥ −f α(t) α0 (t).
−1
dt
ψ[ϕ (z(t))]
Setting t = s, and integrating it from t to ∞ and letting ε → 0 yields
Z ∞
G z(t) ≤ G(c) +
f (s)ds,
t ∈ R+ .
α(t)
From (2.3), (2.5) and the above relation, we obtain the inequality (2.2).
In fact, we can regard Lemma 2.1 as a generalized form of an Ou-Iang type
inequality with advanced argument.
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Xueqin Zhao and Fanwei Meng
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Lemma 2.2. Let u f and g be nonnegative continuous functions defined on R+ ,
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, ∞), α ∈ C 1 (R+ , R+ ) be nondecreasing with α(t) ≥ t on R+ . If
Z ∞
Z ∞
(2.6) ϕ(u(t)) ≤ c +
f (s)w1 (u(s))ds +
g(s)w2 (u(s))ds, t ∈ R+ ,
α(t)
t
then for 0 ≤ T ≤ t < ∞,
(i) For the case w2 (u) ≤ w1 (u),
Z
−1
−1
(2.7)
u(t) ≤ ϕ
G1 G1 (c) +
∞
Z
f (s)ds +
α(t)
(ii) For the case w1 (u) ≤ w2 (u),
Z
−1
−1
(2.8)
u(t) ≤ ϕ
G2 G2 (c) +
∞
A Numerical Method in Terms of
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g(s)ds
.
t
∞
Z
f (s)ds +
α(t)
∞
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g(s)ds
,
t
where
Z
r
Gi (r) =
r0
ds
,
wi (ϕ−1 (s))
r ≥ r0 > 0,
(i = 1, 2)
and ϕ−1 , G−1
i (i = 1, 2) are respectively the inverse of ϕ, Gi , T ∈ R+ is
chosen so that
Z ∞
Z ∞
(2.9) Gi (c) +
f (s)ds +
g(s)ds ∈ Dom(G−1
i ),
α(t)
t
(i = 1, 2),
Xueqin Zhao and Fanwei Meng
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t ∈ [T, ∞).
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Proof. Define the nonincreasing positive function z(t) and make
Z
∞
(2.10) z(t) = c + ε +
Z
f (s)w1 (u(s))ds +
α(t)
∞
g(s)w2 (u(s))ds,
t
0 ≤ T ≤ t < ∞,
where ε is an arbitrary small positive number. From inequality (2.6), we have
(2.11)
u(t) ≤ ϕ−1 [z(t)],
t ∈ R+ .
Differentiating (2.10) and using (2.11) and the monotonicity of ϕ−1 , w1 , w2 , we
deduce that
z 0 (t) = −f α(t) w1 u α(t) α0 (t) − g(t)w2 u(t) ,
≥ −f α(t) w1 ϕ−1 z(α(t)) α0 (t) − g(t)w2 ϕ−1 z(t) ,
≥ −f α(t) w1 ϕ−1 z(t) α0 (t) − g(t)w2 ϕ−1 z(t) .
(i) When w2 (u) ≤ w1 (u)
z 0 (t) ≥ −f α(t) w1 ϕ−1 z(t) α0 (t) − g(t)w1 ϕ−1 z(t) ,
t ∈ R+ .
A Numerical Method in Terms of
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Integral Equation from
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Xueqin Zhao and Fanwei Meng
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For
w1 [ϕ−1 (z(t))] ≥ w1 ϕ−1 (z(∞)) = w1 [ϕ−1 (c + ε)] > 0,
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from the definition of G1 (r), the above relation gives
0
d
z 0 (t)
G1 (z(t)) =
≥
−f
α(t)
α (t) − g(t),
dt
w1 [ϕ−1 (z(t))]
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t ∈ R+ .
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Setting t = s and integrating it from t to ∞ and let ε → 0 yields
Z ∞
Z ∞
G1 z(t) ≤ G1 (c) +
f (s)ds +
g(s)ds, t ∈ R+ ,
α(t)
t
so,
z(t) ≤
G−1
1
Z
G1 (c) +
∞
Z
∞
f (s)ds +
α(t)
Using (2.11), we have
Z
−1
−1
u(t) ≤ ϕ
G1 G1 (c) +
g(s)ds ,
0 ≤ T ≤ t < ∞.
t
∞
α(t)
Z
f (s)ds +
∞
g(s)ds ,
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
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0 ≤ T ≤ t < ∞.
Xueqin Zhao and Fanwei Meng
t
(ii) When w1 (u) ≤ w2 (u), the proof can be completed similarly.
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3.
Main Results
In this section, we obtain our main results as follows:
Theorem 3.1. Let u, f and g be nonnegative continuous functions defined on
R+ 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
function with ψ(u) > 0 on (0, ∞) and α ∈ C 1 (R+ , R+ ) be nondecreasing with
α(t) ≥ t on R+ . If
Z ∞
(3.1)
ϕ(u(t)) ≤ c +
[f (s)u(s)ψ(u(s)) + g(s)u(s)]ds,
t ∈ R+
α(t)
then for 0 ≤ T ≤ t < ∞,
Z
−1
−1
−1
(3.2) u(t) ≤ ϕ
Ω
G
G[Ω(c) +
A Numerical Method in Terms of
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Xueqin Zhao and Fanwei Meng
∞
Z
∞
g(s)ds] +
α(t)
f (s)ds
,
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α(t)
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where
Z
r
Ω(r) =
Zr0z
G(z) =
z0
ds
, r ≥ r0 > 0,
−1
ϕ (s)
ds
, z ≥ z0 > 0,
−1
ψ{ϕ [Ω−1 (s)]}
Ω−1 , ϕ−1 , G−1 are respectively the inverse of Ω, ϕ, G and T ∈ R+ is chosen so
that
Z ∞
Z ∞
G Ω(c) +
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f (s)ds ∈ Dom(G−1 )
g(s)ds +
α(t)
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and
G
−1
Z
G Ω(c) +
∞
Z
∞
g(s)ds +
α(t)
f (s)ds ∈ Dom(Ω−1 )
α(t)
for t ∈ [T, ∞).
Proof. Let us first assume that c > 0. Define the nonincreasing positive function
z(t) by the right-hand side of (3.1). Then z(∞) = c, u(t) ≤ ϕ−1 [z(t)] and
z 0 (t) = − f α(t) u α(t) ψ u α(t) − g α(t) u α(t) α0 (t)
≥ − f α(t) ϕ−1 z(α(t)) ψ ϕ−1 z(α(t)) − g α(t) ϕ−1 z(α(t)) α0 (t)
≥ − f α(t) ϕ−1 z(t) ψ ϕ−1 z(α(t)) − g α(t) ϕ−1 z(t) α0 (t).
A Numerical Method in Terms of
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Xueqin Zhao and Fanwei Meng
Since ϕ−1 (z(t)) ≥ ϕ−1 (c) > 0,
z 0 (t)
≥ − f α(t) ψ ϕ−1 z(α(t)) + g α(t) α0 (t).
−1
ϕ z(t)
Setting t = s and integrating it from t to ∞ yields
Z ∞
Z ∞
Ω(z(t)) ≤ Ω(c) +
g(s)ds +
f (s)ψ[ϕ−1 (z(s))]ds.
α(t)
α(t)
R∞
Let T ≤ T1 be an arbitrary number. We denote p(t) = Ω(c) + α(t) g(s)ds.
From the above relation, we deduce that
Z ∞
Ω(z(t)) ≤ p(T1 ) +
f (s)ψ[ϕ−1 (z(s))]ds,
T1 ≤ t < ∞.
α(t)
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Now an application of Lemma 2.1 gives
Z
−1
−1
z(t) ≤ Ω
G
G(p(T1 )) +
∞
f (s)ds
,
T1 ≤ t < ∞,
α(t)
so,
u(t) ≤ ϕ
−1
Z
−1
−1
Ω
G
G(p(T1 )) +
∞
f (s)ds
,
T1 ≤ t < ∞.
α(t)
Taking t = T1 in the above inequality, since T1 is arbitrary, we can prove the
desired inequality (3.2).
If c = 0 we carry out the above procedure with ε > 0 instead of c and
subsequently let ε → 0.
Corollary 3.2. Let u, f and g be nonnegative continuous functions defined on
R+ and let c be a nonnegative constant. Moreover, let ψ ∈ C(R+ , R+ ) be
a nondecreasing function with ψ(u) > 0 on (0, ∞) and α ∈ C 1 (R+ , R+ ) be
nondecreasing with α(t) ≥ t on R+ . If
Z ∞
2
2
u (t) ≤ c +
[f (s)u(s)ψ(u(s)) + g(s)u(s)]ds, t ∈ R+ ,
α(t)
then for 0 ≤ T ≤ t < ∞,
Z
Z
1 ∞
1 ∞
−1
u(t) ≤ Ω
Ω c+
g(s)ds +
f (s)ds ,
2 α(t)
2 α(t)
where
Z
Ω(r) =
1
r
ds
,
ψ(s)
A Numerical Method in Terms of
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Xueqin Zhao and Fanwei Meng
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r > 0,
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Ω−1 is the inverse of Ω, and T ∈ R+ is chosen so that
Z
Z
1 ∞
1 ∞
Ω c+
g(s)ds +
f (s)ds ∈ Dom(Ω−1 )
2 α(t)
2 α(t)
for all t ∈ [T, ∞).
Corollary 3.3. Let u, f and g be nonnegative continuous functions defined on
R+ and let c be a nonnegative constant. Moreover, let α ∈ C 1 (R+ , R+ ) be
nondecreasing with α(t) ≥ t on R+ . If
Z ∞
2
2
u (t) ≤ c +
[f (s)u2 (s) + g(s)u(s)]ds, t ≥ 0,
α(t)
A Numerical Method in Terms of
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Xueqin Zhao and Fanwei Meng
then
u(t) ≤
1
c+
2
Z ∞
1
g(s)ds exp
f (s)ds ,
2 α(t)
α(t)
Z
∞
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t ≥ 0.
Corollary 3.4. Let u, f and g be nonnegative continuous functions defined on
R+ and let c be a nonnegative constant. Moreover, let p, q be positive constants
with p ≥ q, p 6= 1. Let α ∈ C 1 (R+ , R+ ) be nondecreasing with α(t) ≥ t on
R+ . If
Z
∞
up (t) ≤ c +
[f (s)uq (s) + g(s)u(s)]ds,
α(t)
t ∈ R+ ,
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then for t ∈ R+ ,
 p
p−1
h R
i
R

(1− p1 )
p−1 ∞
1 ∞

+
g(s)ds
exp
f
(s)ds
,
when p = q,
c

p
p α(t)
α(t)


u(t) ≤
1
p−q
p−q

p−1

R
R

(1− p1 )
p−1 ∞
p−q ∞

c
+ p α(t) g(s)ds
+ p α(t) f (s)ds
, when p > q.

Theorem 3.5. Let u, f and g be nonnegative continuous functions defined on
R+ , 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, ∞) and α ∈ C 1 (R+ , R+ )
be nondecreasing with α(t) ≥ t on R+ . If
Z ∞
Z ∞
(3.3) ϕ(u(t)) ≤ c +
f (s)u(s)w1 (u(s))ds +
g(s)u(s)w2 (u(s))ds,
α(t)
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Xueqin Zhao and Fanwei Meng
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t
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then for 0 ≤ T ≤ t < ∞,
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(i) For the case w2 (u) ≤ w1 (u),
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(3.4) u(t)
≤ϕ
−1
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Z
−1
−1
Ω
G1 G1 (Ω(c)) +
∞
α(t)
Z
f (s)ds +
t
∞
,
g(s)ds
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(ii) For the case w1 (u) ≤ w2 (u),
(3.5) u(t)
≤ϕ
−1
Ω
Z
−1
G2 G2 (Ω(c)) +
−1
∞
∞
Z
f (s)ds +
α(t)
g(s)ds
,
t
where
Z
r
Ω(r) =
Zr0z
Gi (z) =
z0
ds
ϕ−1 (s)
,
r ≥ r0 > 0,
ds
wi
{ϕ−1 [Ω−1 (s)]}
z ≥ z0 > 0
,
(i = 1, 2),
Xueqin Zhao and Fanwei Meng
Ω−1 , ϕ−1 , G−1 are respectively the inverse of Ω, ϕ, G, and T ∈ R+ is
chosen so that
Z ∞
Z ∞
Gi Ω(c) +
f (s)ds +
g(s)ds ∈ Dom(G−1
i ),
α(t)
t
Z ∞
Z ∞
−1
Gi Gi Ω(c) +
f (s)ds +
g(s)ds
∈ Dom(Ω−1 ),
α(t)
t
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for all t ∈ [T, ∞).
Close
Proof. Let c > 0, define the nonincreasing positive function z(t) and make
Z ∞
Z ∞
(3.6)
z(t) = c +
f (s)u(s)w1 (u(s))ds +
g(s)u(s)w2 (u(s))ds.
α(t)
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From inequality (3.3), we have
u(t) ≤ ϕ−1 [z(t)].
(3.7)
Differentiating (3.6) and using (3.7) and the monotonicity of ϕ−1 , w1 , w2 , we
deduce that
z 0 (t) = −f α(t) u α(t) w1 u α(t) α0 (t) − g(t)u(t)w2 u(t) ,
≥ −f α(t) ϕ−1 z(α(t)) w1 ϕ−1 z(α(t)) α0 (t)
− g(t)ϕ−1 z(t) w2 ϕ−1 z(t) ,
≥ −f α(t) ϕ−1 z(t) w1 ϕ−1 z(t) α0 (t)
− g(t)ϕ−1 z(t) w2 ϕ−1 z(t) .
A Numerical Method in Terms of
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Xueqin Zhao and Fanwei Meng
(i) When w2 (u) ≤ w1 (u)
z 0 (t)
≥ −f α(t) w1 ϕ−1 z(t) α0 (t) − g(t)w1 ϕ−1 z(t) .
ϕ−1 z(t)
For
w1 [ϕ−1 (z(t))] ≥ w1 [ϕ−1 (z(∞))] = w1 [ϕ−1 (c + ε)] > 0,
setting t = s and integrating from t to ∞ yields
Z ∞
Z
−1
Ω(z(t)) ≤ Ω(c) +
f (s)w1 ϕ z(t) ds +
From Lemma 2.2, we obtain
Z
−1
−1
z(t) ≤ Ω
G1 G1 (Ω(c)) +
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∞
g(s)w1 ϕ−1 z(t) ds.
t
α(t)
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∞
α(t)
Z
f (s)ds +
t
∞
g(s)ds ,
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0 ≤ T ≤ t < ∞.
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Using u(t) ≤ ϕ−1 [z(t)], we get the inequality in (3.4)
If c = 0, we can 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 similarly.
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4.
An Application
We consider an integral equation
(4.1)
Z
p
x (t) = a(t) +
∞
F [s, x(s), x(φ(s))]ds.
t
Assume that:
|F (x, y, u)| ≤ f (x)|u|q + g(x)|u|
(4.2)
and
(4.3)
|a(t)| ≤ c, c > 0 p ≥ q > 0, p 6= 1,
where f, g are nonnegative continuous real-valued functions, and φ ∈ C 1 (R+ , R+ )
is nondecreasing with φ(t) ≥ t on R+ . From (4.1), (4.2) and (4.3) we have
Z ∞
p
|x(t)| ≤ c +
f (s)|x(φ(s))|q + g(s)|x(φ(s))|ds.
t
Making the change of variables from the above inequality and taking
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Xueqin Zhao and Fanwei Meng
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Contents
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M = sup 0 ,
t∈R+ φ (t)
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we have
p
Z
∞
|x(t)| ≤ c + M
φ(t)
f¯(s)|x(s)|q + ḡ(s)|x(s)|ds,
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J. Ineq. Pure and Appl. Math. 6(3) Art. 60, 2005
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in which f¯(s) = f (φ−1 (s)), ḡ(s) = g(φ−1 (s)). From Corollary 3.4, we obtain
 p
p−1
h R
i
∞ ¯
M (p−1) R ∞

(1− p1 )
M

+
ḡ(s)ds
exp
f
(s)ds
,
c

p
p φ(t)
φ(t)




when p = q


|x(t)| ≤
1
p−q
p−q

p−1

R
R
1

∞
∞

ḡ(s)ds
f¯(s)ds
c(1− p ) + M (p−1)
+ M (p−q)
,


p
p
φ(t)
φ(t)



when p > q.
If the integrals of f (s), g(s) are bounded, then we have the bound of the solution
of (4.1).
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Xueqin Zhao and Fanwei Meng
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References
[1] 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.
[2] O. LIPOVAN, A retarded Gronwall-like inequality and its applications, J.
Math. Anal. Appl., 252 (2000), 389–401.
[3] O. LIPOVAN, A retarded Integral inequality and its applications, J. Math.
Anal. Appl., 285 (2003), 436–443.
[4] M. MEDVED, Noninear singular integral inequalities for functions in two
and n independent variables, J. Inequalities and Appl., 5 (2000), 287–308.
[5] 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.
[6] B.G. PACHPATTE, Explicit bounds on certain integral inequalities, J. Math.
Anal. Appl., 267 (2002), 48–61.
[7] 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.
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Xueqin Zhao and Fanwei Meng
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