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The main objective of this paper is to establish a new retarded nonlinear integral inequality with two variables, which provide explicit bound on unknown function.. Introduction Being im

Volume 2010, Article ID 462163, 9 pages

doi:10.1155/2010/462163

Research Article

A New Nonlinear Retarded Integral Inequality and Its Application

1 Department of Mathematics, Hechi University, Guangxi, Yizhou 546300, China

2 School of Mathematics and Computing Science, Guilin University of Electronic Technology,

Guilin 541004, China

Correspondence should be addressed to Wu-Sheng Wang,wang4896@126.com

Received 28 April 2010; Revised 9 July 2010; Accepted 15 August 2010

Academic Editor: L´aszl ´o Losonczi

Copyrightq 2010 Wu-Sheng Wang 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

The main objective of this paper is to establish a new retarded nonlinear integral inequality with two variables, which provide explicit bound on unknown function This inequality given here can

be used as tool in the study of integral equations

1 Introduction

Being important tools in the study of differential equations, integral equations and integro-differential equations, various generalizations of Gronwall inequality and their applications have attracted great interests of many mathematicians Some recent works can be found, for

i1

retarded integral inequality

i1



Cheung3 investigated the inequality in two variables

u p

x, y

p − q

b1x0 

p − q

b2x0 

1.3

ψ

u

x, y

≤ c 

α x0 



γ x0 

1.4



c2

1 2

0



c2

2 2

0



c1

0



c2

0

then, the estimation was used to study the boundedness, asymptotic behavior, slowly growth

of the solutions of the integral equation



0



0

study of certain retarded differential and integral equations It is desirable to establish new inequalities of the above type, which can be used more effectively in the study of certain classes of retarded differential and integral equations

In this paper, we establish a new integral inequality

ψ

u

x, y

≤



c1

x, y



α1x0 

×



c2



x, y



α2x0 

.

1.8

2 Main Result

and suppose that

t → ∞;

Φr :

0

ds

ϕ

ψ−1s , Ψr :

0

ds

ϕ

ψ−1

2.1

Theorem 2.1 Suppose that (H1)–(H5) hold and u x, y is a nonnegative and continuous function

u

x, y

E

for all x, y ∈ x0, X1 × y0, Y1, where

E

x, y

x, y

i1

α i x0 

×

α3−ix0 

β3−iy0f 3−i σ, tdt dσ



ds,

G

x, y



x, y

c2



x, y

i1

α i x0 

β iy0c 3−i s, tf i s, tdt ds,

2.3

ψ−1,Φ−1, and Ψ−1 denote the inverse function of ψ, Φ and Ψ, respectively, and X1, Y1 ∈ Δ is

arbitrarily given on the boundary of the planar region

x, y

E

x, y

Proof From the inequality1.8, for all x, y ∈ x0, X  × J, we have

ψ

u

x, y

≤



c1

X, y



α1x0 

×



c2

X, y



α2x0 

,

2.5

θ

x, y





c1

X, y



α1x0 

×



c2

X, y



α2x0 

.

2.6

fact that ux, y ≤ ψ−1θx, y, we obtain

θ x

x, y

i1



αi x

×



c3−i

X, y



α3−ix0 

θ

x, y

2



i1



αi x

×



c3−i

X, y



α3−ix0 

,

2.7

θ x

x, y

ϕ

ψ−1

θ

i1



αi x

×



c3−i

X, y



α3−ix0 

.

2.8

x, y

x0, y

i1

α i x0 

β i y0 c 3−i

X, y

i1

α i x0 

×

α3−ix0 

β3−iy0f 3−i σ, tϕ3−i



ds



X, y

c2



X, y

i1

α i x0 

β i y0 c 3−i

X, y

i1

α i x0 

×

α3−ix0 

β3−iy0 f 3−i σ, tϕ3−i



ds,

2.9

a positive and nondecreasing function in each variable, θx, y ≤ Φ−1ωx, y and ωx0, y 

i1

α i x0 

β i y0 c 3−i X, yf i s, tdt ds Differentiating ωx, y for x, by

ωx

x, y

i1

αi x

×

α 3−i x

α 3−i x0 

β 3−i y

β 3−iy0f 3−i σ, tϕ 3−i

i1

αi x

×

α 3−i x

α 3−i x0 

β 3−i y

β 3−iy0f 3−i σ, tϕ

ψ−1

ω

x, y

2



i1

αi x

×

α 3−i x

α 3−i x0 

β 3−i y

β 3−iy0f 3−i σ, t dt dσ, ∀



x, y

,

2.10

ω x

x, y

ϕ

ψ−1

ω

i1

αi x

×

α 3−i x

α 3−i x0 

β 3−i y

β 3−i y0 f 3−i σ, tdt dσ,

2.11

x, y

x0, y

i1

α i x0 

×

α 3−i s

α 3−i x0 

β 3−i y

β 3−iy0f 3−i σ, tdt dσ



ds

 Ψ





X, y

c2



X, y

i1

α i x0 

β iy0c 3−i



X, y

i1

α i x0 

×

α 3−i s

α 3−i x0 

β 3−i y

β 3−iy0f 3−i σ, tdt dσ



ds.

2.12

Using the fact ux, y ≤ ψ−1θx, y and θx, y ≤ Φ−1ωx, y, from 2.12 we obtain

u

x, y

θ

x, y

ω

x, y





 Ψ



X, y

c2

X, y

i1

α i x0 

β iy0c 3−i



X, y

i1

α i x0 

×

α 3−i s

α 3−i x0 

β 3−i y

β 3−iy0f 3−i σ, tdt dσ



ds

.

2.13

u

X, y





 Ψ



X, y

c2

X, y

i1

α i x0 

β iy0c 3−i



X, y

i1

α i x0 

×

α 3−i s

α 3−i x0 

β 3−i y

β 3−iy0f 3−i σ, tdt dσ



ds

.

2.14

3 Applications

In this section, we present an application of our result to obtain bound of the solution of a integral equation:

ψ

z

x, y

 k



a1

x, y

−

α1x0 

×



a2

x, y



α2x0 

, ∀x, y

∈ Δ,

3.1

u > 0, i  1, 2.

The integral equation3.1 is obviously more general than 1.7 considered in 10.

Corollary 3.1 Consider integral equation 3.1 and suppose that |g i x − s, t| ≤ f i s, t, i  1, 2,

where f i ∈ C0Δ, R Then all solutions zx, y of 3.1 have the estimate

H

for all x, y ∈ x0, X2 × y0, Y2, where

H

x, y

x, y

i1

α i x0 

×

α3−ix0 

β3−iy0f 3−i σ, tdt dσ



ds,

B

x, y

i1

α i x0 

β iy0a 3−i s, tf i s, tdt ds.

3.3

Functions Φ, Φ−1, Ψ, Ψ−1are defined as in Theorem 2.1 , and X2, Y2 ∈ Δ is arbitrarily given on the

boundary of the planar region

x, y

H

x, y

Proof From the integral equation3.1, we have

ψz

α1x0 

×



α2x0 

≤



α1x0 

×



α2x0 

, ∀x, y

∈ Δ.

3.5

The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions This paper is supported by the Natural Science Foundation of

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The authors are very grateful to the editor and the referees for their helpful comments and valuable...

The integral equation3.1 is obviously more general than 1.7 considered in 10.

Corollary... tϕ3−i



ds,

2.9