Volume 2010, Article ID 462163,9pages doi:10.1155/2010/462163
Research Article
A New Nonlinear Retarded Integral Inequality and
Its Application
Wu-Sheng Wang,
1Ri-Cai Luo,
1and Zizun Li
21Department of Mathematics, Hechi University, Guangxi, Yizhou 546300, China
2School of Mathematics and Computing Science, Guilin University of Electronic Technology,
Guilin 541004, China
Correspondence should be addressed to Wu-Sheng Wang,[email protected]
Received 28 April 2010; Revised 9 July 2010; Accepted 15 August 2010
Academic Editor: L´aszl ´o Losonczi
Copyrightq2010 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 example, in1–7and some references therein. Agarwal et al.1studied the inequality
ut≤at
n
i1 bit
bit0
git, swiusds, t0≤t< t1. 1.1
Agarwal et al. 2 obtained the explicit bound to the unknown function of the following retarded integral inequality
ϕut≤c
n
i1 αit
αit0
uqsfisϕ1us gisϕ2
Cheung3investigated the inequality in two variables
upx, y≤a p p−q
b1x
b1x0 c1y
c1y0
g1s, tuqs, tdt ds
p
p−q
b2x
b2x0 c2y
c2y0
g2s, tuqs, tψus, tdt ds.
1.3
Chen et al.4discussed the following inequality in two variables
ψux, y≤c
αx
αx0
βy
βy0
gs, twus, tdt ds
γx
γx0 δy
δy0
fs, twus, tϕus, tdt ds.
1.4
Pachpatte8obtained an upper bound in the following inequality:
u2t≤
c2 12
t
0
fsusds c2 22
t
0
hsusds . 1.5
Pachpatte9firstly got the estimation of the unknown function of the following inequality:
ut≤
c1 t
0
fsusds c2 t
0
hsusds , 1.6
then, the estimation was used to study the boundedness, asymptotic behavior, slowly growth of the solutions of the integral equation
ut k
c1t− t
0
f1t−susds c2t t
0
f2t−susds , 1.7
1.7was studied by Gripenberg in10.
In this paper, we establish a new integral inequality
ψux, y≤
c1
x, y
α1x
α1x0 β1y
β1y0
f1s, tϕ1us, tdt ds
×
c2
x, y
α2x
α2x0 β2y
β2y0
f2s, tϕ2us, tdt ds .
1.8
We will prove importance of1.8in achieving a desired goal.
2. Main Result
Throughout this paper, x0, x1, y0, y1 ∈ Rare given numbers, and x0 < x1, y0 < y1. I :
x0, x1, J: y0, y1,Δ:I×J, R : 0,∞.For functionshx, gx, y,hxdenotes the derivative ofhx, andgxx, ydenotes the partial derivative gx, yonx. Consider1.8,
and suppose that
H1ψ∈CR,Ris a strictly increasing function withψ0 0 andψt → ∞as
t → ∞;
H2c1, c2:Δ → 0,∞are nondecreasing in each variable;
H3ϕi∈CR,Rare nondecreasing withϕir>0 forr >0, i1,2;
H4αi ∈ C1I, I and βi ∈ C1J, J are nondecreasing such that αix ≤ xand
βiy≤y, i1,2;
H5fi∈CΔ,R, i1,2.
We define functionsΦ,Ψ, andϕby
Φr: r
0
ds ϕψ−1s,
Ψr: r
0
ds
ϕψ−1Φ−1s, r >0,
ϕr:maxϕ1r, ϕ2r
.
2.1
Theorem 2.1. Suppose that (H1)–(H5) hold andux, yis a nonnegative and continuous function
onΔsatisfying1.8. Then one has
for allx, y∈x0, X1×y0, Y1, where
Ex, y ΨGx, y
2
i1 αix
αix0
βiy
βiy0
fis, tdt
×
α3−is
α3−ix0
β3−iy
β3−iy0
f3−iσ, tdt dσ
ds,
Gx, y Φc1
x, yc2
x, y
2
i1 αix
αix0
βiy
βiy0
c3−is, tfis, tdt ds,
2.3
ψ−1,Φ−1, and Ψ−1 denote the inverse function of ψ,Φ and Ψ, respectively, and X
1, Y1 ∈ Δ is
arbitrarily given on the boundary of the planar region
R:x, y∈Δ:Ex, y∈DomΨ−1,Ψ−1Ex, y∈DomΦ−1. 2.4
Proof. From the inequality1.8, for allx, y∈x0, X×J, we have
ψux, y≤
c1
X, y
α1x
α1x0 β1y
β1y0
f1s, tϕ1us, tdt ds
×
c2
X, y
α2x
α2x0 β2y
β2y0
f2s, tϕ2us, tdt ds ,
2.5
wherex0≤X≤X1is chosen arbitrarily, using the assumptionH2. For convenience, we define a functionθx, yby the right-hand side of1.8, that is,
θx, y
c1
X, y
α1x
α1x0 β1y
β1y0
f1s, tϕ1us, tdt ds
×
c2
X, y
α2x
α2x0 β2y
β2y0
f2s, tϕ2us, tdt ds .
2.6
fact thatux, y≤ψ−1θx, y, we obtain θx x, y 2
i1
αix
βiy
βiy0
fiαix, tϕiuαix, tdt
×
c3−i
X, y
α3−ix
α3−ix0
β3−iy
β3−iy0
f3−is, tϕ3−ius, tdt ds
≤ϕψ−1θx, y
2
i1
αix
βiy
βiy0
fiαix, tdt
×
c3−i
X, y
α3−ix
α3−ix0
β3−iy
β3−iy0
f3−is, tϕ3−i
ψ−1θs, tdt ds ,
2.7
for allx, y∈x0, X×J. From2.7, we get
θx
x, y ϕψ−1θx, y ≤
2
i1
αix
βiy
βiy0
fiαix, tdt
×
c3−i
X, y
α3−ix
α3−ix0
β3−iy
β3−iy0
f3−is, tϕ3−i
ψ−1θs, tdt ds .
2.8
By takingsxin2.8and then integrating it fromx0tox, we get
Φθx, y≤Φθx0, y
2
i1 αix
αix0
βiy
βiy0
c3−iX, yfis, tdt ds
2
i1 αix
αix0
βiy
βiy0
fis, tdt
×
α3−is
α3−ix0
β3−iy
β3−iy0
f3−iσ, tϕ3−i
ψ−1θσ, tdt dσ
ds
≤Φc1
X, yc2
X, y
2
i1 αiX
αix0
βiy
βiy0
c3−iX, yfis, tdt ds
2
i1 αix
αix0
βiy
βiy0
fis, tdt
×
α3−is
α3−ix0
β3−iy
β3−iy0
f3−iσ, tϕ3−i
ψ−1θσ, tdt dσ
ds,
2.9
a positive and nondecreasing function in each variable,θx, y≤Φ−1ωx, yandωx 0, y
Φc1X, yc2X, y 2i1 αiX
αix0
βiy
βiy0c3−iX, yfis, tdt ds. Differentiatingωx, yforx, by
the relation amongϕandϕ1, ϕ2, we have
ωx
x, y
2
i1
αix
βiy
βiy0
fiαix, tdt
×
α3−ix
α3−ix0
β3−iy
β3−iy0
f3−iσ, tϕ3−i
ψ−1θσ, tdt dσ
≤2
i1
αix
βiy
βiy0
fiαix, tdt
×
α3−ix
α3−ix0
β3−iy
β3−iy0
f3−iσ, tϕ
ψ−1Φ−1ωσ, tdt dσ
≤ϕψ−1Φ−1ωx, y
2
i1
αix
βiy
βiy0
fiαix, tdt
×
α3−ix
α3−ix0
β3−iy
β3−iy0
f3−iσ, tdt dσ, ∀x,y∈x0, X×
y0, Y1
,
2.10
whereY1is defined by2.4. From2.10, we have
ωx
x, y
ϕψ−1Φ−1ωx, y ≤ 2
i1
αix
βiy
βiy0
fiαix, tdt
×
α3−ix
α3−ix0
β3−iy
β3−iy0f3−iσ, tdt dσ,
2.11
for allx, y∈x0, X×y0, Y1. By takingsxin2.11and then integrating it fromx0tox, using the definition ofΨin2.1, we get
Ψωx, y≤Ψωx0, y
2
i1 αix
αix0
βiy
βiy0
fis, tdt
×
α3−is
α3−ix0
β3−iy
β3−iy0
f3−iσ, tdt dσ
ds
Ψ
Φc1
X, yc2
X, y
2
i1 αiX
αix0
βiy
βiy0
c3−iX, yfis, tdt ds
2
i1 αix
αix0
βiy
βiy0
fis, tdt ×
α3−is
α3−ix0
β3−iy
β3−iy0
f3−iσ, tdt dσ
ds.
Using the factux, y≤ψ−1θx, yandθx, y≤Φ−1ωx, y, from2.12we obtain
ux, y≤ψ−1θx, y≤ψ−1Φ−1ωx, y
≤ψ−1
Φ−1
Ψ−1
Ψ
Φc1
X, yc2
X, y
2
i1 αiX
αix0
βiy
βiy0
c3−iX, yfis, tdt ds
2
i1 αix
αix0
βiy
βiy0
fis, tdt ×
α3−is
α3−ix0
β3−iy
β3−iy0
f3−iσ, tdt dσ
ds
.
2.13
LetxX, from2.13we observe that
uX, y≤ψ−1
Φ−1
Ψ−1
Ψ
Φc1
X, yc2
X, y
2
i1 αiX
αix0
βiy
βiy0
c3−iX, yfis, tdt ds
2
i1 αiX
αix0
βiy
βiy0
fis, tdt ×
α3−is
α3−ix0
β3−iy
β3−iy0
f3−iσ, tdt dσ
ds
.
2.14
SinceX ∈x0, X1is arbitrary, from2.14, we get the required estimation2.2.
3. Applications
In this section, we present an application of our result to obtain bound of the solution of a integral equation:
ψzx, yk
a1
x, y−
α1x
α1x0 β1y
β1y0
g1x−s, tϕ1zs, tdt ds
×
a2
x, y
α2x
α2x0 β2y
β2y0
g2x−s, tϕ2zs, tdt ds , ∀
x, y∈Δ,
3.1
whereψ : R → Ris a strictly increasing function withψ0 0,|ψr| ψ|r| > 0,and
ψt → ∞ast → ∞,kis a given positive constant,|a1|,|a2|:Δ → Rare bounded functions and nondecreasing in each variable, functionsαiandβi satisfy hypothesisH4, i1,2,gi, z ∈
C0Δ,Randϕ
i ∈ C0R,Ris nondecreasing onR such that|ϕiu| ϕi|u|, ϕiu > 0 for
The integral equation3.1is obviously more general than 1.7considered in10. When keepingyfixed, letψzx, y zx, y, ϕizx, y zx, y, αix x, i1,2, x0 0, then integral equation3.1reduces to integral equation1.7in10.
Corollary 3.1. Consider integral equation3.1and suppose that|gix−s, t| ≤ fis, t, i 1,2,
wherefi∈C0Δ,R. Then all solutionszx, yof3.1have the estimate
zx, y≤ψ−1Φ−1Ψ−1Hx, y, 3.2
for allx, y∈x0, X2×y0, Y2, where
Hx, y ΨBx, y
2
i1 αix
αix0
βiy
βiy0
fis, tdt ×
α3−is
α3−ix0
β3−iy
β3−iy0
f3−iσ, tdt dσ
ds,
Bx, y Φa1
x, ya2
x, y
2
i1 αix
αix0
βiy
βiy0
a3−is, tfis, tdt ds.
3.3
FunctionsΦ,Φ−1,Ψ,Ψ−1are defined as inTheorem 2.1, andX
2, Y2∈Δis arbitrarily given on the
boundary of the planar region
R:x, y∈Δ:Hx, y∈DomΨ−1,Ψ−1Hx, y∈DomΦ−1. 3.4
Proof. From the integral equation3.1, we have
ψzx, y≤
a1
x, y
α1x
α1x0 β1y
β1y0
g1x−s, tϕ1|zs, t|dt ds
×
a2
x, y
α2x
α2x0 β2y
β2y0
g2x−s, tϕ2|zs, t|dt ds
≤
a1
x, y
α1x
α1x0 β1y
β1y0
f1s, tϕ1|zs, t|dt ds
×
a2
x, y
α2x
α2x0 β2y
β2y0
f2s, tϕ2|zs, t|dt ds , ∀
x, y∈Δ.
3.5
Clearly, inequality3.5is in the form of1.8. Thus, the estimate3.2of the solutionzx, y
Acknowledgments
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 Guangxi Autonomous Region0991265, the Scientific Research Foundation of the Education Department of Guangxi Autonomous Region 200707MS112, and the Key Discipline of Applied Mathematics of Hechi University of China200725.
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