Volume 2008, Article ID 518646,9pages doi:10.1155/2008/518646
Research Article
On a Generalized Retarded Integral
Inequality with Two Variables
Wu-Sheng Wang1, 2and Cai-Xia Shen1
1Department of Mathematics, Hechi College, Guangxi, Yizhou 546300, China 2Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Wu-Sheng Wang,[email protected]
Received 16 November 2007; Accepted 22 April 2008
Recommended by Wing-Sum Cheung
This paper improves Pachpatte’s results on linear integral inequalities with two variables, and gives an estimation for a general form of nonlinear integral inequality with two variables. This paper does not require monotonicity of known functions. The result of this paper can be applied to discuss on boundedness and uniqueness for a integrodifferential equation.
Copyrightq2008 W.-S. Wang and C.-X. Shen. 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
Gronwall-Bellman inequality1,2is an important tool in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations and integral equations. There can be found a lot of its generalizations in various cases from literaturesee, e.g.,1–12. In11, Pachpatte obtained an estimation for the integral inequality
ux, y≤ax, y x
0 y
0 fs, t
us, t
s
0 t
0
gs, t, σ, τuσ, τdτdσ
dtds. 1.1
His results were applied to a partial integrodifferential equation:
uxyx, y F
x, y, ux, y, x
0 y
0
hx, y, τ, σ, ux, ydτdσ
,
ux, y0
αx, ux0, y
βy,
1.2
In this paper, we discuss a more general form of integral inequality:
ψux, y
≤ax, y
bx
bx0 cy
cy0
fx, y, s, t
ϕ1
us, t s
bx0 t
cy0
gs, t, σ, τϕ2
uσ, τdτdσ
dtds
1.3 for all x, y ∈ x0, x1×y0, y1. Obviously, u appears linearly in 1.1, but in our 1.3 it
is generalized to nonlinear terms: ϕ1us, t and ϕ2us, t. Our strategy is to monotonize
functionsϕis with other two nondecreasing ones such that one has stronger monotonicity than
the other. We apply our estimation to an integrodifferential equation, which looks similar to
1.2but includes delays, and give boundedness and uniqueness of solutions.
2. Main result
Throughout this paper,x0, x1, y0, y1 ∈Rare given numbers. LetR : 0,∞, I : x0, x1, J :
y0, y1,andΛ:I×J⊂R2. Consider inequality1.3, where we suppose thatψ ∈C0R,R is strictly increasing such thatψ∞ ∞,b∈C1I, I,andc∈C1J, Jare nondecreasing, such
thatbx ≤ xandcy ≤ y,a ∈ C1Λ,R
, f ∈ C0Λ2,R, andgx, y, s, t ∈ C0Λ2,Rare given, and ϕi ∈ C0R,R i 1,2are functions satisfyingϕi0 0 andϕiu > 0 for all
u >0.
Define functions
w1s: max
τ∈0,s ϕ1τ
,
w2s: max
τ∈0,s ϕ2τ/w1τ
w1s,
φs:w2s/w1s.
2.1
Obviously,w1, w2, andφin2.1are all nondecreasing and nonnegative functions and satisfy wis≥ϕis, i1,2. Let
W1u u
1
ds w1
ψ−1s, 2.2
W2u u
1
ds w2
ψ−1s, 2.3
Φu u
W11
ds
φψ−1W1−1s. 2.4
Obviously,W1, W2, andΦare strictly increasing inu >0, and therefore the inversesW1−1, W2−1,
andΦ−1are well defined, continuous, and increasing. We note that
Φu u
W11
dx
φψ−1W1−1x
u
W11 w1
ψ−1W1−1xdx w2
ψ−1W1−1x
W−1
1 u dx −1
W2
W1−1u.
Furthermore, letfx, y, s, t:maxτ∈x0,xfτ, y, s, t,which is also nondecreasing inxfor each
fixedy, s, andtand satisfiesfx, y, s, t≥fx, y, s, t≥0.
Theorem 2.1. If inequality1.3holds for the nonnegative functionux, y, then
ux, y≤ψ−1 W2−1Ξx, y 2.6
for allx, y∈x0, X1×y0, Y1, where
Ξx, y:W2
W1−1r2x, y
bx
bx0 cy
cy0
fx, y, s, t s
bx0 t
cy0
gs, t, τ, σdτdσ
dtds,
r2x, y:W1
r1x, y
bx
bx0 cy
cy0
fx, y, s, tdtds,
r1x, y:a
x0, y
x
x0
axs, yds,
2.7
andX1, Y1∈Λis arbitrarily given on the boundary of the planar region
R: x, y∈Λ:Ξx, y∈DomW2−1, r2x, y∈Dom
W1−1. 2.8
Here Domdenotes the domain of a function.
Proof. By the definition of functionswiandfi, from1.3we get
ψux, y
≤ax, y bx
bx0 cy
cy0
fx, y, s, t
w1
us, t s
bx0 t
cy0
gs, t, σ, τw2
uσ, τdτdσ
dtds
2.9
for allx, y∈Λ.
Firstly, we discuss the case thatax, y> 0 for allx, y∈ Λ. It means that r1x, y> 0
for allx, y∈Λ. In such a circumstance,r1x, yis positive and nondecreasing onΛand
r1x, y≥a
x0, y
x
x0
axt, ydt. 2.10
Regarding1.3, we consider the auxiliary inequality
ψux, y
≤r1x, y bx
bx0 cy
cy0
fX, y, s, t
w1
us, t s
bx0 t
cy0
gs, t, σ, τw2
uσ, τdτdσ
dtds
for allx, y∈x0, X×J, wherex0≤X≤X1is chosen arbitrarily. We claim that
ux, y≤ψ−1
W2−1
W2
W1−1
W1
r1x, y
bx
bx0 cy
cy0
fX, y, s, tdtds
bx
bx0 cy
cy0
f1X, y, s, t s
bx0 t
cy0
gs, t, τ, σdτdσ
dtds
2.12
for allx, y∈x0, X×y0, Y1, whereY1is defined by2.8.
Let ηx, y denote the right-hand side of 2.11, which is a nonnegative and nondecreasing function onx0, X×J. Then,2.11is equivalent to
ux, y≤ψ−1ηx, y ∀x, y∈x0, Y×J. 2.13
By the fact thatbx≤xforx∈x0, Xand the monotonicity ofwi, ψ, η, andbx, we have
∂/∂xηx, y w1
ψ−1ηx, y
≤ ∂/∂xr1x, y w1
ψ−1r1x, y
bx
w1
ψ−1ηx, y
× cy
cy0 f1
X, y, bx, tw1
ubx, t bx
bx0 t
cy0
gbx, t, τ, σw2
uτ, σdτdσ
dt
≤ ∂/∂xr1x, y w1
ψ−1r1x, y
bx cy
cy0
f1X, y, bx, tdt
bx cy
cy0
f1X, y, bx, t bx
bx0 t
cy0
gbx, t, τ, σφuτ, σdτdσ
dt
2.14
for allx, y∈x0, X×J. Integrating the above fromx0tox, we get
W1
ηx, y≤W1
r1x, y
bx
bx0 cy
cy0
f1X, y, s, tdtds
bx
bx0 cy
cy0
f1X, y, s, t s
bx0 t
cy0
gs, t, τ, σφuτ, σdτdσ
dtds
2.15
for allx, y∈x0, X×J. Let
ψξx, y:W1
ηx, y,
r2x, y:W1
r1x, y
bxcy
f1X, y, s, tdtds.
From2.15,2.16, we obtain
ψξx, y
≤r2x, y
bx
bx0 cy
cy0
f1X, y, s, t s
bx0 t
cy0
gs, t, τ, σφuτ, σdτdσ
dtds 2.17
for all x0 ≤ x < X, y0 ≤ y < y1. Let βx, ydenote the right-hand side of2.17, which is a
nonnegative and nondecreasing function onx0, Y×J. Then,2.17is equivalent to
ψξx, y≤βx, y ∀x, y∈x0, Y×J. 2.18
From2.13,2.16, and2.18, we have
ux, y≤ψ−1ηx, yψ−1W1−1ψξx, y≤ψ−1W1−1βx, y 2.19
for allx0≤x < X, y0≤y < Y1, whereY1is defined by2.8. By the definitions ofφ, ψ, andW1, φψ−1W1−1sis continuous and nondecreasing on0,∞and satisfies φψ−1W1−1s > 0 fors >0. Leths ψ−1W1−1s. Sincebx≥0 andbx≤xforx ∈x0, X, from2.19we
have
∂/∂xβx, y φhβx, y
≤ ∂/∂xr2x, y φhr2x, y
bx
φhβx, y cy
cy0
f1X, y, bx, t bx
bx0 t
cy0
gbx, t, τ, σφuτ, σdτdσ
dtds
≤ ∂/∂xr2x, y φhr2x, y
bx cy
cy0
f1X, y, bx, t bx
bx0 t
cy0
gbx, t, τ, σdτdσ
dtds
2.20
for allx, y∈x0, X×y0, Y1. Integrating the above fromx0tox, by2.4we get
Φβx, y≤Φr2x, y
bx
bx0 cy
cy0
f1X, y, s, t s
bx0 t
cy0
gs, t, τ, σdτdσ
dtds 2.21
for allx, y∈x0, X×y0, y1. By2.19and the above inequality, we obtain
ux, y
≤ψ−1
W1−1
Φ−1Φr 2x, y
bx
bx0 cy
cy0
f1X, y, s, t s
bx0 t
cy0
gs, t, τ, σdτdσ
dtds
for allx, y∈x0, X×y0, Y1, whereY1is defined by2.8. It follows from2.5that
ux, y≤ψ−1
W2−1
W2
W1−1
W1
r1x, y
bx
bx0 cy
cy0
f1X, y, s, tdtds
bx
bx0 cy
cy0
f1X, y, s, t s
bx0 t
cy0
gs, t, τ, σdτdσ
dtds
,
2.23
which proves the claimed2.12.
We start from the original inequality1.3and see that
ψuX, y
≤r1X, y bX
bx0 cy
cy0
fX, y, s, t
ϕ1
us, t s
bx0 t
cy0
gs, t, σ, τϕ2
uσ, τdτdσ
dtds
2.24
for all y ∈ y0, Y1; namely, the auxiliary inequality 2.11holds for x X, y ∈ y0, Y1. By
2.12, we get
uX, y≤ψ−1
W2−1
W2
W1−1
W1
r1X, y
bX
bx0 cy
cy0
f1X, y, s, tdtds
bX
bx0 cy
cy0
f1X, y, s, t s
bx0 t
cy0
gs, t, τ, σdτdσ
dtds
2.25
for allx0≤X≤X1, y0≤y≤Y1. This proves2.6.
The remainder case is thatax, y 0 for somex, y∈Λ. Let
r1,εx, y:r1x, y ε, 2.26
whereε >0 is an arbitrary small number. Obviously,r1,εx, y>0 for allx, y∈Λ. Using the
same arguments as above, wherer1x, yis replaced withr1,εx, y, we get
ux, y≤ψ−1
W2−1
W2
W1−1
W1
r1,εx, y
bx
bx0 cy
cy0
f1x, y, s, tdtds
bx
bx0 cy
cy0
f1x, y, s, t s
bx0 t
cy0
gs, t, τ, σdτdσ
dtds
2.27
for allx0 ≤X ≤X1, y0≤y≤Y1. Lettingε→0, we obtain2.6because of continuity ofr1,εin
3. Applications
In 11, the partial integrodifferential equation 1.2 was discussed for boundedness and uniqueness of the solutions under the assumptions that
Fx, y, u, v≤fx, y
|u||v|, h
x, y, s, t, us, t≤gx, y, s, tus, t, F
x, y, u1, v1
−Fx, y, u2, v2≤fx, yu1−u2v1−v2, h
x, y, s, t, u1
−hx, y, s, t, u2≤gx, y, s, tu1−u2,
3.1
respectively. In this section, we further consider the nonlinear delay partial integrodifferential equation
uxyx, y F
x, y, ubx, cy, bx
bbx0 cy
ccy0
hbx, cy, τ, σ, uτ, σdτdσ
,
ux, y0
αx, ux0, y
βy
3.2
for all x, y ∈ Λ, where b, c, and u are supposed to be as in Theorem 2.1; h : Λ2 ×R→R, F : Λ×R2→R,α : I→R, andβ : J→Rare all continuous functions such that α0 β0 0.
Obviously, the estimation obtained in11cannot be applied to3.2. We first give an estimation for solutions of3.2under the condition
Fx, y, u, v≤fx, y ϕ1
|u||v|, h
x, y, s, t, us, t≤gx, y, s, tϕ2
us, t. 3.3
Corollary 3.1. If|αxβy|is nondecreasing inxandyand3.3holds, then every solutionum, n
of 3.2satisfies
ux, y≤W2−1Ξx, y ∀x, y∈x0, X1
×y0, Y1
, 3.4
where
Ξx, y:W2
W1−1
W1αx βy bx
bx0 cy
cy0
fb−1s, c−1t bb−1scc−1tdtds
bx
bx0 cy
cy0
fb−1s, c−1t bb−1scc−1t
s
bx0 t
cy0
gs, t, τ, σdτdσ
dtds,
3.5
Corollary 3.1actually gives a condition of boundedness for solutions. Concretely, if there is a positive constantMsuch that
αx βy< M, bx bx0
cy
cy0
fb−1s, c−1t
bb−1scc−1tdtds < M, bx
bx0 cy
cy0
fb−1s, c−1t bb−1scc−1t
s
bx0 t
cy0
gs, t, τ, σdτdσ
dtds < M
3.6
onx0, X1×y0, Y1, then every solutionux, yof3.2is bounded onx0, X1×y0, Y1.
Next, we give the condition of the uniqueness of solutions for3.2. Corollary 3.2. Suppose
F
x, y, u1, v1
−Fx, y, u2, v2≤fx, y
ϕ1u1−u2v1−v2, h
x, y, s, t, u1
−hx, y, s, t, u2≤gx, y, s, tϕ2u1−u2,
3.7
wheref, g, ϕ1, ϕ2are defined as inTheorem 2.1. There is a positive numberMsuch that bx
bx0 cy
cy0
fb−1s, c−1t
bb−1scc−1tdtds < M, bx
bx0 cy
cy0
fb−1s, c−1t bb−1scc−1t
s
bx0 t
cy0
gs, t, τ, σdτdσ
dtds < M
3.8
on x0, X1×y0, Y1. Then, 3.2has at most one solution onx0, X1×y0, Y1, where X1, Y1 are
defined as inTheorem 2.1.
Acknowledgments
This work is supported by the Scientific Research Fund of Guangxi Provincial Education Department no. 200707MS112, the Natural Science Foundation no. 2006N001, and the Applied Mathematics Key Discipline Foundation of Hechi College of China.
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