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On certain integral inequalities in two independent variables for retarded equations
Applied Mathematics and Computation 203 (2008) 608–616
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On certain integral inequalities in two independent variables for retarded equations q Hongxia Zhang, Fanwei Meng * Department of Mathematics, Qufu Normal University, Qufu 273165, Shandong, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t In this paper, we establish some new retarded integral inequalities in two independent variables which can be used as tools in the theory of integral equations with time delays. Applications are given to illustrate the usefulness of the inequalities. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Two independent variables Retarded integral inequalities Partial differential Bounded Uniqueness
1. Introduction Integral inequalities play an important role in the qualitative analysis of differential and integral equations. Many retarded inequalities have been discovered (see [1–3,6,9,11]), recently Lipovan [5] presented a retarded inequality which has very well characters. Others results may be found in [4,7,8,10,12]. Very recently [13], we obtained some new integral inequalities with two independent variables. In this paper, we will establish some new retarded Volterra integral inequalities with two independent variables, which generalize the main results of [5,10,13], which can be used as tools in the theory of partial differential and integral equations with time delays, which provide explicit bounds on the unknown functions.
2. Main results Theorem 2.1. If uðx; yÞ; pðx; yÞ are real valued nonnegative continuous functions defined for x P 0; y P 0; aðx; y; t; sÞ; bðx; y; t; sÞ be continuous nondecreasing in x and y for each t; s. 0 6 aðxÞ 6 x; 0 6 bðyÞ 6 y; a0 ðxÞ; b0 ðyÞ P 0 are real valued continuous functions defined for x P 0; y P 0 satisfies
uðx; yÞ 6 pðx; yÞ þ
Z
aðxÞ 0
q
Z
bðyÞ
aðx; y; t; sÞuðt; sÞds dt þ
0
Z 0
x
Z
y
bðx; y; t; sÞuðt; sÞds dt;
0
This research was supported by the NNSF of China (10771118) and NSF of Shandong Province (Y2005A06). * Corresponding author. E-mail address: [email protected] (F. Meng).
0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.05.006
ð2:1Þ
609
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
then
R R Z Rx Ry aðxÞ bðyÞ aðx;y;t;sÞds dtþ bðx;y;t;sÞds dt 0 0 0 uðx; yÞ 6 pðx; yÞ þ e 0 Z
o ol
Z
aðlÞ
0
Z
bðyÞ
aðl; y; t; sÞpðt; sÞds dt þ
0
0
e
aðlÞ 0
R bðyÞ 0
Rl Ry
aðl;y;t;sÞds dtþ
0
Z
l
R
x
0
0
bðl;y;t;sÞds dt
!
y
bðl; y; t; sÞpðt; sÞds dt dl:
ð2:2Þ
0
Proof. Let
Z
zðx; yÞ ¼
Z
aðxÞ
0
Z
bðyÞ
aðx; y; t; sÞuðt; sÞds dt þ
0
Z
x
0
y
bðx; y; t; sÞuðt; sÞds dt:
0
Then zð0; yÞ ¼ zðx; 0Þ ¼ 0 our assumption on a; b; u; a and b imply that z is positive function that is nondecreasing in each of the variables. We have
zx ðx; yÞ ¼ a0 ðxÞ
bðyÞ
aðx; y; aðxÞ; sÞuðaðxÞ; sÞds þ
0
Z
þ
Z
y
bðx; y; x; sÞuðx; sÞds þ
0
Z
þ
Z0 x Z
þ
0
x
aðxÞ
Z
0
Z
0
Z
aðxÞ
Z
Z
bðyÞ
ox aðx; y; t; sÞuðt; sÞds dt
0
y
ox bðx; y; t; sÞuðt; sÞds dt 6 a0 ðxÞ
0
bðyÞ
ox aðx; y; t; sÞðpðt; sÞ þ zðt; sÞÞds dt þ
0
Z
bðyÞ
aðx; y; aðxÞ; sÞpðaðxÞ; sÞ þ zðaðxÞ; sÞds
0
Z
y
bðx; y; x; sÞðpðx; sÞ þ zðx; sÞÞds 0
y
ox bðx; y; t; sÞðpðt; sÞ þ zðt; sÞÞds dt:
0
Then we have
Z bðyÞ Z aðxÞ Z bðyÞ zx ðx; yÞ zðx; yÞ a0 ðxÞ aðx; y; aðxÞ; sÞds þ ox aðx; y; t; sÞds dt 0 0 0 Z y Z xZ y ox bðx; y; t; sÞds dt þ bðx; y; x; sÞds þ 0
0
0
Z aðxÞ Z bðyÞ aðx; y; aðxÞ; sÞpðaðxÞ; sÞds þ ox aðx; y; t; sÞpðt; sÞds dt 6 a0 ðxÞ 0 0 0 Z y Z xZ y þ bðx; y; x; sÞpðx; sÞds þ ox bðx; y; t; sÞpðt; sÞds dt; Z
bðyÞ
0
0
i.e.
o 6 ox
Z
Z
aðxÞ 0
Z
o ox
zx ðx; yÞ zðx; yÞ
aðyÞ
Z
0
0
aðx; y; t; sÞds dt þ
0
bðyÞ
aðx; y; t; sÞpðt; sÞds dt þ
R R aðxÞ bðyÞ
0
Multiplying (2.3) by e
0
0
Z
bðyÞ
Rx Ry
aðx;y;t;sÞds dtþ
R R aðxÞ bðyÞ
0
0
Z 0
x
Z
x
Z
0
y
y
bðx; y; t; sÞds dt
0
bðx; y; t; sÞpðt; sÞds dt :
ð2:3Þ
0
bðx;y;t;sÞds dt
we obtain
!
Rx Ry
aðx;y;t;sÞds dtþ bðx;y;t;sÞds dt o 0 0 0 0 zðx; yÞe ox R R Rx Ry Z aðxÞ Z bðyÞ Z xZ y aðxÞ bðyÞ aðx;y;t;sÞds dtþ bðx;y;t;sÞds dt o 0 0 0 0 aðx; y; t; sÞpðt; sÞds dt þ bðx; y; t; sÞpðt; sÞds dt : 6e ox 0 0 0 0
ð2:4Þ Now integrating on the interval ½0; x for x, we get
R R aðxÞ bðyÞ zðx; yÞ 6 e
0
0
Z
R R aðlÞ bðyÞ
x
e 0
þ
Z 0
l
Z
aðx;y;t;sÞds dtþ
y
0
0
Rx Ry 0
0
bðx;y;t;sÞds dt
aðl;y;t;sÞds dtþ
Rl Ry 0
0
bðl;y;t;sÞds dt
! !
o ol
Z 0
aðlÞ
Z
bðyÞ
aðl; y; t; sÞpðt; sÞds dt
0
bðl; y; t; sÞpðt; sÞds dt dl :
ð2:5Þ
0
Combine (2.5) with uðx; yÞ 6 pðx; yÞ þ zðx; yÞ to get (2.2) and with this the proof is complete.
h
610
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
Corollary 2.1. Assume that a; b; a; b are as in Theorem 2.1 and pðx; yÞ p > 0, if u 2 ðRþ Rþ ; Rþ Þ satisfies (2.1) then
Rx Ry R aðxÞ R bðyÞ aðx;y;t;sÞds dtþ bðx;y;t;sÞds dt 0 0 0 uðx; yÞ 6 pe 0 ;
x P 0; y P 0:
ð2:6Þ
Corollary 2.2. Assume a; b; a; b as in Theorem 2.1 and pðx; yÞ p > 0 suppose u 2 CðRþ Rþ ; Rþ Þ is a solution of the Volterra integral equation
Z
uðx; yÞ ¼ p þ
aðxÞ
Z
0
bðyÞ
aðx; y; t; sÞuðt; sÞds dt þ
Z
0
Z
x
0
y
bðx; y; t; sÞuðt; sÞds dt;
ð2:7Þ
0
x P 0; y P 0. If
Z lim lim
x!1
y!1
Z
aðxÞ
0
Z
bðyÞ
aðx; y; t; sÞds dt þ
0
Z
x
0
y
bðx; y; t; sÞds dt
< 1;
ð2:8Þ
0
then u is bounded. Theorem 2.2. Let m; a; b; p 2 ðRþ Rþ ; Rþ Þa; b 2 C 1 ðRþ ; Rþ Þ and let a; b is nondecreasing with aðxÞ 6 x; bðyÞ 6 y if u 2 ðRþ Rþ ; Rþ Þ satisfies
uðx; yÞ 6 pðx; yÞ þ mðx; yÞ
Z
aðxÞ
Z
0
bðyÞ
aðt; sÞuðt; sÞds dt þ
Z
0
Z
x
0
y
bðt; sÞuðt; sÞds dt ;
ð2:9Þ
0
then
Rl Ry Z x R aðlÞ R bðyÞ Rx Ry R aðxÞ R bðyÞ aðt;sÞmðt;sÞds dtþ bðt;sÞmðt;sÞds dt aðt;sÞmðt;sÞds dtþ bðt;sÞmðt;sÞds dt 0 0 0 0 0 0 0 0 uðx; yÞ 6 pðx; yÞ þ mðx; yÞe e 0 Z bðyÞ Z y aðaðlÞ; sÞpðaðlÞ; sÞds þ bðl; sÞpðl; sÞ dl; x P 0; y P 0: a0 ðlÞ 0
ð2:10Þ
0
Proof. Let
zðx; yÞ ¼
Z
Z
aðxÞ
0
bðyÞ
aðt; sÞuðt; sÞds dt þ
Z
0
x
Z
0
y
bðt; sÞuðt; sÞds dt:
0
Then
zx ¼ a0 ðxÞ
Z
bðyÞ
aðaðxÞ; sÞuðaðxÞ; sÞds þ
Z
0
6 a0 ðxÞ
Z
y
bðx; sÞuðx; sÞds
0 bðyÞ
aðaðxÞ; sÞ½pðaðxÞ; sÞ þ mðaðxÞ; sÞzðaðxÞ; sÞds þ
Z
0
y
bðx; sÞ½pðx; sÞ þ mðx; sÞzðx; sÞds
0
in view of zðaðxÞ; bðyÞÞ 6 zðx; yÞ;
zx 6 a0 ðxÞ
Z
bðyÞ
aðaðxÞ; sÞ½pðaðxÞ; sÞ þ mðaðxÞ; sÞzðx; yÞds þ
0
Z
y
bðx; sÞ½pðx; sÞ þ mðx; sÞzðx; yÞds:
0
Hence
zx zðx; yÞ 6 a0 ðxÞ
Z
Z
bðyÞ
aðaðxÞ; sÞmðaðxÞ; sÞds þ
0 bðyÞ
aðaðxÞ; sÞpðaðxÞ; sÞds þ
R R aðxÞ bðyÞ
0
Multiplying above inequality by e o zðx; yÞe ox
R R aðxÞ bðyÞ 0
0
0
0
Rx Ry
aðt;sÞmðt;sÞds dtþ
0
0
Z
y
bðx; sÞmðx; sÞds
0
Z
y
bðx; sÞpðx; sÞds: 0
Rx Ry
aðt;sÞmðt;sÞds dtþ
bðt;sÞmðt;sÞds dt
0
0
bðt;sÞmðt;sÞds dt
!
R
6e
aðxÞ 0
ð2:11Þ
we get
R bðyÞ 0
Z a0 ðxÞ 0
aðt;sÞmðt;sÞds dtþ
bðyÞ
Rx Ry 0
0
bðt;sÞmðt;sÞds dt
Z aðaðxÞ; sÞpðaðxÞ; sÞds þ
y
bðx; sÞpðx; sÞds:
0
ð2:12Þ
611
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
Integrating on the interval ½0; x for x we obtain
zðx; yÞ 6 e
R aðxÞ R bðyÞ 0
0
a0 ðlÞ
aðt;sÞmðt;sÞds dtþ
Z
Rx Ry 0
bðt;sÞmðt;sÞds dt
0
bðyÞ
aðaðlÞ; sÞpðaðlÞ; sÞds þ
0
Z
Z
R
x
e
aðlÞ
R bðyÞ
0
0
aðt;sÞmðt;sÞds dtþ
Rl Ry 0
0
bðt;sÞmðt;sÞds dt
0
y
bðl; sÞpðl; sÞ dl x P 0; y P 0:
ð2:13Þ
0
Combine (2.13) with uðx; yÞ 6 pðx; yÞ þ mðx; yÞzðx; yÞ we obtain inequality (2.10). h Corollary 2.3. Let m; a; b; p; a; b defined as in Theorem 2.2 suppose u 2 CðRþ Rþ ; Rþ Þ is a solution of the integral equation
Z uðx; yÞ ¼ pðx; yÞ þ mðx; yÞ
aðxÞ
Z
0
bðyÞ
aðt; sÞuðt; sÞds dt þ
Z
0
Z
x
0
y
bðt; sÞmðt; sÞds dt :
ð2:14Þ
0
If m; p are bounded and
lim lim
Z
x!1 y!1
aðxÞ
Z
0
bðyÞ
aðt; sÞds dt þ
Z
0
x
0
Z
y
bðt; sÞds dt
< 1;
ð2:15Þ
0
then u is bounded. Corollary 2.4. Let m; a; b; a; b defined as in the Theorem 2.2 with pðx; yÞ ! 0 as x ! 1; y ! 1 suppose u 2 CðRþ Rþ ; Rþ Þ is a solution of the integral equation
Z uðx; yÞ ¼ pðx; yÞ þ mðx; yÞ
aðxÞ
Z
0
If
R að1Þ R bð1Þ 0
0
Z lim lim mðx; yÞ
x!1 y!1
aðxÞ
Z
0
aðt; sÞuðt; sÞds dt þ
Z
0
R1 R1
aðt; sÞmðt; sÞds dt þ
bðyÞ
0
0
Z
x
0
y
bðt; sÞuðt; sÞds dt :
0
bðt; sÞmðt; sÞds dt < 1 and
bðyÞ
aðt; sÞpðt; sÞds dt þ
Z
0
x
Z
0
y
bðt; sÞpðt; sÞds dt
¼ 0;
ð2:16Þ
0
then uðx; yÞ ! 0 as x ! 1; y ! 1 in particular, if mðx; yÞ; pðx; yÞ ! 0 R að1Þ R bð1Þ R1 R1 aðt; sÞds dt < 1; bðt; sÞds dt < 1 then uðx; yÞ ! 0 as x ! 1; y ! 1: 0 0 0 0
as
x ! 1; y ! 1
and
Theorem 2.3. Assume p; a; b; a; b be as in Theorem 2.1 xðuÞ is positive continuous nondecreasing for u > 0 with xð0Þ ¼ 0 and R 1 dt ¼ 1, if u 2 CðRþ Rþ ; Rþ Þ satisfies 1 xðtÞ
uðx; yÞ 6 pðx; yÞ þ
Z
aðxÞ
0
Z
Z
bðyÞ
aðx; y; t; sÞxðuðt; sÞÞds dt þ
0
x
Z
0
y
bðx; y; t; sÞxðuðt; sÞÞds dt;
ð2:17Þ
0
x P 0; y P 0, then
Z uðx; yÞ 6 G1 Gðpðx; yÞÞ þ
aðxÞ
0
x P 0; y P 0, where GðuÞ ¼
Ru
Z
bðyÞ
aðx; y; t; sÞds dt þ
Z
0
dt ;u 1 xðtÞ
x 0
Z
y
bðx; y; t; sÞds dt ;
ð2:18Þ
0
> 0:
Proof. Without loss of generality we assume that p > 0, for x P 0; y P 0, otherwise we can be replaced p by p þ e for any e > 0. Assume T 1 ; T 2 > 0 be fixed, and let
zðx; yÞ ¼
Z
aðxÞ
0
Z
bðyÞ
aðx; y; t; sÞxðuðt; sÞÞds dt þ
Z
0
x
Z
0
y
bðx; y; t; sÞxðuðt; sÞÞds dt 0
with assumption on a; b; a; b imply that zðx; yÞ is nondecreasing about x and y. Hence for x 2 ½0; T 1 ; y 2 ½0; T 2 we have
zxy ¼ aðx; y; aðxÞ; bðyÞÞxðuðaðxÞ; bðyÞÞÞa0 ðxÞb0 ðyÞ þ Z
þ a0 ðxÞ Z xZ þ 0
Z
0 bðyÞ
aðxÞ
Z
bðyÞ
ox oy aðx; y; t; sÞxðuðt; sÞÞds dt
0
Z
aðxÞ
oy aðx; y; aðxÞ; sÞxðuðaðxÞ; sÞÞds þ b0 ðyÞ ox aðx; y; t; bðyÞÞxðuðt; bðyÞÞÞdt þ bðx; y; x; yÞxðuðx; yÞÞ 0 0 Z Z x y y ox oy bðx; y; t; sÞxðuðt; sÞÞds dt þ oy bðx; y; x; sÞxðuðx; sÞÞds þ ox bðx; y; t; yÞxðuðt; yÞÞdt
0
0 0
0
0
6 aðx; y; aðxÞ; bðyÞÞxððpðaðxÞ; bðyÞÞ þ zðaðxÞ; bðyÞÞÞÞa ðxÞb ðyÞ þ a ðxÞ þ zðaðxÞ; sÞÞds þ b0 ðyÞ
Z 0
Z
0 bðyÞ
0
aðxÞ
ox aðx; y; t; bðyÞÞxðpðt; bðyÞÞ þ zðt; bðyÞÞÞdt þ
oy aðx; y; aðxÞ; sÞxðpðaðxÞ; sÞ Z 0
aðxÞ
Z 0
bðyÞ
ox oy aðx; y; t; sÞxððpðt; sÞ
612
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
Z xZ y þ zðt; sÞÞÞds dt þ bðx; y; x; yÞxðpðx; yÞ þ zðx; yÞÞ þ ox oy bðx; y; t; sÞxðpðt; sÞ þ zðt; sÞÞds dt 0 0 Z y Z x þ oy bðx; y; x; sÞxðpðx; sÞ þ zðx; sÞÞds þ ox bðx; y; t; yÞxðpðt; yÞ þ zðt; yÞÞdt 0 0 Z aðxÞ Z bðyÞ 6 xðpðT 1 ; T 2 Þ þ zðx; yÞÞ aðx; y; aðxÞ; bðyÞÞa0 ðxÞb0 ðyÞ þ ox oy aðx; y; t; sÞds dt 0 0 Z aðxÞ Z bðyÞ Z xZ y 0 0 þ b ðyÞ ox aðx; y; t; bðyÞÞdt þ a ðxÞ oy aðx; y; aðxÞ; sÞdsþbðx; y; x; yÞ þ ox oy bðx; y; t; sÞds dt 0 0 0 0 Z y Z x þ oy bðx; y; x; sÞds þ ox bðx; y; t; yÞdt ; 0
0
i.e
Z
zxy o2 6 xðpðT 1 ; T 2 Þ þ zðx; yÞÞ ox oy
Z
aðxÞ
0
bðyÞ
aðx; y; t; sÞds dt þ
Z
0
Z
x
0
bðx; y; t; sÞds dt :
y
ð2:19Þ
0
Noting that
o zx zxy 6 : oy xðpðT 1 ; T 2 Þ þ zðx; yÞÞ xðpðT 1 ; T 2 Þ þ zðx; yÞÞ
We have
Z aðxÞ Z bðyÞ Z xZ y o zx o2 aðx; y; t; sÞds dt þ bðx; y; t; sÞds dt : 6 oy xðpðT 1 ; T 2 Þ þ zðx; yÞÞ ox oy 0 0 0 0
ð2:20Þ
Integrating both sides of (2.20) with respect to y from 0 to y we have
zx o 6 xðpðT 1 ; T 2 Þ þ zðx; yÞÞ ox
Z
Z
aðxÞ
0
bðyÞ
aðx; y; t; sÞds dt þ
Z
0
x
Z
0
y
bðx; y; t; sÞds dt ;
ð2:21Þ
0
then integrating above inequality with respect to x from 0 to x we obtain
GðpðT 1 ; T 2 Þ þ zðx; yÞÞ 6 GðpðT 1 ; T 2 ÞÞ þ
Z
aðxÞ
Z
0
bðyÞ
aðx; y; t; sÞds dt þ
Z
0
x 2 ½0; T 1 ; y 2 ½0; T 2 : R1 In view of 1 xdu ðuÞ ¼ 1 we have from (2.22)
Z pðT 1 ; T 2 Þ þ zðx; yÞ 6 G1 GðpðT 1 ; T 2 ÞÞ þ
0
Z
aðxÞ 0
Z
x
bðyÞ
aðx; y; t; sÞds dt þ 0
y
bðx; y; t; sÞds dt;
ð2:22Þ
0
Z 0
x
Z
y
bðx; y; t; sÞds dt :
ð2:23Þ
0
Let x ¼ T 1 ; y ¼ T 2 in (2.23) we have
Z pðT 1 ; T 2 Þ þ zðT 1 ; T 2 Þ 6 G1 GðpðT 1 ; T 2 ÞÞ þ
aðT 1 Þ 0
Z
bðT 2 Þ
aðT 1 ; T 2 ; t; sÞds dt þ
Z
T1
0
0
Z
T2
bðT 1 ; T 2 ; t; sÞds dt :
0
Due to T 1 ; T 2 are arbitrarily and uðx; yÞ 6 pðx; yÞ þ zðx; yÞ we have obtain (2.18). h Corollary 2.5. Let a; b; pa; b; x be as in Theorem 2.3 suppose u 2 CðRþ Rþ ; Rþ Þ is a solution of the nonlinear Volterra equation
uðx; yÞ ¼ pðx; yÞ þ
Z
aðxÞ
Z
0
bðyÞ
aðx; y; t; sÞxðuðt; sÞÞds dt þ
0
Z
x
Z
0
y
bðx; y; t; sÞxðuðt; sÞÞds dt;
0
x P 0; y P 0, if p is bounded and
lim lim
x!1 y!1
Z 0
aðxÞ
Z
bðyÞ
aðx; y; t; sÞds dt þ
0
Z 0
x
Z
y
bðx; y; t; sÞds dt
< 1;
0
then u is bounded. Theorem 2.4. Suppose m; a; b; p 2 CðRþ Rþ ; Rþ Þa; b 2 C 1 ðRþ ; Rþ Þ, and let m; n; p; a; b are nondecreasing function with R aðxÞ 6 xbðyÞ 6 y for x P 0; y P 0 assume x 2 CðRþ ; Rþ Þ be a nondecreasing function such that xðtÞ > 0 for t > 0 and 11 xdtðtÞ ¼ 1 if u 2 CðRþ Rþ ; Rþ Þ satisfies
uðx; yÞ 6 pðx; yÞ þ mðx; yÞ
Z 0
aðxÞ
Z 0
bðyÞ
aðt; sÞxðuðt; sÞÞds dt þ
Z 0
x
Z 0
y
bðx; yÞxðuðt; sÞÞds dt ;
ð2:24Þ
613
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
x P 0; y P 0, then
Z uðx; yÞ 6 G1 Gðpðx; yÞÞ þ mðx; yÞ
aðxÞ
Z
0
x P 0; y P 0; where GðuÞ ¼
Ru
dt ;u 1 xðtÞ
bðyÞ
aðt; sÞds dt þ
Z
0
Z
x
0
y
bðt; sÞds dt
ð2:25Þ
;
0
> 0.
Proof. Without loss of generality we assume that p > 0, for x P 0; y P 0, otherwise we can be replaced p by p þ e for any e > 0. Let T 1 ; T 2 > 0 be fixed, then for x 2 ½0; T 1 ; y 2 ½0; T 2 relation (2.24) and hypotheses on m; p imply
uðx; yÞ 6 pðT 1 ; T 2 Þ þ mðT 1 ; T 2 Þ
Z
aðxÞ
Z
0
bðyÞ
aðt; sÞxðuðt; sÞÞds dt þ
Z
0
x
Z
0
y
bðt; sÞxðuðt; sÞÞds dt :
0
Let
zðx; yÞ ¼
Z
aðxÞ
0
Z
bðyÞ
aðt; sÞxðuðt; sÞÞds dt þ
0
Z
x
0
Z
y
bðt; sÞxðuðt; sÞÞds dt:
0
Then we have
zxy ¼ aðaðxÞ; bðyÞÞxðuðaðxÞ; bðyÞÞÞa0 ðxÞb0 ðyÞ þ bðx; yÞxðuðx; yÞÞ ¼ aðaðxÞ; bðyÞÞxðpðaðxÞ; bðyÞÞ þ mðaðxÞ; bðyÞÞzðaðxÞ; bðyÞÞÞa0 ðxÞb0 ðyÞ þ bðx; yÞxðpðx; yÞ þ mðx; yÞzðx; yÞÞ 6 xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞÞzðx; yÞðaðaðxÞ; bðyÞÞa0 ðxÞb0 ðyÞ þ bðx; yÞÞ; i.e.
zxy 6 aðaðxÞ; bðyÞÞa0 ðxÞb0 ðyÞ þ bðx; yÞ: xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 Þzðx; yÞÞ
ð2:26Þ
Noting that
o zx zxy 6 ; oy xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞÞzðx; yÞ xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞÞzðx; yÞ we have
o zx 6 aðaðxÞ; bðyÞÞa0 ðxÞb0 ðyÞ þ bðx; yÞ: oy xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞÞzðx; yÞ
ð2:27Þ
Integrating both sides of the (2.27) with respect to y from 0 to y we have
Z
zx 6 a0 ðxÞ xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞÞzðx; yÞ
y
aðaðxÞ; bðsÞÞdbðsÞ þ
Z
0
y
bðx; sÞds;
ð2:28Þ
0
R1
then integrating (2.28) with respect to x from 0 to x in view of
GðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 Þzðx; yÞÞ 6 GðpðT 1 ; T 2 ÞÞ þ
Z
Z
aðxÞ
0
dt
¼ 1; we get
xðtÞ
1
bðyÞ
aðt; sÞds dt þ
0
Z
Z
x
0
y
bðt; sÞds dt;
0
x 2 ½0; T 1 ; y 2 ½0; T 2 ; we obtain
Z pðT 1 ; T 2 Þ þ mðT 1 ; T 2 Þzðx; yÞ 6 G1 GðpðT 1 ; T 2 ÞÞ þ
aðxÞ
Z
0
bðyÞ
aðt; sÞds dt þ
0
Z
x
0
Z
y
bðt; sÞds dt :
ð2:29Þ
0
Let x ¼ T 1 ; y ¼ T 2 in (2.12) we have
Z pðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞzðT 1 ; T 2 Þ 6 G1 GðpðT 1 ; T 2 ÞÞ þ
aðT 1 Þ
Z
0
bðT 2 Þ
aðt; sÞds dt þ
Z
T1
Z
0
0
due to T 1 ; T 2 are arbitrarily and uðx; yÞ 6 pðx; yÞ þ mðx; yÞzðx; yÞ; we have (2.25) hold.
T2
bðt; sÞds dt ;
0
h
Corollary 2.6. Let m; a; b; p; a; b; x be as in Theorem 2.4. Suppose u 2 CðRþ Rþ ; Rþ Þ is a solution to the integral equation
Z uðx; yÞ ¼ pðx; yÞ þ mðx; yÞ
aðxÞ
Z
0
x P 0; y P 0; if m; p are bounded and
bðyÞ
aðt; sÞxðuðt; sÞÞds dt þ 0
0
R aðxÞ R bðyÞ 0
Z
0
aðt; sÞds dt þ
Rx Ry 0
0
x
Z
y
bðt; sÞxðuðt; sÞÞds dt ;
0
bðt; sÞds dt < 1 then u is bounded.
614
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
Remark. Using similar method of those in the proof of Theorems 2.2 and 2.4 we can obtain the reversed inequality of (2.9) and (2.24) as follows: Theorem 2.5. Let m; n; p 2 CðRþ Rþ ; Rþ Þa; b 2 C 1 ðRþ ; Rþ Þ and assume that a; b are nondecreasing with aðxÞ P x; bðyÞ P y for x P 0; y P 0 if u 2 CðRþ Rþ ; Rþ Þ satisfies
uðx; yÞ P pðx; yÞ þ mðx; yÞ
Z
aðxÞ
Z
0
bðyÞ
aðt; sÞuðt; sÞds dt þ
Z
0
Z
x
0
y
bðt; sÞuðt; sÞds dt ;
0
then
uðx; yÞ P pðx; yÞ þ mðx; yÞ
Z
Z
x
0
y
e
R aðxÞ R bðyÞ aðlÞ
bðrÞ
aðt;sÞmðt;sÞds dtþ
Rx Ry l
r
bðt;sÞmðt;sÞds dt
0
aðaðlÞ; bðrÞÞðpðaðlÞ; bðrÞÞa0 ðlÞb0 ðrÞ þ bðl; rÞpðl; rÞÞdl dr;
x P 0; y P 0:
ð2:30Þ
Theorem 2.6. Let m; a; b; p 2 CðRþ Rþ ; Rþ Þa; b 2 C 1 ðRþ ; Rþ Þ with m,p are nondecreasing. suppose a; b are nondecreasing and aðxÞ P x; bðyÞ P y for x P 0; y P 0 let x 2 CðRþ ; Rþ Þ be a nondecreasing function such that xðtÞ > 0 for t > 0 if u 2 CðRþ Rþ ; Rþ Þ satisfies
uðx; yÞ P pðx; yÞ þ mðx; yÞ
Z
aðxÞ
Z
0
bðyÞ
aðt; sÞxðuðt; sÞÞds dt þ
Z
0
x
Z
0
y
bðx; yÞxðuðt; sÞÞds dt ;
ð2:24Þ
0
x P 0; y P 0, then
Z uðx; yÞ P G1 Gðpðx; yÞÞ þ mðx; yÞ
aðxÞ 0
Z
bðyÞ
aðt; sÞds dt þ
Z
0
x
Z
0
y
bðt; sÞds dt
;
ð2:25Þ
0
x P 0; y P 0; Ru Ru where GðuÞ ¼ 1 xdtðtÞ ; u P 0 where GðuÞ ¼ 1 xdtðtÞ ; u P 0 and x1 ; y1 are chosen so that
Z Gðpðx; yÞÞ þ mðx; yÞ
aðxÞ
Z
0
bðyÞ
aðt; sÞds dt þ
Z
0
0
x
Z
y
bðt; sÞds dt
2 DomðG1 Þ
0
for all x 2 ½0; x1 ; y 2 ½0; y1 : 3. Some applications In this section we will indicate some applications for our results to obtain the bounds on the solution of some integral equations with time delay. To study the boundedness, uniqueness and continuous of the solution of the initial boundary value problem for retarded Volterra equations In this section we present applications of the inequality (2.1) given in Theorem 2.1 to study the boundness, uniqueness, and continuous dependence of the solution of the initial boundary value problem for retarded Volterra equations of the form
uðx; yÞ ¼ pðx; yÞ þ
Z
x
0
Z
y
f ðx; y; t; s; uðaðxÞ; bðyÞÞ; uðx; yÞÞds dt;
ð3:1Þ
0
where f is continuous, p is continuous on D; a; b 2 C 1 ðRþ ; Rþ Þ, where D ¼ ðRþ Rþ Þ, and aðxÞ 6 x; bðyÞ 6 y; a0 ðxÞ > 0; b0 ðyÞ > 0 for x P 0; y P 0: First we give the bound on the solution of the problem (3.1). Theorem 3.1. Suppose that
jf ðx; y; t; s; w; vÞj 6 aðx; y; t; sÞjwj þ bðx; y; t; sÞjvj;
ð3:2Þ
where a; b are as in Theorem 2.1. Let
M1 ¼ max x2Rþ
1
a0 ðxÞ
;
M 2 ¼ max y2Rþ
1 ; b0 ðyÞ
ðx; y; t; sÞ ¼ aðx; y; a1 ðtÞ; b1 ðsÞÞ: a
ð3:3Þ ð3:4Þ
615
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
If x(t) is any solution of (3.1), then
R juðx; yÞj 6 jpðx; yÞj þ e
o ol
Z
R bðyÞ
aðxÞ 0
0
ðx;y;t;sÞds dtþ M1 M2 a
Rx Ry 0
0
bðx;y;t;sÞds dt
Z
x
R R aðlÞ bðyÞ
e
0
0
0
Z
aðlÞ
0
Rx Ry
ðl;y;t;sÞds dtþ M1 M2 a
0
0
bðl;y;t;sÞds dt
!
bðyÞ
ðl; y; t; sÞpðt; sÞds dt þ bðl; y; t; sÞpðt; sÞds dt dl: M1 M2 a
ð3:5Þ
0
Proof. Using (3.2)–(3.4) in (3.1) and making the change of variables, we have
juðx; yÞj 6 jpðx; yÞj þ
Z
Z
x
0
6 jpðx; yÞj þ
Z
y
½aðx; y; t; sÞjuðaðtÞ; bðsÞÞj þ bðx; y; t; sÞjuðt; sÞjds dt
0
Z
aðxÞ
0
aðyÞ
ðx; y; t; sÞjuðt; sÞjds dt þ M1 M2 a
0
Z
x
0
Z
y
bðx; y; t; sÞjuðt; sÞjds dt:
ð3:6Þ
0
Now a suitable application of the inequality in (2.1) given in Theorem 2.1 to (3.6) yields (3.5). The right hand of (3.5) gives us the bound on the solution xðtÞ of (3.1) in terms of the known functions. Thus, if the right hand of (3.5) is bounded, then we assert that the solution of (3.1) is bounded on D. The next result deals with the uniqueness of the solution of the problem (3.1). h Theorem 3.2. Suppose that the function f in (3.1) satisfies the condition
v Þj 6 aðx; y; t; sÞjw wj þ bðx; y; t; sÞjv v j: jf ðx; y; t; s; w; vÞ f ðx; y; t; s; w;
ð3:7Þ
Let a; b be as defined in Theorem 2.1, M1 ; M 2 be as defined in (3.3). Then the problem (3.1) has at most one solution on D. ðx; yÞ be two solution of (3.1), then we have Proof. Let uðx; yÞ; u
ðx; yÞ ¼ uðx; yÞ u
Z
Z
x
0
y
ðaðtÞ; bðsÞÞ; u ðt; sÞÞds dt; ½f ðx; y; t; s; uðaðtÞ; bðsÞÞ; uðt; sÞÞ f ðx; y; t; s; u
0
using (3.3), (3.4), and (3.7) we have
ðx; yÞj 6 juðx; yÞ u
Z
Z
x
0
Z
y
ðaðtÞ; bðsÞÞjds dt þ aðx; y; t; sÞjuðaðtÞ; bðsÞÞ u 0
aðxÞ
Z
bðyÞ
ðx; y; t; sÞjuðt; sÞ u ðt; sÞjds dt þ M1 M2 a
6 0
0
Z
Z
Z
x
0
Z
x
0
y
ðt; sÞjds bðx; y; t; sÞjuðt; sÞ u
0 y
ðt; sÞjds dt: bðx; y; t; sÞjuðt; sÞ u
ð3:8Þ
0
ðx; yÞj 6 0; so we have uðx; yÞ ¼ u ðx; yÞ; i.e., there is at most Now a suitable application of the Corollary 2.1 we have juðx; yÞ u one solution of the problem (3.1). The following theorem deals with the continuous dependence of the solution on the equation and given initial boundary conditions. Consider the problem (3.1) and the problem
ðx; yÞ þ vðx; yÞ ¼ p
Z
x
Z
0
y
f ðx; y; t; s; vðaðtÞ; bðsÞÞ; vðt; sÞÞds dt;
ð3:9Þ
0
is continuous on D. h where p Theorem 3.3. Suppose that the function f satisfies the condition (3.7) in Theorem 3.2 and further assume that
ðx; yÞj < e; jpðx; yÞ p
ð3:10Þ
be as in Theorem 3.1.Then the solution of (3.1) depends continuously where e > 0 is an arbitrary small constant, and let M 1 ; M 2 ; a on the initial value boundary data. Proof. From (3.1) and (3.9), using (3.7) and (3.10), and making the change of variables, we have
ðx; yÞj þ juðx; yÞ vðx; yÞj 6 jpðx; yÞ p þ
Z
aðxÞ
t
jf ðx; y; t; s; uðaðtÞ; bðsÞÞ; uðt; sÞÞ f ðx; y; t; s; vðaðtÞ; bðsÞÞ; vðt; sÞÞjds < e
0
Z
0
Z
bðyÞ
ðx; y; t; sÞjuðt; sÞ vðt; sÞjds dt þ M1 M2 a
0
Z
x
0
Z
y
bðx; y; t; sÞjuðt; sÞ vðt; sÞjds dt:
ð3:11Þ
0
Now a suitable application of the Corollary 2.1 to (3.11) yields
juðx; yÞ vðx; yÞj 6 ee
R aðxÞ R bðyÞ 0
0
Rx Ry
ðx;y;t;sÞ ds dtþ M1 M 2 a
0
0
bðx;y;t;sÞds dt
;
x P 0; y P 0:
ð3:12Þ
616
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
If
Z lim lim
x!1
y!1
aðxÞ
0
Z 0
bðyÞ
ðx; y; t; sÞds dt þ a
Z 0
x
Z
y
bðx; y; t; sÞds dt
< 1:
ð3:13Þ
0
So the solution to such an initial boundary value problem depends continuously on the initial boundary values, if e ! 0 then juðx; yÞ vðx; yÞj ! 0: Our result is proved. h Acknowledgements The authors thank the referees very much for their valuable suggestions on this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Y.G. Sun, On retarded integral inequalities and their applications, J. Math. Anal. Appl. 301 (2005) 265–275. A. Morro, A Gronwall-like inequality and its applications to continuum thermodynamics, Boll, Un. Mat. Ital. B 6 (1982) 553–562. B.G. Pachpatte, On a certain retarded integral inequality and its applications, J. Inequal. Pure Appl. Math. 5 (2004) (article 19). Run Xu, Yuan Gong Sun, On retarded integral inequalities in two independent variables and their applications, Appl. Math. Comput. 182 (2006) 1260– 1266. O. Lipovan, Integral inequalities for retarded Volterra equations, J. Math. Anal. Appl. 322 (2006) 349–358. O. Lipovan, A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl. 252 (2000) 389–401. Fanewi Meng, Wei Nian, Li On some new integral inequalities and their applications, Appl. Math. Comput. 148 (2004) 381–392. O. Lipovan, A retarded integral inequality and its applications, J. Math. Anal. Appl. 285 (2003) 436–443. T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equation, Ann. Math. 20 (1919) 292–296. B.G. Pachpatte, Explicit bounds on a certain integral inequalities, J. Math. Anal. Appl. 267 (2002) 48–61. B.G. Pachpatte, On a certain retarded integral inequality and applications, J. Inequal. Pure Appl. Math. 3 (2004) (article 18). B.G. Pachpatte, On some new nonlinear retarded integral inequalities, J. Inequal. Pure Appl. Math. 5 (2004) (article 80). Hongxia Zhang, Fanwei Meng, Integral inequalities in two independent variables for retarded Volterra equations, Appl. Math.Comput. 199 (2008) 90– 98.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On certain integral inequalities in two independent variables for retarded equations q Hongxia Zhang, Fanwei Meng * Department of Mathematics, Qufu Normal University, Qufu 273165, Shandong, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t In this paper, we establish some new retarded integral inequalities in two independent variables which can be used as tools in the theory of integral equations with time delays. Applications are given to illustrate the usefulness of the inequalities. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Two independent variables Retarded integral inequalities Partial differential Bounded Uniqueness
1. Introduction Integral inequalities play an important role in the qualitative analysis of differential and integral equations. Many retarded inequalities have been discovered (see [1–3,6,9,11]), recently Lipovan [5] presented a retarded inequality which has very well characters. Others results may be found in [4,7,8,10,12]. Very recently [13], we obtained some new integral inequalities with two independent variables. In this paper, we will establish some new retarded Volterra integral inequalities with two independent variables, which generalize the main results of [5,10,13], which can be used as tools in the theory of partial differential and integral equations with time delays, which provide explicit bounds on the unknown functions.
2. Main results Theorem 2.1. If uðx; yÞ; pðx; yÞ are real valued nonnegative continuous functions defined for x P 0; y P 0; aðx; y; t; sÞ; bðx; y; t; sÞ be continuous nondecreasing in x and y for each t; s. 0 6 aðxÞ 6 x; 0 6 bðyÞ 6 y; a0 ðxÞ; b0 ðyÞ P 0 are real valued continuous functions defined for x P 0; y P 0 satisfies
uðx; yÞ 6 pðx; yÞ þ
Z
aðxÞ 0
q
Z
bðyÞ
aðx; y; t; sÞuðt; sÞds dt þ
0
Z 0
x
Z
y
bðx; y; t; sÞuðt; sÞds dt;
0
This research was supported by the NNSF of China (10771118) and NSF of Shandong Province (Y2005A06). * Corresponding author. E-mail address: [email protected] (F. Meng).
0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.05.006
ð2:1Þ
609
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
then
R R Z Rx Ry aðxÞ bðyÞ aðx;y;t;sÞds dtþ bðx;y;t;sÞds dt 0 0 0 uðx; yÞ 6 pðx; yÞ þ e 0 Z
o ol
Z
aðlÞ
0
Z
bðyÞ
aðl; y; t; sÞpðt; sÞds dt þ
0
0
e
aðlÞ 0
R bðyÞ 0
Rl Ry
aðl;y;t;sÞds dtþ
0
Z
l
R
x
0
0
bðl;y;t;sÞds dt
!
y
bðl; y; t; sÞpðt; sÞds dt dl:
ð2:2Þ
0
Proof. Let
Z
zðx; yÞ ¼
Z
aðxÞ
0
Z
bðyÞ
aðx; y; t; sÞuðt; sÞds dt þ
0
Z
x
0
y
bðx; y; t; sÞuðt; sÞds dt:
0
Then zð0; yÞ ¼ zðx; 0Þ ¼ 0 our assumption on a; b; u; a and b imply that z is positive function that is nondecreasing in each of the variables. We have
zx ðx; yÞ ¼ a0 ðxÞ
bðyÞ
aðx; y; aðxÞ; sÞuðaðxÞ; sÞds þ
0
Z
þ
Z
y
bðx; y; x; sÞuðx; sÞds þ
0
Z
þ
Z0 x Z
þ
0
x
aðxÞ
Z
0
Z
0
Z
aðxÞ
Z
Z
bðyÞ
ox aðx; y; t; sÞuðt; sÞds dt
0
y
ox bðx; y; t; sÞuðt; sÞds dt 6 a0 ðxÞ
0
bðyÞ
ox aðx; y; t; sÞðpðt; sÞ þ zðt; sÞÞds dt þ
0
Z
bðyÞ
aðx; y; aðxÞ; sÞpðaðxÞ; sÞ þ zðaðxÞ; sÞds
0
Z
y
bðx; y; x; sÞðpðx; sÞ þ zðx; sÞÞds 0
y
ox bðx; y; t; sÞðpðt; sÞ þ zðt; sÞÞds dt:
0
Then we have
Z bðyÞ Z aðxÞ Z bðyÞ zx ðx; yÞ zðx; yÞ a0 ðxÞ aðx; y; aðxÞ; sÞds þ ox aðx; y; t; sÞds dt 0 0 0 Z y Z xZ y ox bðx; y; t; sÞds dt þ bðx; y; x; sÞds þ 0
0
0
Z aðxÞ Z bðyÞ aðx; y; aðxÞ; sÞpðaðxÞ; sÞds þ ox aðx; y; t; sÞpðt; sÞds dt 6 a0 ðxÞ 0 0 0 Z y Z xZ y þ bðx; y; x; sÞpðx; sÞds þ ox bðx; y; t; sÞpðt; sÞds dt; Z
bðyÞ
0
0
i.e.
o 6 ox
Z
Z
aðxÞ 0
Z
o ox
zx ðx; yÞ zðx; yÞ
aðyÞ
Z
0
0
aðx; y; t; sÞds dt þ
0
bðyÞ
aðx; y; t; sÞpðt; sÞds dt þ
R R aðxÞ bðyÞ
0
Multiplying (2.3) by e
0
0
Z
bðyÞ
Rx Ry
aðx;y;t;sÞds dtþ
R R aðxÞ bðyÞ
0
0
Z 0
x
Z
x
Z
0
y
y
bðx; y; t; sÞds dt
0
bðx; y; t; sÞpðt; sÞds dt :
ð2:3Þ
0
bðx;y;t;sÞds dt
we obtain
!
Rx Ry
aðx;y;t;sÞds dtþ bðx;y;t;sÞds dt o 0 0 0 0 zðx; yÞe ox R R Rx Ry Z aðxÞ Z bðyÞ Z xZ y aðxÞ bðyÞ aðx;y;t;sÞds dtþ bðx;y;t;sÞds dt o 0 0 0 0 aðx; y; t; sÞpðt; sÞds dt þ bðx; y; t; sÞpðt; sÞds dt : 6e ox 0 0 0 0
ð2:4Þ Now integrating on the interval ½0; x for x, we get
R R aðxÞ bðyÞ zðx; yÞ 6 e
0
0
Z
R R aðlÞ bðyÞ
x
e 0
þ
Z 0
l
Z
aðx;y;t;sÞds dtþ
y
0
0
Rx Ry 0
0
bðx;y;t;sÞds dt
aðl;y;t;sÞds dtþ
Rl Ry 0
0
bðl;y;t;sÞds dt
! !
o ol
Z 0
aðlÞ
Z
bðyÞ
aðl; y; t; sÞpðt; sÞds dt
0
bðl; y; t; sÞpðt; sÞds dt dl :
ð2:5Þ
0
Combine (2.5) with uðx; yÞ 6 pðx; yÞ þ zðx; yÞ to get (2.2) and with this the proof is complete.
h
610
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
Corollary 2.1. Assume that a; b; a; b are as in Theorem 2.1 and pðx; yÞ p > 0, if u 2 ðRþ Rþ ; Rþ Þ satisfies (2.1) then
Rx Ry R aðxÞ R bðyÞ aðx;y;t;sÞds dtþ bðx;y;t;sÞds dt 0 0 0 uðx; yÞ 6 pe 0 ;
x P 0; y P 0:
ð2:6Þ
Corollary 2.2. Assume a; b; a; b as in Theorem 2.1 and pðx; yÞ p > 0 suppose u 2 CðRþ Rþ ; Rþ Þ is a solution of the Volterra integral equation
Z
uðx; yÞ ¼ p þ
aðxÞ
Z
0
bðyÞ
aðx; y; t; sÞuðt; sÞds dt þ
Z
0
Z
x
0
y
bðx; y; t; sÞuðt; sÞds dt;
ð2:7Þ
0
x P 0; y P 0. If
Z lim lim
x!1
y!1
Z
aðxÞ
0
Z
bðyÞ
aðx; y; t; sÞds dt þ
0
Z
x
0
y
bðx; y; t; sÞds dt
< 1;
ð2:8Þ
0
then u is bounded. Theorem 2.2. Let m; a; b; p 2 ðRþ Rþ ; Rþ Þa; b 2 C 1 ðRþ ; Rþ Þ and let a; b is nondecreasing with aðxÞ 6 x; bðyÞ 6 y if u 2 ðRþ Rþ ; Rþ Þ satisfies
uðx; yÞ 6 pðx; yÞ þ mðx; yÞ
Z
aðxÞ
Z
0
bðyÞ
aðt; sÞuðt; sÞds dt þ
Z
0
Z
x
0
y
bðt; sÞuðt; sÞds dt ;
ð2:9Þ
0
then
Rl Ry Z x R aðlÞ R bðyÞ Rx Ry R aðxÞ R bðyÞ aðt;sÞmðt;sÞds dtþ bðt;sÞmðt;sÞds dt aðt;sÞmðt;sÞds dtþ bðt;sÞmðt;sÞds dt 0 0 0 0 0 0 0 0 uðx; yÞ 6 pðx; yÞ þ mðx; yÞe e 0 Z bðyÞ Z y aðaðlÞ; sÞpðaðlÞ; sÞds þ bðl; sÞpðl; sÞ dl; x P 0; y P 0: a0 ðlÞ 0
ð2:10Þ
0
Proof. Let
zðx; yÞ ¼
Z
Z
aðxÞ
0
bðyÞ
aðt; sÞuðt; sÞds dt þ
Z
0
x
Z
0
y
bðt; sÞuðt; sÞds dt:
0
Then
zx ¼ a0 ðxÞ
Z
bðyÞ
aðaðxÞ; sÞuðaðxÞ; sÞds þ
Z
0
6 a0 ðxÞ
Z
y
bðx; sÞuðx; sÞds
0 bðyÞ
aðaðxÞ; sÞ½pðaðxÞ; sÞ þ mðaðxÞ; sÞzðaðxÞ; sÞds þ
Z
0
y
bðx; sÞ½pðx; sÞ þ mðx; sÞzðx; sÞds
0
in view of zðaðxÞ; bðyÞÞ 6 zðx; yÞ;
zx 6 a0 ðxÞ
Z
bðyÞ
aðaðxÞ; sÞ½pðaðxÞ; sÞ þ mðaðxÞ; sÞzðx; yÞds þ
0
Z
y
bðx; sÞ½pðx; sÞ þ mðx; sÞzðx; yÞds:
0
Hence
zx zðx; yÞ 6 a0 ðxÞ
Z
Z
bðyÞ
aðaðxÞ; sÞmðaðxÞ; sÞds þ
0 bðyÞ
aðaðxÞ; sÞpðaðxÞ; sÞds þ
R R aðxÞ bðyÞ
0
Multiplying above inequality by e o zðx; yÞe ox
R R aðxÞ bðyÞ 0
0
0
0
Rx Ry
aðt;sÞmðt;sÞds dtþ
0
0
Z
y
bðx; sÞmðx; sÞds
0
Z
y
bðx; sÞpðx; sÞds: 0
Rx Ry
aðt;sÞmðt;sÞds dtþ
bðt;sÞmðt;sÞds dt
0
0
bðt;sÞmðt;sÞds dt
!
R
6e
aðxÞ 0
ð2:11Þ
we get
R bðyÞ 0
Z a0 ðxÞ 0
aðt;sÞmðt;sÞds dtþ
bðyÞ
Rx Ry 0
0
bðt;sÞmðt;sÞds dt
Z aðaðxÞ; sÞpðaðxÞ; sÞds þ
y
bðx; sÞpðx; sÞds:
0
ð2:12Þ
611
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
Integrating on the interval ½0; x for x we obtain
zðx; yÞ 6 e
R aðxÞ R bðyÞ 0
0
a0 ðlÞ
aðt;sÞmðt;sÞds dtþ
Z
Rx Ry 0
bðt;sÞmðt;sÞds dt
0
bðyÞ
aðaðlÞ; sÞpðaðlÞ; sÞds þ
0
Z
Z
R
x
e
aðlÞ
R bðyÞ
0
0
aðt;sÞmðt;sÞds dtþ
Rl Ry 0
0
bðt;sÞmðt;sÞds dt
0
y
bðl; sÞpðl; sÞ dl x P 0; y P 0:
ð2:13Þ
0
Combine (2.13) with uðx; yÞ 6 pðx; yÞ þ mðx; yÞzðx; yÞ we obtain inequality (2.10). h Corollary 2.3. Let m; a; b; p; a; b defined as in Theorem 2.2 suppose u 2 CðRþ Rþ ; Rþ Þ is a solution of the integral equation
Z uðx; yÞ ¼ pðx; yÞ þ mðx; yÞ
aðxÞ
Z
0
bðyÞ
aðt; sÞuðt; sÞds dt þ
Z
0
Z
x
0
y
bðt; sÞmðt; sÞds dt :
ð2:14Þ
0
If m; p are bounded and
lim lim
Z
x!1 y!1
aðxÞ
Z
0
bðyÞ
aðt; sÞds dt þ
Z
0
x
0
Z
y
bðt; sÞds dt
< 1;
ð2:15Þ
0
then u is bounded. Corollary 2.4. Let m; a; b; a; b defined as in the Theorem 2.2 with pðx; yÞ ! 0 as x ! 1; y ! 1 suppose u 2 CðRþ Rþ ; Rþ Þ is a solution of the integral equation
Z uðx; yÞ ¼ pðx; yÞ þ mðx; yÞ
aðxÞ
Z
0
If
R að1Þ R bð1Þ 0
0
Z lim lim mðx; yÞ
x!1 y!1
aðxÞ
Z
0
aðt; sÞuðt; sÞds dt þ
Z
0
R1 R1
aðt; sÞmðt; sÞds dt þ
bðyÞ
0
0
Z
x
0
y
bðt; sÞuðt; sÞds dt :
0
bðt; sÞmðt; sÞds dt < 1 and
bðyÞ
aðt; sÞpðt; sÞds dt þ
Z
0
x
Z
0
y
bðt; sÞpðt; sÞds dt
¼ 0;
ð2:16Þ
0
then uðx; yÞ ! 0 as x ! 1; y ! 1 in particular, if mðx; yÞ; pðx; yÞ ! 0 R að1Þ R bð1Þ R1 R1 aðt; sÞds dt < 1; bðt; sÞds dt < 1 then uðx; yÞ ! 0 as x ! 1; y ! 1: 0 0 0 0
as
x ! 1; y ! 1
and
Theorem 2.3. Assume p; a; b; a; b be as in Theorem 2.1 xðuÞ is positive continuous nondecreasing for u > 0 with xð0Þ ¼ 0 and R 1 dt ¼ 1, if u 2 CðRþ Rþ ; Rþ Þ satisfies 1 xðtÞ
uðx; yÞ 6 pðx; yÞ þ
Z
aðxÞ
0
Z
Z
bðyÞ
aðx; y; t; sÞxðuðt; sÞÞds dt þ
0
x
Z
0
y
bðx; y; t; sÞxðuðt; sÞÞds dt;
ð2:17Þ
0
x P 0; y P 0, then
Z uðx; yÞ 6 G1 Gðpðx; yÞÞ þ
aðxÞ
0
x P 0; y P 0, where GðuÞ ¼
Ru
Z
bðyÞ
aðx; y; t; sÞds dt þ
Z
0
dt ;u 1 xðtÞ
x 0
Z
y
bðx; y; t; sÞds dt ;
ð2:18Þ
0
> 0:
Proof. Without loss of generality we assume that p > 0, for x P 0; y P 0, otherwise we can be replaced p by p þ e for any e > 0. Assume T 1 ; T 2 > 0 be fixed, and let
zðx; yÞ ¼
Z
aðxÞ
0
Z
bðyÞ
aðx; y; t; sÞxðuðt; sÞÞds dt þ
Z
0
x
Z
0
y
bðx; y; t; sÞxðuðt; sÞÞds dt 0
with assumption on a; b; a; b imply that zðx; yÞ is nondecreasing about x and y. Hence for x 2 ½0; T 1 ; y 2 ½0; T 2 we have
zxy ¼ aðx; y; aðxÞ; bðyÞÞxðuðaðxÞ; bðyÞÞÞa0 ðxÞb0 ðyÞ þ Z
þ a0 ðxÞ Z xZ þ 0
Z
0 bðyÞ
aðxÞ
Z
bðyÞ
ox oy aðx; y; t; sÞxðuðt; sÞÞds dt
0
Z
aðxÞ
oy aðx; y; aðxÞ; sÞxðuðaðxÞ; sÞÞds þ b0 ðyÞ ox aðx; y; t; bðyÞÞxðuðt; bðyÞÞÞdt þ bðx; y; x; yÞxðuðx; yÞÞ 0 0 Z Z x y y ox oy bðx; y; t; sÞxðuðt; sÞÞds dt þ oy bðx; y; x; sÞxðuðx; sÞÞds þ ox bðx; y; t; yÞxðuðt; yÞÞdt
0
0 0
0
0
6 aðx; y; aðxÞ; bðyÞÞxððpðaðxÞ; bðyÞÞ þ zðaðxÞ; bðyÞÞÞÞa ðxÞb ðyÞ þ a ðxÞ þ zðaðxÞ; sÞÞds þ b0 ðyÞ
Z 0
Z
0 bðyÞ
0
aðxÞ
ox aðx; y; t; bðyÞÞxðpðt; bðyÞÞ þ zðt; bðyÞÞÞdt þ
oy aðx; y; aðxÞ; sÞxðpðaðxÞ; sÞ Z 0
aðxÞ
Z 0
bðyÞ
ox oy aðx; y; t; sÞxððpðt; sÞ
612
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
Z xZ y þ zðt; sÞÞÞds dt þ bðx; y; x; yÞxðpðx; yÞ þ zðx; yÞÞ þ ox oy bðx; y; t; sÞxðpðt; sÞ þ zðt; sÞÞds dt 0 0 Z y Z x þ oy bðx; y; x; sÞxðpðx; sÞ þ zðx; sÞÞds þ ox bðx; y; t; yÞxðpðt; yÞ þ zðt; yÞÞdt 0 0 Z aðxÞ Z bðyÞ 6 xðpðT 1 ; T 2 Þ þ zðx; yÞÞ aðx; y; aðxÞ; bðyÞÞa0 ðxÞb0 ðyÞ þ ox oy aðx; y; t; sÞds dt 0 0 Z aðxÞ Z bðyÞ Z xZ y 0 0 þ b ðyÞ ox aðx; y; t; bðyÞÞdt þ a ðxÞ oy aðx; y; aðxÞ; sÞdsþbðx; y; x; yÞ þ ox oy bðx; y; t; sÞds dt 0 0 0 0 Z y Z x þ oy bðx; y; x; sÞds þ ox bðx; y; t; yÞdt ; 0
0
i.e
Z
zxy o2 6 xðpðT 1 ; T 2 Þ þ zðx; yÞÞ ox oy
Z
aðxÞ
0
bðyÞ
aðx; y; t; sÞds dt þ
Z
0
Z
x
0
bðx; y; t; sÞds dt :
y
ð2:19Þ
0
Noting that
o zx zxy 6 : oy xðpðT 1 ; T 2 Þ þ zðx; yÞÞ xðpðT 1 ; T 2 Þ þ zðx; yÞÞ
We have
Z aðxÞ Z bðyÞ Z xZ y o zx o2 aðx; y; t; sÞds dt þ bðx; y; t; sÞds dt : 6 oy xðpðT 1 ; T 2 Þ þ zðx; yÞÞ ox oy 0 0 0 0
ð2:20Þ
Integrating both sides of (2.20) with respect to y from 0 to y we have
zx o 6 xðpðT 1 ; T 2 Þ þ zðx; yÞÞ ox
Z
Z
aðxÞ
0
bðyÞ
aðx; y; t; sÞds dt þ
Z
0
x
Z
0
y
bðx; y; t; sÞds dt ;
ð2:21Þ
0
then integrating above inequality with respect to x from 0 to x we obtain
GðpðT 1 ; T 2 Þ þ zðx; yÞÞ 6 GðpðT 1 ; T 2 ÞÞ þ
Z
aðxÞ
Z
0
bðyÞ
aðx; y; t; sÞds dt þ
Z
0
x 2 ½0; T 1 ; y 2 ½0; T 2 : R1 In view of 1 xdu ðuÞ ¼ 1 we have from (2.22)
Z pðT 1 ; T 2 Þ þ zðx; yÞ 6 G1 GðpðT 1 ; T 2 ÞÞ þ
0
Z
aðxÞ 0
Z
x
bðyÞ
aðx; y; t; sÞds dt þ 0
y
bðx; y; t; sÞds dt;
ð2:22Þ
0
Z 0
x
Z
y
bðx; y; t; sÞds dt :
ð2:23Þ
0
Let x ¼ T 1 ; y ¼ T 2 in (2.23) we have
Z pðT 1 ; T 2 Þ þ zðT 1 ; T 2 Þ 6 G1 GðpðT 1 ; T 2 ÞÞ þ
aðT 1 Þ 0
Z
bðT 2 Þ
aðT 1 ; T 2 ; t; sÞds dt þ
Z
T1
0
0
Z
T2
bðT 1 ; T 2 ; t; sÞds dt :
0
Due to T 1 ; T 2 are arbitrarily and uðx; yÞ 6 pðx; yÞ þ zðx; yÞ we have obtain (2.18). h Corollary 2.5. Let a; b; pa; b; x be as in Theorem 2.3 suppose u 2 CðRþ Rþ ; Rþ Þ is a solution of the nonlinear Volterra equation
uðx; yÞ ¼ pðx; yÞ þ
Z
aðxÞ
Z
0
bðyÞ
aðx; y; t; sÞxðuðt; sÞÞds dt þ
0
Z
x
Z
0
y
bðx; y; t; sÞxðuðt; sÞÞds dt;
0
x P 0; y P 0, if p is bounded and
lim lim
x!1 y!1
Z 0
aðxÞ
Z
bðyÞ
aðx; y; t; sÞds dt þ
0
Z 0
x
Z
y
bðx; y; t; sÞds dt
< 1;
0
then u is bounded. Theorem 2.4. Suppose m; a; b; p 2 CðRþ Rþ ; Rþ Þa; b 2 C 1 ðRþ ; Rþ Þ, and let m; n; p; a; b are nondecreasing function with R aðxÞ 6 xbðyÞ 6 y for x P 0; y P 0 assume x 2 CðRþ ; Rþ Þ be a nondecreasing function such that xðtÞ > 0 for t > 0 and 11 xdtðtÞ ¼ 1 if u 2 CðRþ Rþ ; Rþ Þ satisfies
uðx; yÞ 6 pðx; yÞ þ mðx; yÞ
Z 0
aðxÞ
Z 0
bðyÞ
aðt; sÞxðuðt; sÞÞds dt þ
Z 0
x
Z 0
y
bðx; yÞxðuðt; sÞÞds dt ;
ð2:24Þ
613
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
x P 0; y P 0, then
Z uðx; yÞ 6 G1 Gðpðx; yÞÞ þ mðx; yÞ
aðxÞ
Z
0
x P 0; y P 0; where GðuÞ ¼
Ru
dt ;u 1 xðtÞ
bðyÞ
aðt; sÞds dt þ
Z
0
Z
x
0
y
bðt; sÞds dt
ð2:25Þ
;
0
> 0.
Proof. Without loss of generality we assume that p > 0, for x P 0; y P 0, otherwise we can be replaced p by p þ e for any e > 0. Let T 1 ; T 2 > 0 be fixed, then for x 2 ½0; T 1 ; y 2 ½0; T 2 relation (2.24) and hypotheses on m; p imply
uðx; yÞ 6 pðT 1 ; T 2 Þ þ mðT 1 ; T 2 Þ
Z
aðxÞ
Z
0
bðyÞ
aðt; sÞxðuðt; sÞÞds dt þ
Z
0
x
Z
0
y
bðt; sÞxðuðt; sÞÞds dt :
0
Let
zðx; yÞ ¼
Z
aðxÞ
0
Z
bðyÞ
aðt; sÞxðuðt; sÞÞds dt þ
0
Z
x
0
Z
y
bðt; sÞxðuðt; sÞÞds dt:
0
Then we have
zxy ¼ aðaðxÞ; bðyÞÞxðuðaðxÞ; bðyÞÞÞa0 ðxÞb0 ðyÞ þ bðx; yÞxðuðx; yÞÞ ¼ aðaðxÞ; bðyÞÞxðpðaðxÞ; bðyÞÞ þ mðaðxÞ; bðyÞÞzðaðxÞ; bðyÞÞÞa0 ðxÞb0 ðyÞ þ bðx; yÞxðpðx; yÞ þ mðx; yÞzðx; yÞÞ 6 xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞÞzðx; yÞðaðaðxÞ; bðyÞÞa0 ðxÞb0 ðyÞ þ bðx; yÞÞ; i.e.
zxy 6 aðaðxÞ; bðyÞÞa0 ðxÞb0 ðyÞ þ bðx; yÞ: xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 Þzðx; yÞÞ
ð2:26Þ
Noting that
o zx zxy 6 ; oy xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞÞzðx; yÞ xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞÞzðx; yÞ we have
o zx 6 aðaðxÞ; bðyÞÞa0 ðxÞb0 ðyÞ þ bðx; yÞ: oy xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞÞzðx; yÞ
ð2:27Þ
Integrating both sides of the (2.27) with respect to y from 0 to y we have
Z
zx 6 a0 ðxÞ xðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞÞzðx; yÞ
y
aðaðxÞ; bðsÞÞdbðsÞ þ
Z
0
y
bðx; sÞds;
ð2:28Þ
0
R1
then integrating (2.28) with respect to x from 0 to x in view of
GðpðT 1 ; T 2 Þ þ mðT 1 ; T 2 Þzðx; yÞÞ 6 GðpðT 1 ; T 2 ÞÞ þ
Z
Z
aðxÞ
0
dt
¼ 1; we get
xðtÞ
1
bðyÞ
aðt; sÞds dt þ
0
Z
Z
x
0
y
bðt; sÞds dt;
0
x 2 ½0; T 1 ; y 2 ½0; T 2 ; we obtain
Z pðT 1 ; T 2 Þ þ mðT 1 ; T 2 Þzðx; yÞ 6 G1 GðpðT 1 ; T 2 ÞÞ þ
aðxÞ
Z
0
bðyÞ
aðt; sÞds dt þ
0
Z
x
0
Z
y
bðt; sÞds dt :
ð2:29Þ
0
Let x ¼ T 1 ; y ¼ T 2 in (2.12) we have
Z pðT 1 ; T 2 Þ þ mðT 1 ; T 2 ÞzðT 1 ; T 2 Þ 6 G1 GðpðT 1 ; T 2 ÞÞ þ
aðT 1 Þ
Z
0
bðT 2 Þ
aðt; sÞds dt þ
Z
T1
Z
0
0
due to T 1 ; T 2 are arbitrarily and uðx; yÞ 6 pðx; yÞ þ mðx; yÞzðx; yÞ; we have (2.25) hold.
T2
bðt; sÞds dt ;
0
h
Corollary 2.6. Let m; a; b; p; a; b; x be as in Theorem 2.4. Suppose u 2 CðRþ Rþ ; Rþ Þ is a solution to the integral equation
Z uðx; yÞ ¼ pðx; yÞ þ mðx; yÞ
aðxÞ
Z
0
x P 0; y P 0; if m; p are bounded and
bðyÞ
aðt; sÞxðuðt; sÞÞds dt þ 0
0
R aðxÞ R bðyÞ 0
Z
0
aðt; sÞds dt þ
Rx Ry 0
0
x
Z
y
bðt; sÞxðuðt; sÞÞds dt ;
0
bðt; sÞds dt < 1 then u is bounded.
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H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
Remark. Using similar method of those in the proof of Theorems 2.2 and 2.4 we can obtain the reversed inequality of (2.9) and (2.24) as follows: Theorem 2.5. Let m; n; p 2 CðRþ Rþ ; Rþ Þa; b 2 C 1 ðRþ ; Rþ Þ and assume that a; b are nondecreasing with aðxÞ P x; bðyÞ P y for x P 0; y P 0 if u 2 CðRþ Rþ ; Rþ Þ satisfies
uðx; yÞ P pðx; yÞ þ mðx; yÞ
Z
aðxÞ
Z
0
bðyÞ
aðt; sÞuðt; sÞds dt þ
Z
0
Z
x
0
y
bðt; sÞuðt; sÞds dt ;
0
then
uðx; yÞ P pðx; yÞ þ mðx; yÞ
Z
Z
x
0
y
e
R aðxÞ R bðyÞ aðlÞ
bðrÞ
aðt;sÞmðt;sÞds dtþ
Rx Ry l
r
bðt;sÞmðt;sÞds dt
0
aðaðlÞ; bðrÞÞðpðaðlÞ; bðrÞÞa0 ðlÞb0 ðrÞ þ bðl; rÞpðl; rÞÞdl dr;
x P 0; y P 0:
ð2:30Þ
Theorem 2.6. Let m; a; b; p 2 CðRþ Rþ ; Rþ Þa; b 2 C 1 ðRþ ; Rþ Þ with m,p are nondecreasing. suppose a; b are nondecreasing and aðxÞ P x; bðyÞ P y for x P 0; y P 0 let x 2 CðRþ ; Rþ Þ be a nondecreasing function such that xðtÞ > 0 for t > 0 if u 2 CðRþ Rþ ; Rþ Þ satisfies
uðx; yÞ P pðx; yÞ þ mðx; yÞ
Z
aðxÞ
Z
0
bðyÞ
aðt; sÞxðuðt; sÞÞds dt þ
Z
0
x
Z
0
y
bðx; yÞxðuðt; sÞÞds dt ;
ð2:24Þ
0
x P 0; y P 0, then
Z uðx; yÞ P G1 Gðpðx; yÞÞ þ mðx; yÞ
aðxÞ 0
Z
bðyÞ
aðt; sÞds dt þ
Z
0
x
Z
0
y
bðt; sÞds dt
;
ð2:25Þ
0
x P 0; y P 0; Ru Ru where GðuÞ ¼ 1 xdtðtÞ ; u P 0 where GðuÞ ¼ 1 xdtðtÞ ; u P 0 and x1 ; y1 are chosen so that
Z Gðpðx; yÞÞ þ mðx; yÞ
aðxÞ
Z
0
bðyÞ
aðt; sÞds dt þ
Z
0
0
x
Z
y
bðt; sÞds dt
2 DomðG1 Þ
0
for all x 2 ½0; x1 ; y 2 ½0; y1 : 3. Some applications In this section we will indicate some applications for our results to obtain the bounds on the solution of some integral equations with time delay. To study the boundedness, uniqueness and continuous of the solution of the initial boundary value problem for retarded Volterra equations In this section we present applications of the inequality (2.1) given in Theorem 2.1 to study the boundness, uniqueness, and continuous dependence of the solution of the initial boundary value problem for retarded Volterra equations of the form
uðx; yÞ ¼ pðx; yÞ þ
Z
x
0
Z
y
f ðx; y; t; s; uðaðxÞ; bðyÞÞ; uðx; yÞÞds dt;
ð3:1Þ
0
where f is continuous, p is continuous on D; a; b 2 C 1 ðRþ ; Rþ Þ, where D ¼ ðRþ Rþ Þ, and aðxÞ 6 x; bðyÞ 6 y; a0 ðxÞ > 0; b0 ðyÞ > 0 for x P 0; y P 0: First we give the bound on the solution of the problem (3.1). Theorem 3.1. Suppose that
jf ðx; y; t; s; w; vÞj 6 aðx; y; t; sÞjwj þ bðx; y; t; sÞjvj;
ð3:2Þ
where a; b are as in Theorem 2.1. Let
M1 ¼ max x2Rþ
1
a0 ðxÞ
;
M 2 ¼ max y2Rþ
1 ; b0 ðyÞ
ðx; y; t; sÞ ¼ aðx; y; a1 ðtÞ; b1 ðsÞÞ: a
ð3:3Þ ð3:4Þ
615
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
If x(t) is any solution of (3.1), then
R juðx; yÞj 6 jpðx; yÞj þ e
o ol
Z
R bðyÞ
aðxÞ 0
0
ðx;y;t;sÞds dtþ M1 M2 a
Rx Ry 0
0
bðx;y;t;sÞds dt
Z
x
R R aðlÞ bðyÞ
e
0
0
0
Z
aðlÞ
0
Rx Ry
ðl;y;t;sÞds dtþ M1 M2 a
0
0
bðl;y;t;sÞds dt
!
bðyÞ
ðl; y; t; sÞpðt; sÞds dt þ bðl; y; t; sÞpðt; sÞds dt dl: M1 M2 a
ð3:5Þ
0
Proof. Using (3.2)–(3.4) in (3.1) and making the change of variables, we have
juðx; yÞj 6 jpðx; yÞj þ
Z
Z
x
0
6 jpðx; yÞj þ
Z
y
½aðx; y; t; sÞjuðaðtÞ; bðsÞÞj þ bðx; y; t; sÞjuðt; sÞjds dt
0
Z
aðxÞ
0
aðyÞ
ðx; y; t; sÞjuðt; sÞjds dt þ M1 M2 a
0
Z
x
0
Z
y
bðx; y; t; sÞjuðt; sÞjds dt:
ð3:6Þ
0
Now a suitable application of the inequality in (2.1) given in Theorem 2.1 to (3.6) yields (3.5). The right hand of (3.5) gives us the bound on the solution xðtÞ of (3.1) in terms of the known functions. Thus, if the right hand of (3.5) is bounded, then we assert that the solution of (3.1) is bounded on D. The next result deals with the uniqueness of the solution of the problem (3.1). h Theorem 3.2. Suppose that the function f in (3.1) satisfies the condition
v Þj 6 aðx; y; t; sÞjw wj þ bðx; y; t; sÞjv v j: jf ðx; y; t; s; w; vÞ f ðx; y; t; s; w;
ð3:7Þ
Let a; b be as defined in Theorem 2.1, M1 ; M 2 be as defined in (3.3). Then the problem (3.1) has at most one solution on D. ðx; yÞ be two solution of (3.1), then we have Proof. Let uðx; yÞ; u
ðx; yÞ ¼ uðx; yÞ u
Z
Z
x
0
y
ðaðtÞ; bðsÞÞ; u ðt; sÞÞds dt; ½f ðx; y; t; s; uðaðtÞ; bðsÞÞ; uðt; sÞÞ f ðx; y; t; s; u
0
using (3.3), (3.4), and (3.7) we have
ðx; yÞj 6 juðx; yÞ u
Z
Z
x
0
Z
y
ðaðtÞ; bðsÞÞjds dt þ aðx; y; t; sÞjuðaðtÞ; bðsÞÞ u 0
aðxÞ
Z
bðyÞ
ðx; y; t; sÞjuðt; sÞ u ðt; sÞjds dt þ M1 M2 a
6 0
0
Z
Z
Z
x
0
Z
x
0
y
ðt; sÞjds bðx; y; t; sÞjuðt; sÞ u
0 y
ðt; sÞjds dt: bðx; y; t; sÞjuðt; sÞ u
ð3:8Þ
0
ðx; yÞj 6 0; so we have uðx; yÞ ¼ u ðx; yÞ; i.e., there is at most Now a suitable application of the Corollary 2.1 we have juðx; yÞ u one solution of the problem (3.1). The following theorem deals with the continuous dependence of the solution on the equation and given initial boundary conditions. Consider the problem (3.1) and the problem
ðx; yÞ þ vðx; yÞ ¼ p
Z
x
Z
0
y
f ðx; y; t; s; vðaðtÞ; bðsÞÞ; vðt; sÞÞds dt;
ð3:9Þ
0
is continuous on D. h where p Theorem 3.3. Suppose that the function f satisfies the condition (3.7) in Theorem 3.2 and further assume that
ðx; yÞj < e; jpðx; yÞ p
ð3:10Þ
be as in Theorem 3.1.Then the solution of (3.1) depends continuously where e > 0 is an arbitrary small constant, and let M 1 ; M 2 ; a on the initial value boundary data. Proof. From (3.1) and (3.9), using (3.7) and (3.10), and making the change of variables, we have
ðx; yÞj þ juðx; yÞ vðx; yÞj 6 jpðx; yÞ p þ
Z
aðxÞ
t
jf ðx; y; t; s; uðaðtÞ; bðsÞÞ; uðt; sÞÞ f ðx; y; t; s; vðaðtÞ; bðsÞÞ; vðt; sÞÞjds < e
0
Z
0
Z
bðyÞ
ðx; y; t; sÞjuðt; sÞ vðt; sÞjds dt þ M1 M2 a
0
Z
x
0
Z
y
bðx; y; t; sÞjuðt; sÞ vðt; sÞjds dt:
ð3:11Þ
0
Now a suitable application of the Corollary 2.1 to (3.11) yields
juðx; yÞ vðx; yÞj 6 ee
R aðxÞ R bðyÞ 0
0
Rx Ry
ðx;y;t;sÞ ds dtþ M1 M 2 a
0
0
bðx;y;t;sÞds dt
;
x P 0; y P 0:
ð3:12Þ
616
H. Zhang, F. Meng / Applied Mathematics and Computation 203 (2008) 608–616
If
Z lim lim
x!1
y!1
aðxÞ
0
Z 0
bðyÞ
ðx; y; t; sÞds dt þ a
Z 0
x
Z
y
bðx; y; t; sÞds dt
< 1:
ð3:13Þ
0
So the solution to such an initial boundary value problem depends continuously on the initial boundary values, if e ! 0 then juðx; yÞ vðx; yÞj ! 0: Our result is proved. h Acknowledgements The authors thank the referees very much for their valuable suggestions on this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Y.G. Sun, On retarded integral inequalities and their applications, J. Math. Anal. Appl. 301 (2005) 265–275. A. Morro, A Gronwall-like inequality and its applications to continuum thermodynamics, Boll, Un. Mat. Ital. B 6 (1982) 553–562. B.G. Pachpatte, On a certain retarded integral inequality and its applications, J. Inequal. Pure Appl. Math. 5 (2004) (article 19). Run Xu, Yuan Gong Sun, On retarded integral inequalities in two independent variables and their applications, Appl. Math. Comput. 182 (2006) 1260– 1266. O. Lipovan, Integral inequalities for retarded Volterra equations, J. Math. Anal. Appl. 322 (2006) 349–358. O. Lipovan, A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl. 252 (2000) 389–401. Fanewi Meng, Wei Nian, Li On some new integral inequalities and their applications, Appl. Math. Comput. 148 (2004) 381–392. O. Lipovan, A retarded integral inequality and its applications, J. Math. Anal. Appl. 285 (2003) 436–443. T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equation, Ann. Math. 20 (1919) 292–296. B.G. Pachpatte, Explicit bounds on a certain integral inequalities, J. Math. Anal. Appl. 267 (2002) 48–61. B.G. Pachpatte, On a certain retarded integral inequality and applications, J. Inequal. Pure Appl. Math. 3 (2004) (article 18). B.G. Pachpatte, On some new nonlinear retarded integral inequalities, J. Inequal. Pure Appl. Math. 5 (2004) (article 80). Hongxia Zhang, Fanwei Meng, Integral inequalities in two independent variables for retarded Volterra equations, Appl. Math.Comput. 199 (2008) 90– 98.