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Some new nonlinear retarded sum-difference inequalities with applications

Advances in Difference Equations20112011:41

https://doi.org/10.1186/1687-1847-2011-41

Received: 25 March 2011

Accepted: 10 October 2011

Published: 10 October 2011

Abstract

The main objective of this paper is to establish some new retarded nonlinear sum-difference inequalities with two independent variables, which provide explicit bounds on unknown functions. These inequalities given here can be used as handy tools in the study of boundary value problems in partial difference equations.

2000 Mathematics Subject Classification: 26D10; 26D15; 26D20.

Keywords

  • sum-difference inequalities
  • boundary value problem

1 Introduction

Being important tools in the study of differential, integral, and integro-differential equations, various generalizations of Gronwall inequality [1, 2] and their applications have attracted great interests of many mathematicians (cf. [316], and the references cited therein). Recently, Agarwal et al. [3] studied the inequality
u ( t ) a ( t ) + i = 1 n b i ( t 0 ) b i ( t ) g i ( t , s ) w i ( u ( s ) ) d s , t 0 t < t 1 .
Cheung [17] investigated the inequality
u p ( x , y ) a + p p - q b 1 ( x 0 ) b 1 ( x ) c 1 ( y 0 ) c 1 ( y ) g 1 ( s , t ) u q ( s , t ) d t d s + p p - q b 2 ( x 0 ) b 2 ( x ) c 2 ( y 0 ) c 2 ( y ) g 2 ( s , t ) u q ( s , t ) ψ ( u ( s , t ) ) d t d s .
Agarwal et al. [18] obtained explicit bounds to the solutions of the following retarded integral inequalities:
φ ( u ( t ) ) c + i = 1 n α i ( t 0 ) α i ( t ) u q ( s ) [ f i ( s ) φ u ( s ) + g i ( s ) ] d s , φ ( u ( t ) ) c + i = 1 n α i ( t 0 ) α i ( t ) u q ( s ) [ f i ( s ) φ 1 u ( s ) + g i ( s ) φ 2 l o g u ( s ) ] d s , φ ( u ( t ) ) c + i = 1 n α i ( t 0 ) α i ( t ) u q ( s ) [ f i ( s ) L s , u ( s ) + g i ( s ) u ( s ) ] d s ,
where c is a constant, and Chen et al. [19] did the same for the following inequalities:
ψ ( u ( x , y ) ) c + γ ( x 0 ) γ ( x ) δ ( y 0 ) δ ( y ) f ( s , t ) φ ( u ( s , t ) ) d t d s , ψ ( u ( x , y ) ) c + α ( x 0 ) α ( x ) β ( y 0 ) β ( y ) g ( s , t ) u ( s , t ) d t d s + γ ( x 0 ) γ ( x ) δ ( y 0 ) δ ( y ) f ( s , t ) u ( s , t ) φ ( u ( s , t ) ) d t d s , ψ ( u ( x , y ) ) c + α ( x 0 ) α ( x ) β ( y 0 ) β ( y ) g ( s , t ) w ( u ( s , t ) ) d t d s + γ ( x 0 ) γ ( x ) δ ( y 0 ) δ ( y ) f ( s , t ) w ( u ( s , t ) ) φ ( u ( s , t ) ) d t d s ,

where c is a constant.

Along with the development of the theory of integral inequalities and the theory of difference equations, more attentions are drawn to some discrete versions of Gronwall type inequalities (e.g., [2022] for some early works). Some recent works can be found, e.g., in [6, 2325] and some references therein. Found in [26], the unknown function u in the fundamental form of sum-difference inequality
u ( n ) a ( n ) + s = 0 n - 1 f ( s ) u ( s )
can be estimated by u ( n ) a ( n ) s = 0 n - 1 ( 1 + f ( s ) ) . In [6], the inequality of two variables
u 2 ( m , n ) c 2 + s = m 0 m - 1 t = n 0 n - 1 a ( s , t ) u ( s , t ) + s = m 0 m - 1 t = n 0 n - 1 b ( s , t ) u ( s , t ) w u ( s , t )
was discussed, and the result was generalized in [23] to the inequality
u p ( m , n ) c + s = m 0 m - 1 t = n 0 n - 1 a ( s , t ) u q ( s , t ) + s = m 0 m - 1 t = n 0 n - 1 b ( s , t ) u q ( s , t ) w u ( s , t ) .
In this paper, motivated mainly by the works of Cheung [17, 23], Agarwal et al. [3, 18], and Chen et al. [19], we shall discuss upper bounds of the function u(m, n) satisfying one of the following general sum-difference inequalities
ψ ( u ( m , n ) ) a ( m , n ) + b ( m , n ) i = 1 k s = m 0 m 1 t = n 0 n 1 w ( u ( α i ( s ) , β i ( t ) ) ) [ f i ( s , t ) φ ( u ( α i ( s ) , β i ( t ) ) ) + g i ( s , t ) ] ,
(1.1)
ψ ( u ( m , n ) ) a ( m , n ) + b ( m , n ) i = 1 k s = m 0 m 1 t = n 0 n 1 w ( u ( α i ( s ) , β i ( t ) ) ) [ f i ( s , t ) φ 1 ( u ( α i ( s ) , β i ( t ) ) ) + g i ( s , t ) φ 2 ( l o g u ( α i ( s ) , β i ( t ) ) ) ] ,
(1.2)
ψ ( u ( m , n ) ) a ( m , n ) + b ( m , n ) i = 1 k s = m 0 m 1 t = n 0 n 1 w ( u ( α i ( s ) , β i ( t ) ) ) [ f i ( s , t ) L ( s , t , u ( α i ( s ) , β i ( t ) ) ) + g i ( s , t ) u ( α i ( s ) , β i ( t ) ) ] ,
(1.3)

for (m, n) [m 0, m 1) ∩ + × [n 0, n 1) ∩ +, where a(m, n), b(m, n) are nonnegative and nonde-creasing functions in each variable. Inequalities (1.1), (1.2), and (1.3) are the discrete versions of Agarwal et al. [18] and Chen et al. [19]. They not only generalized the forms with one variable into the ones with two variables but also extended the constant 'c' out of integral into a function 'a(m, n)'. These inequalities will play an important part in the study on difference equation. To illustrate the action of their inequalities, we also gave an example of boundary value problem in partial difference equation.

2 Main result

Throughout this paper, k, m 0, m 1, n 0, n 1 are fixed natural numbers. + := {1, 2, 3, . . .}, I := [m 0, m 1] ∩ +, I m := [m 0, m] ∩ +, J := [n 0, n 1] ∩ +, J n := [n 0, n] ∩ +, + := [0, ∞). For functions s(m), z(m, n), m, n , their first-order (forward) differences are defined by Δs(m) = s(m + 1) - s(m), Δ1 z(m, n) = z(m + 1, n) - z(m, n) and Δ2 z(m, n) = z(m, n + 1) - z(m, n). Obviously, the linear difference equation Δx(m) = b(m) with initial condition x(m 0) = 0 has solution s = m 0 m - 1 b ( s ) . For convenience, in the sequel, we define s = m 0 m 0 - 1 b ( s ) = 0 . We make the following assumptions:

(H 1) ψ C(+, +) is strictly increasing with ψ(0) = 0 and ψ (t) → ∞ as t → ∞;

(H 2) a, b : I × J → (0, ∞) are nondecreasing in each variable;

(H 3) w, φ, φ 1, φ 2 C(+,+) are nondecreasing with w(0) > 0, φ(r) > 0, φ 1(r) > 0 and φ 2(r) > 0 for r > 0;

(H 4) α i : II and β i : JJ are nondecreasing with α i (m) ≤ m and β i (n) ≤ n, i = 1, 2, . . . , k;

(H 5) f i , g i : I × J+, i = 1, 2, . . . , k.

Theorem 1. Suppose (H 1 - H 5) hold and u(m, n) is a nonnegative function on I × J satisfying (1.1). Then, we have
u ( m , n ) ψ - 1 W - 1 Φ - 1 A ( m , n )
(2.1)
for all ( m , n ) I M 1 × J N 1 , where
W ( r ) : = 1 r d s w ( ψ - 1 ( s ) ) f o r r > 0 ; W ( 0 ) : = lim r 0 + W ( r ) ,
(2.2)
Φ ( r ) : = 1 r d s φ ( ψ - 1 ( W - 1 ( s ) ) ) f o r r > 0 ; Φ ( 0 ) : = lim r 0 + Φ ( r ) ,
(2.3)
A ( m , n ) : = Φ W ( a ( m , n ) ) + b ( m , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 g i ( s , t ) + b ( m , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) ,
(2.4)
and (M 1, N 1) I × J is arbitrarily chosen such that
A ( M 1 , N 1 ) D o m ( Φ - 1 ) , Φ - 1 ( A ( M 1 , N 1 ) ) D o m ( W - 1 ) .
(2.5)
Proof. From assumption (H 2) and the inequality (1.1), we have
ψ ( u ( m , n ) ) a ( M , n ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 w ( u ( α i ( s ) , β i ( t ) ) ) [ f i ( s , t ) φ u ( α i ( s ) , β i ( t ) ) + g i ( s , t ) ]
(2.6)
for all (m, n) I M × J, where m 0MM 1 is a natural number chosen arbitrarily. Define a function η(m, n) by the right-hand side of (2.6). Clearly, η(m, n) is positive and nondecreasing in each variable, with η(m 0, n) = a(M, n) > 0. Hence (2.6) is equivalent to
u ( m , n ) ψ - 1 ( η ( m , n ) )
(2.7)
for all (m, n) I M × J. By (H 4) and the monotonicity of w, ψ -1 and η, we have, for all (m, n) I M × J,
Δ 1 η ( m , n ) = b ( M , n ) i = 1 k t = n 0 n - 1 w ( u ( α i ( m ) , β i ( t ) ) ) [ f i ( m , t ) φ ( u ( α i ( m ) , β i ( t ) ) ) + g i ( m , t ) ] w ( ψ - 1 ( η ( m , n ) ) ) b ( M , n ) i = 1 k t = n 0 n - 1 [ f i ( m , t ) φ ( ψ - 1 ( η ( m , t ) ) ) + g i ( m , t ) ] .
(2.8)
On the other hand, by the monotonicity of w and ψ -1,
W ( η ( m + 1 , n ) ) - W ( η ( m , n ) ) = η ( m , n ) η ( m + 1 , n ) d s w ( ψ - 1 ( s ) ) Δ 1 η ( m , n ) w ( ψ - 1 ( η ( m , n ) ) ) .
(2.9)
From (2.8) and (2.9), we have
W ( η ( m + 1 , n ) ) - W ( η ( m , n ) ) b ( M , n ) i = 1 k t = n 0 n - 1 f i ( m , t ) φ ψ - 1 ( η ( m , t ) ) + g i ( m , t )
(2.10)
for (m, n), (m + 1, n) I M × J. Keeping n fixed and substituting m with s in (2.10), and then summing up both sides over s from m 0 to m - 1, we get
W ( η ( m , n ) ) W ( η ( m 0 , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) φ ψ - 1 ( η ( s , t ) ) + g i ( s , t ) = W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 [ f i ( s , t ) φ ψ - 1 ( η ( s , t ) ) + g i ( s , t ) ] c ( M , n ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) φ ψ - 1 ( η ( s , t ) )
(2.11)
for (m, n) I M × J, where
c ( M , n ) = W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 M - 1 t = n 0 n - 1 g i ( s , t ) .
(2.12)
Now, define a function Γ(m, n) by the right-hand side of (2.11). Clearly, Γ(m, n) is positive and nondecreasing in each variable, with Γ(m 0, n) = c(M, n) > 0. Hence (2.11) is equivalent to
η ( m , n ) W - 1 ( Γ ( m , n ) )
(2.13)
for all ( m , n ) I M × J N 1 , where N 1 is defined in (2.5). By (H 4) and the monotonicity of φ, ψ -1, W -1 and Γ , we have, for all ( m , n ) I M × J N 1 ,
Δ 1 Γ ( m , n ) = b ( M , n ) i = 1 k t = n 0 n - 1 f i ( m , t ) φ ( ψ - 1 ( η ( m , t ) ) ) b ( M , n ) φ ( ψ - 1 ( W - 1 ( Γ ( m , n ) ) ) ) i = 1 k t = n 0 n - 1 f i ( m , t ) .
(2.14)
On the other hand, by the monotonicity of φ, ψ -1, and W -1, we have
Φ ( Γ ( m + 1 , n ) ) - Φ ( Γ ( m , n ) ) = Γ ( m , n ) Γ ( m + 1 , n ) d s φ ( ψ - 1 ( W - 1 ( s ) ) ) Δ 1 Γ ( m , n ) φ ( ψ - 1 ( W - 1 ( Γ ( m , n ) ) ) ) .
(2.15)
From (2.14) and (2.15), we obtain
Φ ( Γ ( m + 1 , n ) ) - Φ ( Γ ( m , n ) ) b ( M , n ) i = 1 k t = n 0 n - 1 f i ( m , t )
(2.16)
for ( m , n ) , ( m + 1 , n ) I M × J N 1 . Keeping n fixed and substituting m with s in (2.16), and then summing up both sides over s from m 0 to m - 1, we get
Φ ( Γ ( m , n ) ) Φ ( Γ ( m 0 , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) = Φ ( c ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t )
(2.17)
for ( m , n ) I M × J N 1 . From (2.12) and (2.17), we have
Γ ( m , n ) Φ 1 ( Φ ( c ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 m 1 t = n 0 n 1 f i ( s , t ) ) = Φ 1 [ Φ ( W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 M 1 t = n 0 n 1 g i ( s , t ) ) + b ( M , n ) i = 1 k s = m 0 m 1 t = n 0 n 1 f i ( s , t ) ] .
(2.18)
From (2.7), (2.13), and (2.18), we get
u ( m , n ) ψ 1 ( η ( m , n ) ) ψ 1 ( W 1 ( Γ ( m , n ) ) ) ψ 1 { W 1 [ Φ 1 ( Φ ( W ( a ( M , n ) )
(2.19)
+ b ( M , n ) i = 1 k s = m 0 M - 1 t = n 0 n - 1 g i ( s , t ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t )
(2.20)
for ( m , n ) I M × J N 1 . Let m = M, from (2.20), we observe that
u ( M , n ) ψ - 1 W - 1 Φ - 1 Φ W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 M - 1 t = n 0 n - 1 g i ( s , t ) + b ( M , n ) i = 1 k s = m 0 M - 1 t = n 0 n - 1 f i ( s , t )
(2.21)
for all ( M , n ) I M 1 × J N 1 , where M 1 is defined by (2.5). Since M I M 1 is arbitrary, from (2.21), we get the required estimate
u ( m , n ) ψ - 1 W - 1 Φ - 1 Φ W ( a ( m , n ) ) + b ( m , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 g i ( s , t ) + b ( m , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t )

for all ( m , n ) I M 1 × J N 1 . Theorem 1 is proved.

Theorem 2. Suppose (H 1 - H 5) hold and u(m, n) is a nonnegative function on I × J satisfying (1.2). Then

(i) if φ 1(u) ≥ φ 2(log u), we have
u ( m , n ) ψ - 1 W - 1 Φ 1 - 1 ( D 1 ( m , n ) )
(2.22)

for all ( m , n ) I M 1 × J N 2 ,

(ii) if φ 1(u) ≤ φ 2(log u), we have
u ( m , n ) ψ - 1 W - 1 Φ 2 - 1 ( D 2 ( m , n ) )
(2.23)
for all ( m , n ) I M 3 × J N 3 , where
D j ( m , n ) : = Φ j ( W ( a ( m , n ) ) ) + b ( m , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 [ f i ( s , t ) + g i ( s , t ) ] ; Φ j ( r ) : = 1 r d s φ j ( ψ - 1 ( W - 1 ( s ) ) ) for r > 0 ; Φ j ( 0 ) : = lim r 0 + Φ j ( r ) ;
(2.24)
j = 1, 2; (M 2, N 2) is arbitrarily given on the boundary of the planar region
1 : = { ( m , n ) I × J : D 1 ( m , n ) D o m ( Φ 1 - 1 ) , Φ 1 - 1 ( D 1 ( m , n ) ) D o m ( W - 1 ) } ;
(2.25)
and (M 3, N 3) is arbitrarily given on the boundary of the planar region
2 : = { ( m , n ) I × J : D 2 ( m , n ) D o m ( Φ 2 - 1 ) , Φ 2 - 1 ( D 2 ( m , n ) ) D o m ( W - 1 ) } .
(2.26)
Proof. (i) When φ 1(u) ≥ φ 2(log u), from inequality (1.2), we have
ψ ( u ( m , n ) ) a ( M , n ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 w ( u ( α i ( s ) , β i ( t ) ) ) f i ( s , t ) φ 1 ( u ( α i ( s ) , β i ( t ) ) ) + g i ( s , t ) φ 2 log ( u ( α i ( s ) , β i ( t ) ) )
(2.27)
for all (m, n) I M × J, where m 0MM 2 is chosen arbitrarily. Let Ξ(m, n) denote the right-hand side of (2.27), which is a positive and nondecreasing function in each variable with Ξ (m 0, n) = a(M, n). Hence (2.27) is equivalent to
u ( m , n ) ψ - 1 ( Ξ ( m , n ) ) .
(2.28)
By (H 4) and the monotonicity of w, ψ -1, and Ξ, we have, for all (m, n) I M × J,
Δ 1 Ξ ( m , n ) = b ( M , n ) i = 1 k t = n 0 n - 1 w ( u ( α i ( m ) , β i ( t ) ) ) f i ( m , t ) φ 1 ( u ( α i ( m ) , β i ( t ) ) ) + g i ( m , t ) φ 2 log ( u ( α i ( m ) , β i ( t ) ) ) b ( M , n ) w ψ - 1 ( Ξ ( m , n ) ) i = 1 k t = n 0 n - 1 f i ( m , t ) φ 1 ψ - 1 ( Ξ ( m , t ) ) + g i ( m , t ) φ 2 log ( ψ - 1 ( Ξ ( m , t ) ) )
(2.29)
for all (m, n) I M × J. Similar to the process from (2.9) to (2.11), we obtain
W ( Ξ ( m , n ) ) W ( Ξ ( m 0 , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) φ 1 ( ψ - 1 ( Ξ ( s , t ) ) ) + g i ( s , t ) φ 2 log ( ψ - 1 ( Ξ ( s , t ) ) ) = W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) φ 1 ( ψ - 1 ( Ξ ( s , t ) ) ) + g i ( s , t ) φ 2 log ( ψ - 1 ( Ξ ( s , t ) ) ) W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) + g i ( s , t ) φ 1 ( ψ - 1 ( Ξ ( s , t ) ) )
(2.30)
for all (m, n) I M × J. Now, define a function Θ(m, n) by the right-hand side of (2.30). Clearly, Θ(m, n) is positive and nondecreasing in each variable, with Θ(m 0, n) = W (a(M, n)) > 0. Thus, (2.30) is equivalent to
Ξ ( m , n ) W - 1 ( Θ ( m , n ) ) ( m , n ) I M × J N 2 ,
(2.31)
where N 2 is defined by (2.25). Similar to the process from (2.14) to (2.18), we obtain
Θ ( m , n ) Φ 1 - 1 Φ 1 ( Θ ( m 0 , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 [ f i ( s , t ) + g i ( s , t ) ] = Φ 1 - 1 Φ 1 ( W ( a ( M , n ) ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 [ f i ( s , t ) + g i ( s , t ) ]
(2.32)
for all ( m , n ) I M × J N 2 . From (2.28), (2.31), and (2.32), we conclude that
u ( m , n ) ψ - 1 ( Ξ ( m , n ) ) ψ - 1 W - 1 ( Θ ( m , n ) ) ψ - 1 W - 1 Φ 1 - 1 ( Φ 1 ( W ( a ( M , n ) ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 [ f i ( s , t ) + g i ( s , t ) ] )
(2.33)
for all ( m , n ) I M × J N 2 . Let m = M , from (2.33), we get
u ( M , n ) ψ 1 [ W 1 ( Φ 1 1 ( Φ 1 ( W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 M 1 t = n 0 n 1 [ f i ( s , t ) + g i ( s , t ) ] ) ) ] .
(2.34)

Since M I M 2 is arbitrary, from inequality (2.34), we obtain the required inequality in (2.22).

(ii) When φ 1(u) ≤ φ 2(log u), similar to the process from (2.27) to (2.30), from inequality (1.2), we have
W ( Ξ ( m , n ) ) W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) + g i ( s , t ) φ 2 ( ψ - 1 ( Ξ ( s , t ) ) )
(2.35)
for all ( m , n ) I M × J , M I M 3 , where M 3 is defined in (2.26). Similar to the process from (2.30) to (2.34), we obtain
u ( M , n ) ψ 1 [ W 1 ( Φ 2 1 ( Φ 2 ( W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 M 1 t = n 0 n 1 [ f i ( s , t ) + g i ( s , t ) ] ) ) ] .
(2.36)

Since M I M 3 is arbitrary, from inequality (2.36), we obtain the required inequality in (2.23).

Theorem 3. Suppose (H 1 - H 5) hold and that L, M C ( + 3 , + ) satisfy
0 L ( s , t , u ) - L ( s , t , v ) M ( s , t , v ) ( u - v )
(2.37)
for s, t, u, v + with u > v ≥ 0. If u(m, n) is a nonnegative function on I × J satisfying (1.3) then we have
u ( m , n ) ψ - 1 W - 1 ( Φ 3 - 1 ( E ( m , n ) ) )
(2.38)
for all ( m , n ) I M 4 × J N 4 , where W is defined by (2.2),
Φ 3 ( r ) : = 1 r d s ψ - 1 ( W - 1 ( s ) ) f o r r > 0 ; Φ 3 ( 0 ) : = lim r 0 + Φ 3 ( r ) ,
(2.39)
E ( m , n ) : = Φ 3 ( F ( m , n ) ) + b ( m , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 [ f i ( s , t ) M ( s , t , 0 ) + g i ( s , t ) ] ,
F ( m , n ) : = W ( a ( m , n ) ) + b ( m , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) L ( s , t , 0 ) ,
and (M 4, N 4) I × J is arbitrarily given on the boundary of the planar region
: = { ( m , n ) I × J : E ( m , n ) D o m ( Φ 3 - 1 ) , Φ 3 - 1 ( E ( m , n ) ) D o m ( W - 1 ) } .
(2.40)
Proof. From inequality (1.3), we have
ψ ( u ( m , n ) ) a ( M , n ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 w ( u ( α i ( s ) , β i ( t ) ) ) f i ( s , t ) L s , t , u ( α i ( s ) , β i ( t ) ) + g i ( s , t ) u ( α i ( s ) , β i ( t ) )
(2.41)
for all (m, n) I M × J, where m 0MM 4 is chosen arbitrarily. Let P (m, n) denote the right-hand side of (2.41), which is a positive and nondecreasing function in each variable, with P(m 0, n) = a(M, n). Similar to the process in the proof of Theorem 2 from (2.27) to (2.30), we obtain
W ( P ( m , n ) ) W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) L s , t , ψ - 1 ( P ( s , t ) ) + g i ( s , t ) ψ - 1 ( P ( s , t ) )
(2.42)
for all (m, n) I M × J. From inequality (2.37) and (2.42), we get
W ( P ( m , n ) ) W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) L ( s , t , 0 ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) M ( s , t , 0 ) + g i ( s , t ) ψ - 1 ( P ( s , t ) )
for all (m, n) I M × J. Similar to the process in the proof of Theorem 2 from (2.30) to (2.34), we obtain
u ( m , n ) ψ - 1 W - 1 Φ 3 - 1 Φ 3 W ( a ( M , n ) ) + b ( M , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t ) L ( s , t , 0 ) + b ( m , n ) i = 1 k s = m 0 m - 1 t = n 0 n - 1 [ f i ( s , t ) M ( s , t , 0 ) + g i ( s , t ) ] .
(2.43)

Since M I M 4 is arbitrary, where M 4 is defined in (2.40), from inequality (2.43), we obtain the required inequality in (2.38).

3 Applications to BVP

In this section, we use our result to study certain properties of the solutions of the following boundary value problem (BVP):
Δ 2 Δ 1 ( ψ ( z ( m , n ) ) ) = F m , n , z ( α 1 ( m ) , β 1 ( n ) ) , z ( α 2 ( m ) , β 2 ( n ) ) , , z ( α k ( m ) , β k ( n ) ) , z ( m , n 0 ) = a 1 ( m ) , z ( m 0 , n ) = a 2 ( n ) , z ( m 0 , n 0 ) = a 1 ( m 0 ) = a 2 ( n 0 ) = 0
(3.1)

for m I, n J, where m 0, n 0, m 1, n 1 + are constants, I := [m 0, m 1] ∩ +, J := [n 0, n 1] ∩ +, F : I × J × k , ψ : is strictly increasing on + with ψ(0) = 0, |ψ(r)| = ψ(|r|), and ψ(t) → ∞ as t → ∞; functions α i : II and β i : JJ are nondecreasing such that α i (m) ≤ m and β i (n) ≤ n, i = 1, 2, . . . , k; |a 1| : I+, |a 2| : J+ are both nondecreasing.

We give an upper bound estimate for solutions of BVP (3.1).

Corollary 1. Consider BVP (3.1) and suppose that F satisfies
| F ( m , n , u 1 , u 2 , , u k ) | i = 1 k w ( | u i | ) [ f i ( m , n ) φ ( | u i | ) + g i ( m , n ) ] , ( m , n ) I × J ,
(3.2)
where f i , g i : I × J+ and w, φ C 0(+, +) are nondecreasing with w(u) > 0, φ(u) > 0 for u > 0. Then, all solutions z(m, n) of BVP (3.1) satisfy
| z ( m , n ) | ψ - 1 W - 1 Φ - 1 A ( m , n ) ,
(3.3)
for all ( m , n ) I M 1 × J N 1 , where
A ( m , n ) : = Φ W ( ψ ( | a 1 ( m ) | ) + ψ ( | a 2 ( n ) | ) ) + i = 1 k s = m 0 m - 1 t = n 0 n - 1 g i ( s , t ) + i = 1 k s = m 0 m - 1 t = n 0 n - 1 f i ( s , t )
(3.4)

for all ( m , n ) I M 1 × J N 1 , with W, W -1, Φ, Φ-1 and M 1, N 1 as given in Theorem 1.

Proof. BVP (3.1) is equivalent to
ψ ( z ( m , n ) ) = ψ ( a 1 ( m ) ) + ψ ( a 2 ( n ) ) + s = m 0 m - 1 t = n 0 n - 1 F s , t , z ( α 1 ( s ) , β 1 ( t ) ) , z ( α 2 ( s ) , β 2 ( t ) ) , , z ( α k ( s ) , β k ( t ) )
(3.5)
By (3.2) and (3.5), we get
ψ ( | z ( m , n ) | ) ψ ( | a 1 ( m ) | ) + ψ ( | a 2 ( n ) | ) + s = m 0 m - 1 t = n 0 n - 1 F s , t , z ( α 1 ( s ) , β 1 ( t ) ) , z ( α 2 ( s ) , β 2 ( t ) ) , , z ( α k ( s ) , β k ( t ) ) ψ ( | a 1 ( m ) | ) + ψ ( | a 2 ( n ) | ) + s = m 0 m - 1 t = n 0 n - 1 i = 1 k w | z ( α i ( s ) , β i ( t ) ) | f i ( s , t ) φ ( | z ( α i ( s ) , β i ( t ) ) | ) + g i ( s , t ) .
(3.6)

Clearly, inequality (3.6) is in the form of (1.1). Thus the estimate (3.3) of the solution z(m, n) follows immediately from Theorem 1.

Declarations

Acknowledgements

The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions. This research was supported by National Natural Science Foundation of China(Project No. 11161018), Guangxi Natural Science Foundation(Project No. 0991265), and the Research Grants Council of the Hong Kong SAR, Project No. HKU7016/07P.

Authors’ Affiliations

(1)
Department of Mathematics, Hechi University, Guangxi, Yizhou, People's Republic of China
(2)
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, People's Republic of China
(3)
Department of Mathematics, The University of Hong Kong, Hong Kong, People's Republic of China

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© Wang et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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