Open Access

Some generalized nonlinear retarded integral inequalities with applications

Journal of Inequalities and Applications20122012:31

DOI: 10.1186/1029-242X-2012-31

Received: 13 October 2011

Accepted: 15 February 2012

Published: 15 February 2012

Abstract

In this article we discuss some new generalized nonlinear Gronwall-Bellman-Type integral inequalities with two variables, which include a non-constant term outside the integrals. We use our result to deal with the estimate on the solutions of partial differential equations with the initial and boundary conditions.

Mathematics Subject Classification 2000: 26D10; 26D15; 26D20; 34A40.

Keywords

integral inequality integral inequality technique boundary value problem boundedness uniqueness

1 Introduction

Various generalizations of Gronwall inequality [1, 2] are fundamental tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of differential equations, integral equations, and differential-integral equations. There are a lot of articles investigating its generalizations such as [323]. Recently, Pachpatte [19] provided the explicit estimations of following integral inequalities:
u p ( t ) c + p i = 1 n α i ( t 0 ) α i ( t ) [ a i ( s ) u p ( s ) + b i ( s ) u ( s ) ] d s , u p ( t ) c + p i = 1 n α i ( t 0 ) α i ( t ) [ a i ( s ) u ( s ) w ( u ( s ) ) + b i ( s ) u ( s ) ] d s ,
and
u p ( x , y ) c + p i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) [ a i ( s , t ) u p ( s , t ) + b i ( s , t ) u ( s , t ) ] d t d s , u p ( x , y ) c + p i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) [ a i ( s , t ) u ( s , t ) w ( u ( s , t ) ) + b i ( s , t ) u ( s , t ) ] d t d s ,
where c is a constant. Cheung [7] 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. [3] obtained the 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 ( log ( 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. Chen et al. [6] discussed 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 ( u , s ) 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.

In this article, motivated mainly by the works of Agarwal et al. [3] and Chen et al. [6], Cheung [7], Pachpatte [19], we discuss more general forms of following integral inequalities:
ψ ( u ( x , y ) ) a ( x , y ) + b ( x , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) w ( u ( s , t ) ) [ f i ( s , t ) φ ( u ( s , t ) ) + g i ( s , t ) ] d t d s ,
(1.1)
ψ ( u ( x , y ) ) a ( x , y ) + b ( x , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) w ( u ( s , t ) ) [ f i ( s , t ) φ 1 ( u ( s , t ) ) + g i ( s , t ) φ 2 ( log ( u ( s , t ) ) ) ] d t d s ,
(1.2)
ψ ( u ( x , y ) ) a ( x , y ) + b ( x , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) w ( u ( s , t ) ) [ f i ( s , t ) L ( s , t , u ( s , t ) ) + g i ( s , t ) u ( s , t ) ] d t d s ,
(1.3)

for (x, y) [x0, x1) × [y0, y1), where a(x, y), b(x, y) are nonnegative and nondecreasing functions in each variable. In inequalities (1.1)-(1.3), we generalized the constant c in [1, 5] to the function a(x,y), the function u(x) in [1] to the u(x,y) with two variables.

2 Main result

Throughout this article, x0, x1, y0, y1 are given numbers. I := [x0,x1), J := [y0,y1), Δ:= [x0,x1) × [y0,y1), + := [0,∞). Consider (1.1)-(1.3), and suppose that

(H1) ψ C(+, +) is a strictly increasing function with ψ(0) = 0 and ψ(t) → ∞ as t → ∞;

(H2) a, b: Δ → (0, ∞) are nondecreasing in each variable;

(H3) w, ϕ, ϕ1, ϕ2 C(+,+) are nondecreasing with w(0) > 0, ϕ(r) > 0, ϕ1(r) > 0 and ϕ2(r) > 0 for r > 0;

(H4) α i C1(I,I) and β i C1(J,J) are nondecreasing such that α i (x) ≤ x, α i (x0) = x0, β i (y) ≤ y and β i (y0) = y0, i = 1, 2,..., n;

(H5) f i , g i C(Δ,+), i = 1,2,...,n.

Theorem 1. Suppose that (H1-H5) hold and u(x,y) is a nonnegative and continuous function on Δ satisfying (1.1). Then we have
u ( x , y ) ψ - 1 ( W - 1 ( Φ - 1 ( B ( x , y ) ) ) ) ,
(2.1)
for all (x,y) [x0,X1) × [y0,Y1), where
W ( r ) : = 1 r d s w ( ψ - 1 ( s ) ) , r > 0 , W ( 0 ) : = lim r 0 + W ( r ) ,
(2.2)
Φ ( r ) : = 1 r d s φ ( ψ - 1 ( W - 1 ( s ) ) ) , r > 0 , Φ ( 0 ) : = lim r 0 + Φ ( r ) ,
(2.3)
B ( x , y ) : = Φ ( A ( x , y ) ) + b ( x , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) d t d s ,
(2.4)
A ( x , y ) : = W ( a ( x , y ) ) + b ( x , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) g i ( s , t ) d t d s ,
(2.5)
ψ-1, W-1 and Φ-1 denote the inverse function of ψ, W and Φ, respectively, and (X1,Y1) Δ is arbitrarily given on the boundary of the planar region
: = { ( x , y ) Δ : B ( x , y ) Dom ( Φ - 1 ) , Φ - 1 ( B ( x , y ) ) Dom ( W - 1 ) } .
(2.6)
Proof. From assumption H2 and the inequality (1.1), we have
ψ ( u ( x , y ) ) a ( X , y ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) w ( u ( s , t ) ) [ f i ( s , t ) φ ( u ( s , t ) ) + g i ( s , t ) ] d t d s ,
(2.7)
for all (x,y) [x0,X] × [y0,y1), where x0XX1 is chosen arbitrarily. Define a function η(x, y) by the right-hand side of (2.7). Clearly, η(x, y) is a positive and nondecreasing function in each variable, η(x0,y) = a(X,y) > 0. Then, (2.7) is equivalent to
u ( x , y ) ψ - 1 ( η ( x , y ) ) ,
(2.8)
for all (x,y) [x0,X] × [y0,y1). By the fact that α i (x) ≤ x for x [x0,x1), β i (y) ≤ y for y [y0,y1),i = 1,2,...,n, and the monotonicity of w,ψ-1,η, we have for all (x,y) [x0,X] × [y0,y1),
η x ( x , y ) = b ( X , y ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) w ( u ( α i ( x ) , t ) ) [ f i ( α i ( x ) , t ) φ ( u ( α i ( x ) , t ) ) + g i ( α i ( x ) , t ) ] d t w ( ψ 1 ( η ( x , y ) ) ) b ( x , y ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) [ f i ( α i ( x ) , t ) φ ( ψ 1 ( η ( α i ( x ) , t ) ) ) ] + g i ( α i ( x ) , t ) ] d t .
(2.9)
From (2.9), we get
η x ( x , y ) w ( ψ 1 ( η ( x , y ) ) ) b ( X , y ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) [ f i ( α i ( x ) , t ) φ ( ψ 1 ( η ( α i ( x ) , t ) ) ) + g i ( α i ( x ) , t ) ] d t ,
(2.10)
for all (x,y) [x0,X] × [y0,y1). Integrating (2.10) from x0 to x, by the definition of W in (2.2), we get for all (x,y) [x0,X] × [y0,y1),
W ( η ( x , y ) ) W ( η ( x 0 , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) φ ( ψ - 1 ( η ( s , t ) ) ) + g i ( s , t ) d t d s = W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) φ ( ψ - 1 ( η ( s , t ) ) ) + g i ( s , t ) d t d s W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( X ) β i ( y 0 ) β i ( y ) g i ( s , t ) d t d s + b ( X , Y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) φ ( ψ - 1 ( η ( s , t ) ) ) d t d s = c ( X , y ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) φ ( ψ - 1 ( η ( s , t ) ) ) d t d s ,
(2.11)
where
c ( X , y ) = W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) g i ( s , t ) d t d s .
(2.12)
Now, define a function Γ(x,y) by the right-hand side of (2.11). Clearly, Γ(x,y) is a positive and nondecreasing function in each variable, Γ(x0,y) = c(X, y) > 0. then, (2.11) is equivalent to
η ( x , y ) W - 1 ( Γ ( x , y ) ) ,
(2.13)
for all (x,y) [x0,X] × [y0,Y1), where Y1 is defined in (2.6). By the fact that α i (x) ≤ x for x [x0,x1), β i (y) ≤ y for y [y0,y1), i = 1, 2,...,n, and the monotonicity of ϕ, ψ-1, W-1, Γ, we have for all (x,y) [x0,X] × [y0,Y1),
Γ x ( x , y ) = b ( X , y ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) f i ( α i ( x ) , t ) φ ( ψ - 1 ( η ( α i ( x ) , t ) ) ) d t b ( X , y ) φ ( ψ - 1 ( W - 1 ( Γ ( x , y ) ) ) ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) f i ( α i ( x ) , t ) d t .
(2.14)
From (2.14), we have for all (x,y) [x0,X] × [y0,Y1),
Γ x ( x , y ) φ ( ψ - 1 ( W - 1 ( Γ ( x , y ) ) ) ) b ( X , y ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) f i ( α i ( x ) , t ) d t .
(2.15)
Integrating (2.15) from x0 to x, by the definition of Φ in (2.3), we get
Φ ( Γ ( x , y ) ) Φ ( Γ ( x 0 y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) d t d s = Φ ( c ( X , Y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) d t d s ,
(2.16)
for all (x,y) [x0,X] × [y0,Y1). From (2.12) and (2.16), we find
Γ ( x , y ) Φ - 1 Φ ( c ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) d t d s = Φ - 1 Φ W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( X ) β i ( y 0 ) β i ( y ) g i ( s , t ) d t d s + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) d t d s ,
(2.17)
for all (x, y) [x0, X] × [y0, Y1). From (2.8), (2.13), and (2.17), we get
u ( x , y ) ψ - 1 ( η ( x , y ) ) ψ - 1 ( W - 1 ( Γ ( x , y ) ) ) ψ - 1 W - 1 Φ - 1 Φ W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( X ) β i ( y 0 ) β i ( y ) g i ( s , t ) d t d s + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) d t d s ,
(2.18)
for all (x, y) [x0,X] × [y0,Y1). Let x = X, from (2.18), we observe that
u ( X , y ) ψ - 1 W - 1 Φ - 1 Φ W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( X ) β i ( y 0 ) β i ( y ) g i ( s , t ) d t d s | + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) d t d s ,
(2.19)
for all (X, y) [x0, X1) × [y0, Y1), where X1 is defined by (2.6). Since X [x0, X1) is arbitrary, from (2.19), we get the required estimations
u ( x , y ) ψ - 1 W - 1 Φ - 1 Φ W ( a ( x , y ) ) + b ( x , y ) i = 1 n α i ( x 0 ) α i ( X ) β i ( y 0 ) β i ( y ) g i ( s , t ) d t d s + b ( x , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) d t d s ,

for all (x,y) [x0,X1) × [y0,Y1). Theorem 1 is proved.

Remark that Theorem 1 generalizes Theorem 2.1 in [3].

Theorem 2. Suppose that (H1-H5) hold and u(x,y) is a nonnegative and continuous function on Δ satisfying (1.2). Then

(i) if ϕ1(u) ≥ ϕ2(log(u)), we have
u ( x , y ) ψ - 1 W - 1 ψ 1 - 1 D 1 ( x , y ) ,
(2.20)

for all (x,y) [x0,X2) × [y0,Y2),

(ii) if ϕ1(u) < ϕ2(log(u)), we have
u ( x , y ) ψ - 1 W - 1 Ψ 2 - 1 ( D 2 ( x , y ) ) ,
(2.21)
for all (x,y) [x0,X3) × [y0,Y3), where W is defined by (2.2) in Theorem 1,
D j ( x , y ) : = Ψ j ( W ( a ( x , y ) ) ) + b ( x , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) + g i ( s , t ) d t d s , Ψ j ( r ) : = 1 r d s φ j ( ψ - 1 ( W - 1 ( s ) ) ) , Ψ j ( 0 ) : = lim r 0 + Ψ j ( r ) ,
(2.22)
j = 1, 2, ψ-1, W-1, Ψ 1 - 1 and Ψ 2 - 1 denote the inverse function of ψ, W, Ψ1 and Ψ2, respectively, (X2,Y2) is arbitrarily given on the boundary of the planar region
1 : = ( x , y ) Δ : D 1 ( x , y ) Dom Ψ 1 - 1 , Ψ 1 - 1 ( D 1 ( x , y ) ) Dom ( W - 1 ) ,
(2.23)
and (X3,Y3) is arbitrarily given on the boundary of the planar region
2 : = ( x , y ) Δ : D 2 ( x , y ) Dom Ψ 2 - 1 , Ψ 2 - 1 ( D 2 ( x , y ) ) Dom ( W - 1 ) .
(2.24)
Proof. From the inequality (1.2), we have
ψ ( u ( x , y ) ) a ( X , y ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) w ( u ( s , t ) ) f i ( s , t ) φ 1 ( u ( s , t ) ) + g i ( s , t ) φ 2 ( log ( u ( s , t ) ) ) d t d s ,
(2.25)
for all (x,y) [x0,X] × [y0,y1), where x0XX2 is chosen arbitrarily. Let Ξ(x,y) denote the right-hand side of (2.25), which is a positive and nondecreasing function in each variable, Ξ(x0,y) = a(X,y). Then, (2.25) is equivalent to u(x,y) ≤ ψ-1(Ξ(x,y)). By the fact that α i (x) ≤ x for x [x0, x1), β i (y) ≤ y for y [y0, y1), i = 1, 2,..., n, and the monotonicity of w,ψ-1,Ξ, we have for all (x,y) [x0,X] × [y0,y1),
Ξ x ( x , y ) = b ( X , y ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) w ( u ( α i ( s ) , t ) ) f i ( α i ( x ) , t ) φ 1 ( u ( α i ( x ) , t ) ) + g i ( α i ( x ) , t ) φ 2 ( log ( u ( α i ( x ) , t ) ) ) d t b ( X , y ) w ( ψ - 1 ( Ξ ( x , y ) ) ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) f i ( α i ( x ) , t ) φ 1 ( ψ - 1 ( Ξ ( α i ( x ) , t ) ) ) + g i ( α i ( x ) , t ) φ 2 ( log ( ψ - 1 ( Ξ ( α i ( x ) , t ) ) ) ) d t ,
(2.26)
for all (x,y) [x0,X] × [y0,y1). From (2.26), we have
Ξ x ( x , y ) w ( ψ - 1 ( Ξ ( x , y ) ) ) b ( X , y ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) f i ( α i ( x ) , t ) φ 1 ( ψ - 1 ( Ξ ( α i ( x ) , t ) ) ) + g i ( α i ( x ) , t ) φ 2 ( log ( ψ - 1 ( Ξ ( α i ( x ) , t ) ) ) ) d t ,
(2.27)
for all (x,y) [x0,X] × [y0,y1). Integrating (2.27) from x0 to x, by the definition of W in (2.2), we get
W ( Ξ ( x , y ) ) W ( Ξ ( x 0 , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) φ 1 ( ψ - 1 ( Ξ ( s , t ) ) ) + g i ( s , t ) φ 2 ( log ( ψ - 1 ( Ξ ( s , t ) ) ) ) d t d s = W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) φ 1 ( ψ - 1 ( Ξ ( s , t ) ) ) + g i ( s , t ) φ 2 ( log ( ψ - 1 ( Ξ ( s , t ) ) ) ) d t d s ,
(2.28)

for all (x,y) [x0,X] × [y0,y1).

When ϕ1(u) ≥ ϕ2(log(u)), from the inequality (2.28), we have
W ( Ξ ( x , y ) ) W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) + g i ( s , t ) φ 1 ( ψ - 1 ( Ξ ( s , t ) ) ) d t d s ,
(2.29)
for all (x,y) [x0,X] × [y0,y1). Now, define a function Θ(x,y) by the right-hand side of (2.29). Clearly, Θ(x,y) is a positive and nondecreasing function in each variable, Θ(x0,y) = W(a(X,y)) > 0. Then, (2.29) is equivalent to
Ξ ( x , y ) W - 1 ( Θ ( x , y ) ) , ( x , y ) [ x 0 , X ] × y 0 , Y 2 ,
(2.30)
where Y2 is defined by (2.23). Differentiating Θ(x,y) in x for any fixed y [y0,Y2), we have
Θ x ( x , y ) = b ( X , y ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) f i ( α i ( x ) , t ) + g i ( α i ( x ) , t ) φ 1 ( ψ - 1 ( Ξ ( α i ( x ) , t ) ) ) d t b ( X , y ) φ 1 ( ψ - 1 ( W - 1 ( Θ ( x , y ) ) ) ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) f i ( α i ( x ) , t ) + g i ( α i ( x ) , t ) d t ,
(2.31)
for all (x,y) [x0,X] × [y0,Y2). From (2.31), we have
Θ x ( x , y ) φ 1 ( ψ - 1 ( W - 1 ( Θ ( x , y ) ) ) ) b ( X , y ) i = 1 n α i ( x ) β i ( y 0 ) β i ( y ) f i ( α i ( x ) , t ) + g i ( α i ( x ) , t ) d t ,
(2.32)
for all (x,y) [x0,X] × [y0,Y2). Integrating (2.32) from x0 to x, by the definition of Ψ1 in (2.22), we obtain
Ψ 1 ( Θ ( x , y ) ) Ψ 1 ( Θ ( x 0 , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) + g i ( s , t ) d t d s = Ψ 1 W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) + g i ( s , t ) d t d s .
(2.33)
From (2.30) and (2.33), we conclude
u ( x , y ) ψ - 1 ( Ξ ( x , y ) ) ψ - 1 ( W - 1 ( Θ ( x , y ) ) ) ψ - 1 W - 1 Ψ 1 - 1 Ψ 1 ( W ( a ( X , y ) ) ) + b ( X , y ) i = 1 n α i ( x ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) + g i ( s , t ) d t d s ,
(2.34)
for all (x,y) [x0,X] × [y0,Y2). Let x = X, from (2.34), we get
u ( X , y ) ψ - 1 W - 1 Ψ 1 - 1 Ψ 1 ( W ( a ( X , y ) ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( X ) β i ( y 0 ) β i ( y ) f i ( s , t ) + g i ( s , t ) d t d s .
(2.35)

Since X [x0,X2) is arbitrary, from the inequality (2.35), we obtain the required inequality in (2.20).

When ϕ1(u) ≤ ϕ2(log(u)), from the inequality (2.28), we have
W ( Ξ ( x , y ) ) W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) + g i ( s , t ) φ 2 ( log ( ψ - 1 ( Ξ ( s , t ) ) ) ) d t d s , W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) + g i ( s , t ) φ 2 ( ψ - 1 ( Ξ ( s , t ) ) ) d t d s ,
(2.36)
for all (x,y) [x0,X] × [y0,y1), where x0XX3. Similarly to the above process from (2.29) to (2.35), from (2.36), we obtain
u ( X , y ) ψ - 1 W - 1 Ψ 2 - 1 Ψ 2 ( W ( a ( X , y ) ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( X ) β i ( y 0 ) β i ( y ) f i ( s , t ) + g i ( s , t ) d t d s .
(2.37)

Since X [x0,X3) is arbitrary, where X3 is defined by (2.24), from the inequality (2.37), we obtain the required inequality in (2.21).

Theorem 3. Suppose that (H1-H5) hold and that L, M C + 3 , + satisfy
0 L ( s , t , u ) - L ( s , t , v ) M ( s , t , v ) ( u - v ) ,
(2.38)
for s, t, u, v + with u > v ≥ 0. If u(x,y) is a nonnegative and continuous function on Δ satisfying (1.3), then we have
u ( x , y ) ψ - 1 W - 1 Ψ 3 - 1 E x , y ,
(2.39)
for all (x,y) [x0,X4) × [y0,Y4), where W is defined by (2.2),
Ψ 3 ( r ) : = 1 r d s ψ - 1 ( W - 1 ( s ) ) , r > 0 , Ψ 3 ( 0 ) : = lim r 0 + Ψ 3 ( r ) ,
(2.40)
E ( x , y ) : = Ψ 3 ( F ( x , y ) ) + b ( x , y ) i = 1 n i = 1 α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) M ( s , t , 0 ) + g i ( s , t ) d t d s ,
F ( x , y ) : = W ( a ( x , y ) ) + b ( x , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) L ( s , t , 0 ) d t d s ,
ψ-1,W-1 and Ψ 3 - 1 denote the inverse function of ψ, W and Ψ3, respectively, and (X4,Y4) Δ is arbitrarily given on the boundary of the planar region
: = { ( x , y ) Δ : E ( x , y ) Dom ( Ψ 3 - 1 ) , Ψ 3 - 1 ( E ( x , y ) ) Dom ( W - 1 ) } .
(2.41)
Proof. From the inequality (1.3), we have
ψ ( u ( x , y ) ) a ( X , y ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) w ( u ( s , t ) ) f i ( s , t ) L ( s , t , u ( s , t ) ) + g i ( s , t ) u ( s , t ) d t d s ,
(2.42)
for all (x,y) [x0,X] × [y0,y1), where x0XX4 is chosen arbitrarily. Let P(x,y) denote the right-hand side of (2.42), which is a positive and nondecreasing function in each variable, P(x0,y) = a(X,y). Similarly to the process in the proof of Theorem 2 from (2.25) to (2.28), we obtain
W ( P ( x , y ) ) W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) L ( s , t , ψ - 1 ( P ( s , t ) ) ) + g i ( s , t ) ψ - 1 ( P ( s , t ) ) d t d s , ( x , y ) [ x 0 X ] × y 0 , y 1 .
(2.43)
From the inequality (2.38) and (2.43), we get
W ( P ( x , y ) ) W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( X ) β i ( y 0 ) β i ( y ) f i ( s , t ) L ( s , t , 0 ) d t d s + b ( X , y ) i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i ( s , t ) M ( s , t , 0 ) + g i ( s , t ) ψ - 1 ( P ( s , t ) ) d t d s ,
for all (x, y) [x0,X] × [y0,y1). Similarly to the process in the proof of Theorem 2 from (2.29) to (2.35), we obtain
u ( X , y ) ψ - 1 W - 1 Ψ 3 - 1 Ψ 3 W ( a ( X , y ) ) + b ( X , y ) i = 1 n α i ( x 0 ) α i ( X ) β i ( y 0 ) β i ( y ) f i ( s , t ) L ( s , t , 0 ) d t d s + b ( X , y ) i = 1 n α i ( x 0 ) α i ( X ) β i ( y 0 ) β i ( y ) f i ( s , t ) M ( s , t , 0 ) + g i ( s , t ) d t d s ,
(2.44)

where Ψ3 is defined by (2.40). Since X [x0,X4) is arbitrary, where X4 is defined by (2.41), from the inequality (2.44), we obtain the required inequality in (2.39).

3 Applications to BVP

In this section we use our result to study certain properties of solution of the following boundary value problem (simply called BVP):
2 ψ ( z ( x , y ) ) x y = F ( x , y , z ( α 1 ( x ) , β 1 ( y ) ) , z ( α 2 ( x ) , β 2 ( y ) ) , . . . , z ( α n ( x ) , β n ( y ) ) ) , z ( x , y 0 ) = a 1 ( x ) , z ( x 0 , y ) = a 2 ( y ) , a 1 ( x 0 ) = a 2 ( y 0 ) = 0 ,
(3.1)

for x I,y J, where x0,y0,x1,y1 + are constants, I := [x0,x1), J := [y0,y1), F C(I × J × n ,), ψ: is strictly increasing on + with ψ(0) = 0, |ψ(r)| = ψ(|r|) > 0, for |r| > 0 and ψ(t) → ∞ as t → ∞; functions α i C1(I,I);β i C1(J,J),i = 1,2,...,n are nondecreasing such that α i (x) ≤ x, β i (y) ≤ y,α i (x0) = x0, β i (y0) = y0; |a1| C1(I,+), |a2| C1(J,+) are both nondecreasing. Though this equation is similar to the equation discussed in Section 3 in [3], our results are more general than the results obtained in [3].

We first give an estimate for solutions of the BVP (3.1) so as to obtain a condition for boundedness.

Corollary 1. Consider BVP (3.1) and suppose that F C(I × J × n ,) satisfies
F ( x , y , u 1 , u 2 , . . . , u n ) i = 1 n w u i f i ( x , y ) φ u i + g i ( x , y ) , ( x , y ) I × J ,
(3.2)
where f i ,g i C(I × J,+) and w,ϕ C(+,+) are nondecreasing such that w(u) > 0,ϕ(u) > 0 for u > 0. Then all solutions z(x,y) of BVP (3.1) have the estimate
z ( x , y ) ψ - 1 W - 1 Φ - 1 B ( x , y ) ,
(3.3)
for all (x,y) [x0,X1) × [y0,Y1), where
B ( x , y ) : = Φ ( A ( x , y ) ) + i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i α i - 1 ( s ) , β i - 1 ( t ) α i α i - 1 ( s ) β i β i - 1 ( t ) d t d s , A ( x , y ) : = W ψ a 1 ( x ) + ψ a 2 ( y ) + i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i - 1 g i α i - 1 ( s ) , β i - 1 ( t ) α i α i - 1 ( s ) β i β i - 1 ( t ) d t d s ,

for all (x,y) [x0,X1) × [y0,Y1), where functions W, W-1, Φ, Φ-1 and real numbers X1, Y1 are given as in Theorem 1.

Proof. The equivalent integral equation of BVP (3.1) is
ψ ( z ( x , y ) ) = ψ ( a 1 ( x ) ) + ψ ( a 2 ( y ) ) + x 0 x y 0 y F s , t , z ( α 1 ( s ) , β 1 ( t ) ) , z ( α 2 ( s ) , β 2 ( t ) ) , . . . , z ( α n ( s ) , β n ( t ) ) d t d s .
(3.4)
By (3.2) and (3.4), we get that
ψ z ( x , y ) ψ a 1 ( x ) + ψ a 2 ( y ) + x 0 x y 0 y F s , t , z ( α 1 ( s ) , β 1 ( t ) ) , z ( α 2 ( s ) , β 2 ( t ) ) , . . . , z ( α 2 ( s ) , β n ( t ) ) d t d s ψ a 1 ( x ) + ψ a 2 ( y ) + x 0 x y 0 y i = 1 n w z ( α i ( s ) , β i ( t ) ) f i ( s , t ) φ z ( α i ( s ) , β i ( t ) ) + g i ( s , t ) d t d s = ψ a 1 ( x ) + ψ a 2 ( y ) + i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) w z ( s 1 , t 1 ) f i α i - 1 ( s 1 ) , β i - 1 ( t 1 ) φ z ( s 1 , t 1 ) + g i α i - 1 ( s 1 ) , β i - 1 ( t 1 ) α i α i - 1 ( s 1 ) β i β i - 1 ( t 1 ) d t 1 d s 1 ,
(3.5)

where a change of variables s1 = α i (s), t1 = β i (t),i = 1,2,...,n are made. Clearly, the inequality (3.5) is in the form of (1.1). Thus the estimate (3.3) of the solution z(x,y) in this corollary is obtained immediately by our Theorem 1.

Our Corollary 1 actually gives a condition of boundedness for solutions. Concretely, if
ψ a 1 ( x ) + ψ a 2 ( y ) < , i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) f i α i - 1 ( s ) , β i - 1 ( t ) α i α i - 1 ( s ) β i β i - 1 ( t ) d t d s < , i = 1 n α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) g i α i - 1 ( s ) , β i - 1 ( t ) α i α i - 1 ( s ) β i β i - 1 ( t ) d t d s < ,

on [x0,X1) × [y0,Y1), then every solution z(x,y) of BVP (3.1) is bounded on [x0,X1) × [y0,Y1).

Next, we discuss the uniqueness of solutions for BVP (3.1).

Corollary 2. Consider BVP (3.1) and suppose that F C(I × J × n ,) satisfies
F ( x , y , u 1 , u 2 , , u n ) - F ( x , y , v 1 , v 2 , , v n ) i = 1 n f i ( x , y ) ψ ( u i ) - ψ ( v i ) ,
(3.6)

for all (x,y) I × J and u i , v i , i = 1, 2,..., n, where I = [x0, x1], J = [y0, y1] are two finite intervals, and f i C(I × J, +),i = 1,2,...,n. Then BVP (3.1) has at most one solution on I × J.

Proof. Assume that both z(x,y) and z ̃ ( x , y ) are solutions of BVP (3.1). From the equivalent integral Equations (3.4) and (3.6), we have
ψ ( z ( x , y ) ) - ψ ( z ̃ ( x , y ) ) x 0 x y 0 y F ( s , t , z ( α 1 ( s ) , β 1 ( t ) ) , z ( α 2 ( s ) , β 2 ( t ) ) , , z ( α n ( s ) , β n ( t ) ) ) - F ( s , t , z ̃ ( α 1 ( s ) , β 1 ( t ) ) , z ̃ ( α 2 ( s ) , β 2 ( t ) ) , , z ̃ ( α n ( s ) , β n ( t ) ) ) d t d s
x 0 x y 0 y i = 1 n f i ( s , t ) ψ ( z ( α i ( s ) , β i ( t ) ) ) - ψ ( z ̃ ( α i ( s ) , β i ( t ) ) ) d t d s ε + α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) i = 1 n f i ( α i - 1 ( s 1 ) , β i - 1 ( t 1 ) ) ψ ( z ( s 1 , t 1 ) ) - ψ ( z ̃ ( s 1 , t 1 ) ) α i ( α i - 1 ( s 1 ) ) β i ( β i - 1 ( t 1 ) ) d t 1 d s 1 ,
(3.7)
for all (x,y) I × J, where changes of variables s1 = α i (s), t1 = β i (t) are made, ε > 0 is an arbitrary small number. Clearly, the inequality (3.7) is in the form of (1.1). Suitably applying our Theorem 1 to (3.7), we get an estimate of the form (2.1) for all (x,y) I × J,
ψ ( z ( x , y ) ) - ψ ( z ̃ ( x , y ) ) ε exp α i ( x 0 ) α i ( x ) β i ( y 0 ) β i ( y ) i = 1 n f i α i - 1 ( s ) , β i - 1 ( t ) α i α i - 1 ( s ) β i β i - 1 ( t ) d t d s .
(3.8)

Letting ε → 0+, since α i ( x 0 ) α i ( x ) β i ( y