2 Main result
Throughout this article, x0, x1, y0, y1 ∈ ℝ are given numbers. I := [x0,x1), J := [y0,y1), Δ:= [x0,x1) × [y0,y1), ℝ+ := [0,∞). Consider (1.1)-(1.3), and suppose that
(H1) ψ ∈ C(ℝ+, ℝ+) 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(ℝ+,ℝ+) are nondecreasing with w(0) > 0, ϕ(r) > 0, ϕ1(r) > 0 and ϕ2(r) > 0 for r > 0;
(H4) αi ∈ C1(I,I) and βi ∈ C1(J,J) are nondecreasing such that αi(x) ≤ x, αi(x0) = x0, βi(y) ≤ y and βi(y0) = y0, i = 1, 2,..., n;
(H5) fi, gi ∈ C(Δ,ℝ+), 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
)
)
)
)
,
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
)
,
Φ
(
r
)
:
=
∫
1
r
d
s
φ
(
ψ
-
1
(
W
-
1
(
s
)
)
)
,
r
>
0
,
Φ
(
0
)
:
=
lim
r
→
0
+
Φ
(
r
)
,
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
,
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
,
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
ℛ
:
=
{
(
x
,
y
)
∈
Δ
:
B
(
x
,
y
)
∈
Dom
(
Φ
-
1
)
,
Φ
-
1
(
B
(
x
,
y
)
)
∈
Dom
(
W
-
1
)
}
.
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
,
for all (x,y) ∈ [x0,X] × [y0,y1), where x0 ≤ X ≤ X1 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
)
)
,
for all (x,y) ∈ [x0,X] × [y0,y1). By the fact that αi(x) ≤ x for x ∈ [x0,x1), βi(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
.
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
,
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
,
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
.
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
)
)
,
for all (x,y) ∈ [x0,X] × [y0,Y1), where Y1 is defined in (2.6). By the fact that αi(x) ≤ x for x ∈ [x0,x1), βi(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
.
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
.
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
,
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
,
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
,
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
,
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
)
,
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
)
)
,
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
)
,
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
ℛ
1
:
=
(
x
,
y
)
∈
Δ
:
D
1
(
x
,
y
)
∈
Dom
Ψ
1
-
1
,
Ψ
1
-
1
(
D
1
(
x
,
y
)
)
∈
Dom
(
W
-
1
)
,
and (X3,Y3) is arbitrarily given on the boundary of the planar region
ℛ
2
:
=
(
x
,
y
)
∈
Δ
:
D
2
(
x
,
y
)
∈
Dom
Ψ
2
-
1
,
Ψ
2
-
1
(
D
2
(
x
,
y
)
)
∈
Dom
(
W
-
1
)
.
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
,
for all (x,y) ∈ [x0,X] × [y0,y1), where x0 ≤ X ≤ X2 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 αi(x) ≤ x for x ∈ [x0, x1), βi(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
,
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
,
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
,
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
,
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
,
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
,
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
,
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
.
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
,
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
.
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
,
for all (x,y) ∈ [x0,X] × [y0,y1), where x0 ≤ X ≤ X3. 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
.
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
ℝ
+
3
,
ℝ
+
satisfy
0
≤
L
(
s
,
t
,
u
)
-
L
(
s
,
t
,
v
)
≤
M
(
s
,
t
,
v
)
(
u
-
v
)
,
for s, t, u, v ∈ ℝ+ 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
,
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
)
,
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
ℛ
:
=
{
(
x
,
y
)
∈
Δ
:
E
(
x
,
y
)
∈
Dom
(
Ψ
3
-
1
)
,
Ψ
3
-
1
(
E
(
x
,
y
)
)
∈
Dom
(
W
-
1
)
}
.
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
,
for all (x,y) ∈ [x0,X] × [y0,y1), where x0 ≤ X ≤ X4 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
.
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
,
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
,
for x ∈ I,y ∈ J, where x0,y0,x1,y1 ∈ ℝ+ are constants, I := [x0,x1), J := [y0,y1), F ∈ C(I × J × ℝn,ℝ), ψ: ℝ → ℝ is strictly increasing on ℝ+ with ψ(0) = 0, |ψ(r)| = ψ(|r|) > 0, for |r| > 0 and ψ(t) → ∞ as t → ∞; functions αi ∈ C1(I,I);βi ∈ C1(J,J),i = 1,2,...,n are nondecreasing such that αi(x) ≤ x, βi(y) ≤ y,αi(x0) = x0, βi(y0) = y0; |a1| ∈ C1(I,ℝ+), |a2| ∈ C1(J,ℝ+) 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 × ℝn,ℝ) 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
,
where fi,gi ∈ C(I × J,ℝ+) and w,ϕ ∈ C(ℝ+,ℝ+) 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
)
,
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
.
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
,
where a change of variables s1 = αi (s), t1 = βi(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 × ℝn,ℝ) 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
)
,
for all (x,y) ∈ I × J and ui, vi ∈ ℝ, i = 1, 2,..., n, where I = [x0, x1], J = [y0, y1] are two finite intervals, and fi ∈ C(I × J,ℝ+),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
,
for all (x,y) ∈ I × J, where changes of variables s1 = αi (s), t1 = βi(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
.
Letting ε → 0+, since
∫
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
is finite on finite intervals I and J, ψ is a strictly increasing function, from (3.8), we conclude that
ψ
(
z
(
x
,
y
)
)
-
ψ
(
z
̃
(
x
,
y
)
)
≤
0
, implying that
z
(
x
,
y
)
=
z
̃
(
x
,
y
)
for all (x,y) ∈ I × J. The uniqueness is proved.
Remark Suppose that F ∈ C(I × J × ℝn,ℝ) in BVP (3.1) satisfies
F
(
x
,
y
,
u
1
,
u
2
,
…
,
u
n
)
≤
∑
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
.
By using Theorem 2, we can give an estimate for solutions of the BVP (3.1).
Suppose that F ∈ C(I × J × ℝn,ℝ) in BVP (3.1) satisfies
F
(
x
,
y
,
u
1
,
u
2
,
…
,
u
n
)
≤
∑
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
.
By using Theorem 3, we can give an estimate for solutions of the BVP (3.1) too.