# On Some Vector-Valued Inequalities of Gronwall Type

Authors: Dah-Chin Luor
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

In this paper we established some vector-valued inequalities of Gronwall type in ordered Banach spaces. Our results can be applied to investigate systems of real-valued Gronwall-type inequalities. We also show that the classical Gronwall-Bellman-Bihari integral inequality can be generalized from composition operators to a variety of operators, which include integral operators, maximal operators, geometric mean operators, and geometric maximal operators.


---

## Body

## 1. Introduction

It is well known that the Gronwall-type inequalities play an important role in the study of qualitative properties of solutions to differential equations and integral equations. The Gronwall inequality was established in 1919 by Gronwall [1] and then it was generalized by Bellman [2]. In fact, if
(1)
u
(
t
)
≤
η
+
∫
0
t
‍
g
(
s
)
u
(
s
)
d
s
,
0
≤
t
≤
b
,
where η
≥
0 and u, and g are nonnegative continuous functions on [
0
,
b
], then
(2)
u
(
t
)
≤
η
exp
⁡
(
∫
0
t
‍
g
(
s
)
d
s
)
,
0
≤
t
≤
b
.
This result plays a key role in studying stability and asymptotic behavior of solutions to differential equations and integral equations. One of the important nonlinear generalizations of (1) and (2) was established by Bihari [3]. Assume that 0
≤
k
≤
η
<
C, u and g are nonnegative continuous functions on [
a
,
b
], and r is a positive increasing continuous function on [
0
,
∞
). Bihari showed that if
(3)
u
(
t
)
≤
η
+
∫
a
t
‍
g
(
s
)
r
(
u
(
s
)
)
d
s
,
a
≤
t
≤
b
,
and ∫
a
b
‍
g
(
s
)
d
s
<
∫
η
C
‍
r
(
s
)
-
1
d
s, then
(4)
u
(
t
)
≤
G
-
1
(
G
(
η
)
+
∫
a
t
‍
g
(
s
)
d
s
)
<
C
,
a
≤
t
≤
b
,
where G
(
x
)
=
∫
k
x
‍
r
(
s
)
-
1
d
s, x
≥
k. By choosing r
(
s
)
=
s, inequality (4) can be reduced to the form (2). Many results on the various generalizations of real-valued Gronwall-Bellman-Bihari type inequalities are established. See [4–12], [13, CH.XII], [14–16], and the references given in this literature.Another direction of generalizations is the development of the abstract Gronwall lemma. These results are closely related to the fixed points of operators. See [17, 18], [13, CH.XIV], [19], [20, Proposition 7.15], and the references given in this literature.Inequality (3) can be written in a general form
(5)
u
(
t
)
≤
η
+
∫
a
t
‍
g
(
s
)
A
[
u
]
(
s
)
d
s
,
a
≤
t
≤
b
,
where A is a positive operator on continuous functions. If A is a composition operator defined by A
[
u
]
(
s
)
=
r
(
u
(
s
)
), then (5) is reduced to (3). We show that if A belongs to the class ℱ of operators which is defined in Section 5, then we have an upper estimate of u which is similar to the form (4). It is worth pointing out that the class ℱ includes integral operators
(6)
𝕋
[
f
]
(
t
)
=
∫
a
t
‍
k
(
t
,
s
)
r
(
f
(
s
)
)
d
s
,
maximal operators
(7)
𝕄
[
f
]
(
t
)
=
sup
⁡
a
<
x
<
t
∫
x
t
‍
k
(
t
,
s
)
r
(
f
(
s
)
)
d
s
,
geometric mean operators
(8)
𝔾
[
f
]
(
t
)
=
exp
⁡
(
1
∫
a
t
‍
k
(
t
,
s
)
d
s
∫
a
t
‍
k
(
t
,
s
)
log
⁡
r
(
f
(
s
)
)
d
s
)
,
and geometric maximal operators
(9)
𝒢
[
f
]
(
t
)
=
sup
⁡
a
<
x
<
t
exp
⁡
(
1
∫
x
t
‍
k
(
t
,
s
)
d
s
∫
x
t
‍
k
(
t
,
s
)
log
⁡
r
(
f
(
s
)
)
d
s
)
.
We discuss these operators and the class ℱ in Section 5. We extend the Gronwall-Bellman-Bihari inequality (3) and (4) to the form (5) and the operator A in (5) is generalized from a composition operator to the class ℱ of operators.The aim of this paper is to investigate vector-valued inequalities in ordered Banach spaces. Under suitable conditions we have estimates foru which are similar to (4). By restricting our results to Euclidean spaces we study systems of real-valued Gronwall-type inequalities. For the one-dimensional case, real-valued inequalities of the form (5) with A
∈
ℱ are discussed.Throughout this paper, letℝ
+
=
[
0
,
∞
), I
=
[
a
,
b
], where -
∞
<
a
<
b
<
∞, let Y
1
,
…
,
Y
n be Banach spaces, 𝕐
=
Y
1
×
⋯
×
Y
n; and Y be an ordered Banach space with an order cone K (see [20, Definition 7.1]). We denote by C
(
X
1
,
X
2
) and C
1
(
X
1
,
X
2
) the space of continuous operators and the space of continuously Fréchet differentiable operators, respectively, from X
1 into X
2.


## 2. Preliminaries

Forf
∈
C
(
I
,
Y
), the integral of f on I and the derivative of f at t
0
∈
I, which are denoted by ∫
a
b
‍
f
(
t
)
d
t and f
′
(
t
0
), respectively, can be defined as generalizations of the usual definitions of integral and derivative for real-valued functions. One may see [20, Sections 3.1 and 3.2] for the definitions and properties of the integral and derivative. In particular, if f
′
(
t
) exists for all t
∈
I, where f
′
(
a
) and f
′
(
b
) are defined by one-sided limits, and if f
′ is continuous on I, then f
(
b
)
-
f
(
a
)
=
∫
a
b
‍
f
′
(
t
)
d
t. If we define F
(
t
)
=
∫
a
t
‍
f
(
s
)
d
s for all t
∈
[
a
,
b
], then F
′
(
t
) exists for all t
∈
(
a
,
b
) and F
′
(
t
)
=
f
(
t
). See [20, Propositions 3.5 and 3.7]. By defining
(10)
𝕂
=
{
f
:
f
∈
C
(
I
,
Y
)
,
f
(
t
)
∈
K
∀
t
∈
I
}
⊆
C
(
I
,
Y
)
,
we see that C
(
I
,
Y
) is an ordered Banach space with sup-norm ∥
·
∥
∞ and the order cone 𝕂. If f
≤
g in C
(
I
,
Y
), then ∫
a
b
‍
f
(
t
)
d
t
≤
∫
a
b
‍
g
(
t
)
d
t.By [21, Theorem 7.1.9] we see that, for t
∈
(
a
,
b
), the Fréchet derivative D
ℱ
f
(
t
) exists if and only if f
′
(
t
) exists and
(11)
(
D
ℱ
f
(
t
)
)
(
h
)
=
h
f
′
(
t
)
∀
h
∈
ℝ
.
Here we denote by (
D
ℱ
f
(
t
)
)
(
h
) the value of D
ℱ
f
(
t
) at h.Fory
i
∈
Y
i, i
=
1
,
…
,
n, let y
=
(
y
1
,
…
,
y
n
)
∈
𝕐 and we define ∥
y
∥
𝕐
=
∥
y
1
∥
Y
1
+
⋯
+
∥
y
n
∥
Y
n. Then ∥
·
∥
𝕐 is a norm and 𝕐, with coordinatewise linear operations, is a Banach space. If Y
1
,
…
,
Y
n are ordered Banach spaces with ordered cones K
1
,
…
,
K
n, respectively, then 𝕐 is an ordered Banach space with an order cone K
1
×
⋯
×
K
n. It is easy to see that a sequence of points y
(
m
)
=
(
y
1
(
m
)
,
…
,
y
n
(
m
)
) in 𝕐 converges to a point (
c
1
,
…
,
c
n
)
∈
𝕐 if and only if y
i
(
m
) converges to c
i in Y
i for each i
=
1
,
…
,
n. For f
i
∈
C
(
I
,
Y
i
), i
=
1
,
…
,
n, we define f
(
t
)
=
(
f
1
(
t
)
,
…
,
f
n
(
t
)
). Then f
∈
C
(
I
,
𝕐
). If f
1
′
(
t
)
,
…
,
f
n
′
(
t
) all exist for t
∈
I, then the derivative of f at t exists and  f
′
(
t
)
=
(
f
1
′
(
t
)
,
…
,
f
n
′
(
t
)
).LetX, Y, and Z be Banach spaces, and U, V are nonempty open subsets of X, Y, respectively, and let g
:
U
→
V and f
:
V
→
Z. Suppose that g is continuous and Fréchet differentiable at a point z
∈
U and that f is continuous and  Fréchet differentiable at the point g
(
z
)
∈
V. Then f
∘
g is continuous and Fréchet  differentiable at z, and
(12)
D
ℱ
(
f
∘
g
)
(
z
)
=
D
ℱ
f
(
g
(
z
)
)
∘
D
ℱ
g
(
z
)
.LetX
1, X
2, and Y be Banach spaces, and A is a nonempty open subset of X
1
×
X
2, and let f
:
A
→
Y be given by (
x
1
,
x
2
)
↦
f
(
x
1
,
x
2
). For (
z
1
,
z
2
)
∈
A, let A
1
=
{
x
1
∈
X
1
:
(
x
1
,
z
2
)
∈
A
} and let g
(
x
1
)
=
f
(
x
1
,
z
2
) for all x
1
∈
A
1. It is clear that A
1
⊆
X
1 is open and g
:
A
1
→
Y. If g has a Fréchet derivative at z
1, then we define the partial Fréchet derivative of f at (
z
1
,
z
2
) with respect to the variable x
1 to be D
ℱ
,
1
f
(
z
1
,
z
2
)
=
D
ℱ
g
(
z
1
); it is a linear operator of X
1 into Y. The derivative D
ℱ
,
2
f
(
z
1
,
z
2
) is defined similarly. If f is Fréchet differentiable at (
z
1
,
z
2
), then f is Fréchet differentiable with respect to both variables at (
z
1
,
z
2
) and
(13)
(
D
ℱ
f
(
z
1
,
z
2
)
)
(
(
x
1
,
x
2
)
)
=
(
D
ℱ
,
1
f
(
z
1
,
z
2
)
)
(
x
1
)
+
(
D
ℱ
,
2
f
(
z
1
,
z
2
)
)
(
x
2
)
for all (
x
1
,
x
2
)
∈
X
1
×
X
2. Moreover, f is continuously Fréchet differentiable in a neighborhood of (
z
1
,
z
2
) if and only if all partial  Fréchet derivatives are continuous in a neighborhood of (
z
1
,
z
2
). Similar results hold for maps of the form (
x
1
,
…
,
x
n
)
↦
f
(
x
1
,
…
,
x
n
).Notation. Here we give notations used in this paper for reader’s convenience. u
∈
C
(
I
,
Y
), Φ
:
C
(
I
,
Y
)
→
C
(
I
,
Y
), F
∈
C
1
(
I
×
𝕐
,
Y
), S
i
:
C
(
I
,
Y
)
→
C
1
(
I
,
Y
i
), i
=
1
,
…
,
n, and 𝕊
:
C
(
I
,
Y
)
→
C
1
(
I
,
I
×
𝕐
) is defined by
(14)
𝕊
[
u
]
(
t
)
=
(
t
,
S
1
[
u
]
(
t
)
,
…
,
S
n
[
u
]
(
t
)
)
,
T
i
:
C
(
I
,
Y
)
⟶
C
(
I
,
Y
)
,
i
=
0
,
…
,
n
,
ℓ
∈
C
1
(
Y
,
Y
)
,
U
i
∈
C
(
I
,
Y
)
,
i
=
0
,
…
,
n
.


## 3. Some Vector-Valued Gronwall-Type Inequalities

In this section we consider vector-valued inequalities of the form(15)
Φ
[
u
]
≤
F
∘
𝕊
[
u
]
in
C
(
I
,
Y
)
.
Theorem 1 gives an estimate of u which is similar to the form (4).Theorem 1.
LetL
0
∈
Y. Suppose that F
(
𝕊
[
u
]
(
a
)
)
≤
L
0 and Φ is bijective and is of monotone type, and suppose that there exist monotone increasing operators T
i
:
C
(
I
,
Y
)
→
C
(
I
,
Y
), i
=
0
,
…
,
n, such that for all t
∈
I,
(16)
(
D
ℱ
,
1
F
(
𝕊
[
u
]
(
t
)
)
)
(
1
)
≤
T
0
[
u
]
(
t
)
,
(17)
(
D
ℱ
,
i
+
1
F
(
𝕊
[
u
]
(
t
)
)
)
(
S
i
[
u
]
′
(
t
)
)
≤
T
i
[
u
]
(
t
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
i
=
1
,
…
,
n
.
We also suppose that there exist ℓ
∈
C
1
(
Y
,
Y
) and U
i
∈
C
(
I
,
Y
), i
=
0
,
…
,
n, such that ℓ is bijective and monotone increasing and is of monotone type, and for i
=
0
,
1
,
…
,
n,
(18)
(
D
ℱ
ℓ
(
(
F
∘
𝕊
[
u
]
)
(
t
)
)
)
(
T
i
[
Φ
-
1
[
F
∘
𝕊
[
u
]
]
]
(
t
)
)
≤
U
i
(
t
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
t
∈
I
.
If u satisfies (15), then
(19)
u
≤
Φ
-
1
[
V
]
i
n
C
(
I
,
Y
)
,
where V
∈
C
(
I
,
Y
) is defined by
(20)
V
(
t
)
=
ℓ
-
1
(
ℓ
(
L
0
)
+
∑
i
=
0
n
‍
∫
a
t
‍
U
i
(
y
)
d
y
)
,
t
∈
I
.
Moreover, if there exists M
∈
Y such that ℓ
(
V
(
t
)
)
≤
M for all t
∈
I, then
(21)
u
≤
Φ
-
1
[
ℓ
-
1
(
M
)
]
i
n
C
(
I
,
Y
)
,
where ℓ
-
1
(
M
) in (21) is the constant function in C
(
I
,
Y
) with value ℓ
-
1
(
M
).Proof of Theorem1.
Letα
u
=
F
∘
𝕊
[
u
]
∈
C
1
(
I
,
Y
). By (15) we have Φ
[
u
]
(
t
)
≤
α
u
(
t
) for all t
∈
I. By the chain rule we see that, for t
∈
(
a
,
b
) and h
∈
ℝ,
(22)
h
α
u
′
(
t
)
=
(
D
ℱ
α
u
(
t
)
)
(
h
)
=
(
D
ℱ
(
F
∘
𝕊
[
u
]
)
(
t
)
)
(
h
)
=
(
D
ℱ
F
(
𝕊
[
u
]
(
t
)
)
∘
D
ℱ
𝕊
[
u
]
(
t
)
)
(
h
)
.
Since 𝕊
[
u
]
∈
C
1
(
I
,
I
×
𝕐
), (
D
ℱ
𝕊
[
u
]
(
t
)
)
(
h
)
=
h
𝕊
[
u
]
′
(
t
) and we see that
(23)
h
α
u
′
(
t
)
=
(
D
ℱ
F
(
𝕊
[
u
]
(
t
)
)
)
(
h
𝕊
[
u
]
′
(
t
)
)
=
(
D
ℱ
,
1
F
(
𝕊
[
u
]
(
t
)
)
)
(
h
)
+
(
D
ℱ
,
2
F
(
𝕊
[
u
]
(
t
)
)
)
(
h
S
1
[
u
]
′
(
t
)
)
+
⋯
+
(
D
ℱ
,
n
+
1
F
(
𝕊
[
u
]
(
t
)
)
)
(
h
S
n
[
u
]
′
(
t
)
)
=
h
(
D
ℱ
,
1
F
(
𝕊
[
u
]
(
t
)
)
)
(
1
)
+
h
(
D
ℱ
,
2
F
(
𝕊
[
u
]
(
t
)
)
)
(
S
1
[
u
]
′
(
t
)
)
+
⋯
+
h
(
D
ℱ
,
n
+
1
F
(
𝕊
[
u
]
(
t
)
)
)
(
S
n
[
u
]
′
(
t
)
)
.
This implies
(24)
α
u
′
(
t
)
=
(
D
ℱ
,
1
F
(
𝕊
[
u
]
(
t
)
)
)
(
1
)
+
(
D
ℱ
,
2
F
(
𝕊
[
u
]
(
t
)
)
)
(
S
1
[
u
]
′
(
t
)
)
+
⋯
+
(
D
ℱ
,
n
+
1
F
(
𝕊
[
u
]
(
t
)
)
)
(
S
n
[
u
]
′
(
t
)
)
.
Since Φ is bijective, we write Φ
-
1
[
α
u
] to be the solution of the equation Φ
[
x
]
=
α
u. By Φ
[
u
]
≤
α
u in C
(
I
,
Y
) and [20, Proposition 7.37] we see that u
≤
Φ
-
1
[
α
u
]. This shows that T
i
[
u
]
≤
T
i
[
Φ
-
1
[
α
u
]
] for i
=
0,1
,
…
,
n and hence
(25)
α
u
′
(
t
)
≤
∑
i
=
0
n
‍
T
i
[
u
]
(
t
)
≤
∑
i
=
0
n
‍
T
i
[
Φ
-
1
[
α
u
]
]
(
t
)
.
Since ℓ is monotone increasing, we see that D
ℱ
ℓ
(
y
) is positive and hence monotone increasing for each y
∈
Y. Therefore for h
>
0,
(26)
h
(
ℓ
∘
α
u
)
′
(
t
)
=
(
D
ℱ
(
ℓ
∘
α
u
)
(
t
)
)
(
h
)
=
(
D
ℱ
ℓ
(
α
u
(
t
)
)
∘
D
ℱ
α
u
(
t
)
)
(
h
)
=
(
D
ℱ
ℓ
(
α
u
(
t
)
)
)
(
h
α
u
′
(
t
)
)
≤
h
(
D
ℱ
ℓ
(
α
u
(
t
)
)
)
(
∑
i
=
0
n
‍
T
i
[
Φ
-
1
[
α
u
]
]
(
t
)
)
=
h
∑
i
=
0
n
‍
(
D
ℱ
ℓ
(
α
u
(
t
)
)
)
(
T
i
[
Φ
-
1
[
α
u
]
]
(
t
)
)
≤
h
∑
i
=
0
n
‍
U
i
(
t
)
.
For a
<
d
<
t
<
b we obtain
(27)
(
ℓ
∘
α
u
)
(
t
)
-
(
ℓ
∘
α
u
)
(
d
)
≤
∑
i
=
0
n
‍
∫
d
t
‍
U
i
(
y
)
d
y
.
By letting d
→
a and the condition α
u
(
a
)
≤
L
0,
(28)
ℓ
(
α
u
(
t
)
)
≤
ℓ
(
L
0
)
+
∑
i
=
0
n
‍
∫
a
t
‍
U
i
(
y
)
d
y
=
ℓ
(
V
(
t
)
)
for all t
∈
(
a
,
b
). Since ℓ
∘
α
u and ℓ
∘
V are continuous on I, inequality (28) holds for all t
∈
I. This implies α
u
(
t
)
≤
V
(
t
) for all t
∈
I. Therefore Φ
[
u
]
≤
α
u
≤
V and u
≤
Φ
-
1
[
V
] in C
(
I
,
Y
).
Sinceℓ
-
1 and Φ
-
1 are monotone increasing, it follows that if ℓ
(
V
(
t
)
)
≤
M for all t
∈
I then V
≤
ℓ
-
1
(
M
) in C
(
I
,
Y
) and we obtain (21). This completes the proof.In the following we consider two particular cases as examples of Theorem1. We see that under these cases, conditions in Theorem 1 can be reduced to more simpler forms.In the caseY
1
=
Y
2
=
⋯
=
Y
n
=
Y and F
(
t
,
y
1
,
…
,
y
)
=
y
1
+
⋯
+
y
n, we see that F
∘
𝕊
[
u
]
∈
C
1
(
I
,
Y
) is given by
(29)
F
∘
𝕊
[
u
]
(
t
)
=
F
(
𝕊
[
u
]
(
t
)
)
=
∑
i
=
1
n
‍
S
i
[
u
]
(
t
)
,
t
∈
I
.
Moreover, D
ℱ
,
1
F
(
·
) is the zero operator of I into Y and D
ℱ
,
i
+
1
F
(
·
) is the identity operator of Y into Y. We have the following corollary.Corollary 2.
LetL
0
∈
Y and let u, Φ, ℓ, S
i, T
i, U
i, i
=
1
,
…
,
n, and 𝕊 be given as in Theorem 1 with Y
1
=
Y
2
=
⋯
=
Y
n
=
Y. Suppose that ∑
i
=
1
n
‍
S
i
[
u
]
(
a
)
≤
L
0, and
(30)
S
i
[
u
]
′
(
t
)
≤
T
i
[
u
]
(
t
)
,
(31)
(
D
ℱ
ℓ
(
∑
j
=
1
n
‍
S
j
[
u
]
(
t
)
)
)
(
T
i
[
Φ
-
1
[
∑
j
=
1
n
‍
S
j
[
u
]
]
]
(
t
)
)
≤
U
i
(
t
)
for each i
=
1
,
…
,
n and for all t
∈
I. If u satisfies
(32)
Φ
[
u
]
≤
∑
i
=
1
n
‍
S
i
[
u
]
i
n
C
(
I
,
Y
)
,
then we have (19)–(21).Remark 3.
IfK
i
:
C
(
I
,
Y
)
→
C
(
I
,
Y
) and S
i
[
u
]
(
t
)
=
∫
a
t
‍
K
i
[
u
]
(
z
)
d
z for each i
=
1
,
…
,
n, then (32) is reduced to the following integral inequality:
(33)
Φ
[
u
]
(
t
)
≤
∑
i
=
1
n
‍
∫
a
t
‍
K
i
[
u
]
(
z
)
d
z
,
t
∈
I
.
By [20, Proposition 3.7] the item S
i
[
u
]
′
(
t
) in (17) and (30) can be replaced by K
i
[
u
]
(
t
). Moreover, if K
i is monotone increasing for each i, then we can choose T
i
=
K
i, i
=
1
,
…
,
n, in Corollary 2 and condition (30) is redundant.Consider the case thatY
1
,
…
,
Y
n are ordered Banach spaces, Y
=
𝕐, and F
(
t
,
y
1
,
…
,
y
n
)
=
(
y
1
,
…
,
y
n
). We see that F
∘
𝕊
[
u
]
∈
C
1
(
I
,
𝕐
) is given by
(34)
(
F
∘
𝕊
[
u
]
)
(
t
)
=
F
(
𝕊
[
u
]
(
t
)
)
=
(
S
1
[
u
]
(
t
)
,
…
,
S
n
[
u
]
(
t
)
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
t
∈
I
.
Moreover, D
ℱ
,
1
F
(
·
) is the zero operator of I into 𝕐 and D
ℱ
,
i
+
1
F
(
·
) is the operator of Y
i into 𝕐 such that for h
∈
Y
i, the ith element of (
D
ℱ
,
i
+
1
F
(
·
)
)
(
h
) is h and the other elements are zero.Remark 4.
LetΦ
i
:
C
(
I
,
𝕐
)
→
C
(
I
,
Y
i
), i
=
1
,
…
,
n, and Φ
:
C
(
I
,
𝕐
)
→
C
(
I
,
𝕐
) is defined by Φ
[
u
]
(
t
)
=
(
Φ
1
[
u
]
(
t
)
,
…
,
Φ
n
[
u
]
(
t
)
) for t
∈
I. Suppose that each Φ
i is injective and is of monotone type. Suppose that for any w
=
(
w
1
,
…
,
w
n
)
∈
C
(
I
,
𝕐
), where w
i
∈
C
(
I
,
Y
i
), there exist v
∈
C
(
I
,
𝕐
) such that Φ
i
[
v
]
=
w
i for each i
=
1
,
…
,
n. Then Φ is bijective and is of monotone type.The following corollary can be obtained by Theorem1.Corollary 5.
LetL
0
=
(
L
1
,
…
,
L
n
)
∈
𝕐 and let u, ℓ, S
i, i
=
1
,
…
,
n, and 𝕊 be given as in Theorem 1 with Y
=
𝕐. Let Φ
1
,
…
,
Φ
n, Φ be given as in Remark 4. Suppose that S
i
[
u
]
(
a
)
≤
L
i, i
=
1
,
…
,
n, and there exist U
i
∈
C
(
I
,
𝕐
) and monotone increasing operators G
i
:
C
(
I
,
𝕐
)
→
C
(
I
,
Y
i
), i
=
1
,
…
,
n, such that
(35)
S
i
[
u
]
′
(
t
)
≤
G
i
[
u
]
(
t
)
,
(36)
(
D
ℱ
ℓ
(
(
F
∘
𝕊
[
u
]
)
(
t
)
)
)
(
T
i
[
Φ
-
1
[
F
∘
𝕊
[
u
]
]
]
(
t
)
)
≤
U
i
(
t
)
,
for i
=
1
,
…
,
n and for all t
∈
I, where T
i
[
·
]
(
t
)
∈
𝕐 and the ith element of T
i
[
·
]
(
t
) is G
i
[
·
]
(
t
) and the other elements are zero. If u satisfies the systems
(37)
Φ
i
[
u
]
(
t
)
≤
S
i
[
u
]
(
t
)
,
i
=
1
,
…
,
n
,
for all t
∈
I, then we have (19)–(21).Remark 6.
IfK
i
:
C
(
I
,
𝕐
)
→
C
(
I
,
Y
i
) and S
i
[
u
]
(
t
)
=
∫
a
t
‍
K
i
[
u
]
(
z
)
d
z for each i
=
1
,
…
,
n, then (37) is reduced to the system of integral inequalities
(38)
Φ
i
[
u
]
(
t
)
≤
∫
a
t
‍
K
i
[
u
]
(
z
)
d
z
,
t
∈
I
,
i
=
1
,
…
,
n
.
The item S
i
[
u
]
′
(
t
) in (35) can be replaced by K
i
[
u
]
(
t
). Moreover, if K
i is monotone increasing for each i, then we can choose G
i
=
K
i, i
=
1
,
…
,
n, in Corollary 5 and condition (35) is redundant.


## 4. Systems of Real-Valued Gronwall-Type Inequalities

In this section we apply results in Section3 to obtain systems of real-valued Gronwall-type inequalities. Consider the case Y
=
ℝ
m, Y
i
=
ℝ, i
=
1
,
…
,
n, 𝕐
=
ℝ
n, and K
=
{
(
x
1
,
…
,
x
m
)
∈
ℝ
m
:
x
i
≥
0
,
i
=
1
,
…
,
m
}. Let u
i
∈
C
(
I
,
ℝ
), Φ
i
:
C
(
I
,
ℝ
m
)
→
C
(
I
,
ℝ
), F
i
∈
C
1
(
I
×
ℝ
n
,
ℝ
), ℓ
i
∈
C
1
(
ℝ
m
,
ℝ
), i
=
1
,
…
,
m, and u
∈
C
(
I
,
ℝ
m
), u
(
t
)
=
(
u
1
(
t
)
,
…
,
u
m
(
t
)
), F
∈
C
1
(
I
×
ℝ
n
,
ℝ
m
), F
=
(
F
1
,
…
,
F
m
). Let S
i
:
C
(
I
,
ℝ
m
)
→
C
1
(
I
,
ℝ
), i
=
1
,
…
,
n.DefineΦ
:
C
(
I
,
ℝ
m
)
→
C
(
I
,
ℝ
m
) by
(39)
Φ
[
u
]
(
t
)
=
(
Φ
1
[
u
]
(
t
)
,
…
,
Φ
m
[
u
]
(
t
)
)
,
t
∈
I
.
Then (15) can be reduced to the system
(40)
Φ
i
[
u
]
(
t
)
≤
(
F
i
∘
𝕊
[
u
]
)
(
t
)
,
t
∈
I
,
i
=
1
,
…
,
m
.Remark 7.
Suppose thatΦ
i is injective and is of monotone type for each i
=
1
,
…
,
m. Suppose that for any w
=
(
w
1
,
…
,
w
m
)
∈
C
(
I
,
ℝ
m
), where w
i
∈
C
(
I
,
ℝ
), there exist υ
∈
C
(
I
,
ℝ
m
) such that Φ
i
[
υ
]
=
w
i for each i
=
1
,
…
,
m. Then Φ is bijective and is of monotone type.Since(41)
(
D
ℱ
,
1
F
(
t
,
y
1
,
…
,
y
n
)
)
(
1
)
=
∂
F
∂
t
(
t
,
y
1
,
…
,
y
n
)
,
and, for i
=
1
,
…
,
n, the linear transform D
ℱ
,
i
+
1
F
(
t
,
y
1
,
…
,
y
n
)
:
ℝ
→
ℝ
m can be represented by the m
×
1 matrix [
(
∂
F
k
/
∂
y
i
)
(
t
,
y
1
,
…
,
y
n
)
], conditions (16)-(17) are reduced to
(42)
∂
F
∂
t
(
𝕊
[
u
]
(
t
)
)
≤
T
0
[
u
]
(
t
)
,
(43)
[
∂
F
k
∂
y
i
(
𝕊
[
u
]
(
t
)
)
]
S
i
[
u
]
′
(
t
)
≤
T
i
[
u
]
(
t
)
,
i
=
1
,
…
,
n
,
where T
i
:
C
(
I
,
ℝ
m
)
→
C
(
I
,
ℝ
m
), i
=
0
,
…
,
n, are monotone increasing operators. Here the vector T
i
[
u
]
(
t
) in (43) is written as a column matrix. In particular, if m
=
1, then F
∈
C
1
(
I
×
ℝ
n
,
ℝ
) and (43) is reduced to
(44)
∂
F
∂
y
i
(
𝕊
[
u
]
(
t
)
)
·
S
i
[
u
]
′
(
t
)
≤
T
i
[
u
]
(
t
)
,
i
=
1
,
…
,
n
.Remark 8.
Defineℓ
∈
C
1
(
ℝ
m
,
ℝ
m
) by ℓ
=
(
ℓ
1
,
…
,
ℓ
m
). Suppose that ℓ
k is injective and monotone increasing and is of monotone type for each k, and, for any z
=
(
z
1
,
…
,
z
m
)
∈
ℝ
m, there exist y
∈
ℝ
m such that ℓ
k
(
y
)
=
z
k for each k
=
1
,
…
,
m. Then ℓ is bijective and monotone increasing and is of monotone type. Moreover, the linear transform D
ℱ
ℓ
(
ξ
1
,
…
,
ξ
m
) can be represented by the m
×
m Jacobian matrix [
D
j
ℓ
k
(
ξ
1
,
…
,
ξ
m
)
], 1
≤
j, k
≤
m. Hence (18) is reduced to
(45)
[
D
j
ℓ
k
(
(
F
∘
𝕊
[
u
]
)
(
t
)
)
]
T
i
[
Φ
-
1
[
F
∘
𝕊
[
u
]
]
]
(
t
)
≤
U
i
(
t
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
i
=
0
,
…
,
n
,
where U
i
∈
C
(
I
,
ℝ
m
). Here the vectors T
i
[
Φ
-
1
[
F
∘
𝕊
[
u
]
]
]
(
t
) and U
i
(
t
) in (45) are written as column matrices.Theorem 9.
LetL
0
∈
ℝ
m and the conditions of Φ
1
,
…
,
Φ
m, Φ be given in Remark 7. Suppose that F
(
𝕊
[
u
]
(
a
)
)
≤
L
0, and there exist monotone increasing operators T
i
:
C
(
I
,
ℝ
m
)
→
C
(
I
,
ℝ
m
), i
=
0
,
…
,
n, such that (42)-(43) are satisfied. We also suppose that conditions for ℓ
1
,
…
,
ℓ
m,  ℓ in Remark 8 hold and (45) is satisfied. If u satisfies the system (40), then one has (19)–(21) with Y
=
ℝ
m.LetΨ
i
:
C
(
I
,
ℝ
)
→
C
(
I
,
ℝ
), i
=
1
,
…
,
m. If we define Φ
:
C
(
I
,
ℝ
m
)
→
C
(
I
,
ℝ
m
) by
(46)
Φ
[
u
]
(
t
)
=
(
Ψ
1
[
u
1
]
(
t
)
,
…
,
Ψ
m
[
u
m
]
(
t
)
)
,
t
∈
I
,
then (15) can be reduced to the system
(47)
Ψ
i
[
u
i
]
(
t
)
≤
(
F
i
∘
𝕊
[
u
]
)
(
t
)
,
t
∈
I
,
i
=
1
,
…
,
m
.Remark 10.
Suppose thatΨ
i is bijective and is of monotone type for each i
=
1
,
…
,
m. Then Φ defined by (46) is bijective and is of monotone type, and
(48)
Φ
-
1
[
f
]
(
t
)
=
(
Ψ
1
-
1
[
f
1
]
(
t
)
,
…
,
Ψ
m
-
1
[
f
m
]
(
t
)
)
∀
t
∈
I
,
where f
=
(
f
1
,
…
,
f
m
)
∈
C
(
I
,
ℝ
m
). In particular, the function Φ
-
1
[
F
∘
𝕊
[
u
]
]
∈
C
(
I
,
ℝ
m
) is given by
(49)
Φ
-
1
[
F
∘
𝕊
[
u
]
]
(
t
)
=
(
Ψ
1
-
1
[
F
1
∘
𝕊
[
u
]
]
(
t
)
,
…
,
Ψ
m
-
1
[
F
m
∘
𝕊
[
u
]
]
(
t
)
)
2222222222222222222222222222222222
∀
t
∈
I
.Example 11.
DefineΨ
j
[
u
j
]
(
t
)
=
ψ
j
(
u
j
(
t
)
), where ψ
j
∈
C
(
ℝ
,
ℝ
). If ψ
j is strictly increasing from ℝ onto  ℝ, then Ψ
j is bijective and is of monotone type, and (48) can be reduced to
(50)
Φ
-
1
[
f
]
(
t
)
=
(
ψ
1
-
1
(
f
1
(
t
)
)
,
…
,
ψ
m
-
1
(
f
m
(
t
)
)
)
∀
t
∈
I
,
where f
=
(
f
1
,
…
,
f
m
)
∈
C
(
I
,
ℝ
m
).Remark 12.
Letl
i
∈
C
1
(
ℝ
,
ℝ
), i
=
1
,
…
,
m. Define ℓ
∈
C
1
(
ℝ
m
,
ℝ
m
) by
(51)
ℓ
(
ξ
1
,
…
,
ξ
m
)
=
(
l
1
(
ξ
1
)
,
…
,
l
m
(
ξ
m
)
)
.
Suppose that l
i is strictly increasing from ℝ  onto ℝ for i
=
1
,
…
,
m; then it is easy to see that ℓ is bijective and monotone increasing and is of monotone type, and
(52)
ℓ
-
1
(
ξ
1
,
…
,
ξ
m
)
=
(
l
1
-
1
(
ξ
1
)
,
…
,
l
m
-
1
(
ξ
m
)
)
.
Moreover, the Jacobian matrix [
D
j
l
k
(
ξ
1
,
…
,
ξ
m
)
] is a diagonal matrix with diagonal entries l
k
′
(
ξ
k
), k
=
1
,
…
,
m. Hence D
ℱ
ℓ
(
(
F
∘
𝕊
[
u
]
)
(
t
)
) in (18) can be represented by the m
×
m diagonal matrix with diagonal entries l
k
′
(
(
F
k
∘
𝕊
[
u
]
)
(
t
)
), k
=
1
,
…
,
m.The following theorem can be obtained by Theorem1.Theorem 13.
LetL
1
,
…
,
L
m
∈
ℝ and the conditions of Ψ
1
,
…
,
Ψ
m, Φ be given as in Remark 10. Suppose that F
j
(
𝕊
[
u
]
(
a
)
)
≤
L
j, j
=
1
,
…
,
m, and there exist monotone increasing operators T
i
:
C
(
I
,
ℝ
m
)
→
C
(
I
,
ℝ
m
), i
=
0
,
…
,
n, such that (42)-(43) are satisfied. We also suppose that there exist l
j
∈
C
1
(
ℝ
,
ℝ
), j
=
1
,
…
,
m, and U
i
j
∈
C
(
I
,
ℝ
), i
=
0
,
…
,
n, j
=
1
,
…
,
m, such that conditions for l
j and ℓ in Remark 12 hold, and (18) with U
i
=
(
U
i
1
,
…
,
U
i
m
) are satisfied. If u satisfies (47), then we have
(53)
u
j
(
t
)
≤
Ψ
j
-
1
[
V
j
]
(
t
)
,
t
∈
I
,
j
=
1
,
…
,
m
,
where
(54)
V
j
(
t
)
=
l
j
-
1
(
l
j
(
L
j
)
+
∑
i
=
0
n
‍
∫
a
t
‍
U
i
j
(
y
)
d
y
)
,
t
∈
I
.
Moreover, if there exists M
1
,
…
,
M
m
∈
ℝ such that l
j
(
V
j
(
t
)
)
≤
M
j for all t
∈
I, j
=
1
,
…
,
m, then
(55)
u
j
(
t
)
≤
Ψ
j
-
1
[
l
j
-
1
(
M
j
)
]
∀
t
∈
I
,
j
=
1
,
…
,
m
,
where l
j
-
1
(
M
j
) in (55) is the constant function in C
(
I
,
ℝ
) with value l
j
-
1
(
M
j
).Consider the particular casem
=
n and F
j
(
t
,
y
1
,
…
,
y
m
)
=
y
j, j
=
1
,
…
,
m of Theorem 13. In this case (
F
i
∘
𝕊
[
u
]
)
(
t
)
=
S
i
[
u
]
(
t
) and (47) is reduced to
(56)
Ψ
i
[
u
i
]
(
t
)
≤
S
i
[
u
]
(
t
)
,
t
∈
I
,
i
=
1
,
…
,
m
.
Assume that
(57)
S
i
[
u
]
(
a
)
≤
L
i
,
i
=
1
,
…
,
m
.
We also assume that there exist monotone increasing operators G
i
:
C
(
I
,
ℝ
m
)
→
C
(
I
,
ℝ
), i
=
1
,
…
,
m, such that
(58)
S
i
[
u
]
′
(
t
)
≤
G
i
[
u
]
(
t
)
,
i
=
1
,
…
,
m
.
We also suppose that there exist l
i
∈
C
1
(
ℝ
,
ℝ
) and W
i
∈
C
(
I
,
ℝ
), i
=
1
,
…
,
m, such that conditions for l
i in Remark 12 hold and satisfy the following condition:
(59)
l
i
′
(
S
i
[
u
]
(
t
)
)
G
i
[
Φ
-
1
[
F
∘
𝕊
[
u
]
]
]
(
t
)
≤
W
i
(
t
)
,
i
=
1
,
…
,
m
,
where
(60)
Φ
-
1
[
F
∘
𝕊
[
u
]
]
(
t
)
=
(
Ψ
1
-
1
[
S
1
[
u
]
]
(
t
)
,
…
,
Ψ
m
-
1
[
S
m
[
u
]
]
(
t
)
)
22222222222222222222222222222222222222
∀
t
∈
I
.
Then Theorem 13 implies that if u satisfies (56) then we have (53) with
(61)
V
j
(
t
)
=
l
j
-
1
(
l
j
(
L
j
)
+
∫
a
t
‍
W
j
(
y
)
d
y
)
,
t
∈
I
.Remark 14.
Condition (59) can be satisfied if there exist r
i
∈
C
(
ℝ
,
(
0
,
∞
)
), i
=
1
,
…
,
m, such that
(62)
G
i
[
Φ
-
1
[
F
∘
𝕊
[
u
]
]
]
(
t
)
≤
W
i
(
t
)
r
i
(
S
i
[
u
]
(
t
)
)
,
t
∈
I
,
and we choose
(63)
l
i
(
t
)
=
∫
c
i
t
‍
1
r
i
(
s
)
d
s
,
i
=
1
,
…
,
m
,
where c
i is some constant. Moreover, V
j defined by (61) satisfies l
j
(
V
j
(
t
)
)
≤
M
j, where
(64)
M
j
=
max
⁡
a
≤
t
≤
b
{
l
j
(
L
j
)
+
∫
a
t
‍
W
j
(
y
)
d
y
}
,
j
=
1
,
…
,
m
.By a little modification of Theorem13 we have Corollary 15.Corollary 15.
LetL
1
,
…
,
L
m
∈
ℝ and the conditions of Ψ
1
,
…
,
Ψ
m be given as in Remark 10. Suppose that (57) holds, and there exist monotone increasing operators G
i
:
C
(
I
,
ℝ
m
)
→
C
(
I
,
ℝ
), i
=
1
,
…
,
m, such that (58) is satisfied. For i
=
1
,
…
,
m, we suppose that there exist W
i
∈
C
(
I
,
ℝ
) and r
i
∈
C
(
ℝ
,
(
0
,
∞
)
) such that (62) holds. Define l
i by (63). Assume that l
i
(
-
∞
)
<
l
i
(
V
i
(
t
)
)
<
l
i
(
∞
) for all t
∈
I, where V
i is defined by (61). If u satisfies (56), then we have (53) and (55) with (64).As an example we consider the system of integral inequalities(65)
ψ
j
(
u
j
(
t
)
)
≤
L
j
+
∫
a
t
‍
g
j
(
s
)
A
j
[
u
]
(
s
)
d
s
,
22222222222222
t
∈
I
,
j
=
1
,
…
,
m
.
Here ψ
j
∈
C
(
ℝ
,
ℝ
) is strictly increasing from ℝ onto ℝ, L
j
∈
ℝ, g
j
∈
C
(
I
,
ℝ
+
), and A
j
:
C
(
I
,
ℝ
m
)
→
C
(
I
,
ℝ
). Assume that for each j
=
1
,
…
,
m, A
j is monotone increasing and there exist w
j
∈
C
(
I
,
ℝ
) and r
j
∈
C
(
ℝ
,
(
0
,
∞
)
) such that
(66)
A
j
[
(
ψ
1
-
1
∘
f
1
,
…
,
ψ
m
-
1
∘
f
m
)
]
(
t
)
≤
w
j
(
t
)
r
j
(
f
j
(
t
)
)
,
222222222222222222222222222222222222
t
∈
I
,
for all f
1
,
…
,
f
m
∈
C
(
I
,
ℝ
). Moreover, we suppose that l
j
(
-
∞
)
<
l
j
(
L
j
)
+
∫
a
t
‍
g
j
(
y
)
w
j
(
y
)
d
y
<
l
j
(
∞
) for all t
∈
I, where l
j is given in (63). By Example 11 and Corollary 15 with Ψ
j
[
u
j
]
(
t
)
=
ψ
j
(
u
j
(
t
)
), S
j
[
u
]
(
t
)
=
L
j
+
∫
a
t
‍
g
j
(
s
)
A
j
[
u
]
(
s
)
d
s, G
j
[
u
]
(
t
)
=
g
j
(
t
)
A
j
[
u
]
(
t
), and W
j
(
t
)
=
g
j
(
t
)
w
j
(
t
), we have
(67)
u
j
(
t
)
≤
ψ
j
-
1
(
l
j
-
1
(
l
j
(
L
j
)
+
∫
a
t
g
j
(
y
)
w
j
(
y
)
d
y
)
‍
)
,
22222222222222222222222
t
∈
I
,
j
=
1
,
…
,
m
.


## 5. A Real-Valued Gronwall-Type Inequality

In this section we establish a real-valued Gronwall-type inequality which extends (3) from composition operators A
[
u
]
(
s
)
=
r
(
u
(
s
)
) to A
∈
ℱ. Consider the case m
=
1 of (65):
(68)
ψ
(
u
(
t
)
)
≤
L
+
∫
a
t
‍
g
(
s
)
A
[
u
]
(
s
)
d
s
,
t
∈
I
.
Here u
∈
C
(
I
,
ℝ
+
), ψ
∈
C
(
ℝ
+
,
ℝ
+
) is strictly increasing and ψ
(
∞
)
=
∞, L
≥
0, g
∈
C
(
I
,
ℝ
+
), and A
:
C
(
I
,
ℝ
)
→
C
(
I
,
ℝ
). Let ℱ be the class of operators A
:
C
(
I
,
ℝ
)
→
C
(
I
,
ℝ
) such that A is positive and monotone increasing, and there exist w
∈
C
(
I
,
ℝ
+
) and r
∈
C
(
ℝ
+
,
(
0
,
∞
)
) such that for all increasing f
∈
C
(
I
,
ℝ
+
), we have
(69)
A
[
f
]
(
t
)
≤
w
(
t
)
r
(
f
(
t
)
)
,
t
∈
I
.
By some modification of results in Section 4 we have Theorem 16.Theorem 16.
LetL
≥
0 and u, ψ, g, and A be given as above. Suppose A
∈
ℱ. Let l
(
t
)
=
∫
c
t
‍
(
1
/
r
(
ψ
-
1
(
s
)
)
)
d
s, where 0
≤
c
≤
L and t
≥
c. Assume that l
(
∞
)
>
l
(
L
)
+
∫
a
b
‍
g
(
y
)
w
(
y
)
d
y. If u satisfies (68), then for all t
∈
I we have
(70)
u
(
t
)
≤
ψ
-
1
(
l
-
1
(
l
(
L
)
+
∫
a
t
‍
g
(
y
)
w
(
y
)
d
y
)
)
≤
ψ
-
1
(
l
-
1
(
l
(
L
)
+
∫
a
b
‍
g
(
y
)
w
(
y
)
d
y
)
)
.It is easy to see that ifA
∈
ℱ, then A
m
∈
ℱ for m
>
0. If A
1
,
…
,
A
k
∈
ℱ, then A
=
∏
i
=
1
k
‍
A
i
∈
ℱ. Moreover, we also have A
=
∑
i
=
1
k
‍
b
i
A
i
∈
ℱ, where b
i
≥
0. Hence the class of operators ℱ is closed under multiplication and linear combination with nonnegative coefficients. This and the following examples show that Theorem 16 can be applied to a variety of operators A in (68).Example 17.
Let(71)
A
[
f
]
(
t
)
=
r
(
f
(
t
)
)
,
where r
∈
C
(
ℝ
+
,
(
0
,
∞
)
) is increasing. Then A
∈
ℱ with w
(
t
)
=
1 in (69). In this case and ψ
≡
1, Theorem 16 is reduced to (3)-(4).Example 18.
Let(72)
A
[
f
]
(
t
)
=
sup
⁡
a
≤
s
≤
t
f
(
s
)
.
Then A
∈
ℱ with w
(
t
)
=
1 and r
(
t
)
=
1 in (69).Example 19.
Let(73)
𝕋
[
f
]
(
t
)
=
∫
a
t
‍
k
(
t
,
s
)
r
(
f
(
s
)
)
d
s
,
where k
∈
C
(
I
2
,
ℝ
+
) and r
∈
C
(
ℝ
+
,
(
0
,
∞
)
) is increasing. Then 𝕋
∈
ℱ with w
(
t
)
=
∫
a
t
‍
k
(
t
,
s
)
d
s in (69).Example 20.
Let(74)
𝔾
[
f
]
(
t
)
=
exp
⁡
(
1
∫
a
t
‍
k
(
t
,
s
)
d
s
∫
a
t
‍
k
(
t
,
s
)
log
⁡
r
(
f
(
s
)
)
d
s
)
,
where k
∈
C
(
I
2
,
(
0
,
∞
)
) and r
∈
C
(
ℝ
+
,
(
0
,
∞
)
) is increasing. Then 𝔾
∈
ℱ with w
(
t
)
=
1 in (69).Example 21.
Let(75)
𝕄
[
f
]
(
t
)
=
sup
⁡
a
<
x
<
t
∫
x
t
‍
k
(
t
,
s
)
r
(
f
(
s
)
)
d
s
,
where k and r are given as in Example 19. Then  𝕄
∈
ℱ with w
(
t
)
=
∫
a
t
‍
k
(
t
,
s
)
d
s in (69).Example 22.
Let(76)
𝒢
[
f
]
(
t
)
=
sup
⁡
a
<
x
<
t
exp
⁡
(
1
∫
x
t
‍
k
(
t
,
s
)
d
s
∫
x
t
‍
k
(
t
,
s
)
log
⁡
r
(
f
(
s
)
)
d
s
)
,
where k and r are given as in Example 20. Then 𝒢
∈
ℱ with w
(
t
)
=
1 in (69).



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*Source: 973586-2014-01-22.xml*