# Hyers-Ulam Stability of the First-Order Matrix Differential Equations

Authors: Soon-Mo Jung
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

We prove the generalized Hyers-Ulam stability of the first-order linear homogeneous matrix differential equationsy
→
'
(
t
)
=
A
(
t
)
y
→
(
t
). Moreover, we apply this result to prove the generalized Hyers-Ulam stability of the nth order linear differential equations with variable coefficients.


---

## Body

## 1. Introduction

For a given positive integern, let I be a nondegenerate interval of R. We will consider the (linear) differential equation of nth order(1)
F
y
n
,
y
n
-
1
,
…
,
y
′
,
y
,
x
=
0defined on I, where y
:
I
→
C is an n times continuously differentiable function.For an arbitraryɛ
>
0, assume that an n times continuously differentiable function y
:
I
→
C satisfies the differential inequality(2)
F
y
n
,
y
n
-
1
,
…
,
y
′
,
y
,
x
≤
ɛfor all x
∈
I. If for each function y
:
I
→
C, satisfying inequality (2), there exists a solution y
0
:
I
→
C of the differential equation (1) such that (3)
y
x
-
y
0
x
≤
K
ɛfor any x
∈
I, where K
(
ɛ
) depends on ɛ only and satisfies lim
ɛ
→
0
⁡
K
(
ɛ
)
=
0, then we say that the differential equation (1) satisfies (or has) the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain I is not the whole space R). If the above statement also holds when we replace ɛ and K
(
ɛ
) with some appropriate φ
(
x
) and Φ
(
x
), respectively, then we say that the differential equation (1) satisfies the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability). For more detailed definition of the Hyers-Ulam stability, refer to [1].Obłoza seems to be the first author who investigated the Hyers-Ulam stability of linear differential equations (see [2, 3]). Given real-valued constants a and b, let g
,
r
:
(
a
,
b
)
→
R be continuous functions with ∫
a
b
‍
g
x
d
x
<
∞. Assume that ɛ
>
0 is an arbitrary real number. Obłoza proved that if a differentiable function y
:
(
a
,
b
)
→
R satisfies the inequality y
′
x
+
g
x
y
x
-
r
x
≤
ɛ for all x
∈
(
a
,
b
) and if a function y
0
:
(
a
,
b
)
→
R satisfies y
0
′
(
x
)
+
g
(
x
)
y
0
(
x
)
=
r
(
x
) for all x
∈
(
a
,
b
) and y
(
τ
)
=
y
0
(
τ
) for some τ
∈
(
a
,
b
), then there exists a constant δ
>
0 such that y
x
-
y
0
x
≤
δ for all x
∈
(
a
,
b
).Thereafter, Alsina and Ger [4] proved that if a differentiable function y
:
(
a
,
b
)
→
R satisfies the differential inequality y
′
x
-
y
x
≤
ɛ, then there exists a function y
0
:
(
a
,
b
)
→
R such that y
0
′
(
x
)
=
y
0
(
x
) and y
x
-
y
0
x
≤
3
ɛ for all x
∈
(
a
,
b
). This result of Alsina and Ger was generalized by Takahasi et al. [5]. Indeed, they proved the Hyers-Ulam stability of the Banach space valued differential equation y
′
(
x
)
=
λ
y
(
x
) (see also [6–12]).In this paper, we prove the generalized Hyers-Ulam stability of the first-order linear homogeneous matrix differential equations:(4)
y
→
′
t
=
A
t
y
→
t
.Moreover, we apply this result to prove the generalized Hyers-Ulam stability of the nth order linear differential equations with variable coefficients.


## 2. Preliminaries

In what follows, letI
=
(
a
,
b
) be an arbitrary open interval with -
∞
≤
a
<
b
≤
∞ and let n be a fixed integer larger than 1. The first-order linear inhomogeneous differential equation with matrix coefficient is of the form(5)
y
→
′
t
=
A
t
y
→
t
+
f
→
t
,where y
→
:
I
→
C
n is an unknown vector function to be determined, A
(
t
) is an (
n
×
n
) matrix whose entries are complex-valued continuous functions on I, and f
→
:
I
→
C
n is a continuous column vector function.According to [13, Section 15], if the coefficient matrix A
(
t
) is an (
n
×
n
) matrix whose entries are complex-valued continuous functions, then there exist n linearly independent solutions y
→
1
,
…
,
y
→
n to the homogeneous matrix differential equation (4) and each of its solutions can be written in the form(6)
y
→
t
=
Y
t
η
→
,where η
→
=
η
1
η
2
⋯
η
n
tr
⁡ is a constant column vector and Y
(
t
) is a fundamental matrix whose columns consist of n linearly independent solutions y
→
1
(
t
)
,
…
,
y
→
n
(
t
) to the homogeneous differential equation (4);that is, Y
(
t
) is an (
n
×
n
) matrix defined by (7)
Y
t
=
y
→
1
t
y
→
2
t
⋯
y
→
n
t
.In view of [13, Section 16], the general solution to the inhomogeneous differential equation (5) is given by(8)
y
→
t
=
Y
t
η
→
+
Y
t
∫
c
t
Y
-
1
τ
f
→
τ
d
τ
,where Y
(
t
) is a fundamental matrix for the corresponding homogeneous differential equation (4) and c is an arbitrarily given real number included in I.Let(
C
n
,
·
n
) be a complex normed space, where each of its elements is a column vector, and let C
n
×
n be a vector space consisting of all (
n
×
n
) complex matrices. We choose a norm ·
n
×
n on C
n
×
n which is compatible with ·
n;that is, both norms obey(9)
A
B
n
×
n
≤
A
n
×
n
B
n
×
n
,
A
y
→
n
≤
A
n
×
n
y
→
nfor all A
,
B
∈
C
n
×
n and y
→
∈
C
n.Some of the most important matrix norms are induced byp-norms. For 1
≤
p
≤
∞, the matrix norm induced by the p-norm, (10)
A
p
≔
sup
x
→
≠
0
→
⁡
A
x
→
p
x
→
p
,is called the matrix p-norm. For example, we get (11)
A
1
=
max
1
≤
j
≤
n
⁡
∑
i
=
1
n
a
i
j
,
A
∞
=
max
1
≤
i
≤
n
⁡
∑
j
=
1
n
a
i
j
.It is known that the matrix p-norm, together with the p-norm, satisfies the conditions in (9), where(12)
x
→
1
=
∑
j
=
1
n
x
j
,
x
→
∞
=
max
1
≤
j
≤
n
⁡
x
jfor any x
→
∈
C
n.


## 3. Generalized Hyers-Ulam Stability of (4)

We prove the main theorem of this paper concerning the generalized Hyers-Ulam stability of the first-order linear homogeneous matrix differential equation (4).Theorem 1.
Assume thatI
=
(
a
,
b
) is an arbitrary open interval with -
∞
≤
a
<
b
≤
∞. Given a fixed integer n
>
1, let (
C
n
,
·
n
) and (
C
n
×
n
,
·
n
×
n
) be complex normed spaces with the property (9), whose elements are column vectors, respectively, (
n
×
n
) complex matrices. Assume that the coefficient matrix A
(
t
) is an (
n
×
n
) matrix whose entries are complex-valued continuous functions on I. Let c be an arbitrary real number included in I and φ
:
I
→
0
,
∞ a continuous function. If a continuously differentiable vector function y
→
:
I
→
C
n satisfies the inequality(13)
y
→
′
t
-
A
t
y
→
t
n
≤
φ
tfor all t
∈
I, then there exists a solution y
→
c
:
I
→
C
n to the homogeneous matrix differential equation (4) such that (14)
y
→
t
-
y
→
c
t
n
≤
Y
t
n
×
n
∫
c
t
Y
-
1
τ
n
×
n
φ
τ
d
τfor all t
∈
I, where Y
(
t
) is a fundamental matrix of the homogeneous matrix differential equation (4).Proof.
If we define a continuous vector functionf
→
:
I
→
C
n by f
→
(
t
)
≔
y
→
′
(
t
)
-
A
(
t
)
y
→
(
t
), then it follows from (13) that(15)
f
→
t
n
=
y
→
′
t
-
A
t
y
→
t
n
≤
φ
t
,
(16)
y
→
′
t
=
A
t
y
→
t
+
f
→
tfor all t
∈
I.
In view of (5) and (8), we have (17)
y
→
t
=
Y
t
η
→
+
Y
t
∫
c
t
Y
-
1
τ
f
→
τ
d
τfor all t
∈
I, where η
→
=
η
1
⋯
η
n
tr
⁡ is a constant column vector and Y
(
t
) is a fundamental matrix whose columns consist of n linearly independent solutions to the homogeneous equation (4) (see [13, Section 16]).
If we define a continuously differentiable vector functiony
→
c
:
I
→
C
n by y
→
c
(
t
)
≔
Y
(
t
)
η
→ for each t
∈
I, then y
→
c is a solution to the homogeneous matrix differential equation (4) in view of (6). (We know that every function y
→
c
(
t
)
=
Y
(
t
)
η
→ is a solution to (4).) Moreover, by (9) and (15), we have (18)
y
→
t
-
y
→
c
t
n
=
Y
t
∫
c
t
Y
-
1
τ
f
→
τ
d
τ
n
≤
Y
t
n
×
n
∫
c
t
Y
-
1
τ
f
→
τ
d
τ
n
≤
Y
t
n
×
n
∫
c
t
Y
-
1
τ
n
×
n
f
→
τ
n
d
τ
≤
Y
t
n
×
n
∫
c
t
Y
-
1
τ
n
×
n
φ
τ
d
τfor any t
∈
I.If we setn
=
2, A
(
t
)
≡
A (a constant matrix), and I
=
(
a
,
b
) with -
∞
≤
a
<
0
<
b
≤
∞, then Theorem 1 yields the following corollary.Corollary 2.
Let(
C
2
,
·
2
) and (
C
2
×
2
,
·
2
×
2
) be complex normed spaces with property (9), whose elements are column vectors, respectively, (
2
×
2
) complex matrices. Assume that the coefficient matrix A is given by (19)
A
≔
a
11
a
12
a
21
a
22
,where a
i
j’s are complex numbers with(20)
a
11
a
22
-
a
12
a
21
≠
0
,
a
11
+
a
22
2
a
11
a
22
-
a
12
a
21
≠
4
.Assume that I
=
(
a
,
b
) is an arbitrary open interval with -
∞
≤
a
<
b
≤
∞ and c is an arbitrary real number included in I. Let φ
:
I
→
0
,
∞ be a continuous function. If a continuously differentiable vector function y
→
:
I
→
C
2 satisfies the inequality(21)
y
→
′
t
-
A
y
→
t
2
≤
φ
tfor all t
∈
I, then there exists a solution y
→
c
:
I
→
C
2 to the homogeneous matrix differential equation (4) with A
(
t
)
≡
A such that (22)
y
→
t
-
y
→
c
t
2
≤
P
2
×
2
P
-
1
2
×
2
Z
t
2
×
2
∫
c
t
Z
-
1
τ
2
×
2
φ
τ
d
τfor all t
∈
I, where Z
(
t
) and Z
-
1
(
t
) are given in (33) and (34).Proof.
The complex numbers(23)
λ
1
=
a
11
+
a
22
+
a
11
-
a
22
2
+
4
a
12
a
21
2
,
λ
2
=
a
11
+
a
22
-
a
11
-
a
22
2
+
4
a
12
a
21
2are the eigenvalues of A and the vectors (24)
v
→
1
=
2
a
12
-
a
11
+
a
22
+
a
11
-
a
22
2
+
4
a
12
a
21
=
2
a
12
2
λ
1
-
2
a
11
,
v
→
2
=
2
a
12
-
a
11
+
a
22
-
a
11
-
a
22
2
+
4
a
12
a
21
=
2
a
12
2
λ
2
-
2
a
11are the eigenvectors of A corresponding to λ
1 and λ
2, respectively.
In view of [14, Theorems 1.4.5 and 7.1.4], the conditions in (20) imply that A is invertible and 0
≠
λ
1
≠
λ
2
≠
0. Hence, the eigenvectors v
→
1 and v
→
2 are linearly independent.
If we set(25)
P
≔
v
→
1
v
→
2
,then P is invertible and (26)
P
-
1
=
1
4
a
12
a
11
-
a
22
2
+
4
a
12
a
21
a
11
-
a
22
+
a
11
-
a
22
2
+
4
a
12
a
21
2
a
12
-
a
11
+
a
22
+
a
11
-
a
22
2
+
4
a
12
a
21
-
2
a
12
=
1
2
a
12
2
λ
1
-
a
11
-
a
22
λ
1
-
a
22
a
12
λ
1
-
a
11
-
a
12
.Due to a well known theorem of linear algebra, it holds that(27)
P
-
1
A
P
=
λ
1
0
0
λ
2
≕
D
,
or
A
=
P
D
P
-
1
.
If we define a vector functionz
→
:
I
→
C
2 by(28)
z
→
t
≔
P
-
1
y
→
tfor all t
∈
I, then z
→
(
t
) is continuously differentiable, and it follows from (21) and (27) that (29)
z
→
′
t
-
D
z
→
t
2
=
P
-
1
y
→
′
t
-
P
-
1
P
D
P
-
1
y
→
t
2
=
P
-
1
y
→
′
t
-
P
-
1
A
y
→
t
2
≤
P
-
1
2
×
2
y
→
′
t
-
A
y
→
t
2
≤
P
-
1
2
×
2
φ
tfor all t
∈
I.
According to Theorem1, there exists a vector function z
→
c
:
I
→
C
2 satisfying(30)
z
→
c
′
t
=
D
z
→
c
t
,
(31)
z
→
t
-
z
→
c
t
2
≤
Z
t
2
×
2
∫
c
t
Z
-
1
τ
2
×
2
P
-
1
2
×
2
φ
τ
d
τfor every t
∈
I, where Z
(
t
) is a fundamental matrix of the homogeneous matrix differential equation (30).
If we set(32)
z
→
1
t
≔
e
λ
1
t
0
,
z
→
2
t
≔
0
e
λ
2
t
,then z
→
1 and z
→
2 are the solutions to (30) which are linearly independent. Therefore, the fundamental matrix Z
(
t
) of the matrix differential equation (30) may be expressed as(33)
Z
t
=
z
→
1
t
z
→
2
t
=
e
λ
1
t
0
0
e
λ
2
t
.Then, we have(34)
Z
-
1
t
=
e
-
λ
1
t
0
0
e
-
λ
2
tfor any t
∈
I.
If we sety
→
c
(
t
)
≔
P
z
→
c
(
t
) for all t
∈
I, then it follows from (27) and (30) that (35)
y
→
c
′
t
=
P
z
→
c
′
t
=
P
D
z
→
c
t
=
P
D
P
-
1
P
z
→
c
t
=
A
y
→
c
tfor all t
∈
I;that is, y
→
c
(
t
) is a solution to the matrix differential equation (4) with A
(
t
)
≡
A.
Finally, it follows from (28) and (31) that (36)
y
→
t
-
y
→
c
t
2
=
P
z
→
t
-
P
z
→
c
t
2
≤
P
2
×
2
z
→
t
-
z
→
c
t
2
≤
P
2
×
2
P
-
1
2
×
2
Z
t
2
×
2
∫
c
t
Z
-
1
τ
2
×
2
φ
τ
d
τfor all t
∈
I.


## 4. Applications

We now consider the linearnth order differential equations with variable coefficients(37)
y
n
t
+
a
n
-
1
t
y
n
-
1
t
+
⋯
+
a
1
t
y
′
t
+
a
0
t
y
t
=
0
,where a
0
,
a
1
,
…
,
a
n
-
1
:
I
→
C are continuous functions.In this section, we set(38)
A
t
≔
0
1
0
0
⋯
0
0
0
0
1
0
⋯
0
0
⋮
⋮
⋮
⋮
⋱
⋮
⋮
0
0
0
0
⋯
0
1
-
a
0
t
-
a
1
t
-
a
2
t
-
a
3
t
⋯
-
a
n
-
2
t
-
a
n
-
1
tfor all t
∈
I.Theorem 3.
Assume thatI
=
(
a
,
b
) is an arbitrary open interval with -
∞
≤
a
<
b
≤
∞. For a fixed integer n
>
1, assume that continuous functions a
0
,
a
1
,
…
,
a
n
-
1
:
I
→
C are given. Let c be an arbitrary real number included in I and φ
:
I
→
0
,
∞ a continuous function. If an n times continuously differentiable function y
:
I
→
C satisfies the inequality(39)
y
n
t
+
a
n
-
1
t
y
n
-
1
t
+
a
n
-
2
t
y
n
-
2
t
+
⋯
+
a
1
t
y
′
t
+
a
0
t
y
t
≤
φ
tfor all t
∈
I, then there exists a solution y
c
:
I
→
C to the homogeneous matrix differential equation (37) such that (40)
y
t
-
y
c
t
≤
U
t
∞
∫
c
t
U
-
1
τ
∞
φ
τ
d
τfor all t
∈
I, where U
(
t
) is a fundamental matrix of the first-order linear homogeneous matrix differential equation (46).Proof.
If we define continuous functionsu
1
,
…
,
u
n
:
I
→
C by(41)
u
1
t
=
y
t
,
u
2
t
=
y
′
t
,
⋮
u
n
t
=
y
n
-
1
t
,then it follows from (39) and (41) that (42)
u
1
′
t
-
u
2
t
=
0
,
u
2
′
t
-
u
3
t
=
0
,
⋮
u
n
-
1
′
t
-
u
n
t
=
0and moreover (43)
u
n
′
t
+
a
n
-
1
t
u
n
t
+
a
n
-
2
t
u
n
-
1
t
+
⋯
+
a
1
t
u
2
t
+
a
0
t
u
1
t
=
y
n
t
+
a
n
-
1
t
y
n
-
1
t
+
a
n
-
2
t
y
n
-
2
t
+
⋯
+
a
1
t
y
′
t
+
a
0
t
y
t
≤
φ
tfor all t
∈
I. That is, we have (44)
u
→
′
t
-
A
t
u
→
t
∞
≤
φ
tfor all t
∈
I, where the coefficient matrix A
(
t
) is defined in (38), the vector norm ·
∞ is defined in (12), and the continuously differentiable vector function u
→
:
I
→
C
n is defined by(45)
u
→
t
≔
u
1
t
u
2
t
⋮
u
n
t
=
y
t
y
′
t
⋮
y
n
-
1
t
.
According to Theorem1, there exists a continuously differentiable vector function u
→
c
:
I
→
C
n satisfying(46)
u
→
c
′
t
=
A
t
u
→
c
t
,
(47)
u
→
t
-
u
→
c
t
∞
≤
U
t
∞
∫
c
t
U
-
1
τ
∞
φ
τ
d
τfor all t
∈
I, where U
(
t
) is a fundamental matrix of the first-order linear homogeneous matrix differential equation (46).
We now set(48)
u
→
c
t
≔
u
c
,
1
t
u
c
,
2
t
⋮
u
c
,
n
tand then we consider the first n
-
1 elements of (46) to show that(49)
u
c
,
1
′
t
=
u
c
,
2
t
,
u
c
,
2
′
t
=
u
c
,
3
t
=
u
c
,
1
′
′
t
,
⋮
u
c
,
n
-
1
′
t
=
u
c
,
n
t
=
u
c
,
1
n
-
1
t
.Moreover, we use the last equality of (49) and then the last element of (46) to see(50)
u
c
,
n
′
t
=
u
c
,
1
n
t
=
-
a
0
t
u
c
,
1
t
-
a
1
t
u
c
,
2
t
-
⋯
-
a
n
-
2
t
u
c
,
n
-
1
t
-
a
n
-
1
t
u
c
,
n
tfor all t
∈
I. The continuous differentiability of u
c
,
n implies that u
c
,
1 is n times continuously differentiable. Using (49) and (50), we further have(51)
u
c
,
1
n
t
+
a
n
-
1
t
u
c
,
1
n
-
1
t
+
⋯
+
a
1
t
u
c
,
1
′
t
+
a
0
t
u
c
,
1
t
=
0for every t
∈
I.
If we define a functiony
c
:
I
→
C by (52)
y
c
t
≔
u
c
,
1
t
,then it follows from (51) that the n times continuously differentiable function y
c is a solution to the homogeneous matrix differential equation (37). Moreover, by (45), (47), and (48), we get (53)
y
t
-
y
c
t
=
u
1
t
-
u
c
,
1
t
≤
u
→
t
-
u
→
c
t
∞
≤
U
t
∞
∫
c
t
U
-
1
τ
∞
φ
τ
d
τfor all t
∈
I.Remark 4.
Popa and Raşa [11] proved the Hyers-Ulam stability of the linear nth order differential equations with variable coefficients (37). For any continuous function λ
:
I
→
C, they introduced the differential operators D
λ and D
n by (54)
D
λ
y
t
≔
y
′
t
+
λ
y
t
,
D
n
y
t
≔
y
n
t
+
a
n
-
1
t
y
n
-
1
t
+
⋯
+
a
1
t
y
′
t
+
a
0
t
y
tfor any n times continuously differentiable function y
:
I
→
C. Under the assumptions that there exist continuous functions r
1
,
r
2
,
…
,
r
n
:
I
→
C such that (55)
D
n
=
D
r
1
∘
D
r
2
∘
⋯
∘
D
r
nand that inf
t
∈
I
⁡
r
k
t
>
0 for every k
∈
1,2
,
…
,
n, they proved the Hyers-Ulam stability of the linear nth order differential equations (37) (see [11, Theorem  3.1]).
We proved the generalized Hyers-Ulam stability of the linearnth order differential equations (37) in Theorem 3 without additional assumptions. Based on this argument, Theorem 3 gains an advantage over [11, Theorem  3.1].
However, we remark that the hypotheses in [11, Theorem  3.1] are stronger, but the conclusions are also stronger: the Hyers-Ulam constant is described in a more explicit fashion.Finally, we introduce an example to illustrate the usefulness of Theorem3 for proving the generalized Hyers-Ulam stability of the second-order linear differential equation(56)
y
′
′
t
+
ω
2
y
t
=
0for a simple harmonic oscillator.Example 5.
Assume thatI
=
(
a
,
b
) is an arbitrary open interval with -
∞
≤
a
<
b
≤
∞. Let ω be a positive real number, let c be an arbitrary real number included in I, and let φ
:
I
→
0
,
∞ be a continuous function such that (57)
∫
c
t
φ
τ
d
τexists for each t
∈
I. If a twice continuously differentiable function y
:
I
→
C satisfies the inequality(58)
y
′
′
t
+
ω
2
y
t
≤
φ
tfor all t
∈
I, then there exists a solution y
c
:
I
→
C to the second-order linear differential equation (56) such that (59)
y
t
-
y
c
t
≤
max
⁡
1
,
ω
ω
+
1
ω
∫
c
t
φ
τ
d
τfor all t
∈
I.Proof.
If we set(60)
A
≔
0
1
-
ω
2
0
,
u
→
t
≔
y
t
y
′
t
, then it follows from (58) that(61)
u
→
′
t
-
A
u
→
t
∞
≤
φ
tfor all t
∈
I.
In view of Theorem3, there exists a twice continuously differentiable solution y
c
:
I
→
C to (56) such that(62)
y
t
-
y
c
t
≤
U
t
∞
∫
c
t
U
-
1
τ
∞
φ
τ
d
τfor all t
∈
I, where U
(
t
) is a fundamental matrix of the matrix differential equation u
→
′
(
t
)
=
A
u
→
(
t
).
We know that the eigenvalues of the matrixA are λ
1
=
-
i
ω and λ
2
=
i
ω and the eigenvectors of A corresponding to λ
1 and λ
2 are (63)
1
-
i
ω
,
1
i
ω
,respectively. If we set (64)
P
≔
1
1
-
i
ω
i
ω
,
D
≔
-
i
ω
0
0
i
ω
, then A
=
P
D
P
-
1 or D
=
P
-
1
A
P.
If we setz
→
(
t
)
≔
P
-
1
u
→
(
t
), then it follows from (61) that (65)
z
→
′
t
-
D
z
→
t
∞
=
P
-
1
u
→
′
t
-
P
-
1
A
P
P
-
1
u
→
t
∞
≤
P
-
1
∞
φ
tfor all t
∈
I. It is not difficult to find a fundamental matrix Z
(
t
) of the homogeneous matrix differential equation z
→
′
(
t
)
=
D
z
→
(
t
): (66)
Z
t
=
e
-
i
ω
t
0
0
e
i
ω
t
.Using this fundamental matrix, we further get (67)
U
t
=
P
Z
t
=
e
-
i
ω
t
e
i
ω
t
-
i
ω
e
-
i
ω
t
i
ω
e
i
ω
t
,
U
-
1
t
=
1
2
i
ω
i
ω
e
i
ω
t
-
e
i
ω
t
i
ω
e
-
i
ω
t
e
-
i
ω
t
.
Since(68)
U
t
∞
=
max
⁡
2,2
ω
,
U
-
1
t
∞
=
ω
+
1
2
ω
,it follows from (62) that (69)
y
t
-
y
c
t
≤
max
⁡
1
,
ω
ω
+
1
ω
∫
c
t
φ
τ
d
τ for all t
∈
I.



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*Source: 614745-2015-11-30.xml*