# Some Properties of Solutions of a Functional-Differential Equation of Second Order with Delay

Authors: Veronica Ana Ilea; Diana Otrocol
Journal: The Scientific World Journal
 (2014)

---

## Abstract

Existence, uniqueness, data dependence (monotony, continuity, and differentiability with respect to parameter), and Ulam-Hyers stability results for the solutions of a system of functional-differential equations with delays are proved. The techniques used are Perov’s fixed point theorem and weakly Picard operator theory.


---

## Body

## 1. Introduction

Functional-differential equations with delay arise when modeling biological, physical, engineering, and other processes whose rate of change of state at any moment of timet is determined not only by the present state but also by past state.The description of certain phenomena in physics has to take into account that the rate of propagation is finite. For example, oscillation in a vacuum tube can be described by the following equation in dimensionless variables [1, 2]:
(1)
x
′′
(
t
)
+
2
r
x
′
(
t
)
+
ω
2
x
(
t
)
+
2
q
x
′
(
t
-
1
)
=
ϵ
x
′
3
(
t
-
1
)
.
In this equation, time delay is due to the fact that the time necessary for electrons to pass from the cathode to the anode in the tube is finite. The same equation has been used in the theory of stabilization of ships [2]. The dynamics of an autogenerator with delay and second-order filter was described in [3] by the equation
(2)
x
′′
(
t
)
+
2
δ
x
′
(
t
)
+
x
(
t
)
=
f
(
x
(
t
-
h
)
)
.
The model of ship course stabilization under conditions of uncertainty may be described by the following equation [4]:
(3)
I
x
′′
(
t
)
+
H
x
′
(
t
)
=
-
K
Ψ
(
t
)
+
b
0
ξ
0
′
(
t
)
,
h
h
h
h
h
h
h
h
h
h
x
(
t
0
)
=
x
0
,
x
′
(
t
0
)
=
0
,
with x
(
t
) being the angle of the deviation from course, Ψ
(
t
) the turning angle of the rudder, and ξ
0
(
t
) the stochastic disturbance. In the process of mathematical modeling, often small delays are neglected; that is why sometimes false conclusions appear. As an example we can give the following equation [1]:
(4)
x
′′
(
t
)
+
x
′
(
t
)
+
x
(
t
)
=
a
[
x
′′
(
t
-
h
)
+
x
′
(
t
-
h
)
+
x
(
t
-
h
)
]
,
which is asymptotically stable for h
=
0 but unstable for arbitrary h
>
0. Here a
>
1. If h
=
0 the above system is asymptotically stable. The characteristic equation is Δ
(
z
)
=
(
z
2
+
z
+
1
)
(
1
-
a
e
-
h
z
) and has the following zeros with positive real part if h
>
0
:
z
k
=
1
/
h
(
In
⁡
a
+
2
k
π
i
), i
2
=
-
1, k
=
0
,
±
1
,
…. So the trivial solution is unstable for any h
>
0.In this paper, we continue the research in this field and develop the study of the following general functional differential equation with delay:(5)
x
′′
(
t
)
=
f
(
t
,
x
(
t
)
,
x
′
(
t
)
,
x
(
t
-
h
)
,
x
′
(
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
t
∈
[
a
,
b
]
,
x
(
t
)
=
φ
(
t
)
,
t
∈
[
a
-
h
,
a
]
.
Existence, uniqueness, data dependence (monotony, continuity, and differentiability with respect to parameter), and Ulam-Hyers stability results of solution for the Cauchy problem are obtained. Our results are essentially based on Perov's fixed point theorem and weakly Picard operator technique, which will be presented in Section 2. More results about functional and integral differential equations using these techniques can be found in [5–8]. The problem (5) is equivalent to the following system:
(6)
x
′
(
t
)
=
z
(
t
)
,
t
∈
[
a
,
b
]
,
z
′
(
t
)
=
f
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
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
t
∈
[
a
,
b
]
,
with the initial conditions
(7)
x
(
t
)
=
φ
(
t
)
,
t
∈
[
a
-
h
,
a
]
,
z
(
t
)
=
φ
′
(
t
)
,
t
∈
[
a
-
h
,
a
]
.
By a solution of the system (6) we understand a function (
x
z
)
∈
C
(
[
a
-
h
,
b
]
,
ℝ
2
)
∩
C
1
(
[
a
,
b
]
,
ℝ
2
) that verifies the system.We suppose that(C1)
a
<
b, h
>
0;
(C2)
f
∈
C
(
[
a
,
b
]
×
ℝ
4
,
ℝ
), φ
∈
C
1
(
[
a
-
h
,
a
]
,
ℝ
);
(C3)
there existsL
1
,
L
2
>
0 such that ∀
t
∈
[
a
,
b
], u
i
,
v
i
,
u
~
i
,
v
~
i
∈
ℝ, i
=
1,2, we have
(8)
|
f
(
t
,
u
1
,
v
1
,
u
2
,
v
2
)
-
f
(
t
,
u
~
1
,
v
~
1
,
u
~
2
,
v
~
2
)
|
≤
L
1
max
(
|
u
1
-
u
~
1
|
,
|
u
2
-
u
~
2
|
)
+
L
2
max
(
|
v
1
-
v
~
1
|
,
|
v
2
-
v
~
2
|
)
.If(
x
z
)
∈
C
(
[
a
-
h
,
b
]
,
ℝ
2
)
∩
C
1
(
[
a
,
b
]
,
ℝ
2
) is a solution of the problem (6)-(7), then (
x
z
) is a solution of the following integral system:(9)
(
x
z
)
(
t
)
=
{
(
φ
φ
′
)
(
t
)
,
for
t
∈
[
a
-
h
,
a
]
,
(
φ
(
a
)
+
∫
a
t
‍
z
(
s
)
d
s
φ
′
(
a
)
+
∫
a
t
‍
f
(
s
,
x
(
s
)
,
z
(
s
)
,
x
(
s
-
h
)
,
z
(
s
-
h
)
)
d
s
)
for
t
∈
[
a
,
b
]
.If(
x
z
)
∈
C
(
[
a
-
h
,
b
]
,
ℝ
2
) is a solution of (9), then (
x
z
)
∈
C
(
[
a
-
h
,
b
]
,
ℝ
2
)
∩
C
1
(
[
a
,
b
]
,
ℝ
2
) and (
x
z
) is a solution of (6)-(7).Moreover, the system (6) is equivalent to the functional integral system(10)
(
x
z
)
(
t
)
=
{
(
x
z
)
(
t
)
,
for
t
∈
[
a
-
h
,
a
]
,
(
x
(
a
)
+
∫
a
t
‍
z
(
s
)
d
s
z
(
a
)
+
∫
a
t
‍
f
(
s
,
x
(
s
)
,
z
(
s
)
,
x
(
s
-
h
)
,
z
(
s
-
h
)
)
d
s
)
for
t
∈
[
a
,
b
]
.We consider the operatorsB
,
E
:
C
(
[
a
-
h
,
b
]
,
ℝ
2
)
→
C
(
[
a
-
h
,
b
]
,
ℝ
2
), B
=
(
B
1
(
x
z
)
B
2
(
x
z
)
),E
=
(
E
1
(
x
z
)
E
2
(
x
z
)
) defined by B
(
x
z
)
(
t
)
≔ the right hand side of (9), for t
∈
[
a
-
h
,
b
] and E
(
x
z
)
(
t
)
≔ the right hand side of (10), for t
∈
[
a
-
h
,
b
].


## 2. Preliminaries

In this section, we introduce notations, definitions, and preliminary results which are used throughout this paper; see [9–17]. Let (
X
,
d
) be a metric space and A
:
X
→
X an operator. We will use the following notations:F
A
≔
{
x
∈
X
∣
A
(
x
)
=
x
}—the fixed points set of A;
I
(
A
)
≔
{
Y
⊂
X
∣
A
(
Y
)
⊂
Y
,
Y
≠
∅
}—the family of the nonempty invariant subset of A;
A
n
+
1
≔
A
∘
A
n, A
0
=
1
X, A
1
=
A, n
∈
ℕ.Definition 1.
Let(
X
,
d
) be a metric space. An operator A
:
X
→
X is a Picard operator (PO) if there exists x
*
∈
X such that(i)
F
A
=
{
x
*
};
(ii)
the sequence(
A
n
(
x
0
)
)
n
∈
ℕ converges to x
* for all x
0
∈
X.Definition 2.
Let(
X
,
d
) be a metric space. An operator A
:
X
→
X is a weakly Picard operator (WPO) if the sequence (
A
n
(
x
)
)
n
∈
ℕ converges for all x
∈
X and its limit (which may depend on x) is a fixed point of A.Definition 3.
IfA is a weakly Picard operator, then we consider the operator A
∞ defined by
(11)
A
∞
:
X
⟶
X
,
A
∞
(
x
)
≔
lim
⁡
n
→
∞
A
n
(
x
)
.Remark 4.
It is clear thatA
∞
(
X
)
=
F
A.Definition 5.
LetA be a weakly Picard operator and c
>
0. The operator Ais c-weakly Picard operator if
(12)
d
(
x
,
A
∞
(
x
)
)
≤
c
d
(
x
,
A
(
x
)
)
,
∀
x
∈
X
.The following concept is important for our further considerations.Definition 6.
Let(
X
,
d
) be a metric space and f
:
X
→
X an operator. The fixed point equation
(13)
x
=
f
(
x
)
is Ulam-Hyers stable if there exists a real number c
f
>
0 such that for each ɛ
>
0 and each solution y
* of the inequation
(14)
d
(
y
,
f
(
y
)
)
≤
ɛ
,
there exists a solution x
* of (13) such that
(15)
d
(
y
*
,
x
*
)
≤
c
f
ɛ
.Now we have the following.Theorem 7 (see [17]).
Iff
:
X
→
X is c-WPO, then the equation
(16)
x
=
f
(
x
)
is Ulam-Hyers stable.Another result from the WPO theory is the following (see, e.g., [11]).Theorem 8 (fibre contraction principle).
Let(
X
,
d
) and (
Y
,
ρ
) be two metric spaces and A
:
X
×
Y
→
X
×
Y, A
=
(
B
,
C
), (
B
:
X
→
X
,
C
:
X
×
Y
→
Y
) a triangular operator. One supposes that(i)
(
Y
,
ρ
) is a complete metric space;
(ii)
the operatorB is Picard operator;
(iii)
there existsl
∈
[
0,1
) such that C
(
x
,
·
)
:
Y
→
Y is a l-contraction, for all x
∈
X;
(iv)
if(
x
*
,
y
*
)
∈
F
A, then C
(
·
,
y
*
) is continuous in x
*.Then the operator A is Picard operator.Throughout this paper we denote byM
m
m
(
ℝ
+
) the set of all m
×
m matrices with positive elements and by I the identity m
×
m matrix. A square matrix Q with nonnegative elements is said to be convergent to zero if Q
k
→
0 as k
→
∞. It is known that the property of being convergent to zero is equivalent to each of the following three conditions (see [9, 10]):(a)
I
-
Q is nonsingular and (
I
-
Q
)
-
1
=
I
+
Q
+
Q
2
+
⋯ (where I stands for the unit matrix of the same order as Q);
(b)
the eigenvalues ofQ are located inside the open unit disc of the complex plane;
(c)
I
-
Q is nonsingular and (
I
-
Q
)
-
1 has nonnegative elements.We finish this section by recalling the following fundamental result (see [9, 18]).Theorem 9 (Perov’s fixed point theorem).
Let(
X
,
d
) with d
(
x
,
y
)
∈
ℝ
m be a complete generalized metric space and A
:
X
→
X an operator. One supposes that there exists a matrix Q
∈
M
m
m
(
ℝ
+
), such that(i)
d
(
A
(
x
)
,
A
(
y
)
)
≤
Q
d
(
x
,
y
), for all x
,
y
∈
X;
(ii)
Q
n
→
0 as n
→
∞.
Then(a)
F
A
=
{
x
*
};
(b)
A
n
(
x
)
=
x
*  as  n
→
∞, ∀
x
∈
X;
(c)
d
(
A
n
(
x
)
,
x
*
)
≤
(
I
-
Q
)
-
1
Q
n
d
(
x
0
,
A
(
x
0
)
).


## 3. Main Results

In this section, we present existence, uniqueness, and data dependence (monotony, continuity, and differentiability with respect to parameter) results of solution for the Cauchy problem (6)-(7).


### 3.1. Existence and Uniqueness

Using Perov’s fixed point theorem, we obtain existence and uniqueness theorem for the solution of the problem (6)-(7).Theorem 10.
One supposes that(i)
the conditions (C1)–(C3) are satisfied;
(ii)
Q
n
→
0 as n
→
∞, where Q
=
(
b
-
a
)
(
0
1
L
1
L
2
).
Then,(a)
the problem (6)-(7) has a unique solution (
x
*
z
*
)
∈
C
1
(
[
a
,
b
]
,
ℝ
2
);
(b)
for all(
x
0
z
0
)
∈
C
1
(
[
a
,
b
]
,
ℝ
2
), the sequence (
x
n
z
n
)
n
∈
ℕ defined by (
x
n
+
1
z
n
+
1
)
=
B
(
x
n
z
n
) converges uniformly to (
x
*
z
*
), for all t
∈
[
a
,
b
], and
(17)
∥
(
x
n
z
n
)
-
(
x
*
z
*
)
∥
≤
(
I
-
Q
)
-
1
Q
n
∥
(
x
0
z
0
)
-
(
x
1
z
1
)
∥
;
(c)
the operatorB is Picard operator in (
C
(
[
a
-
h
,
b
]
,
ℝ
2
)
,
→
u
n
i
f
);
(d)
the operatorE is weakly Picard operator in (
C
(
[
a
-
h
,
b
]
,
ℝ
2
)
,
→
u
n
i
f
).Proof.
Consider on the spaceX
≔
C
(
[
a
-
h
,
b
]
,
ℝ
2
) the norm
(18)
∥
(
u
1
v
1
)
-
(
u
2
v
2
)
∥
≔
max
⁡
(
|
u
1
-
u
2
|
|
v
1
-
v
2
|
)
,
which endows X with the uniform convergence.
LetX
(
χ
χ
′
)
=
{
(
x
z
)
∈
X
∣
(
x
z
)
|
[
a
-
h
,
a
]
=
(
χ
χ
′
), for (
χ
χ
′
)
∈
C
(
[
a
-
h
,
a
]
,
ℝ
2
)
}. Then X
=
∪
(
χ
χ
′
)
∈
C
(
[
a
-
h
,
a
]
,
ℝ
2
)
X
(
χ
χ
′
) is a partition of X, and from [12] we have(1)
B
(
X
)
⊂
X
(
χ
χ
′
)
,
B
(
X
(
χ
χ
′
)
)
⊂
X
(
χ
χ
′
);
(2)
B
|
X
(
χ
χ
′
)
=
E
|
X
(
χ
χ
′
).
On the other hand, fort
∈
[
a
-
h
,
a
]
∪
[
a
,
b
]
(19)
∥
(
B
1
(
x
1
z
1
)
B
2
(
x
1
z
1
)
)
-
(
B
1
(
x
2
z
2
)
B
2
(
x
2
z
2
)
)
∥
≤
(
b
-
a
)
(
0
1
L
1
L
2
)
∥
(
x
1
z
1
)
-
(
x
2
z
2
)
∥
,
whence B is a contraction in (
X
,
∥
·
∥
) with Q
=
(
b
-
a
)
(
0
1
L
1
L
2
). Applying Perov's theorem we obtain (a), (b), and (c). Moreover, the operator B is c-PO and E is c-WPO with c
=
[
1
-
(
b
-
a
)
(
0
1
L
1
L
2
)
]
-
1.


### 3.2. Inequalities of Čaplygin Type

Now we establish the Čaplygin type inequalities.Theorem 11.
One supposes that(i)
the conditions (a), (b), and (c) in Theorem10 are satisfied;
(ii)
u
i
,
v
i
,
u
~
i
,
v
~
i
∈
ℝ,  (
u
i
v
i
)
≤
(
u
~
i
v
~
i
),  i
=
1,2 imply that
(20)
f
(
t
,
u
1
,
v
1
,
u
2
,
v
2
)
≤
f
(
t
,
u
~
1
,
v
~
1
,
u
~
2
,
v
~
2
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
t
∈
[
a
,
b
]
.
Let(
x
*
z
*
) be a solution of (6) and (
y
*
w
*
) a solution of the system
(21)
y
′
(
t
)
≤
w
(
t
)
,
t
∈
[
a
,
b
]
,
w
′
(
t
)
≤
f
(
t
,
y
(
t
)
,
w
(
t
)
,
y
(
t
-
h
)
,
w
(
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
t
∈
[
a
,
b
]
.
Then
(22)
(
y
*
w
*
)
|
[
a
-
h
,
a
]
≤
(
x
*
z
*
)
|
[
a
-
h
,
a
]
i
m
p
l
i
e
s
t
h
a
t
(
y
*
w
*
)
≤
(
x
*
z
*
)
.Proof.
We have that(23)
(
x
*
z
*
)
=
E
(
x
*
z
*
)
,
(
y
*
w
*
)
≤
E
(
y
*
w
*
)
.
From Theorem 10, (c), E is a weakly Picard operator. From condition (ii), we obtain that E
∞ is increasing [11]. So
(24)
(
y
*
w
*
)
≤
E
∞
(
y
*
w
*
)
=
E
∞
(
y
~
*
w
~
*
)
≤
E
∞
(
x
~
*
z
~
*
)
=
(
x
*
z
*
)
,
where (
x
~
*
z
~
*
)
∈
X
(
x
z
)
|
[
a
-
h
,
a
].


### 3.3. Data Dependence: Monotony

In this subsection, we study the monotony of the solution of the problem (6)-(7) with respect to φ and f.Theorem 12 (comparison theorem).
Letf
i
∈
C
(
[
a
,
b
]
×
ℝ
4
,
ℝ
),  i
=
1,2
,
3, be as in Theorem 10. One supposes that(i)
f
1
≤
f
2
≤
f
3;
(ii)
f
2
(
t
,
·
,
·
,
·
,
·
)
:
ℝ
4
→
ℝ is increasing, ∀
t
∈
[
a
,
b
].
Let(
x
i
*
z
i
*
) be a solution of the system
(25)
x
′
(
t
)
=
z
(
t
)
,
t
∈
[
a
,
b
]
,
z
′
(
t
)
=
f
i
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
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
t
∈
[
a
,
b
]
.
Then(26)
(
x
1
*
z
1
*
)
|
[
a
-
h
,
a
]
≤
(
x
2
*
z
2
*
)
|
[
a
-
h
,
a
]
≤
(
x
3
*
z
3
*
)
|
[
a
-
h
,
a
]
imply that (
x
1
*
z
1
*
)
|
[
a
-
h
,
a
]
≤
(
x
2
*
z
2
*
)
|
[
a
-
h
,
a
]
≤
(
x
3
*
z
3
*
)
|
[
a
-
h
,
a
]
.Proof.
We consider the operatorsE
i corresponding to each system (25). The operators E
i, i
=
1,2
,
3 are weakly Picard operators. Taking into consideration the condition (ii), E
2 is increasing. From (i) we have E
1
≤
E
2
≤
E
3. On the other hand, we have that
(27)
(
x
i
*
z
i
*
)
=
E
i
∞
(
x
~
i
*
z
~
i
*
)
,
i
=
1,2
,
3
,
where (
x
~
*
z
~
*
)
∈
X
(
x
z
)
|
[
a
-
h
,
a
]. The proof follows from the abstract comparison Lemma (see [11]).


### 3.4. Data Dependence: Continuity

Consider the problem (6)-(7) with the dates f
i
∈
C
(
[
a
,
b
]
×
ℝ
4
,
ℝ
),  i
=
1,2and suppose that f
i satisfy the conditions from Theorem 10 with the same Lipshitz constants. We obtain the data dependence result.Theorem 13.
Letf
i
∈
C
(
[
a
,
b
]
×
ℝ
4
,
ℝ
), φ
i
∈
C
(
[
a
-
h
,
a
]
,
ℝ
),  i
=
1,2, be as in Theorem 10. One supposes that(i)
there existsη
1
,
η
2
>
0 such that
(28)
|
(
φ
1
φ
1
′
)
(
t
)
-
(
φ
2
φ
2
′
)
(
t
)
|
≤
(
η
1
η
2
)
,
∀
t
∈
[
a
-
h
,
a
]
;
(ii)
there existsη
3
>
0 such that
(29)
|
f
1
(
t
,
u
1
,
u
2
,
u
3
,
u
4
)
-
f
2
(
t
,
u
1
,
u
2
,
u
3
,
u
4
)
|
≤
η
3
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
t
∈
[
a
,
b
]
,
u
i
∈
ℝ
,
i
=
1,4
¯
.
Then(30)
∥
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
-
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
∥
≤
(
I
-
Q
)
-
1
(
η
1
η
2
+
(
b
-
a
)
η
3
)
,
where (
x
*
z
*
)
(
·
,
φ
,
φ
′
,
f
) denote the unique solution of (6)-(7).Proof.
Consider the operatorsB
φ
i
,
f
i, i
=
1,2. From Theorem 10, it follows that
(31)
∥
B
φ
1
,
f
1
(
x
1
z
1
)
-
B
φ
1
,
f
1
(
x
2
z
2
)
∥
≤
Q
∥
(
x
1
z
1
)
-
(
x
2
z
2
)
∥
,
∀
(
x
i
z
i
)
∈
X
.
Additionally,
(32)
∥
B
φ
1
,
f
1
(
x
z
)
-
B
φ
2
,
f
2
(
x
z
)
∥
≤
(
η
1
η
2
+
(
b
-
a
)
η
3
)
.
Thus,(33)
∥
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
-
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
∥
=
∥
B
φ
1
,
f
1
(
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
)
-
B
φ
2
,
f
2
(
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
)
∥
≤
∥
B
φ
1
,
f
1
(
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
)
-
B
φ
1
,
f
1
(
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
)
∥
+
∥
B
φ
1
,
f
1
(
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
)
-
B
φ
2
,
f
2
(
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
)
∥
≤
Q
∥
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
-
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
∥
+
(
η
1
η
2
+
(
b
-
a
)
η
3
)
,
and since Q
n
→
∞, as n
→
∞ implies that (
I
-
Q
)
-
1
∈
M
22
(
ℝ
+
), we finally obtain
(34)
∥
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
-
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
∥
≤
(
I
-
Q
)
-
1
(
η
1
η
2
+
(
b
-
a
)
η
3
)
.


### 3.5. Data Dependence: Differentiability

Consider the following differential system with parameter:(35)
x
′
(
t
)
=
z
(
t
)
,
t
∈
[
a
,
b
]
,
z
′
(
t
)
=
f
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
t
-
h
)
;
λ
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
d
t
∈
[
a
,
b
]
,
λ
∈
J
,
with the initial conditions
(36)
x
(
t
)
=
φ
(
t
)
,
t
∈
[
a
-
h
,
a
]
,
z
(
t
)
=
φ
′
(
t
)
,
t
∈
[
a
-
h
,
a
]
,
where J
⊂
ℝ is a compact interval.Suppose that the following conditions are satisfied:(C1)
a
<
b, h
>
0, J
⊂
ℝ a compact interval;
(C2)
f
∈
C
(
[
a
,
b
]
×
ℝ
4
×
J
,
ℝ
);
(C3)
φ
∈
C
1
(
[
a
-
h
,
a
]
,
ℝ
);
(C4)
there existsL
f
>
0, such that
(37)
|
∂
f
(
t
,
u
1
,
u
2
,
u
3
,
u
4
;
λ
)
∂
u
i
|
≤
L
1
,
u
1
,
u
3
∈
ℝ
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
i
=
1,3
,
λ
∈
J
;
|
∂
f
(
t
,
u
1
,
u
2
,
u
3
,
u
4
;
λ
)
∂
u
i
|
≤
L
2
,
u
2
,
u
4
∈
ℝ
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
i
=
2,4
,
λ
∈
J
;
(C5)
forQ
=
(
b
-
a
)
(
0
1
L
1
L
2
), we have Q
n
→
0 as n
→
∞.Then, from Theorem10, we have that the problem (6)-(7) has a unique solution, (
x
*
z
*
)
∈
C
1
(
[
a
,
b
]
,
ℝ
2
). We prove that (
x
*
z
*
)
∈
C
1
(
J
,
ℝ
2
), ∀
t
∈
[
a
-
h
,
b
].For this we consider the system(38)
x
′
(
t
;
λ
)
=
z
(
t
;
λ
)
,
t
∈
[
a
,
b
]
,
λ
∈
J
,
z
′
(
t
;
λ
)
=
f
(
t
,
x
(
t
;
λ
)
,
z
(
t
;
λ
)
,
x
(
t
-
h
;
λ
)
,
z
(
t
-
h
;
λ
)
;
λ
)
,
t
∈
[
a
,
b
]
,
λ
∈
J
,
with (
x
*
z
*
)
∈
C
(
[
a
-
h
,
b
]
×
J
,
ℝ
2
)
∩
C
1
(
[
a
,
b
]
×
J
,
ℝ
2
).Theorem 14.
Consider the problem (38)–(36) and suppose the conditions (C1)–(C5) hold. Then,(i)
Equations (38)–(36) have a unique solution (
x
*
z
*
)
(
·
,
λ
), in C
(
[
a
-
h
,
b
]
×
J
,
ℝ
2
);
(ii)
(
x
*
z
*
)
(
·
,
λ
)
∈
C
1
(
J
,
ℝ
2
),
∀
t
∈
[
a
-
h
,
b
].Proof.
The problem (38)–(36) is equivalent with the following functional-integral system:(39)
(
x
z
)
(
t
;
λ
)
=
(
φ
(
a
)
+
∫
a
t
‍
z
(
s
;
λ
)
d
s
φ
′
(
a
)
+
∫
a
t
‍
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
d
s
)
,for t
∈
[
a
,
b
] and
(40)
(
x
z
)
(
t
;
λ
)
=
(
φ
φ
′
)
(
t
)
,
for
t
∈
[
a
-
h
,
a
]
.
Now let us take the operator A
:
C
(
[
a
-
h
,
b
]
×
J
,
ℝ
2
)
→
C
(
[
a
-
h
,
b
]
×
J
,
ℝ
2
), defined by
(41)
A
(
x
z
)
(
t
;
λ
)
≔
(
A
1
(
x
z
)
A
2
(
x
z
)
)
≔
the
right
hand
side
of
(
39
)
,
h
h
h
h
h
h
h
h
h
h
h
h
for
t
∈
[
a
,
b
]
,
A
(
x
z
)
(
t
;
λ
)
≔
(
A
1
(
x
z
)
A
2
(
x
z
)
)
≔
the
right
hand
side
of
(
40
)
,
h
h
h
h
h
h
h
h
h
h
for
t
∈
[
a
-
h
,
a
]
.
LetX
=
C
(
[
a
-
h
,
b
]
×
J
,
ℝ
2
).
It is clear, from the proof of the Theorem10, that in the condition (C1)–(C5), the operator A
:
(
X
,
∥
·
∥
)
→
(
X
,
∥
·
∥
) is Picard operator.
Let(
x
*
z
*
) be the unique fixed point of A.
Supposing that there exists(
∂
x
*
/
∂
λ
∂
z
*
/
∂
λ
), from (39)-(40), we obtain that
(42)
∂
x
*
∂
λ
=
∫
a
t
‍
∂
z
(
s
;
λ
)
∂
λ
d
s
,
∂
z
*
∂
λ
=
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
1
)
-
1
)
∂
x
(
s
;
λ
)
∂
λ
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
2
)
-
1
)
∂
z
(
s
;
λ
)
∂
λ
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
3
)
-
1
)
∂
x
(
s
-
h
;
λ
)
∂
λ
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
4
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
4
)
-
1
)
∂
z
(
s
-
h
;
λ
)
∂
λ
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
λ
)
-
1
)
d
s
,
for all t
∈
[
a
,
b
], λ
∈
J.
This relation suggests that we consider the following operator:(43)
C
:
X
×
X
⟶
X
,
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
⟶
C
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
,
where C
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
(
t
;
λ
)
=
0 for t
∈
[
a
-
h
,
a
], λ
∈
J, and
(44)
C
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
(
t
;
λ
)
≔
(
C
1
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
C
2
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
)
,
where
(45)
C
1
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
(
t
;
λ
)
≔
∫
a
t
‍
∂
z
(
s
;
λ
)
∂
λ
d
s
,
C
2
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
(
t
;
λ
)
≔
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
1
)
-
1
)
u
1
(
s
;
λ
)
d
s
+
∫
a
t
(

z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1
(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
2
)
-
1
)
u
2
(
s
;
λ
)
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
3
)
-
1
)
u
3
(
s
-
h
;
λ
)
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
4
)
-
1
)
u
4
(
s
-
h
;
λ
)
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
λ
)
-
1
)
d
s
,
for t
∈
[
a
,
b
]
,
λ
∈
J. Here we use the notations u
1
(
s
;
λ
)
≔
∂
x
(
s
;
λ
)
/
∂
λ, u
2
(
s
;
λ
)
≔
∂
z
(
s
;
λ
)
/
∂
λ, u
3
(
s
-
h
;
λ
)
≔
∂
x
(
s
-
h
;
λ
)
/
∂
λ, and u
4
(
s
-
h
;
λ
)
≔
∂
z
(
s
-
h
;
λ
)
/
∂
λ.
In this way, we have the triangular operatorD
:
X
×
X
→
X
×
X, (
(
x
12
z
12
)
,
(
u
13
u
24
)
)
→
(
A
(
x
12
z
12
)
,
C
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
), where A is Picard operator and C
(
(
x
12
z
12
)
,
(
·
·
)
)
:
X
→
X is Q
C-contraction with Q
C
=
(
b
-
a
)
(
0
1
L
1
L
2
).
From Theorem8 the operator D is Picard operator; that is, the sequences
(46)
(
x
12
n
+
1
z
12
n
+
1
)
≔
A
(
x
12
n
z
12
n
)
,
(
u
13
n
+
1
u
24
n
+
1
)
≔
C
(
(
x
12
n
z
12
n
)
,
(
u
13
n
u
24
n
)
)
,
n
∈
ℕ, converge uniformly, with respect to t
∈
[
a
-
h
,
b
]
,
λ
∈
J, to (
(
x
12
n
z
12
n
)
,
(
u
13
n
u
24
n
)
)
∈
F
D, for all (
x
12
0
z
12
0
)
,
(
u
13
0
u
24
0
)
∈
X.
If we take(47)
(
x
12
0
z
12
0
)
=
(
0
0
)
,
(
u
13
0
u
24
0
)
=
(
∂
x
12
0
∂
λ
∂
z
12
0
∂
λ
)
=
(
0
0
)
,
then
(48)
(
u
13
1
u
24
1
)
=
(
∂
x
12
1
∂
λ
∂
z
12
1
∂
λ
)
.
By induction we prove that(49)
(
u
13
n
u
24
n
)
=
(
∂
x
12
n
∂
λ
∂
z
12
n
∂
λ
)
,
∀
n
∈
ℕ
.
So, (
x
12
n
z
12
n
)
⟶
unif
(
x
12
*
z
12
*
), as n
→
∞, and (
∂
x
12
n
/
∂
λ
∂
z
12
n
/
∂
λ
)
⟶
unif
(
u
13
*
u
24
*
), as n
→
∞.
From a Weierstrass argument we get that there exists(50)
(
∂
x
12
*
∂
λ
∂
z
12
*
∂
λ
)
,
i
=
1,2
,
(
∂
x
12
*
∂
λ
∂
z
12
*
∂
λ
)
=
(
u
13
*
u
24
*
)
.


### 3.6. Ulam-Hyers Stability

We start this section by presenting the Ulam-Hyers stability concept (see [15, 16]). For f
∈
C
(
[
a
,
b
]
×
ℝ
4
,
ℝ
), ɛ
>
0, and ψ
∈
C
(
[
a
-
h
,
b
]
,
ℝ
+
), h
>
0, we consider the system
(51)
x
′
(
t
)
=
z
(
t
)
,
t
∈
[
a
,
b
]
,
z
′
(
t
)
=
f
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
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
t
∈
[
a
,
b
]
and the following inequations
(52)
|
x
′
(
t
)
-
z
(
t
)
|
≤
ɛ
,
t
∈
[
a
,
b
]
,
|
z
′
(
t
)
-
f
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
t
-
h
)
)
|
≤
ɛ
,
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
∈
[
a
,
b
]
,
(53)
|
x
′
(
t
)
-
z
(
t
)
|
≤
ψ
(
t
)
,
t
∈
[
a
,
b
]
,
|
z
′
(
t
)
-
f
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
t
-
h
)
)
|
≤
ψ
(
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
t
∈
[
a
,
b
]
.Definition 15.
The system (51) is Ulam-Hyers stable if there exists a real number c
>
0 such that for each ɛ
>
0 and for each solution (
y
w
)
∈
C
2
(
[
a
-
h
,
b
]
,
ℝ
2
) of (52) there exists a solution (
x
z
)
∈
C
2
(
[
a
-
h
,
b
]
,
ℝ
2
) of (51) with
(54)
|
(
y
w
)
(
t
)
-
(
x
z
)
(
t
)
|
≤
c
ɛ
,
∀
t
∈
[
a
-
h
,
b
]
.Theorem 16.
One supposes that(i)
the conditions (C1)–(C3) are satisfied;
(ii)
Q
n
→
0 as n
→
∞, where Q
=
(
b
-
a
)
(
0
1
L
1
L
2
).
Then the system (6) is Ulam-Hyers stable.Proof.
The system (6) is equivalent with the functional integral system (10). We consider the operator E
:
C
(
[
a
-
h
,
b
]
,
ℝ
2
)
→
C
(
[
a
-
h
,
b
]
,
ℝ
2
), defined by E
(
x
z
)
(
t
)
≔ the right hand side of (10), for t
∈
[
a
-
h
,
b
]. So
(55)
(
x
z
)
=
E
(
x
z
)
(
t
)
,
t
∈
[
a
-
h
,
b
]
.
From Theorem10, E is c-WPO with c
=
[
1
-
(
b
-
a
)
(
0
1
L
1
L
2
)
]
-
1. Applying Theorem 7 we obtain that (6) is Ulam-Hyers stable.Remark 17.
Another proof for the above theorem can be done using Gronwall lemma [8, 14–17].


## 3.1. Existence and Uniqueness

Using Perov’s fixed point theorem, we obtain existence and uniqueness theorem for the solution of the problem (6)-(7).Theorem 10.
One supposes that(i)
the conditions (C1)–(C3) are satisfied;
(ii)
Q
n
→
0 as n
→
∞, where Q
=
(
b
-
a
)
(
0
1
L
1
L
2
).
Then,(a)
the problem (6)-(7) has a unique solution (
x
*
z
*
)
∈
C
1
(
[
a
,
b
]
,
ℝ
2
);
(b)
for all(
x
0
z
0
)
∈
C
1
(
[
a
,
b
]
,
ℝ
2
), the sequence (
x
n
z
n
)
n
∈
ℕ defined by (
x
n
+
1
z
n
+
1
)
=
B
(
x
n
z
n
) converges uniformly to (
x
*
z
*
), for all t
∈
[
a
,
b
], and
(17)
∥
(
x
n
z
n
)
-
(
x
*
z
*
)
∥
≤
(
I
-
Q
)
-
1
Q
n
∥
(
x
0
z
0
)
-
(
x
1
z
1
)
∥
;
(c)
the operatorB is Picard operator in (
C
(
[
a
-
h
,
b
]
,
ℝ
2
)
,
→
u
n
i
f
);
(d)
the operatorE is weakly Picard operator in (
C
(
[
a
-
h
,
b
]
,
ℝ
2
)
,
→
u
n
i
f
).Proof.
Consider on the spaceX
≔
C
(
[
a
-
h
,
b
]
,
ℝ
2
) the norm
(18)
∥
(
u
1
v
1
)
-
(
u
2
v
2
)
∥
≔
max
⁡
(
|
u
1
-
u
2
|
|
v
1
-
v
2
|
)
,
which endows X with the uniform convergence.
LetX
(
χ
χ
′
)
=
{
(
x
z
)
∈
X
∣
(
x
z
)
|
[
a
-
h
,
a
]
=
(
χ
χ
′
), for (
χ
χ
′
)
∈
C
(
[
a
-
h
,
a
]
,
ℝ
2
)
}. Then X
=
∪
(
χ
χ
′
)
∈
C
(
[
a
-
h
,
a
]
,
ℝ
2
)
X
(
χ
χ
′
) is a partition of X, and from [12] we have(1)
B
(
X
)
⊂
X
(
χ
χ
′
)
,
B
(
X
(
χ
χ
′
)
)
⊂
X
(
χ
χ
′
);
(2)
B
|
X
(
χ
χ
′
)
=
E
|
X
(
χ
χ
′
).
On the other hand, fort
∈
[
a
-
h
,
a
]
∪
[
a
,
b
]
(19)
∥
(
B
1
(
x
1
z
1
)
B
2
(
x
1
z
1
)
)
-
(
B
1
(
x
2
z
2
)
B
2
(
x
2
z
2
)
)
∥
≤
(
b
-
a
)
(
0
1
L
1
L
2
)
∥
(
x
1
z
1
)
-
(
x
2
z
2
)
∥
,
whence B is a contraction in (
X
,
∥
·
∥
) with Q
=
(
b
-
a
)
(
0
1
L
1
L
2
). Applying Perov's theorem we obtain (a), (b), and (c). Moreover, the operator B is c-PO and E is c-WPO with c
=
[
1
-
(
b
-
a
)
(
0
1
L
1
L
2
)
]
-
1.


## 3.2. Inequalities of Čaplygin Type

Now we establish the Čaplygin type inequalities.Theorem 11.
One supposes that(i)
the conditions (a), (b), and (c) in Theorem10 are satisfied;
(ii)
u
i
,
v
i
,
u
~
i
,
v
~
i
∈
ℝ,  (
u
i
v
i
)
≤
(
u
~
i
v
~
i
),  i
=
1,2 imply that
(20)
f
(
t
,
u
1
,
v
1
,
u
2
,
v
2
)
≤
f
(
t
,
u
~
1
,
v
~
1
,
u
~
2
,
v
~
2
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
t
∈
[
a
,
b
]
.
Let(
x
*
z
*
) be a solution of (6) and (
y
*
w
*
) a solution of the system
(21)
y
′
(
t
)
≤
w
(
t
)
,
t
∈
[
a
,
b
]
,
w
′
(
t
)
≤
f
(
t
,
y
(
t
)
,
w
(
t
)
,
y
(
t
-
h
)
,
w
(
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
t
∈
[
a
,
b
]
.
Then
(22)
(
y
*
w
*
)
|
[
a
-
h
,
a
]
≤
(
x
*
z
*
)
|
[
a
-
h
,
a
]
i
m
p
l
i
e
s
t
h
a
t
(
y
*
w
*
)
≤
(
x
*
z
*
)
.Proof.
We have that(23)
(
x
*
z
*
)
=
E
(
x
*
z
*
)
,
(
y
*
w
*
)
≤
E
(
y
*
w
*
)
.
From Theorem 10, (c), E is a weakly Picard operator. From condition (ii), we obtain that E
∞ is increasing [11]. So
(24)
(
y
*
w
*
)
≤
E
∞
(
y
*
w
*
)
=
E
∞
(
y
~
*
w
~
*
)
≤
E
∞
(
x
~
*
z
~
*
)
=
(
x
*
z
*
)
,
where (
x
~
*
z
~
*
)
∈
X
(
x
z
)
|
[
a
-
h
,
a
].


## 3.3. Data Dependence: Monotony

In this subsection, we study the monotony of the solution of the problem (6)-(7) with respect to φ and f.Theorem 12 (comparison theorem).
Letf
i
∈
C
(
[
a
,
b
]
×
ℝ
4
,
ℝ
),  i
=
1,2
,
3, be as in Theorem 10. One supposes that(i)
f
1
≤
f
2
≤
f
3;
(ii)
f
2
(
t
,
·
,
·
,
·
,
·
)
:
ℝ
4
→
ℝ is increasing, ∀
t
∈
[
a
,
b
].
Let(
x
i
*
z
i
*
) be a solution of the system
(25)
x
′
(
t
)
=
z
(
t
)
,
t
∈
[
a
,
b
]
,
z
′
(
t
)
=
f
i
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
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
t
∈
[
a
,
b
]
.
Then(26)
(
x
1
*
z
1
*
)
|
[
a
-
h
,
a
]
≤
(
x
2
*
z
2
*
)
|
[
a
-
h
,
a
]
≤
(
x
3
*
z
3
*
)
|
[
a
-
h
,
a
]
imply that (
x
1
*
z
1
*
)
|
[
a
-
h
,
a
]
≤
(
x
2
*
z
2
*
)
|
[
a
-
h
,
a
]
≤
(
x
3
*
z
3
*
)
|
[
a
-
h
,
a
]
.Proof.
We consider the operatorsE
i corresponding to each system (25). The operators E
i, i
=
1,2
,
3 are weakly Picard operators. Taking into consideration the condition (ii), E
2 is increasing. From (i) we have E
1
≤
E
2
≤
E
3. On the other hand, we have that
(27)
(
x
i
*
z
i
*
)
=
E
i
∞
(
x
~
i
*
z
~
i
*
)
,
i
=
1,2
,
3
,
where (
x
~
*
z
~
*
)
∈
X
(
x
z
)
|
[
a
-
h
,
a
]. The proof follows from the abstract comparison Lemma (see [11]).


## 3.4. Data Dependence: Continuity

Consider the problem (6)-(7) with the dates f
i
∈
C
(
[
a
,
b
]
×
ℝ
4
,
ℝ
),  i
=
1,2and suppose that f
i satisfy the conditions from Theorem 10 with the same Lipshitz constants. We obtain the data dependence result.Theorem 13.
Letf
i
∈
C
(
[
a
,
b
]
×
ℝ
4
,
ℝ
), φ
i
∈
C
(
[
a
-
h
,
a
]
,
ℝ
),  i
=
1,2, be as in Theorem 10. One supposes that(i)
there existsη
1
,
η
2
>
0 such that
(28)
|
(
φ
1
φ
1
′
)
(
t
)
-
(
φ
2
φ
2
′
)
(
t
)
|
≤
(
η
1
η
2
)
,
∀
t
∈
[
a
-
h
,
a
]
;
(ii)
there existsη
3
>
0 such that
(29)
|
f
1
(
t
,
u
1
,
u
2
,
u
3
,
u
4
)
-
f
2
(
t
,
u
1
,
u
2
,
u
3
,
u
4
)
|
≤
η
3
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
t
∈
[
a
,
b
]
,
u
i
∈
ℝ
,
i
=
1,4
¯
.
Then(30)
∥
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
-
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
∥
≤
(
I
-
Q
)
-
1
(
η
1
η
2
+
(
b
-
a
)
η
3
)
,
where (
x
*
z
*
)
(
·
,
φ
,
φ
′
,
f
) denote the unique solution of (6)-(7).Proof.
Consider the operatorsB
φ
i
,
f
i, i
=
1,2. From Theorem 10, it follows that
(31)
∥
B
φ
1
,
f
1
(
x
1
z
1
)
-
B
φ
1
,
f
1
(
x
2
z
2
)
∥
≤
Q
∥
(
x
1
z
1
)
-
(
x
2
z
2
)
∥
,
∀
(
x
i
z
i
)
∈
X
.
Additionally,
(32)
∥
B
φ
1
,
f
1
(
x
z
)
-
B
φ
2
,
f
2
(
x
z
)
∥
≤
(
η
1
η
2
+
(
b
-
a
)
η
3
)
.
Thus,(33)
∥
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
-
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
∥
=
∥
B
φ
1
,
f
1
(
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
)
-
B
φ
2
,
f
2
(
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
)
∥
≤
∥
B
φ
1
,
f
1
(
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
)
-
B
φ
1
,
f
1
(
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
)
∥
+
∥
B
φ
1
,
f
1
(
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
)
-
B
φ
2
,
f
2
(
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
)
∥
≤
Q
∥
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
-
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
∥
+
(
η
1
η
2
+
(
b
-
a
)
η
3
)
,
and since Q
n
→
∞, as n
→
∞ implies that (
I
-
Q
)
-
1
∈
M
22
(
ℝ
+
), we finally obtain
(34)
∥
(
x
*
z
*
)
(
·
,
φ
1
,
φ
1
′
,
f
1
)
-
(
x
*
z
*
)
(
·
,
φ
2
,
φ
2
′
,
f
2
)
∥
≤
(
I
-
Q
)
-
1
(
η
1
η
2
+
(
b
-
a
)
η
3
)
.


## 3.5. Data Dependence: Differentiability

Consider the following differential system with parameter:(35)
x
′
(
t
)
=
z
(
t
)
,
t
∈
[
a
,
b
]
,
z
′
(
t
)
=
f
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
t
-
h
)
;
λ
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
d
t
∈
[
a
,
b
]
,
λ
∈
J
,
with the initial conditions
(36)
x
(
t
)
=
φ
(
t
)
,
t
∈
[
a
-
h
,
a
]
,
z
(
t
)
=
φ
′
(
t
)
,
t
∈
[
a
-
h
,
a
]
,
where J
⊂
ℝ is a compact interval.Suppose that the following conditions are satisfied:(C1)
a
<
b, h
>
0, J
⊂
ℝ a compact interval;
(C2)
f
∈
C
(
[
a
,
b
]
×
ℝ
4
×
J
,
ℝ
);
(C3)
φ
∈
C
1
(
[
a
-
h
,
a
]
,
ℝ
);
(C4)
there existsL
f
>
0, such that
(37)
|
∂
f
(
t
,
u
1
,
u
2
,
u
3
,
u
4
;
λ
)
∂
u
i
|
≤
L
1
,
u
1
,
u
3
∈
ℝ
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
i
=
1,3
,
λ
∈
J
;
|
∂
f
(
t
,
u
1
,
u
2
,
u
3
,
u
4
;
λ
)
∂
u
i
|
≤
L
2
,
u
2
,
u
4
∈
ℝ
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
i
=
2,4
,
λ
∈
J
;
(C5)
forQ
=
(
b
-
a
)
(
0
1
L
1
L
2
), we have Q
n
→
0 as n
→
∞.Then, from Theorem10, we have that the problem (6)-(7) has a unique solution, (
x
*
z
*
)
∈
C
1
(
[
a
,
b
]
,
ℝ
2
). We prove that (
x
*
z
*
)
∈
C
1
(
J
,
ℝ
2
), ∀
t
∈
[
a
-
h
,
b
].For this we consider the system(38)
x
′
(
t
;
λ
)
=
z
(
t
;
λ
)
,
t
∈
[
a
,
b
]
,
λ
∈
J
,
z
′
(
t
;
λ
)
=
f
(
t
,
x
(
t
;
λ
)
,
z
(
t
;
λ
)
,
x
(
t
-
h
;
λ
)
,
z
(
t
-
h
;
λ
)
;
λ
)
,
t
∈
[
a
,
b
]
,
λ
∈
J
,
with (
x
*
z
*
)
∈
C
(
[
a
-
h
,
b
]
×
J
,
ℝ
2
)
∩
C
1
(
[
a
,
b
]
×
J
,
ℝ
2
).Theorem 14.
Consider the problem (38)–(36) and suppose the conditions (C1)–(C5) hold. Then,(i)
Equations (38)–(36) have a unique solution (
x
*
z
*
)
(
·
,
λ
), in C
(
[
a
-
h
,
b
]
×
J
,
ℝ
2
);
(ii)
(
x
*
z
*
)
(
·
,
λ
)
∈
C
1
(
J
,
ℝ
2
),
∀
t
∈
[
a
-
h
,
b
].Proof.
The problem (38)–(36) is equivalent with the following functional-integral system:(39)
(
x
z
)
(
t
;
λ
)
=
(
φ
(
a
)
+
∫
a
t
‍
z
(
s
;
λ
)
d
s
φ
′
(
a
)
+
∫
a
t
‍
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
d
s
)
,for t
∈
[
a
,
b
] and
(40)
(
x
z
)
(
t
;
λ
)
=
(
φ
φ
′
)
(
t
)
,
for
t
∈
[
a
-
h
,
a
]
.
Now let us take the operator A
:
C
(
[
a
-
h
,
b
]
×
J
,
ℝ
2
)
→
C
(
[
a
-
h
,
b
]
×
J
,
ℝ
2
), defined by
(41)
A
(
x
z
)
(
t
;
λ
)
≔
(
A
1
(
x
z
)
A
2
(
x
z
)
)
≔
the
right
hand
side
of
(
39
)
,
h
h
h
h
h
h
h
h
h
h
h
h
for
t
∈
[
a
,
b
]
,
A
(
x
z
)
(
t
;
λ
)
≔
(
A
1
(
x
z
)
A
2
(
x
z
)
)
≔
the
right
hand
side
of
(
40
)
,
h
h
h
h
h
h
h
h
h
h
for
t
∈
[
a
-
h
,
a
]
.
LetX
=
C
(
[
a
-
h
,
b
]
×
J
,
ℝ
2
).
It is clear, from the proof of the Theorem10, that in the condition (C1)–(C5), the operator A
:
(
X
,
∥
·
∥
)
→
(
X
,
∥
·
∥
) is Picard operator.
Let(
x
*
z
*
) be the unique fixed point of A.
Supposing that there exists(
∂
x
*
/
∂
λ
∂
z
*
/
∂
λ
), from (39)-(40), we obtain that
(42)
∂
x
*
∂
λ
=
∫
a
t
‍
∂
z
(
s
;
λ
)
∂
λ
d
s
,
∂
z
*
∂
λ
=
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
1
)
-
1
)
∂
x
(
s
;
λ
)
∂
λ
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
2
)
-
1
)
∂
z
(
s
;
λ
)
∂
λ
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
3
)
-
1
)
∂
x
(
s
-
h
;
λ
)
∂
λ
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
4
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
4
)
-
1
)
∂
z
(
s
-
h
;
λ
)
∂
λ
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
λ
)
-
1
)
d
s
,
for all t
∈
[
a
,
b
], λ
∈
J.
This relation suggests that we consider the following operator:(43)
C
:
X
×
X
⟶
X
,
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
⟶
C
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
,
where C
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
(
t
;
λ
)
=
0 for t
∈
[
a
-
h
,
a
], λ
∈
J, and
(44)
C
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
(
t
;
λ
)
≔
(
C
1
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
C
2
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
)
,
where
(45)
C
1
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
(
t
;
λ
)
≔
∫
a
t
‍
∂
z
(
s
;
λ
)
∂
λ
d
s
,
C
2
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
(
t
;
λ
)
≔
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
1
)
-
1
)
u
1
(
s
;
λ
)
d
s
+
∫
a
t
(

z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1
(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
2
)
-
1
)
u
2
(
s
;
λ
)
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
3
)
-
1
)
u
3
(
s
-
h
;
λ
)
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
u
4
)
-
1
)
u
4
(
s
-
h
;
λ
)
d
s
+
∫
a
t
(
z
(
s
-
h
;
λ
)
;
λ
)
)
×
(
∂
u
1
)
-
1

(
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
z
(
s
-
h
;
λ
)
;
λ
)
∂
f
(
s
,
x
(
s
;
λ
)
,
z
(
s
;
λ
)
,
x
(
s
-
h
;
λ
)
,
)
×
(
∂
λ
)
-
1
)
d
s
,
for t
∈
[
a
,
b
]
,
λ
∈
J. Here we use the notations u
1
(
s
;
λ
)
≔
∂
x
(
s
;
λ
)
/
∂
λ, u
2
(
s
;
λ
)
≔
∂
z
(
s
;
λ
)
/
∂
λ, u
3
(
s
-
h
;
λ
)
≔
∂
x
(
s
-
h
;
λ
)
/
∂
λ, and u
4
(
s
-
h
;
λ
)
≔
∂
z
(
s
-
h
;
λ
)
/
∂
λ.
In this way, we have the triangular operatorD
:
X
×
X
→
X
×
X, (
(
x
12
z
12
)
,
(
u
13
u
24
)
)
→
(
A
(
x
12
z
12
)
,
C
(
(
x
12
z
12
)
,
(
u
13
u
24
)
)
), where A is Picard operator and C
(
(
x
12
z
12
)
,
(
·
·
)
)
:
X
→
X is Q
C-contraction with Q
C
=
(
b
-
a
)
(
0
1
L
1
L
2
).
From Theorem8 the operator D is Picard operator; that is, the sequences
(46)
(
x
12
n
+
1
z
12
n
+
1
)
≔
A
(
x
12
n
z
12
n
)
,
(
u
13
n
+
1
u
24
n
+
1
)
≔
C
(
(
x
12
n
z
12
n
)
,
(
u
13
n
u
24
n
)
)
,
n
∈
ℕ, converge uniformly, with respect to t
∈
[
a
-
h
,
b
]
,
λ
∈
J, to (
(
x
12
n
z
12
n
)
,
(
u
13
n
u
24
n
)
)
∈
F
D, for all (
x
12
0
z
12
0
)
,
(
u
13
0
u
24
0
)
∈
X.
If we take(47)
(
x
12
0
z
12
0
)
=
(
0
0
)
,
(
u
13
0
u
24
0
)
=
(
∂
x
12
0
∂
λ
∂
z
12
0
∂
λ
)
=
(
0
0
)
,
then
(48)
(
u
13
1
u
24
1
)
=
(
∂
x
12
1
∂
λ
∂
z
12
1
∂
λ
)
.
By induction we prove that(49)
(
u
13
n
u
24
n
)
=
(
∂
x
12
n
∂
λ
∂
z
12
n
∂
λ
)
,
∀
n
∈
ℕ
.
So, (
x
12
n
z
12
n
)
⟶
unif
(
x
12
*
z
12
*
), as n
→
∞, and (
∂
x
12
n
/
∂
λ
∂
z
12
n
/
∂
λ
)
⟶
unif
(
u
13
*
u
24
*
), as n
→
∞.
From a Weierstrass argument we get that there exists(50)
(
∂
x
12
*
∂
λ
∂
z
12
*
∂
λ
)
,
i
=
1,2
,
(
∂
x
12
*
∂
λ
∂
z
12
*
∂
λ
)
=
(
u
13
*
u
24
*
)
.


## 3.6. Ulam-Hyers Stability

We start this section by presenting the Ulam-Hyers stability concept (see [15, 16]). For f
∈
C
(
[
a
,
b
]
×
ℝ
4
,
ℝ
), ɛ
>
0, and ψ
∈
C
(
[
a
-
h
,
b
]
,
ℝ
+
), h
>
0, we consider the system
(51)
x
′
(
t
)
=
z
(
t
)
,
t
∈
[
a
,
b
]
,
z
′
(
t
)
=
f
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
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
t
∈
[
a
,
b
]
and the following inequations
(52)
|
x
′
(
t
)
-
z
(
t
)
|
≤
ɛ
,
t
∈
[
a
,
b
]
,
|
z
′
(
t
)
-
f
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
t
-
h
)
)
|
≤
ɛ
,
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
∈
[
a
,
b
]
,
(53)
|
x
′
(
t
)
-
z
(
t
)
|
≤
ψ
(
t
)
,
t
∈
[
a
,
b
]
,
|
z
′
(
t
)
-
f
(
t
,
x
(
t
)
,
z
(
t
)
,
x
(
t
-
h
)
,
z
(
t
-
h
)
)
|
≤
ψ
(
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
t
∈
[
a
,
b
]
.Definition 15.
The system (51) is Ulam-Hyers stable if there exists a real number c
>
0 such that for each ɛ
>
0 and for each solution (
y
w
)
∈
C
2
(
[
a
-
h
,
b
]
,
ℝ
2
) of (52) there exists a solution (
x
z
)
∈
C
2
(
[
a
-
h
,
b
]
,
ℝ
2
) of (51) with
(54)
|
(
y
w
)
(
t
)
-
(
x
z
)
(
t
)
|
≤
c
ɛ
,
∀
t
∈
[
a
-
h
,
b
]
.Theorem 16.
One supposes that(i)
the conditions (C1)–(C3) are satisfied;
(ii)
Q
n
→
0 as n
→
∞, where Q
=
(
b
-
a
)
(
0
1
L
1
L
2
).
Then the system (6) is Ulam-Hyers stable.Proof.
The system (6) is equivalent with the functional integral system (10). We consider the operator E
:
C
(
[
a
-
h
,
b
]
,
ℝ
2
)
→
C
(
[
a
-
h
,
b
]
,
ℝ
2
), defined by E
(
x
z
)
(
t
)
≔ the right hand side of (10), for t
∈
[
a
-
h
,
b
]. So
(55)
(
x
z
)
=
E
(
x
z
)
(
t
)
,
t
∈
[
a
-
h
,
b
]
.
From Theorem10, E is c-WPO with c
=
[
1
-
(
b
-
a
)
(
0
1
L
1
L
2
)
]
-
1. Applying Theorem 7 we obtain that (6) is Ulam-Hyers stable.Remark 17.
Another proof for the above theorem can be done using Gronwall lemma [8, 14–17].



---

*Source: 878395-2014-02-10.xml*