D ifferential
E quations
& A pplications
Volume 5, Number 4 (2013), 501–518
doi:10.7153/dea-05-29
RANDOM SET–VALUED FUNCTIONAL DIFFERENTIAL EQUATIONS
WITH THE SECOND TYPE HUKUHARA DERIVATIVE
H O V U AND L E S I D ONG
(Communicated by Marek T. Malinowski)
Abstract. This paper concerns with the initial value problem for random set-valued functional
differential equations with the second type Hukuhara derivative (RSFDEs). By using the techniques of successive approximations, the existence and uniqueness of solutions are established.
Two kinds of boundedness of the solution are also established. In addition, the problem at least
one solution under some conditions is proven and two examples illustrate the results.
1. Introduction
Functional differential equation (FDE) show the fact that the velocity of the system depends not only on the state of the system at a given instant but depends upon the
history of the trajectory until this instant. The class of differential equations with delay
encompasses a large variety of differential equations. FDE plays an important role in
an increasing number of system models in biology, engineering, physics and other sciences. There exists an extensive literature dealing with functional differential equations
and their applications. We refer to the monograph [14], and references therein.
The study of set differential equations in a semilinear metric space has gained
much attention [20]. Many interesting results in this direction one can find e.g. in
[1, 2, 3, 4, 5, 6, 9, 10, 15, 19, 21, 29, 30, 31, 32, 42, 44, 49, 48, 46, 11]. The set
differential equations have a significant influence in fuzzy differential equations (see
e.g. [18, 21, 23, 7, 25, 16, 43, 8, 17]), random fuzzy differential equations (see e.g.
[26, 27, 28, 33]), set-valued and fuzzy stochastic differential equations (see e.g. [12, 13,
34, 35, 36, 37, 38, 39, 40, 41]). A solution to set differential equations with Hukuhara
derivative defined as in [20], is the Hukuhara differentiable mapping. This implies
that the diameter of the solution values is a nondecreasing function of time (see e.g.
[20]). From the application’s point of view this property can be sometimes inconvenient because it means practically that uncertainty, contained in a model of a physical
system which is described by set differential equations, can only grow as times goes by.
Hence the successive values of modelled phenomenon are covered by nondecreasing
Mathematics subject classification (2010): 34F05, 34G20, 34A12.
Keywords and phrases: Random set-valued variable, set-valued functional differential equations, successive approximation, at least one solution, second type Hukuhara derivative.
This research is supported by Ton Duc Thang University, Ho Chi Minh city, Vietnam.
c
, Zagreb
Paper DEA-05-29
501
502
H O V U AND L E S I D ONG
sets of tolerance. In [47], authors introduced a concept of generalized Hukuhara differentiability of interval-valued mapping, which allows them to obtain the solutions of
interval differential equations with decreasing diameter of solutions values. The papers
[29, 30, 31, 32] present some investigations of the interval differential equations and
set differential equations with the second type Hukuhara derivative. In [18], the authors
proved the global existence of solutions for interval-valued integro-differential equations with initial conditions under generalized H-differentiability. In [2], the authors
established the existence of solutions and some properties of set solutions for a class of
set functional differential equations in a separable Banach space.
In this paper, inspired and motivated by Malinowski [26, 27, 28, 29, 30, 31, 33],
Lupulescu [2, 22, 23, 24],Park and Jeong [45]. We consider the random set-valued functional differential equations with second type Hukuhara derivative. The paper will be
organized as follows. As preliminaries we recall some basic results set-valued mapping,
set-valued stochastic process. In section 3, we concerns with the initial value problem for random set-valued functional differential equations with second type Hukuhara
derivative. By using the techniques of successive approximations, the existence and
uniqueness of solutions are established. Two kinds of boundedness of the solution are
also established. In addition, the problem at least one solution under some conditions
is proven and two examples illustrate the results.
2. Preliminaries
Let Kc (Rd ) denoted the collection of nonempty, compact and convex subsets of
Rd . The following operations can be naturally defined on it:
X + Y = {x + y : x ∈ X, y ∈ Y }; λ X = {λ x : x ∈ X}, λ ∈ R+ .
The Hausdorff metric is defined as
D[X,Y ] = max{supinf d(y, x); sup inf d(x, y)}
y∈Y,x∈X
x∈X,y∈Y
where X,Y are bounded subsets of Rd . It is clear that the Hausdorff metric satisfies
the relations of the ordinary metric.
It is known that (Kc (Rd ), D) is a complete metric space. Moreover, Kc (Rd )
equipped with the above-mentioned natural algebraic operations of addition and nonnegative scalar multiplication becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space. On the other hand, the
Hausdorff metric D is compatible with the operations defined on it as described by the
following properties: for any X,Y, Z,W ∈ Kc (Rd )
D[X + Z,Y + Z] = D[X,Y ],
D[kX, kY ] = |k|D[X,Y ],
D[X + Z,Y + W ] D[X,Y ] + D[Z,W ].
Differ. Equ. Appl. 5 (2013), 501–518.
503
Let X,Y ∈ Kc (Rd ). If there exists a subset Z ∈ Kc (Rd ) such that X = Y + Z ,
then we call Z the Hukuhara difference of X and Y . The set Z we denote by X Y .
Note that X Y = X + (−1)Y .
The problem of the existence of Hukuhara difference X Y is often inconvenient
although it is known ([20]) that for X,Y ∈ Kc (Rd ) the Hukuhara difference X Y
exists iff the following if satisfied: if x belongs to the boundary of X then there exists
a point z such that x ∈ Y + z ⊂ A.
Next, we recall the definition of a derivative which will be used in the paper:
D EFINITION 1. ([29]) A mapping F : [a, b] → Kc (Rd ) is second type Hukuhara
differentiable at t0 ∈ [a, b] if there exists F (t0 ) ∈ Kc (Rd ) such that the limits
lim (−h−1 )(F(t0 − h) F(t0 )), and lim (−h−1 )(F(t0 ) F(t0 + h)),
h→0+
h→0+
exist and are equal to F (t0 ). The set F (t0 ) is said to be the second type Hukuhara
derivative of set-mapping F at the point t0 .
In this definition it is implicit that for all h > 0 (sufficiently small) the Hukuhara differences F(t0 − h) F(t0 ) and F(t0 ) F(t0 + h) have to exist.
D EFINITION 2. The function F : [a, b] → Kc (Rd ) is called second type Hukuhara
differentiable on [a, b] if F is second type Hukuhara differentiable at every point t0 ∈
[a, b].
R EMARK 1. ([29]) Let F : [a, b] → Kc (Rd ) be a second type Hukuhara differentiable on [a, b]. Then
i) F is continuous on [a, b];
ii) the function diam(F) : [a, b] → [0, ∞) is nonincreasing on [a, b].
The mapping F : [a, b] → Kc (Rd ) is said to be integrable if the set S(F) of integrable selectors of F is nonempty. Then
b
a
F(t)dt =
b
a
f (t)dt | f ∈ S(F) .
It is also known that for the integrable set-valued mapping F, G : [a, b] → Kc (Rd ) we
have D[F(·), G(·)] is integrable and the following property is valid
D
a
b
F(t)dt,
b
a
b
G(t)dt D[F(t), G(t)]dt.
a
If a set-valued function F is second type Hukuhara differentiable on [a, b] and F is integrable, then for t ∈ [a, b] ,
F(a) = F(t) + (−1)
t
a
F (s)ds.
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H O V U AND L E S I D ONG
Let (Ω, A , P) be a complete probability space. A set-valued mapping F : Ω → Kc (Rd )
is called a set-valued random variable if
/ ∈ A , for every closed set O ⊂ Rd .
{ω ∈ Ω : F(ω ) ∩ O = 0}
D EFINITION 3. ([26]) A set-valued mapping F : [0, ∞) × Ω → Kc (Rd ) is said to
be set-valued stochastic process if F(·, ω ) is a set-valued function with any fixed ω ∈ Ω
and F(t, ·) is a set-valued random variable for any fixed t ∈ [0, ∞).
For a positive number σ , we denote by Cσ the space C([−σ , 0], Kc (Rd )). Also
we denote by
Dσ [X,Y ] = sup D[X(t),Y (t)],
t∈[−σ ,0]
the metric on the space Cσ . Let X ∈ C([−σ , b], Kc (Rd )). Then for each t ∈ [0, b] we
denote by Xt ∈ Cσ defined by Xt = X(t + s), s ∈ [−σ , 0].
Assume that F : [0, b] × Ω × Cσ → Kc (Rd ) satisfies the following hypotheses:
(F1) F· (t, ϕ ) : Ω → Kc (Rd ) is a set-valued random variable for t ∈ [0, b], ϕ ∈ Cσ .
(F2) with P.1 the mapping Fω (·, ·) : [0, b] ×Cσ → Kc (Rd ) is a continuous set-valued
mapping at every at (t0 , ϕ0 ) ∈ [0, b] ×Cσ , i.e., there exists Ω∗ ⊂ Ω with P(Ω∗ ) =
1 such that for every ω ∈ Ω∗ the following is true : for every ε > 0 there exists
δ > 0 such that for every t ∈ [0, b] and ϕ , ϕ0 ∈ Cσ it holds
max{|t − t0 |, Dσ [ϕ , ϕ0 ]} < δ =⇒ D[Fω (t, ϕ ), Fω (t0 , ϕ0 )] < ε .
For convenience, from now on, the fact that there that exists Ω∗ ⊂ Ω such that
P(Ω∗ ) = 1 and for every ω ∈ Ω∗ it holds X(ω ) = Y (ω ), where X,Y random eleP.1
ments, will be written as X(ω ) = Y (ω ). Similarly, for the inequalities. Also if there
exists Ω∗ ⊂ Ω such that P(Ω∗ ) = 1 and for every fixed ω ∈ Ω∗ it holds X(t, ω ) =
Y (t, ω ) for every t ∈ [−σ , b], where X,Y are stochastic process, then we will write
[−σ ,b] P.1
X(t, ω ) = Y (t, ω ) in short, or X(t, ω ) = Y (t, ω ) for every t ∈ [−σ , b] with P.1 .
Similarly, for the inequality.
3. Main results
In this paper, we will consider the random set-valued functional differential
equation as follows:
⎧
P.1
⎨ X (t, ω ) [0,b],
= Fω (t, Xt )
(3.1)
,0], P.1
⎩ X(t, ω ) [−σ =
ϕ (t, ω )
where F : [0, b] × Ω ×Cσ → Kc (Rd ) is a set-valued stochastic process and the symbol
denotes the derivative from Definition 1.
505
Differ. Equ. Appl. 5 (2013), 501–518.
D EFINITION 4. By the solution of (3.1) we mean a continuous set-valued stochastic process X : [−σ , b] × Ω → Kc (Rd ) that satisfies (3.1). A solution X is unique if it
holds
[−σ ,b] P.1
D[X(t, ω ),Y (t, ω )] = 0
for any set-valued stochastic process Y : [−σ , b] × Ω → Kc (Rd ) which is a solution of
(3.1).
L EMMA 1. A set-valued mapping X : [−σ , b] × Ω → Kc (Rd ) is a solution to the
problem (3.1) if and only if X is a continuous set-valued stochastic process and X
satisfies the following set-valued stochastic integral equation
⎧
,0], P.1
⎨ X(t, ω ) [−σ =
ϕ (t, ω ),
(3.2)
P.1
⎩ ϕ (0, ω ) [0,b],
= X(t, ω ) + (−1) t F (s, X )ds.
0 ω
s
T HEOREM 1. Assume that F : [0, b] × Ω ×Cσ → Kc (Rd ) satisfies (F1) - (F2) and
with P.1 for every t ∈ [0, b] and every ϕ , ψ ∈ Cσ it holds
D[Fω (t, ϕ ), Fω (t, ψ )] L(t, ω )Dσ [ϕ , ψ ],
(3.3)
where L : [0, b] × Ω → (0, ∞) such that L(·, ω ) is continuous with P.1 . Suppose that
there exist non-negative constants γ and Q such that
D[Fω (t, ϕ ), {0}]
[0,b]×Cσ , P.1
Q,
(3.4)
n
d
and the sequence {X}∞
n=0 , X : [−σ , γ ] × Ω → Kc (R ) given by
ϕ (t, ω ) for [−σ , 0],
0
X (t, ω ) =
ϕ (0, ω ) for [0, γ ],
and for n = 1, 2, ...
n
X (t, ω ) =
ϕ (t, ω )
ϕ (0, ω ) (−1) 0t Fω (s, Xsn−1 )ds
for [−σ , 0],
for [0, γ ],
(3.5)
(3.6)
is well defined (i.e. the foregoing H-differences do exist). Then there exists a constant
ϑ > 0 such that the random set-valued functional differential equation (3.1) has a
unique local solution X(t, ω ) on the interval [0, ϑ ].
Proof. To prove the theorem we shall use the method of successive approxima1
} . Then for t ∈ [0, ϑ ]
tions. Let us define θ ∈ Cσ by θ ≡ {0} and set ϑ = min{b, γ ,
2Q
we have
D[Fω (t, Xt0 ), {0}] D[Fω (t, Xt0 ), Fω (t, θ )] + D[Fω (t, θ ), {0}]
506
H O V U AND L E S I D ONG
L(t, ω )Dσ [Xt0 (·, ω ), θ ] + D[Fω (t, θ ), {0}]
L(t, ω ) sup D[X 0 (t, ω ), θ ] + D[Fω (t, θ ), {0}]
t∈[0,ϑ ]
LM + K < ∞,
where
P.1
M = sup D[X 0 (t, ω ), θ ], K
[0,ϑ ], P.1
=
t∈[0,ϑ ]
P.1
sup D[F(t, θ ), {0}] and L = sup L(t, ω ).
t∈[0,ϑ ]
t∈[0,ϑ ]
For t ∈ [0, ϑ ], from (3.4) and (3.5), it is easy to see that
t
D[X 1 (t, ω ), X 0 (t, ω )] = D ϕ (0, ω ) (−1) Fω (s, Xs0 )ds, ϕ (0, ω )
0
t
Fω (s, Xs0 )ds, {0}
D
t
0
D[Fω (s, Xs0 ), {0}]ds
0
[0,ϑ ], P.1
Qt
Qϑ .
where Q = LM + K .
Using the conditions (3.3)-(3.6), we have
t
t
D[X 2 (t, ω ), X 1 (t, ω )] = D (−1) Fω (s, Xs1 )ds, (−1) Fω (s, Xs0 )ds
0
0
t
t
1
0
Fω (s, Xs )ds, Fω (s, Xs )ds
=D
0
[0,ϑ ],P.1 t
0
[0,ϑ ],P.1 t
[0,ϑ ],P.1
=
[0,ϑ ],P.1
=
[0,ϑ ],P.1
0
0
D[Fω (s, Xs1 ), Fω (s, Xs0 )]ds
L(s, ω )Dσ [Xs1 (·, ω ), Xs0 (·, ω )]ds
L(ω )
L(ω )
t
sup D[X 1 (s + r, ω ), X 0 (s + r, ω )]ds
0 r∈[−σ ,0]
t
sup D[X 1 (ι , ω ), X 0 (ι , ω )]ds
0 ι ∈[s−σ ,s]
Q [L(ω )t]2
·
,
L(ω )
2!
where L(ω ) = sup L(t, ω ).
t∈[0,ϑ ]
Further for every n > 2 and t ∈ [0, ϑ ] we obtain
t
t
D[X n (t, ω ), X n−1 (t, ω )] = D (−1) Fω (s, Xsn−1 )ds, (−1) Fω (s, Xsn−2 )ds
0
0
507
Differ. Equ. Appl. 5 (2013), 501–518.
t
D[Fω (s, Xsn−1 ), Fω (s, Xsn−2 )]ds
0
[0,ϑ ],P.1
L(ω )
= L(ω )
= L(ω )
= L(ω )
t
t
0
Dσ [Xsn−1 (·, ω ), Xsn−2 (·, ω )]ds
sup D[Xsn−1 (r, ω ), Xsn−2 (r, ω )]ds
0 r∈[−σ ,0]
t
sup D[X n−1 (r + s, ω ), X n−2 (r + s, ω )]ds
0 r∈[−σ ,0]
t
sup D[X n−1 (ι , ω ), X n−2 (ι , ω )]ds.
0 ι ∈[s−σ ,s]
If we assume that
D[X n−1(t, ω ), X n−2 (t, ω )]
[0,ϑ ],P.1
Q [L(ω )t]n−1
·
.
L(ω ) (n − 1)!
Thus, by mathematical induction, for n ∈ N and t ∈ [0, ϑ ]
D[X n (t, ω ), X n−1 (t, ω )]
[0,ϑ ],P.1 t
Q [L(ω )s]n−1
·
ds
L(ω ) (n − 1)!
0
Q [L(ω )ϑ ]n
·
.
L(ω )
n!
L(ω )
(3.7)
It is easy to see that for n ∈ N the functions X n (·, ω ) : [−σ , ϑ ] → Kc (Rd ) are continuous with P.1 .
Now, for any n ∈ N and t ∈ [0, ϑ ] we shall show that the sequence {X n(t, ω )} is a
Cauchy sequence uniformly in t with P.1 and then {X n (·, ω )} is uniformly convergent
with P.1 . For n > m > 0 , from (3.7) we obtain
n
∑
D[X n (t, ω ), X m (t, ω )] D[X k+1 (t, ω ), X k (t, ω )]
[0,ϑ ],P.1
k=m
Q
n
∑
k=m
Lk (ω )
ϑ k+1
(k + 1)!
(3.8)
ϑn
with P.1 , then for each
n!
n=1
ε > 0 there exists n0 ∈ N large enough such that n, m n0
∞
The almost sure convergence of the series ∑ Ln−1 (ω )
sup D[X n (t, ω ), X m (t, ω )] ε .
(3.9)
t∈[0,ϑ ]
Since (Kc (Rd ), D) is a complete metric space, it follows that there exists Ω∗ ⊂ Ω such
that P(Ω∗ ) = 1 and for every ω ∈ Ω∗ the sequence {X n (·, ω )} is uniformly convergent.
We shall show that X(t, ω ) is a solution of (3.1). For any ε > 0 , there is n0 large
enough such that for every n n0 , n ∈ N we derive
t
t
D
Fω (s, Xsn )ds, Fω (s, Xs )ds
0
0
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H O V U AND L E S I D ONG
t
D[Fω (s, Xsn ), Fω (s, Xs )]ds
0
[0,ϑ ],P.1
[0,ϑ ],P.1
=
[0,ϑ ],P.1
=
[0,ϑ ],P.1
=
L(ω )
L(ω )
L(ω )
L(ω )
t
0
Dσ [Xsn (·, ω ), Xs (·, ω )]ds
t
sup D[Xsn (r, ω ), Xs (r, ω )]ds
0 r∈[−σ ,0]
t
sup D[X n (r + s, ω ), X(r + s, ω )]ds
0 r∈[−σ ,0]
t
P.1
sup D[X n (ι , ω ), X(ι , ω )]ds → 0.
0 ι ∈[s−σ ,s]
Hence, by virtue of Lebesgue Dominated Convergence theorem,
t
t
D
Fω (s, Xsn )ds, Fω (s, Xs )ds → 0,
0
0
as n → ∞ for any t ∈ [0, ϑ ], with P.1 .
Consequently, we have
t
D ϕ (0, ω ), X(t, ω ) + (−1) Fω (s, Xs )ds
0
[0,ϑ ],P.1
[0,ϑ ],P.1
D[X n (t, ω ), X(t, ω )] + D
0
t
Fω (s, Xsn−1 )ds,
sup D[X n (t, ω ), X(t, ω )] + D
t∈[0,b]
0
t
t
0
Fω (s, Xs )ds
Fω (s, Xsn−1 )ds,
t
0
Fω (s, Xs )ds
Thus, in view of the two previous convergence results and the fact that the second term
of the right-hand side is equal to zero, we have
t
[0,ϑ ],P.1
= 0.
D ϕ (0, ω ), X(t, ω ) + (−1) Fω (s, Xs )ds
0
Hence, X(t, ω ) is the solution of (3.1). By Lemma 1, we have that X(t, ω ) is a
solution of (3.1).
Finally, we prove the uniqueness of the solution (3.2). Let us assume that X,Y :
[−σ , ϑ ] × Ω → Kc (Rd ) are two continuous set-valued stochastic process which are
solutions of (3.1).
Then we have
t
D[Fω (s, Xs ), Fω (s,Ys )]ds
D X(t, ω ),Y (t, ω ) 0
[0,ϑ ],P.1
[0,ϑ ],P.1
=
L(ω )
L(ω )
t
0
t
Dσ [Xs (·, ω ),Ys (·, ω )]ds
sup D[X(r + s, ω ),Y (r + s, ω )]ds
0 r∈[−σ ,0]
509
Differ. Equ. Appl. 5 (2013), 501–518.
[0,ϑ ],P.1
L(ω )
=
t
sup D[X(ι , ω ),Y (ι , ω )]ds
0 ι ∈[s−σ ,s]
(3.10)
If we take
ξ (s, ω ) =
sup D[X(ι , ω ),Y (ι , ω )]
ι ∈[s−σ ,s]
for any s ∈ [0,t] , then from (3.10), we have
ξ (t, ω )
[0,ϑ ],P.1
L(ω )
t
0
ξ (s, ω )ds
Applying Lemma 2 in ([26]), we obtain ξ (t, ω ) = 0 on t ∈ [0, ϑ ] with P.1. Hence,
X(t, ω )
[0,ϑ ],P.1
=
Y (t, ω ).
This completes the proof.
T HEOREM 2. Let F : [0, b] × Ω ×Cσ → Kc (Rd ) satisfies the conditions of Theorem 1, ϕ ∈ Cσ and let X : [−σ , ϑ ] × Ω → Kc (Rd ) be the solution of (3.1). Then we
have
sup D[X(t, ω ), {0}]
t∈[0,ϑ ]
[0,ϑ ],P.1 D[ϕ (0, ω ), {0}] + K ϑ + bL(ω )D[ϕ (0, ω ), {0}] exp(tL(ω )),
where θ ∈ Cσ is such that θ ≡ {0} and K = sup D[Fω (t, θ ), {0}].
t∈[0,ϑ ]
Proof. Since X(t, ω ) is the solution of (3.1), by Lemma 1, for t ∈ [0, ϑ ] we have
t
D[X(t, ω ), {0}] = D ϕ (0, ω ) (−1) Fω (s, Xs )ds, {0}
0
[0,ϑ ],P.1
D[ϕ (0, ω ), {0}] +
D[ϕ (0, ω ), {0}] +
+
[0,ϑ ],P.1
[0,ϑ ],P.1
=
[0,ϑ ],P.1
=
t
0
t
0
t
0
D[Fω (s, Xs ), {0}]ds
D[Fω (s, Xs ), Fω (s, θ )]ds
D[Fω (s, θ ), {0}]ds
D[ϕ (0, ω ), {0}] + Kt + L(ω )
D[ϕ (0, ω ), {0}] + Kt + L(ω )
D[ϕ (0, ω ), {0}] + Kt + L(ω )
t
0
t
Dσ [Xs (·, ω ), θ ]ds
sup D[Xs (r, ω ), θ ]ds
0 r∈[−σ ,0]
t
sup D[X(ι , ω ), θ ]ds
0 ι ∈[s−σ ,s]
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H O V U AND L E S I D ONG
[0,ϑ ],P.1
D[ϕ (0, ω ), {0}] + Kt + L(ω )
+ L(ω )
[0,ϑ ],P.1
t
t
sup
0 ι ∈[s−σ ,0]
D[X(ι , ω ), θ ]ds
sup D[X(ι , ω ), θ ]ds
0 ι ∈[0,s]
D[ϕ (0, ω ), {0}] + K ϑ + ϑ L(ω )D[ϕ (0, ω ), θ ]
+ L(ω )
t
sup D[X(ι , ω ), θ ]ds.
0 ι ∈[0,s]
Thus we infer that for t ∈ [0, ϑ ] it holds
sup D[X(ι , ω ), {0}]
[0,ϑ ],P.1
ι ∈[0,t]
D[ϕ (0, ω ), {0}] + K ϑ + ϑ L(ω )D[ϕ (0, ω ), θ ]
+ L(ω )
t
sup D[X(ι , ω ), θ ]ds.
0 ι ∈[0,s]
Applying Lemma 2 in ([26]), we obtain
sup D[X(ι , ω ), {0}]
ι ∈[0,t]
[0,ϑ ],P.1 D[ϕ (0, ω ), {0}] + K ϑ + ϑ L(ω )D[ϕ (0, ω ), θ ] exp(tL(ω )).
This completes the proof.
T HEOREM 3. Assume that F : [0, b] × Ω × Cσ → Kc (Rd ) satisfies the conditions
of Theorem 1, ϕ , ψ ∈ Cσ and let X,Y : [−σ , ϑ ] × Ω → Kc (Rd ) be the solutions of
(3.1) with X(t, ω ) = ϕ (t, ω ) and Y (t, ω ) = ψ (t, ω ) for t ∈ [−σ , 0]. Then we have
sup D[X(t, ω ),Y (t, ω )]
t∈[0,ϑ ]
[0,ϑ ],P.1 D[ϕ (0, ω ), ψ (0, ω )] + ϑ L(ω )D[ϕ (0, ω ), ψ (0, ω )] exp(tL(ω )).
Proof. Since X(t, ω ) and Y (t, ω ) are the solutions of (3.1), we obtain
D[X(t,ω ),Y (t, ω )]
t
t
= D ϕ (0, ω ) (−1) Fω (s, Xs )ds, ψ (0, ω ) (−1) Fω (s,Ys )ds
0
[0,ϑ ],P.1
[0,ϑ ],P.1
D[ϕ (0, ω ), ψ (0, ω )] +
t
0
0
D Fω (s, Xs ), Fω (s,Ys ) ds
D[ϕ (0, ω ), ψ (0, ω )] + L(ω )
t
0
Dσ [Xs (·, ω ),Ys (·, ω )]ds
Differ. Equ. Appl. 5 (2013), 501–518.
[0,ϑ ],P.1
[0,ϑ ],P.1
=
[0,ϑ ],P.1
=
[0,ϑ ],P.1
=
D[ϕ (0, ω ), ψ (0, ω )] + L(ω )
D[ϕ (0, ω ), ψ (0, ω )] + L(ω )
D[ϕ (0, ω ), ψ (0, ω )] + L(ω )
D[ϕ (0, ω ), ψ (0, ω )] + L(ω )
+ L(ω )
[0,ϑ ],P.1
t
t
511
sup D[Xs (r, ω ),Ys (r, ω )]ds
0 r∈[−σ ,0]
t
sup D[X(r + s, ω ),Y (r + s, ω )]ds
0 r∈[−σ ,0]
t
sup D[X(ι , ω ),Y (ι , ω )]ds
0 ι ∈[s−σ ,s]
t
sup
0 ι ∈[s−σ ,0]
D[X(ι , ω ),Y (ι , ω )]ds
sup D[X(ι , ω ),Y (ι , ω )]ds
0 ι ∈[0,s]
D[ϕ (0, ω ), ψ (0, ω )] + bL(ω )D[ϕ (0, ω ), ψ (0, ω )]
+ L(ω )
t
sup D[X(ι , ω ),Y (ι , ω )]ds.
0 ι ∈[0,s]
Applying Lemma 2 in ([26]), we obtain
sup D[X(ι , ω ),Y (ι , ω )]
ι ∈[0,t]
[0,ϑ ],P.1 D[ϕ (0, ω ), ψ (0, ω )] + ϑ L(ω )D[ϕ (0, ω ), ψ (0, ω )] exp(tL(ω )),
for t ∈ [0, ϑ ] . This completes the proof.
T HEOREM 4. Suppose that the function F : [0, b] × Ω → Kc (Rd ) satisfies the
conditions of Theorem 1. Assume that there exists a real-valued stochastic process
f : [0, b] × Ω → [0, ∞) satisfying
b
0
P.1
f (t, ω )dt K with K > 0 , and such that with P.1
P.1
D[Fω (t, ϕ ), {0}] f (t, ω )
for every t ∈ [0, b] and every ϕ ∈ Cσ . Suppose that there exists a non-negative constant
n
d
r < b such that for t ∈ [0, r] the sequence {X n }∞
n=0 , X : [−σ , r] × Ω → Kc (R ) given
by
if t ∈ [−σ , 0],
P.1 ϕ (t, ω )
0
X (t, ω ) =
ϕ (0, ω ) if t ∈ [0, r],
and for n = 1, 2, . . .
⎧
⎪
⎨ϕ (t, ω )
P.1
X n (t, ω ) = ϕ (0, ω )
⎪
⎩
ϕ (0, ω ) (−1) 0t Fω (s, Xsn )ds
if t ∈ [−σ , 0],
if t ∈ [0, r]n1 ,
if t ∈ [0, r]n2 ∪ . . . ∪ [0, r]nn ,
512
H O V U AND L E S I D ONG
is well defined (i.e. the foregoing H-differences do exist). Then there exists at least one
solution X(t, ω ) to the random set-valued differential equation (3.1) on the interval
[0, r].
Proof. Let us now define for n ∈ N the interval
[0, r]nk =
We have
n
k r; r , k = 1, n.
n
n
k −1
[0, r]nk = [0, r] for n ∈ N..
k=1
Let us observe that
P.1
D[X n (t, ω ), {0}] sup D[ϕ (t, ω ), {0}] +
t∈[−σ ,0]
P.1
sup D[ϕ (t, ω ), {0}] +
t∈[−σ ,0]
P.1
sup D[ϕ (t, ω ), {0}] +
t∈[−σ ,0]
t−r/n
0
t−r/n
0
r
0
D[Fω (s, Xsn ), {0}]ds
f (s, ω )ds
f (s, ω )ds < ∞.
where θ ∈ Cσ is such that θ ≡ {0} . Therefore the sequence {X n (·, ω )} is uniformly
bounded with P.1 .
For t1 ,t2 ∈ [0, r], ω ∈ Ω and n 2 we have
n
n
P.1
D[X (t1 , ω ), X (t2 , ω )] P.1
max{t1 ,t2 }−r/n
min{t1 ,t2 }−r/n
max{t1 ,t2 }−r/n
min{t1 ,t2 }−r/n
D[Fω (s, Xsn ), {0}]ds
f (s, ω )ds.
Hence, if |t1 − t2 | → 0 then D[X n (t1 , ω ), X n (t2 , ω )] → 0 with P.1 . This implies the
sequence {X n (·, ω )} is equi-continuous with P.1 .
By Arzela- Ascoli theorem, then there exists a subsequence {X nm } ⊂ X n which
is uniformly convergent to some X : [−σ , r] × Ω → Kc (Rd ) with P.1 , i.e, there exists
Ω∗ ⊂ Ω with P(Ω∗ ) = 1 such that for every ω ∈ Ω it holds
sup D[X nm (t, ω ), X(t, ω )] → 0, as m → ∞.
t∈[0,r]
We shall show that X : [−σ , r] × Ω → Kc (Rd ) is a solution to (3.1). Let {nm } ⊂ N be
r
the sequence defined in the preceding steps. For (t, ω ) ∈ [ ; r] × Ω we have
nm
P.1
X nm (t, ω ) = ϕ (0, ω ) (−1)
t−r/nm
0
Fω (s, Xsnm )ds.
513
Differ. Equ. Appl. 5 (2013), 501–518.
Let us notice the following: for every (t, ω ) ∈ [0, r] × Ω, there exists m0 ∈ N such
that for every m m0 , we can write
D[ϕ (0, ω ),X(t, ω ) + (−1)
t
Fω (s, Xs )ds]
0
nm
D[ϕ (0, ω ), X (t, ω ) + (−1)
+ D X nm (t, ω ) + (−1)
+ (−1)
t
0
t−r/nm
0
t−r/nm
0
Fω (s, Xsnm )ds]
Fω (s, Xsnm )ds, X(t, ω )
Fω (s, Xs )ds
nm
D[X (t, ω ), X(t, ω )] + D
t−r/nm
0
Fω (s, Xsnm )ds,
t
0
Fω (s, Xs )ds .
The first term of the right-hand side of the inequality uniformly converges to zero with
P.1 . It remains to show that the second summand converges to zero. Let us observe
that
t
t−r/nm
D
Fω (s, Xsnm )ds, Fω (s, Xs )ds
0
D
0
t−r/nm
0
+D
[0,r], P.1
[0,r], P.1
[0,r], P.1
[0,r], P.1
=
L(ω )
L(ω )
L(ω )
L(ω )
Fω (s, Xsnm )ds,
t−r/nm
0
t−r/nm
0
t−r/nm
0
t−r/nm
0
t−r/nm
0
t−r/nm
0
Fω (s, Xs )ds,
Fω (s, Xs )ds
t
0
Fω (s, Xs )ds
Dσ [Fω (s, Xsnm ), Fω (s, Xs )]ds + D
Dσ [Xsnm (·, ω ), Xs (·, ω )]ds +
t
t
t−r/nm
t−r/nm
sup D[X nm (ι , ω ), X(ι , ω )]ds +
ι ∈[s−σ ,s]
As
D[X nm (t, ω ), X(t, ω )]
[0,r], P.1
D[Fω (s, Xs ), {0}]ds
sup D[X nm (r + s, ω ), X(r + s, ω )]ds +
r∈[−σ ,0]
Fω (s, Xs )ds, {0}
t
t
t−r/nm
t−r/nm
f (t, ω )ds
f (t, ω )ds.
2 f (t, ω )
we have Lebesgue Dominated Convergence Theorem that
t
sup D[X nm (ι , ω ), X(ι , ω )]ds → 0, as m → ∞, for every t ∈ [0, r] with P.1,
0 ι ∈[s−σ ,s]
and
t
t−r/nm
f (t, ω )ds → 0 as m → ∞, with P.1,
514
H O V U AND L E S I D ONG
for every t ∈ [0, r], ω ∈ Ω. Hence, we obtain
[0,r],P.1
ϕ (0, ω ) = X(t, ω ) + (−1)
t
0
Fω (s, Xsn )ds.
This completes the proof.
4. Illustrative examples
In this section we consider two examples which illustrate the main results of the
paper.
E XAMPLE 1. Let us consider a class of random set-valued functional differential equations with distributed delay. For m ∈ N and times 0 < σ1 < . . . < σm < σ .
Consider the problem initial condition as follows:
⎧
m
⎨X (t, ω ) P.1
= −0 σ G0 (s, ω , X(t + s, ω ) + ∑ Gi (s, ω , X(t − σi , ω ) for t ∈ [0, b],
⎩
i=1
P.1
X(t, ω ) = ϕ (t, ω )
for t ∈ [−σ , 0]
(4.1)
where (Ω, A , P) is a complete probability space, and
Cσ = C([−σ , 0] × Ω; Kc(Rd )),
X : [−σ , b] × Ω → Kc (Rd ),
Gi : [0, b] × Ω × Cσ → Kc (Rd ), i = 1, m
are some set-valued random mappings.
Assume that Gi,ω : [0, b] × Ω → Kc (Rd ), i = 1, m satisfy the following hypotheses:
(G1) Gi,· : Ω → Kc (Rd ), i = 1, m are set-valued random variables for t ∈ [0, b], ϕ ∈
Cσ and ϕ is a set-valued stochastic process,
(G2) with P.1 the mapping Gi,ω : [0, b] × Ω → Kc (Rd ), i = 1, m are continuous setvalued mapping for every (t0 , ϕ0 ) ∈ [0, b] × Cσ and ω ∈ Ω,
(G3) there exist stochastic processes Li : [0, b] × Ω → (0, ∞) such that Li (·, ω ) with
P.1 and for i = 1, m, t ∈ [0, b], ϕ , ψ ∈ Cσ ,
P.1
D[Gi,ω (t, ϕ ), Gi,ω (t, ψ )] Li (t, ω )Dσ [ϕ , ψ ]
(G4) for ϕ ∈ Cσ ,
P.1
D[Gi,ω (t, ϕ ), {0}] Qi ,
where Qi > 0 , i = 1, m.
Differ. Equ. Appl. 5 (2013), 501–518.
515
P ROPOSITION 1. Assume that Gi,ω : [0, b] × Ω → Kc (Rd ), i = 1, m satisfy assumptions (G1)-(G4), then the problem (4.1) has a unique solution.
Proof. It is easy to prove assumptions (F1)-(F2) are satisfied from (G1)-(G2). By
(G3) we have the right-hand side of (4.1) satisfies (3.3), and (G4) satisfies (3.4). This
completes the proof.
E XAMPLE 2. Suppose that X : [−1, 1] × Ω → Kc (R) and Ω = (0, π /2). Let us
consider the following random interval-valued functional differential equation:
⎧
P.1
⎨X (t, ω ) [0,1],
= −X(t − 1, ω ),
(4.2)
P.1
⎩X(t, ω ) [−1,0],
= [−1, 1 − 2t] cos ω .
P.1
Let us denote X(t, ω ) = [X(t, ω ), X(t, ω )] for each t . Then it holds
P.1
X (t, ω ) = [X (t, ω ), X (t, ω )].
Therefore we arrive to the system random interval-valued functional differential equation:
⎧
[0,1], P.1
⎪
X (t, ω ) = −X(t − 1, ω ),
⎪
⎪
⎪
⎪
P.1
⎨ X (t, ω ) [0,1],
= −X(t − 1, ω ),
(4.3)
[−1,0], P.1
⎪
⎪
X(t,
ω
)
=
−
cos
ω
,
⎪
⎪
⎪
⎩
[−1,0], P.1
X(t, ω ) = (1 − 2t) cos ω .
By solving (4.3), then the solution of the system random interval-valued functional
differential equation (4.2) is derived as follows:
[−1, 1 − 2t] cos ω ,
for t ∈ [−1, 0],
(4.4)
X(t, ω ) =
2
[−1 − t, 1 − 3t + t ] cos ω , for t ∈ [0, 1].
The boundaries of X together with the solution (4.2) are illustrated in Figure 1 on the
next page.
Acknowledgements. The author would like to express his gratitude to the Editorin-Chief, the Associate Editor and the anonymous referees for their helpful comments.
516
H O V U AND L E S I D ONG
2.5
\.underline{x}
\.overline{x}
2
1.5
x(t,pi/4)
1
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
t
0.5
1
Figure 1: Graph of the solution to (4.4).
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(Received April 16, 2013)
(Revised September 3, 2013)
Ho Vu
Division of Computational Mathematics and Engineering
Ton Duc Thang University
Ho Chi Minh city
Vietnam
e-mail: [email protected],[email protected]
Le Si Dong
Faculty of Basic Education
Banking University
Ho Chi Minh city
Vietnam
e-mail: [email protected]
Differential Equations & Applications
www.ele-math.com
[email protected]