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Interval-valued functional integro-differential equations

  • Ngo Van Hoa1,
  • Nguyen Dinh Phu2,
  • Tran Thanh Tung3Email author and
  • Le Thanh Quang2
Advances in Difference Equations20142014:177

https://doi.org/10.1186/1687-1847-2014-177

Received: 16 February 2014

Accepted: 28 June 2014

Published: 22 July 2014

Abstract

This paper is devoted to studying the local and global existence and uniqueness results for interval-valued functional integro-differential equations (IFIDEs). In the paper, for the local existence and uniqueness, the method of successive approximations is used and for the global existence and uniqueness, the contraction principle is a good tool in investigating. Some examples are given to illustrate the results.

MSC:34G20, 34A12, 34K30.

Keywords

interval-valued differential equationsgeneralized Hukuhara derivativefunctional integro-differential equations

1 Introduction

Functional differential equations (or, as they are called, delay differential equations) play an important role in an increasing number of system models in biology, engineering, physics and other sciences. There exists an extensive amount of literature dealing with functional differential equations and their applications; the reader is referred to the monographs [16] and the references therein.

The set-valued differential and integral equations are an important part of the theory of set-valued analysis. They have an important value in theory and application in control theory; and they were studied in 1969 by De Blasi and Iervolino [7]. Recently, set-valued differential equations have been studied by many authors due to their application in many areas. For many results in the theory of set-valued differential and integral equations, the readers can be referred to the following books and papers [823] and the references therein. The interval-valued analysis and interval-valued differential equations (IDEs) are the particular cases of the set-valued analysis and set differential equations, respectively. In many cases, when modeling real-world phenomena, information about the behavior of a dynamic system is uncertain, and we have to consider these uncertainties to gain more models. The interval-valued differential and integro-differential equations can be used to model dynamic systems subject to uncertainties. Recently, many works have been done by several authors in the theory of interval-valued differential equations (see, e.g., [2426]). These equations can be studied with a framework of the Hukuhara derivative [27]. However, it causes that the solutions have increasing length of their values. Stefanini and Bede [26] proposed to consider the so-called strongly generalized derivative of interval-valued functions. The interval-valued differential equations with this derivative can have solutions with decreasing length of their values. This approach was the starting point for the topic of interval-valued differential equations (see [24, 25]). Besides that, some very important extensions of the interval-valued differential equations are the set differential equations (see [6, 8, 1114, 16, 18, 20, 23, 2831]).

The connection between the fuzzy analysis and the interval analysis is very well known (Moore and Lodwick [32]). Interval analysis and fuzzy analysis were introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena. Based on the results in [33], there are some very important extensions, and the development related to the subject of the present paper is in the field of fuzzy sets, i.e., fuzzy calculus and fuzzy differential equations under generalized Hukuhara derivative. Recently, several works, e.g., [5, 9, 10, 30, 3445], have been done on fuzzy differential equations and fuzzy integro-differential equations, the fuzzy stochastic differential equations [4651], fractional fuzzy differential equations [30, 5255], and some methods for solving fuzzy differential equations [56, 57].

In the papers [2426], one can find the studies on interval-valued differential equations under generalized Hukuhara differentiability, i.e., equations of the form
D H g X ( t ) = F ( t , X ( t ) ) , X ( t 0 ) = X 0 K C ( R ) , t [ t 0 , t 0 + p ] ,
(1.1)

where D H g denotes two kinds of derivatives, namely the classical Hukuhara derivative and the second-type Hukuhara derivative (generalized Hukuhara differentiability). The existence and uniqueness of a Cauchy problem is then obtained under an assumption that the coefficients satisfy a condition with the Lipschitz constant (see [26]). The proof is based on the application of the Banach fixed point theorem. In [25], under the generalized Lipschitz condition, Malinowski obtained the existence and uniqueness of solutions to both kinds of IDEs. In this paper, we study two kinds of solutions to IFIDEs. The different types of solutions to IFIDEs are generated by the usage of two different concepts of interval-valued derivative. Furthermore, in [5], Lupulescu established the local and global existence and uniqueness results for fuzzy functional differential equations. Malinowski [6] studied the existence and uniqueness result of solution to the delay set-valued differential equations under the condition that the right-hand side of an equation is Lipschitzian in the functional variable. Inspired and motivated by the results of Stefanini and Bede [26], Malinowski [24, 25] and Lupulescu [5], we consider the interval-valued functional integro-differential equations under generalized Hukuhara derivative. The paper is organized as follows. As preliminaries, we recall some basic concepts and notations about interval analysis and interval-valued differential equations. In Section 3, we present the local and global existence and uniqueness theorem of solution of IFIDEs under generalized Hukuhara derivatives. In the last section, we give some examples as simple illustrations of the theory of interval-valued functional integro-differential equations.

2 Preliminaries

Let us denote by K C ( R ) the set of any nonempty compact intervals of the real line . The addition and scalar multiplication in K C ( R ) are defined as usual, i.e., for A , B K C ( R ) , A = [ A ̲ , A ¯ ] , B = [ B ̲ , B ¯ ] , where A ̲ A ¯ , B ̲ B ¯ , and λ 0 , then we have
A + B = [ A ̲ + B ̲ , A ¯ + B ¯ ] , λ A = [ λ A ̲ , λ A ¯ ] ( λ A = [ λ A ¯ , λ A ̲ ] ) .
Furthermore, let A K C ( R ) , λ 1 , λ 2 , λ 3 , λ 4 , R and λ 3 λ 4 0 , then we have λ 1 ( λ 2 A ) = ( λ 1 λ 2 ) A and ( λ 3 + λ 4 ) A = λ 3 A + λ 4 A . Let A , B K C ( R ) as above, the Hausdorff metric H in K C ( R ) is defined as follows:
H [ A , B ] = max { | A ̲ B ̲ | , | A ¯ B ¯ | } .
(2.1)
It is known that ( K C ( R ) , H ) is a complete, separable and locally compact metric space. We define the magnitude and the length of A K C ( R ) by
H [ A , 0 ] = A = max { | A ̲ | , | A ¯ | } , len ( A ) = A ¯ A ̲ ,

respectively, where 0 is the zero element of K C ( R ) which is regarded as one point.

The Hausdorff metric (2.1) satisfies the following properties:
H [ A + C , B + C ] = H [ A , B ] and H [ A , B ] = H [ B , A ] , H [ A + B , C + D ] H [ A , C ] + H [ B , D ] , H [ λ A , λ B ] = | λ | H [ A , B ]
for all A , B , C , D K C ( R ) and λ R . Let A , B K C ( R ) . If there exists an interval C K C ( R ) such that A = B + C , then we call C the Hukuhara difference of A and B. We denote the interval C by A B . Note that A B A + ( ) B . It is known that A B exists in the case len ( A ) len ( B ) . Besides that, we can see the following properties for A , B , C , D K C ( R ) (see [24]):
  • If A B , A C exist, then H [ A B , A C ] = H [ B , C ] ;

  • If A B , C D exist, then H [ A B , C D ] = H [ A + D , B + C ] ;

  • If A B , A ( B + C ) exist, then there exist ( A B ) C and ( A B ) C = A ( B + C ) ;

  • If A B , A C , C B exist, then there exist ( A B ) ( A C ) and ( A B ) ( A C ) = C B .

Definition 2.1 We say that the interval-valued mapping X : [ a , b ] R + K C ( R ) is continuous at the point t [ a , b ] if for every ε > 0 there exists δ = δ ( t , ε ) > 0 such that
H [ X ( t ) , X ( s ) ] ε

for all s [ a , b ] with | t s | < δ .

The strongly generalized differentiability was introduced in [26] and studied in [6, 24, 25, 31, 4143].

Definition 2.2 Let X : ( a , b ) K C ( R ) and t ( a , b ) . We say that X is strongly generalized differentiable at t if there exists D H g X ( t ) K C ( R ) such that
  1. (i)
    for all h > 0 sufficiently small, X ( t + h ) X ( t ) , X ( t ) X ( t h ) and
    lim h 0 H [ X ( t + h ) X ( t ) h , D H g X ( t ) ] = 0 , lim h 0 H [ X ( t ) X ( t h ) h , D H g X ( t ) ] = 0 ,

    or

     
  2. (ii)
    for all h > 0 sufficiently small, X ( t ) X ( t + h ) , X ( t h ) X ( t ) and
    lim h 0 H [ X ( t ) X ( t + h ) h , D H g X ( t ) ] = 0 , lim h 0 H [ X ( t h ) X ( t ) h , D H g X ( t ) ] = 0 ,

    or

     
  3. (iii)
    for all h > 0 sufficiently small, X ( t + h ) X ( t ) , X ( t h ) X ( t ) and
    lim h 0 H [ X ( t + h ) X ( t ) h , D H g X ( t ) ] = 0 , lim h 0 H [ X ( t h ) X ( t ) h , D H g X ( t ) ] = 0 ,

    or

     
  4. (iv)
    for all h > 0 sufficiently small, X ( t ) X ( t + h ) , X ( t ) X ( t h ) and the limits
    lim h 0 H [ X ( t ) X ( t + h ) h , D H g X ( t ) ] = 0 , lim h 0 H [ X ( t ) X ( t h ) h , D H g X ( t ) ] = 0

    (h at denominators means 1 h ).

     

In this definition, case (i) ((i)-differentiability for short) corresponds to the classical H-derivative, so this differentiability concept is a generalization of the Hukuhara derivative. In this paper we consider only the two first of Definition 2.2. In the other cases, the derivative is trivial because it is reduced to a crisp element (more precisely, D H g X ( t ) R ). Further, we say that X is (i)-differentiable or (ii)-differentiable on [ a , b ] , if it is differentiable in the sense (i) or (ii) of Definition 2.2, respectively.

Theorem 2.1 Let X : ( a , b ) K C ( R ) be (i)-differentiable or (ii)-differentiable on ( a , b ) , and assume that the derivative D H g X is integrable over ( a , b ) . We have
  1. (a)

    if X is (i)-differentiable on ( a , b ) , then a b D H g X ( t ) d t = X ( b ) X ( a ) ;

     
  2. (b)

    if X is (ii)-differentiable on ( a , b ) , then a b D H g X ( t ) d t = ( 1 ) ( X ( a ) X ( b ) ) .

     

Provided that, the above Hukuhara differences exist.

Lemma 2.1 (see [2426])

Assume that F : [ t 0 , t 0 + p ] × K C ( R ) K C ( R ) is continuous. The interval-valued differential equation (1.1) is equivalent to one of the following integral equations:
X ( t ) = X ( t 0 ) + t 0 t F ( s , X ( s ) ) d s , t [ t 0 , t 0 + p ] ,
if X is (i)-differentiable, and
X ( t ) = X ( t 0 ) ( 1 ) t 0 t F ( s , X ( s ) ) d s , t [ t 0 , t 0 + p ]

if X is (ii)-differentiable, provided that the H-difference exists.

The following well-known result is useful in the next section.

Lemma 2.2 Let a ( t ) , b ( t ) and c ( t ) be real-valued nonnegative continuous functions defined on R + , d 0 is a constant for which the inequality
a ( t ) d + 0 t [ b ( s ) a ( s ) + b ( s ) 0 s c ( r ) a ( r ) d r ] d s
(2.2)
holds for all t R + . Then
a ( t ) d [ 1 + 0 t b ( s ) exp ( 0 s ( b ( r ) + c ( r ) ) d r ) d s ] .

Corollary 2.1 (see [2426])

Let X : [ t 0 , t 0 + p ] K C ( R ) be given. Denote X ( t ) = [ X ̲ ( t ) , X ¯ ( t ) ] for t [ t 0 , t 0 + p ] , where X ̲ , X ¯ : [ t 0 , t 0 + p ] R .
  1. (i)

    If the mapping X is (i)-differentiable (i.e., classical Hukuhara differentiable) at t [ t 0 , t 0 + p ] , then the real-valued functions X ̲ , X ¯ are differentiable at t and D H g X ( t ) = [ X ̲ ( t ) , X ¯ ( t ) ] .

     
  2. (ii)

    If the mapping X is (ii)-differentiable at t [ t 0 , t 0 + p ] , then the real-valued functions X ̲ , X ¯ are differentiable at t and D H g X ( t ) = [ X ¯ ( t ) , X ̲ ( t ) ] .

     

3 Main results

For a positive number σ, we denote by C σ = C ( [ σ , 0 ] , K C ( R ) ) the space of continuous mappings from [ σ , 0 ] to K C ( R ) . Define a metric H σ in C σ by
H σ [ X , Y ] = sup t [ σ , 0 ] H [ X ( t ) , Y ( t ) ] .

Let p > 0 . Denote I = [ t 0 , t 0 + p ] , J = [ t 0 σ , t 0 ] I = [ t 0 σ , t 0 + p ] . For any t I , denote X t by the element of C σ defined by X t ( s ) = X ( t + s ) for s [ σ , 0 ] .

Let us consider the interval-valued functional integro-differential equations (IFIDEs) with the generalized Hukuhara derivative under the form
{ D H g X ( t ) = F ( t , X t ) + t 0 t G ( t , s , X s ) d s , t t 0 , X ( t ) = φ ( t t 0 ) = φ 0 , t 0 t t 0 σ ,
(3.1)

where F : I × C σ K C ( R ) , G : I × I × C σ K C ( R ) , φ C σ and the symbol D H g denotes the generalized Hukuhara derivative from Definition 2.2. By a solution to equation (3.1) we mean an interval-valued mapping X C ( J , K C ( R ) ) that satisfies X ( t ) = φ ( t t 0 ) for t [ t 0 σ , t 0 ] , X is differentiable on [ t 0 , t 0 + p ] and D H g X ( t ) = F ( t , X t ) + t 0 t G ( t , s , X s ) d s for t I . We note that the solution in this sense is considered just one-side differentiable at t = t 0 (specifically, right-differentiable at t = t 0 ).

Lemma 3.1 Assume that F C ( I × C σ , K C ( R ) ) , G C ( I × I × C σ , K C ( R ) ) and X C ( J , K C ( R ) ) . Then the interval-valued mapping t F ( t , X t ) + t 0 t G ( t , s , X s ) d s belongs to C ( I , K C ( R ) ) .

Remark 3.1 Under assumptions of the lemma above, the mapping t F ( t , X t ) + t 0 t G ( t , s , X s ) d s is integrable over the interval I.

Remark 3.2 If F : I × C σ K C ( R ) , G : I × I × C σ K C ( R ) are continuous and X C ( J , K C ( R ) ) , then the mapping t F ( t , X t ) + t 0 t G ( t , s , X s ) d s is bounded on I. Also, the function t F ( t , 0 ) + t 0 t G ( t , s , 0 ) d s is bounded on I.

Lemma 3.2 Assume that F : I × C σ K C ( R ) , G : I × I × C σ K C ( R ) are continuous. An interval-valued mapping X : J K C ( R ) is called a local solution to problem (3.1) on J if and only if X is a continuous interval-valued mapping and it satisfies one of the following interval-valued integral equations:
( S 1 ) { X ( t ) = φ ( t t 0 ) for  t [ t 0 σ , t 0 ] , X ( t ) = φ ( 0 ) + t 0 t ( F ( s , X s ) + t 0 t G ( t , s , X s ) d s ) d s , t I ,
(3.2)
if X is (i)-differentiable,
( S 2 ) { X ( t ) = φ ( t t 0 ) for  t [ t 0 σ , t 0 ] , X ( t ) = φ ( 0 ) ( 1 ) t 0 t ( F ( s , X s ) + t 0 t G ( t , s , X s ) d s ) d s , t I ,
(3.3)

if X is (ii)-differentiable. We remark that in (3.3), the following statement is hidden: there exists the Hukuhara difference φ ( 0 ) ( 1 ) t 0 t ( F ( s , X s ) + t 0 t G ( t , s , X s ) d s ) d s .

Proof We prove the case of (ii)-differentiability, the proof of the other case being similar. Assume that X : [ t 0 , t 0 + p ] K C ( R ) is a solution to problem (3.1). Hence X is (ii)-differentiable on [ t 0 , t 0 + p ] and D H g X is integrable as a continuous function. Applying Theorem 2.1, we obtain that
X ( t 0 ) = X ( t ) + ( 1 ) t 0 t D H g X ( s ) d s
for every t [ t 0 , t 0 p ] . Since X ( t 0 ) = φ ( 0 ) and D H g X ( s ) = F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ for s [ t 0 , t 0 + p ] , we easily obtain
{ X ( t ) = φ ( t t 0 ) for  t [ t 0 σ , t 0 ] , X ( t ) = φ ( 0 ) ( 1 ) t 0 t ( F ( s , X s ) + t 0 t G ( t , s , X s ) d s ) d s , t I .
To show that the opposite implication is true, let us assume that X : [ t 0 , t 0 + p ] K C ( R ) is a continuous interval-valued mapping and it satisfies equation (3.3). Equation (3.3) allows us to claim that φ ( 0 ) = X ( t 0 ) and that there exists the Hukuhara difference
φ ( 0 ) ( 1 ) t 0 t ( F ( s , X s ) + t 0 t G ( t , s , X s ) d s ) d s for every  t [ t 0 , t 0 + p ] .
Now, let t [ t 0 , t 0 + p ) and h be such that t + h [ t 0 , t 0 + p ] . We observe that
X ( t ) X ( t + h ) = ( 1 ) t t + h ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s .
(3.4)
Indeed, we have by direct computation
X ( t + h ) + ( 1 ) t t + h ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s = φ ( 0 ) ( 1 ) t 0 t + h ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s + ( 1 ) t t + h ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s = φ ( 0 ) ( 1 ) t 0 t + h ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s + ( 1 ) t 0 t + h ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s ( 1 ) t 0 t ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s = φ ( 0 ) ( 1 ) t 0 t ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s = X ( t ) .
Similarly to (3.4), we can obtain
X ( t h ) X ( t ) = ( 1 ) t h t ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s
(3.5)
for t ( t 0 , t 0 + p ] . Multiplying (3.4)-(3.5) by 1 h and passing to limit with h 0 , we have by Definition 2.2 that X is (ii)-differentiable, and consequently
D H g X ( t ) = F ( t , X t ) + t 0 t G ( t , s , X s ) d s for  t [ t 0 , t 0 + p ] .
Indeed, we have, for every t [ t 0 , t 0 + p ] ,
lim h 0 + X ( t ) X ( t + h ) h = lim h 0 + 1 h t t + h ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s
and
lim h 0 + X ( t h ) X ( t ) h = lim h 0 + 1 h t h t ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s .
Since F, G are continuous, for h 0 + , we obtain
lim h 0 + X ( t ) X ( t + h ) h = F ( t , X t ) + t 0 t G ( t , s , X s ) d s .
Proceeding as above, we can obtain
lim h 0 + X ( t h ) X ( t ) h = F ( t , X t ) + t 0 t G ( t , s , X s ) d s .

The proof is complete. □

Definition 3.1 Let X : J K C ( R ) be an interval-valued function which is (i)-differentiable. If X and its derivative satisfy problem (3.1), we say that X is (i)-solution of problem (3.1). (i)-solution X : J K C ( R ) is unique if it holds H [ X ( t ) , Y ( t ) ] = 0 for any Y : J K C ( R ) which is (i)-solution of (3.1).

Definition 3.2 Let X : J K C ( R ) be an interval-valued function which is (ii)-differentiable. If X and its derivative satisfy problem (3.1), we say that X is (ii)-solution of problem (3.1). (ii)-solution X : J K C ( R ) is unique if it holds H [ X ( t ) , Y ( t ) ] = 0 for any X : J K C ( R ) which is (ii)-solution of (3.1).

Theorem 3.1 Let φ ( t t 0 ) C σ and suppose that F C ( I × C σ , K C ( R ) ) , G C ( I × I × C σ , K C ( R ) ) satisfy the conditions: there exists a constant L > 0 such that
max { H [ F ( t , X ) , F ( t , Y ) ] , H [ G ( t , s , X ) , G ( t , s , Y ) ] } L H σ [ X , Y ]
for every t [ t 0 , t 0 + p ] , ( t , s ) [ t 0 , t 0 + p ] × [ t 0 , t 0 + p ] and X , Y C σ . Moreover, there exists M > 0 such that max { H [ F ( t , X ) , 0 ] , H [ G ( t , s , X ) , 0 ] } M . Then the following successive approximations given by
X ˆ 0 ( t ) = { φ ( t t 0 ) , t [ t 0 σ , t 0 ] , φ ( 0 ) , t I , X ˆ n + 1 ( t ) = { φ ( t t 0 ) , t [ t 0 σ , t 0 ] , φ ( 0 ) + t 0 t ( F ( s , X ˆ s n ) + t 0 s G ( s , τ , X ˆ τ n ) d τ ) d s
(3.6)
for the case of (i)-differentiability, and
X ˜ 0 ( t ) = { φ ( t t 0 ) , t [ t 0 σ , t 0 ] , φ ( 0 ) , t [ t 0 , t 0 + d ] , X ˜ n + 1 ( t ) = { φ ( t t 0 ) , t [ t 0 σ , t 0 ] , φ ( 0 ) ( 1 ) t 0 t ( F ( s , X ˜ s n ) + t 0 s G ( s , τ , X ˜ τ n ) d τ ) d s
(3.7)

for the case of (ii)-differentiability (where 0 < d p such that equation (3.7) is well defined, i.e., the foregoing Hukuhara differences do exist), converge uniformly to two unique solutions X ˆ ( t ) and X ˜ ( t ) of (3.1), respectively, on [ a , a + r ] where r = min { p , d } .

Proof We prove that for the case of (ii)-differentiability, the proof of the other case is similar. From assumptions of the theorem, we have
H [ X ˜ 1 ( t ) , X ˜ 0 ( t ) ] = H [ φ ( 0 ) ( 1 ) t 0 t ( F ( s , X ˜ s 0 ) + t 0 s G ( s , τ , X ˜ τ 0 ) d τ ) d s , φ ( 0 ) ] t 0 t ( H [ F ( s , X ˜ s 0 ) , 0 ] + t 0 s H [ G ( s , τ , X ˜ τ 0 ) , 0 ] d τ ) d s M ( t t 0 ) + M ( t t 0 ) 2 2 !
for t [ t 0 , t 0 + r ] . Further, for every n 2 and t [ t 0 , t 0 + r ] , we get
H [ X ˜ n + 1 ( t ) , X ˜ n ( t ) ] = H [ t 0 t ( F ( s , X ˜ s n ) + t 0 s G ( s , τ , X ˜ τ n ) d τ ) d s , t 0 t ( F ( s , X ˜ s n 1 ) + t 0 s G ( s , τ , X ˜ τ n 1 ) d τ ) d s ] L t 0 t ( H σ [ X ˜ s n , X ˜ s n 1 ] + t 0 s H σ [ X ˜ τ n , X ˜ τ n 1 ] d τ ) d s L t 0 t ( sup θ [ σ , 0 ] H [ X ˜ n ( s + θ ) , X ˜ n 1 ( s + θ ) ] + t 0 s sup θ [ σ , 0 ] H [ X ˜ n ( τ + θ ) , X ˜ n 1 ( τ + θ ) ] d τ ) d s = L t 0 t ( sup r [ s σ , s ] H [ X ˜ n ( r ) , X ˜ n 1 ( r ) ] + t 0 s sup υ [ τ σ , τ ] H [ X ˜ n ( υ ) , X ˜ n 1 ( υ ) ] d υ ) d r .
In particular, from (3.7) it follows that
H [ X ˜ 2 ( t ) , X ˜ 1 ( t ) ] L M ( ( t t 0 ) 2 2 ! + 2 ( t t 0 ) 3 3 ! + ( t t 0 ) 4 4 ! ) .
Therefore, by mathematical induction, for every n N and t [ t 0 , t 0 + r ] ,
H [ X ˜ n + 1 ( t ) , X ˜ n ( t ) ] M L n ( ( t t 0 ) n + 1 ( n + 1 ) ! + n λ 1 ( t t 0 ) n + 2 ( n + 2 ) ! + + n λ n ( t t 0 ) 2 n + 1 ( 2 n + 1 ) ! + ( t t 0 ) 2 n + 2 ( 2 n + 2 ) ! ) .
(3.8)
In inequality (3.8), λ 1 , , λ n are balancing constants. We observe that for every n { 0 , 1 , 2 , } , the function X ˜ n ( ) : [ t 0 σ , t 0 + r ] K C ( R ) is continuous. Indeed, since φ C σ , X ˜ 0 ( t ) is continuous on t [ t 0 σ , t 0 ] . We see that
H [ X ˜ 1 ( t + h ) , X ˜ 1 ( t ) ] = H [ φ ( 0 ) ( 1 ) t 0 t + h ( F ( s , X ˜ s 0 ) + t 0 s G ( s , τ , X ˜ τ 0 ) d τ ) d s , φ ( 0 ) ( 1 ) t 0 t ( F ( s , X ˜ s 0 ) + t 0 s G ( s , τ , X ˜ τ 0 ) d τ ) d s ] t t + h ( H [ F ( s , X ˜ s 0 ) , 0 ] + t 0 s H [ G ( s , τ , X ˜ τ 0 ) , 0 ] d τ ) d s M h + M h 2 2 ! 0 as  h 0 + .
Thus, by mathematical induction, for every n 2 , we deduce that
H [ X ˜ n ( t + h ) , X ˜ n ( t ) ] 0
as h 0 + . A similar inequality is obtained for H [ X ˜ n ( t h ) , X ˜ n ( t ) ] 0 as h 0 + . In the sequel, we shall show that for { X ˜ n ( t ) } the Cauchy convergence condition is satisfied uniformly in t, and as a consequence { X ˜ n ( ) } is uniformly convergent. For n > m > 0 , from (3.8) we obtain
sup t I H [ X ˜ n ( t ) , X ˜ m ( t ) ] = sup t J H [ X ˜ n ( t ) , X ˜ m ( t ) ] k = m n 1 sup t J H [ X ˜ k + 1 ( t ) , X ˜ k ( t ) ] M k = m n 1 ( ( t t 0 ) k + 1 ( k + 1 ) ! + k λ 1 ( t t 0 ) k + 2 ( k + 2 ) ! + + k λ k ( t t 0 ) 2 k + 1 ( 2 k + 1 ) ! + ( t t 0 ) 2 k + 2 ( 2 k + 2 ) ! ) .
The convergence of this series implies that for any ε > 0 we find n 0 N large enough such that for n , m > n 0 ,
H [ X ˜ n ( t ) , X ˜ m ( t ) ] ε .
(3.9)
Since ( K C ( R ) , H ) is a complete metric space and (3.9) holds, the sequence { X ˜ n ( ) } is uniformly convergent to a mapping X ˜ C ( [ t 0 σ , t 0 + r ] , K C ( R ) ) . We shall show that X ˜ is (ii)-solution to (3.1). Since X ˜ n ( t ) = φ ( t t 0 ) for every n = 0 , 1 , 2 , and every t [ t 0 σ , t 0 ] , we easily have X ˜ ( t ) = φ ( t t 0 ) . For s [ t 0 , t 0 + r ] and n N ,
H [ t 0 t F ( s , X ˜ s n ) d s , t 0 t F ( s , X ˜ s ) d s ] L t 0 t sup θ [ s σ , s ] H [ X ˜ n ( θ ) , X ˜ ( θ ) ] d θ 0 , H [ t 0 t t 0 s G ( s , τ , X ˜ τ n ) d τ d s , t 0 t t 0 s G ( s , τ , X ˜ τ ) d τ d s ] L t 0 t t 0 s sup θ [ τ σ , τ ] H [ X ˜ n ( υ ) , X ˜ ( υ ) ] d υ d s 0
as n for any t [ t 0 , t 0 + r ] . Consequently, we have
H [ φ ( 0 ) , X ˜ ( t ) + ( 1 ) t 0 t ( F ( s , X ˜ s ) + t 0 t G ( t , s , X ˜ s ) d s ) d s ] H [ X ˜ n ( t ) , X ˜ ( t ) ] + t 0 t ( H [ F ( s , X ˜ s n 1 ) , F ( s , X ˜ s ) ] + t 0 s H [ G ( s , τ , X ˜ τ n 1 ) , G ( s , τ , X ˜ τ ) ] d τ ) d s .
We infer that
H [ φ ( 0 ) , X ˜ ( t ) + ( 1 ) t 0 t ( F ( s , X ˜ s ) + t 0 t G ( t , s , X ˜ s ) d s ) d s ] = 0
for every t [ t 0 , t 0 + r ] . Therefore, X ˜ is the (ii)-solution of (3.1), due to Lemma 3.2 it follows that X ˜ is the (ii)-solution of (3.1). For the uniqueness of the (ii)-solution X ˜ , let us assume that X ˜ , Y ˜ C ( [ t 0 σ , t 0 + r ] , K C ( R ) ) are two solutions of (3.3). By definition of the solution, X ˜ ( t ) = Y ˜ ( t ) if t [ t 0 σ , t 0 ] . Note that for t [ t 0 , t 0 + r ] ,
H [ X ˜ ( t ) , Y ˜ ( t ) ] L t 0 t ( sup θ [ s σ , s ] H [ X ˜ ( θ ) , Y ˜ ( θ ) ] + t 0 s sup υ [ τ σ , τ ] H [ X ˜ ( υ ) , Y ˜ ( υ ) ] d τ ) d s .
If we put a ( s ) = sup r [ s σ , s ] H [ X ˜ ( r ) , Y ˜ ( r ) ] , s [ t 0 , t ] [ t 0 , t 0 + r ] , then we obtain
a ( t ) L t 0 t ( a ( s ) + t 0 s a ( τ ) d τ ) d s ,

and by Lemma 2.2 we obtain that a ( t ) = 0 on [ t 0 , t 0 + r ] . This proves the uniqueness of the (ii)-solution for (3.1) □

Remark 3.3 The existence and uniqueness results for solutions of problem (3.1) can be obtained by using the contraction principle.

Now, we present the studies and results concerning the global existence and uniqueness of two solutions for (3.1), each one corresponding to a different type of differentiability, by using the contraction principle, which was studied in [5] for fuzzy functional differential equations. In the following, for a given k > 0 , we consider the set S k of all continuous interval-valued functions X C ( [ t 0 σ , ) , K C ( R ) ) such that X ( t ) = φ ( t t 0 ) on [ t 0 σ , t 0 ] and sup t t 0 σ { H [ X ( t ) , 0 ] exp ( k t ) } < . On S k we can define the following metric:
H k [ X , Y ] = sup t t 0 σ { H [ X ( t ) , Y ( t ) ] exp ( k t ) } ,
(3.10)

where k > 0 is chosen suitably later. It is easy to prove that the space ( S k , H k ) of continuous interval-valued functions X : [ t 0 , ) K C ( R ) is a complete metric space with distance (3.10).

Theorem 3.2 Assume that
  1. (i)
    F C ( [ t 0 , ) × C σ , K C ( R ) ) , G C ( [ t 0 , ) × [ t 0 , ) × C σ , K C ( R ) ) and there exists a constant L > 0 such that
    max { H [ F ( t , X ) , F ( t , Y ) ] , H [ G ( t , s , X ) , G ( t , s , Y ) ] } L H σ [ X , Y ]

    for all X , Y C σ and t , s t 0 ;

     
  2. (ii)
    there exist M > 0 and b > 0 such that
    max { H [ F ( t , 0 ) , 0 ] , H [ G ( t , s , 0 ) , 0 ] } M exp ( b t )
     

for all t t 0 , where b < k .

Then
  1. (a)

    the interval-valued functional integro-differential equation (3.1) has (i)-solution on [ t 0 , ) ;

     
  2. (b)
    the interval-valued functional integro-differential equation (3.1) has (ii)-solution on [ t 0 , ) if the following condition holds:
    t 0 t ( len ( F ( s , X s ) ) + t 0 t len ( G ( t , s , X s ) ) d s ) d s len ( φ ( 0 ) ) , t t 0 .
    (3.11)
     
Proof Since the way of the proof is similar for both cases, we only consider the case of (ii)-differentiability for X. Note that the space ( S k , H k ) under inequality (3.11) depends on the positive constant k, the functions F, G and the initial condition φ. In ( S k , H k ) , the continuity of F, G guarantees that S k under inequality (3.11) is a closed set in C ( [ t 0 , ) , K C ( R ) ) , so that S k under inequality (3.11) is a complete metric space considering the distance H k . We consider the complete metric space ( S k , H k ) and define an operator
T : S k S k , X T X
given by
( T X ) ( t ) = { φ ( t t 0 ) if  t [ t 0 σ , t 0 ] , φ ( 0 ) ( 1 ) t 0 t ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s if  t t 0 .

We can choose a big enough value for k such that is a contraction, so the Banach fixed point theorem provides the existence of a unique fixed point for , that is, a unique solution for (3.1).

First, we shall prove that T ( S k ) S k , i.e., the operator T is well defined, with assumption k > b . Indeed, let X S k . For each t t 0 , we get
H k [ ( T X ) ( t ) , 0 ] = sup t t 0 { H [ φ ( 0 ) ( 1 ) t 0 t ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s , 0 ] exp ( k t ) } sup t t 0 { ( H [ φ ( 0 ) , 0 ] + t 0 t { H [ F ( s , X s ) , F ( s , 0 ) ] + H [ F ( s , 0 ) , 0 ] } d s + t 0 t ( t 0 s { H [ G ( s , τ , X τ ) , G ( s , τ , 0 ) ] + H [ G ( s , τ , 0 ) , 0 ] } d τ ) d s ) exp ( k t ) } sup t t 0 { ( H [ φ ( 0 ) , 0 ] + L t 0 t H σ [ X s , 0 ] d s + M b exp ( b t ) + L t 0 t ( t 0 s H σ [ X τ , 0 ] d τ ) d s + M b 2 exp ( b t ) ) exp ( k t ) } sup t t 0 { ( H [ φ ( 0 ) , 0 ] + L t 0 t sup θ [ σ , 0 ] H [ X ( s + θ ) , 0 ] d s + M b exp ( b t ) + L t 0 t ( t 0 s sup θ [ σ , 0 ] H [ X ( τ + θ ) , 0 ] d τ ) d s + M b 2 exp ( b t ) ) exp ( k t ) } .
Since X S k , there exists ρ > 0 such that H [ X ( t ) , 0 ] < ρ exp ( k t ) for all t t 0 σ . Therefore, for all t t 0 , we obtain
H k [ ( T X ) ( t ) , 0 ] sup t t 0 { ( H [ φ ( 0 ) , 0 ] + ( 1 + 1 k ) ρ L k exp ( k t ) + ( 1 + 1 b ) M b exp ( b t ) ) exp ( k t ) } H [ φ ( 0 ) , 0 ] + ( 1 + 1 b ) 1 b ( M + ρ L ) K + ( 1 + 1 b ) 1 b ( M + ρ L ) < .

We infer that T X S k .

Next, we shall prove that is a contraction by metric H k . Let X , Y S k . Then, for θ [ σ , 0 ] , H [ ( T X ) ( t 0 + θ ) , ( T Y ) ( t 0 + θ ) ] = 0 . For each t t 0 , we have
H k [ ( T X ) ( t ) , ( T Y ) ( t ) ] = sup t t 0 { H [ ( T X ) ( t ) , ( T Y ) ( t ) ] exp ( k t ) } = sup t t 0 { H [ φ ( 0 ) ( 1 ) t 0 t ( F ( s , X s ) + t 0 s G ( s , τ , X τ ) d τ ) d s , φ ( 0 ) ( 1 ) t 0 t ( F ( s , Y s ) + t 0 s G ( s , τ , Y τ ) d τ ) d s ] exp ( k t ) } sup t t 0 { ( L t 0 t ( H σ [ X s , Y s ] + t 0 s H σ [ X τ , Y τ ] d τ ) d s ) exp ( k t ) } = sup t t 0 { ( L t 0 t sup θ [ σ , 0 ] H [ X ( s + θ ) , Y ( s + θ ) ] d s + L t 0 t ( t 0 s sup θ [ σ , 0 ] H [ X ( τ + θ ) , Y ( τ + θ ) ] ) d s ) exp ( k t ) } = sup t t 0 { ( L t 0 t sup r [ s σ , s ] H [ X ( r ) , Y ( r ) ] d r + L t 0 t ( t 0 s sup υ [ τ σ , τ ] H [ X ( υ ) , Y ( υ ) ] d υ ) d s ) exp ( k t ) } L H k [ X , Y ] sup t t 0 ( t 0 t ( exp ( k ( r t ) ) + t 0 s exp ( k ( υ t ) ) d υ ) d r ) ( 1 + k ) L H k [ x , y ] k 2 .

Choosing k > b and ( 1 + k ) L k 2 < 1 , it follows that the operator on S k is a contraction. Using the Banach fixed point theorem provides the existence of a unique fixed point for , and the unique fixed point of is in the space S k , that is, a unique solution for (3.1) in the case of (ii)-differentiability. □

4 Illustrations

In this part, some simple examples are given to illustrate the theory of IFIDEs. We shall consider IFIDEs (3.1) with (i) and (ii) derivatives, respectively. Let us start the illustrations with considering the following interval-valued functional integro-differential equation:
{ D H g X ( t ) = F ( t , X t ) + t 0 t k ( t , s ) X s d s , t J , X ( t ) = φ ( t t 0 ) , t [ σ , t 0 ] ,
(4.1)

where F : I × C σ K C ( R ) , k ( t , s ) : I × I R . Let X ( t ) = [ X ̲ ( t ) , X ¯ ( t ) ] . By using Corollary 2.1, we have the following two cases.

If we consider the derivative of X ( t ) by using (i)-differentiability, then from Corollary 2.1, we have D H g X ( t ) = [ X ̲ ( t ) , X ¯ ( t ) ] for t t 0 . Therefore, (4.1) is translated into the following delay integro-differential system:
{ X ̲ ( t ) = F ̲ ( t , X ̲ t , X ¯ t ) + t 0 t k ( t , s ) X s ̲ d s , t t 0 , X ¯ ( t ) = F ¯ ( t , X ̲ t , X ¯ t ) + t 0 t k ( t , s ) X s ¯ d s , t t 0 , X ̲ ( t ) = φ ̲ ( t t 0 ) , σ t t 0 , X ¯ ( t ) = φ ¯ ( t t 0 ) , σ t t 0 .
(4.2)
If we consider the derivative of X ( t ) by using (ii)-differentiability, then from Corollary 2.1, we have D H g X ( t ) = [ X ¯ ( t ) , X ̲ ( t ) ] for t t 0 . Therefore, (4.1) is translated into the following delay integro-differential system:
{ X ¯ ( t ) = F ̲ ( t , X ̲ t , X ¯ t ) + t 0 t k ( t , s ) X s ̲ d s , t t 0 , X ̲ ( t ) = F ¯ ( t , X ̲ t , X ¯ t ) + t 0 t k ( t , s ) X s ¯ d s , t t 0 , X ̲ ( t ) = φ ̲ ( t t 0 ) , σ t t 0 , X ¯ ( t ) = φ ¯ ( t t 0 ) , σ t t 0 ,
(4.3)
where
k ( t , s ) X s ̲ = { k ( t , s ) X ̲ s , k ( t , s ) 0 , k ( t , s ) X ¯ s , k ( t , s ) < 0 , k ( t , s ) X s ¯ = { k ( t , s ) X ¯ s , k ( t , s ) 0 , k ( t , s ) X ̲ s , k ( t , s ) < 0 .

Remark 4.1 If we ensure that the solutions ( X ̲ ( t ) , X ¯ ( t ) ) of systems (4.2) and (4.3) respectively are valid sets of interval-valued functions and if the derivatives ( X ̲ ( t ) , X ¯ ( t ) ) are valid sets of interval-valued functions with two kinds of differentiability respectively, then we can construct the solution of interval-valued functional differential equation (4.1).

Next, we shall consider two examples being a simple illustration for the theory of interval-valued functional integro-differential equations.

Example 4.1 Let us consider the linear interval-valued functional integro-differential equation (with k ( t , s ) 0 ) under two kinds of Hukuhara derivatives
{ D H g X ( t ) = λ X ( t 1 2 ) , X ( t ) = φ ( t ) , t [ 1 2 , 0 ] ,
(4.4)

where φ ( t ) = [ 1 , 1 ] , λ > 0 . In this example we shall solve (4.4) on [ 0 , 1 ] .

Case 1. Considering (i)-differentiability, problem (4.4) is translated into the following delay system:
{ X ̲ ( t ) = λ X ¯ ( t 1 2 ) , t 0 , X ¯ ( t ) = λ X ̲ ( t 1 2 ) , t 0 , X ̲ ( t ) = 1 , 1 2 t 0 , X ¯ ( t ) = 1 , 1 2 t 0 .
(4.5)
Solving delay system (4.5) by using the method of steps, we obtain a unique (i)-solution to (4.4) defined on [ 0 , 1 ] and it is of the form
X ( t ) = { [ 1 ( 1 + λ t ) , ( 1 + λ t ) ] for  t [ 0 , 1 2 ] , [ 1 ( 1 + λ t + λ 2 ( t 1 ) 2 2 ) , 1 + λ t + λ 2 ( t 1 ) 2 2 ] for  t [ 1 2 , 1 ] .
The (i)-solution is illustrated in Figure 1.
Figure 1
Figure 1

(i)-solution to ( 4.4 ) ( λ = 0.5 ).

Case 2. Considering (ii)-differentiability, problem (4.4) is translated into the following delay system:
{ X ¯ ( t ) = λ X ¯ ( t 1 2 ) , t 0 , X ̲ ( t ) = λ X ̲ ( t 1 2 ) , t 0 , X ̲ ( t ) = 1 , 1 2 t 0 , X ¯ ( t ) = 1 , 1 2 t 0 .
(4.6)
We obtain a unique (ii)-solution to (4.4) defined on [ 0 , 1 ] and it is of the form
X ( t ) = { [ λ t 1 , 1 λ t ] for  t [ 0 , 1 2 ] , [ 1 ( 1 λ t + λ 2 ( t 1 ) 2 2 ) , 1 λ t + λ 2 ( t 1 ) 2 2 ] for  t [ 1 2 , 1 ] .
The (ii)-solution is illustrated in Figure 2.
Figure 2
Figure 2

(ii)-solution to ( 4.4 ) ( λ = 0.5 ).

Example 4.2 Let us consider the linear interval-valued functional integro-differential equation under two kinds of Hukuhara derivatives
{ D H g X ( t ) = X ( t 1 2 ) + α t 0 t e ( s t ) X ( s 1 2 ) d s , X ( t ) = φ ( t ) , t [ 1 2 , 0 ] ,
(4.7)

where φ ( t ) = [ 1 , 2 t ] , α R { 0 } . In this example we shall solve (4.7) on [ 0 , 1 / 2 ] .

Case 1. ( α > 0 ) From (4.2), we get
{ X ̲ ( t ) = X ̲ ( t 1 2 ) + α 0 t e ( s t ) X ̲ ( s 1 2 ) d s , t 0 , X ¯ ( t ) = X ¯ ( t 1 2 ) + α 0 t e ( s t ) X ¯ ( s 1 2 ) d s , t 0 , X ̲ ( t ) = 1 , 1 2 t 0 , X ¯ ( t ) = 2 t , 1 2 t 0 .
(4.8)
Following the method of steps, we obtain the (i)-solution to (4.7) defined on [ 0 , 1 / 2 ] and it is of the form
X ( t ) = [ 1 + t 2 + α 2 ( 1 e t ) , 2 + 2 t t 2 2 + α ( 3 t 3 e t ) ] , t [ 0 , 1 / 2 ] .
From (4.3) we obtain
{ X ¯ ( t ) = X ̲ ( t 1 2 ) + α 0 t e ( s t ) X ̲ ( s 1 2 ) d s , t 0 , X ̲ ( t ) = X ¯ ( t 1 2 ) + α 0 t e ( s t ) X ¯ ( s 1 2 ) d s , t 0 , X ̲ ( t ) = 1 , 1 2 t 0 , X ¯ ( t ) = 2 t , 1 2 t 0 .
(4.9)
The (ii)-solution to (4.7) defined on [ 0 , 1 / 2 ] is of the form
X ( t ) = [ 1 + 2 t t 2 2 + α ( 3 t 3 e t ) , 2 + α 2 ( 1 e t ) ] , t [ 0 , 1 / 2 ] .
In Figures 3 and 4, (i)-solution and (ii)-solution curves of (4.7) are given.
Figure 3
Figure 3

Graphs of X ̲ ( t ) , X ¯ ( t ) for t [ 1 2 , 1 2 ] , α = 0.1 .

Figure 4
Figure 4

Graphs of X ̲ ( t ) , X ¯ ( t ) for t [ 1 2 , 1 2 ] , α = 0.1 .

Case 2. ( α < 0 ) From (4.2) we get
{ X ̲ ( t ) = X ̲ ( t 1 2 ) + α 0 t e ( s t ) X ¯ ( s 1 2 ) d s , t 0 , X ¯ ( t ) = X ¯ ( t 1 2 ) + α 0 t e ( s t ) X ̲ ( s 1 2 ) d s , t 0 , X ̲ ( t ) = 1 , 1 2 t 0 , X ¯ ( t ) = 2 t , 1 2 t 0 .
(4.10)
By solving delay integro-differential system (4.10), we obtain (i)-solution
X ( t ) = [ 1 + t 2 + α ( 3 t 3 e t ) , 2 + 2 t t 2 2 + α 2 ( 1 e t ) ] , t [ 0 , 1 / 2 ] .
The (i)-solution of (4.7) on [ 1 / 2 , 1 / 2 ] is illustrated in Figure 5.
Figure 5
Figure 5

Graphs of X ̲ ( t ) , X ¯ ( t ) for t [ 1 2 , 1 2 ] , α = 0.1 .

From (4.3) we obtain
{ X ¯ ( t ) = X ̲ ( t 1 2 ) + α 0 t e ( s t ) X ¯ ( s 1 2 ) d s , t 0 , X ̲ ( t ) = X ¯ ( t 1 2 ) + α 0 t e ( s t ) X ̲ ( s 1 2 ) d s , t 0 , X ̲ ( t ) = 1 , 1 2 t 0 , X ¯ ( t ) = 2 t , 1 2 t 0 .
(4.11)
By solving delay integro-differential systems (4.11), we obtain (ii)-solution
X ( t ) = [ 1 + 2 t t 2 2 + α 2 ( 1 e t ) , 2 + t 2 + α ( 3 t 3 e t ) ] , t [ 0 , 1 / 2 ] .
The (ii)-solution of (4.7) on [ 1 / 2 , 1 / 2 ] is illustrated in Figure 6.
Figure 6
Figure 6

Graphs of X ̲ ( t ) , X ¯ ( t ) for t [ 1 2 , 1 2 ] , α = 0.1 .

5 Conclusion

In this study, we have established the local and global existence and uniqueness results of two solutions for interval-valued functional integro-differential equations. For the local existence and uniqueness, we use the method of successive approximations under the Lipschitz condition, and for global existence and uniqueness, we use the contraction principle under suitable conditions. In our further work, we would like to use these results to study the local and global existence and uniqueness results of solutions for interval-valued functional integro-differential equations under Caputo-type interval-valued fractional derivatives.

Declarations

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper.

Authors’ Affiliations

(1)
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
(2)
Faculty of Mathematics and Computer Science, University of Science, VNU, Ho Chi Minh City, Vietnam
(3)
Faculty of Natural Science and Technology, Tay Nguyen University, Buon Ma Thuot City, Vietnam

References

  1. Hale JK: Theory of Functional Differential Equations. Springer, New York; 1977.View ArticleMATHGoogle Scholar
  2. Kolmanovskii VB, Myshkis A: Applied Theory of Functional Differential Equations. Kluwer Academic, Dordrecht; 1992.View ArticleMATHGoogle Scholar
  3. Kuang Y: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston; 1993.MATHGoogle Scholar
  4. Lupulescu V: On a class of functional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. 2010., 2010: Article ID 64Google Scholar
  5. Lupulescu V: On a class of fuzzy functional differential equations. Fuzzy Sets Syst. 2009, 160: 1547–1562. 10.1016/j.fss.2008.07.005MathSciNetView ArticleMATHGoogle Scholar
  6. Malinowski MT: Second type Hukuhara differentiable solutions to the delay set-valued differential equations. Appl. Math. Comput. 2012, 218(18):9427–9437. 10.1016/j.amc.2012.03.027MathSciNetView ArticleMATHGoogle Scholar
  7. De Blasi FS, Iervolino F: Equazioni differenziali con soluzioni a valore compatto convesso. Boll. Unione Mat. Ital. 1969, 4(2):491–501.MathSciNetMATHGoogle Scholar
  8. Agarwal RP, O’Regan D: Existence for set differential equations via multivalued operator equations. Differ. Equ. Appl. 2007, 5: 1–5.View ArticleMathSciNetGoogle Scholar
  9. Agarwal RP, O’Regan D, Lakshmikantham V: Viability theory and fuzzy differential equations. Fuzzy Sets Syst. 2005, 151(3):563–580. 10.1016/j.fss.2004.08.008MathSciNetView ArticleMATHGoogle Scholar
  10. Agarwal RP, O’Regan D, Lakshmikantham V: A stacking theorem approach for fuzzy differential equations. Nonlinear Anal. TMA 2003, 55(3):299–312. 10.1016/S0362-546X(03)00241-4MathSciNetView ArticleMATHGoogle Scholar
  11. Bhaskar TG, Lakshmikantham V: Set differential equations and flow invariance. J. Appl. Anal. 2003, 82(2):357–368.MathSciNetView ArticleMATHGoogle Scholar
  12. De Blasi FS, Lakshmikantham V, Bhaskar TG: An existence theorem for set differential inclusions in a semilinear metric space. Control Cybern. 2007, 36(3):571–582.MathSciNetMATHGoogle Scholar
  13. Devi JV: Generalized monotone iterative technique for set differential equations involving causal operators with memory. Int. J. Adv. Eng. Sci. Appl. Math. 2011, 8(3):74–83.View ArticleMathSciNetGoogle Scholar
  14. Hoa NV, Phu ND: On maximal and minimal solutions for set-valued differential equations with feedback control. Abstr. Appl. Anal. 2012., 2012: Article ID 816218Google Scholar
  15. Kaleva O: Fuzzy differential equations. Fuzzy Sets Syst. 1987, 24: 301–317. 10.1016/0165-0114(87)90029-7MathSciNetView ArticleMATHGoogle Scholar
  16. Lakshmikantham V, Bhaskar TG, Devi JV: Theory of Set Differential Equations in Metric Spaces. Cambridge Scientific Publishers, Cambridge; 2006.MATHGoogle Scholar
  17. Lakshmikantham V, Mohapatra R: Theory of Fuzzy Differential Equations and Inclusions. Taylor & Francis, London; 2003.View ArticleMATHGoogle Scholar
  18. Lakshmikantham V, Tolstonogov AA: Existence and interrelation between set and fuzzy differential equations. Nonlinear Anal. TMA 2003, 55(3):255–268. 10.1016/S0362-546X(03)00228-1MathSciNetView ArticleMATHGoogle Scholar
  19. Phu ND, Quang LT, Tung TT: Stability criteria for set control differential equations. Nonlinear Anal. TMA 2008, 69: 3715–3721. 10.1016/j.na.2007.10.007MathSciNetView ArticleMATHGoogle Scholar
  20. Phu ND, Tung TT: Some results on sheaf-solutions of sheaf set control problems. Nonlinear Anal. TMA 2007, 67: 1309–1315. 10.1016/j.na.2006.07.018MathSciNetView ArticleMATHGoogle Scholar
  21. Puri M, Ralescu D: Differentials of fuzzy functions. J. Math. Anal. Appl. 1983, 91: 552–558. 10.1016/0022-247X(83)90169-5MathSciNetView ArticleMATHGoogle Scholar
  22. Song S, Wu C: Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations. Fuzzy Sets Syst. 2000, 110: 55–67. 10.1016/S0165-0114(97)00399-0MathSciNetView ArticleMATHGoogle Scholar
  23. Tu NN, Tung TT: Stability of set differential equations and applications. Nonlinear Anal. TMA 2009, 71: 1526–1533. 10.1016/j.na.2008.12.045MathSciNetView ArticleMATHGoogle Scholar
  24. Malinowski MT: Interval Cauchy problem with a second type Hukuhara derivative. Inf. Sci. 2012, 213: 94–105.MathSciNetView ArticleMATHGoogle Scholar
  25. Malinowski MT: Interval differential equations with a second type Hukuhara derivative. Appl. Math. Lett. 2011, 24: 2118–2123. 10.1016/j.aml.2011.06.011MathSciNetView ArticleMATHGoogle Scholar
  26. Stefanini L, Bede B: Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal. TMA 2009, 71: 1311–1328. 10.1016/j.na.2008.12.005MathSciNetView ArticleMATHGoogle Scholar
  27. Hukuhara M: Intégration des applications mesurables dont la valeur est un compact convexe. Funkc. Ekvacioj 1967, 10: 205–229.MathSciNetMATHGoogle Scholar
  28. Agarwal RP, Arshad S, O’Regan D, Lupulescu V: A Schauder fixed point theorem in semilinear spaces and applications. Fixed Point Theory Appl. 2013., 2013: Article ID 306Google Scholar
  29. Chalco-Cano Y, Rufián-Lizana A, Román-Flores H, Jiménez-Gamero MD: Calculus for interval-valued functions using generalized Hukuhara derivative and applications. Fuzzy Sets Syst. 2013, 219: 49–67.View ArticleMathSciNetMATHGoogle Scholar
  30. Khastan A, Nieto JJ, Rodríguez-López R: Schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty. Fixed Point Theory Appl. 2014., 2014: Article ID 21Google Scholar
  31. Malinowski MT: On set differential equations in Banach spaces - a second type Hukuhara differentiability approach. Appl. Math. Comput. 2012, 219(1):289–305. 10.1016/j.amc.2012.06.019MathSciNetView ArticleMATHGoogle Scholar
  32. Moore R, Lodwick W: Interval analysis and fuzzy set theory. Fuzzy Sets Syst. 2003, 135(1):5–9. 10.1016/S0165-0114(02)00246-4MathSciNetView ArticleMATHGoogle Scholar
  33. Bede B, Gal SG: Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst. 2005, 151: 581–599. 10.1016/j.fss.2004.08.001MathSciNetView ArticleMATHGoogle Scholar
  34. Alikhani R, Bahrami F, Jabbari A: Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations. Nonlinear Anal. TMA 2012, 75(4):810–1821.MathSciNetView ArticleMATHGoogle Scholar
  35. Allahviranloo T, Hajighasemi S, Khezerloo M, Khorasany M, Salahshour S: Existence and uniqueness of solutions of fuzzy Volterra integro-differential equations. Communications in Computer and Information Science 81. Information Processing and Management of Uncertainty in Knowledge-Based Systems. Applications 2010, 491–500.Google Scholar
  36. Bede B: A note on ‘Two-point boundary value problems associated with non-linear fuzzy differential equations’. Fuzzy Sets Syst. 2006, 157(7):986–989. 10.1016/j.fss.2005.09.006MathSciNetView ArticleMATHGoogle Scholar
  37. Bede B, Rudas IJ, Bencsik AL: First order linear fuzzy differential equations under generalized differentiability. Inf. Sci. 2007, 177: 1648–1662. 10.1016/j.ins.2006.08.021MathSciNetView ArticleMATHGoogle Scholar
  38. Bede B, Stefanini L: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 2013, 230: 119–141.MathSciNetView ArticleMATHGoogle Scholar
  39. Chalco-Cano Y, Román-Flores H: On new solutions of fuzzy differential equations. Chaos Solitons Fractals 2008, 38: 112–119. 10.1016/j.chaos.2006.10.043MathSciNetView ArticleMATHGoogle Scholar
  40. Lupulescu V: Initial value problem for fuzzy differential equations under dissipative conditions. Inf. Sci. 2008, 178(23):4523–4533. 10.1016/j.ins.2008.08.005MathSciNetView ArticleMATHGoogle Scholar
  41. Malinowski MT: Random fuzzy differential equations under generalized Lipschitz condition. Nonlinear Anal., Real World Appl. 2012, 13(2):860–881. 10.1016/j.nonrwa.2011.08.022MathSciNetView ArticleMATHGoogle Scholar
  42. Malinowski MT: Existence theorems for solutions to random fuzzy differential equations. Nonlinear Anal. TMA 2010, 73(6):1515–1532. 10.1016/j.na.2010.04.049MathSciNetView ArticleMATHGoogle Scholar
  43. Malinowski MT: On random fuzzy differential equations. Fuzzy Sets Syst. 2009, 160(21):3152–3165. 10.1016/j.fss.2009.02.003MathSciNetView ArticleMATHGoogle Scholar
  44. Stefanini L, Sorini L, Guerra ML: Parametric representation of fuzzy numbers and application to fuzzy calculus. Fuzzy Sets Syst. 2006, 157: 2423–2455. 10.1016/j.fss.2006.02.002MathSciNetView ArticleMATHGoogle Scholar
  45. Mizukoshi MT, Barros LC, Chalco-Cano Y, Román-Flores H, Bassanezi RC: Fuzzy differential equations and the extension principle. Inf. Sci. 2007, 177: 3627–3635. 10.1016/j.ins.2007.02.039View ArticleMathSciNetMATHGoogle Scholar
  46. Malinowski MT: Strong solutions to stochastic fuzzy differential equations of Itô type. Math. Comput. Model. 2012, 55: 918–928. 10.1016/j.mcm.2011.09.018MathSciNetView ArticleMATHGoogle Scholar
  47. Malinowski MT: Itô type stochastic fuzzy differential equations with delay. Syst. Control Lett. 2012, 61: 692–701. 10.1016/j.sysconle.2012.02.012MathSciNetView ArticleMATHGoogle Scholar
  48. Malinowski MT: Some properties of strong solutions to stochastic fuzzy differential equations. Inf. Sci. 2013, 252: 62–80.MathSciNetView ArticleMATHGoogle Scholar
  49. Malinowski MT: Approximation schemes for fuzzy stochastic integral equations. Appl. Math. Comput. 2013, 219: 11278–11290. 10.1016/j.amc.2013.05.040MathSciNetView ArticleMATHGoogle Scholar
  50. Malinowski MT: On a new set-valued stochastic integral with respect to semimartingales and its applications. J. Math. Anal. Appl. 2013, 408: 669–680. 10.1016/j.jmaa.2013.06.054MathSciNetView ArticleMATHGoogle Scholar
  51. Malinowski MT: Modeling with stochastic fuzzy differential equations. In Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. Edited by: Chakraverty S. IGI Global, Hershey; 2014:150–172.View ArticleGoogle Scholar
  52. Agarwal RP, Arshad S, O’Regan D, Lupulescu V: Fuzzy fractional integral equations under compactness type condition. Fract. Calc. Appl. Anal. 2012, 15: 572–590. 10.2478/s13540-012-0040-1MathSciNetView ArticleMATHGoogle Scholar
  53. Arshad S, Lupulescu V: On the fractional differential equations with uncertainty. Nonlinear Anal. TMA 2011, 74(11):3685–3693. 10.1016/j.na.2011.02.048MathSciNetView ArticleMATHGoogle Scholar
  54. Arshad S, Lupulescu V: Fractional differential equation with fuzzy initial condition. Electron. J. Differ. Equ. 2011., 2011: Article ID 34Google Scholar
  55. Allahviranloo T, Salahshour S, Abbasbandy S: Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun. Nonlinear Sci. Numer. Simul. 2012, 17(3):1372–1381. 10.1016/j.cnsns.2011.07.005MathSciNetView ArticleMATHGoogle Scholar
  56. Allahviranloo T, Abbasbandy S, Sedaghatfar O, Darabi P: A new method for solving fuzzy integro-differential equation under generalized differentiability. Neural Comput. Appl. 2012, 21(1):191–196.View ArticleGoogle Scholar
  57. Allahviranloo T, Amirteimoori A, Khezerloo M, Khezerloo S: A new method for solving fuzzy Volterra integro-differential equations. Aust. J. Basic Appl. Sci. 2011, 5(4):154–164.MATHGoogle Scholar

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© Hoa et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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