1. Introduction

The majority of the theories describing the world we live in, are based on differential and integral equations. Such equations appear not only in the physical sciences, but in biology, medicine, chemistry, sociology, and all scientific disciplines that attempt to understand the world surrounding us. Recently the set-valued differential or integral equations have attracted the increasing interest of scientists. The development of the theory of set-valued differential equations (SDEs) begun from works of F.S. de Blasi and F. Iervolino ^[1]. After that the properties of solutions of SDEs [2-14]^[2], the set-valued integro-differential equations (SIDEs) ^[15], the impulse set-valued differential equations ^{[9, 11, 16]}, set-valued control systems ^{[12, 17, 18, 19]}, the set-valued integral equations (SIEs) ^{[20, 21]} and asymptotic methods ^{[9, 11, 12, 19]} were considered. On the other hand, SDEs and SIEs are useful in other areas of mathematics. For example, SDEs and SIEs are used as an auxiliary tool to prove the existence results for differential and integral inclusions ^{[9, 13, 14]}. Also, one can employ SDEs and SIEs in the investigation of fuzzy differential and integral equations ^{[5, 7, 8, 11, 23, 24, 25]}. Moreover, SDEs and SIEs are a natural generalization of usual ordinary differential and integral equations in finite (or infinite) dimensional Banach spaces. Therefore, in this article we consider set-valued Volterra integral equations and prove the existence and uniqueness theorem.

2. Main Result

Let be the family of all nonempty convex compact subsets of with the Hausdorff metric

where be a -neighborhood of the set .

Lemma 2.1 ^[26]. The following properties hold:

1.  is a complete metric space,

2. ,

3.  for all and .

Definition 2.1 ^[27]. Let . A set such that is called a Hukuhara difference of the sets and and is denoted by .

Consider the set-valued Volterra integral equation

(2.1)

where , , , is a set-valued mapping, . The integral is understood in the sense of ^[27].

Definition 2.2. A set-valued mapping is called a solution of integral equation (2.1) if it is continuous and satisfies integral equation (2.1) on interval [0, T].

Remark 2.1. There exist such that but there exists no such that . For example, , . In this case there exists such that , but does not exist such that .

Let be a space of all nonempty strictly convex closed sets of and all element of ^[28].

Remark 2.2. If and then ^[28].

Definition 2.3. It is said that the set can be embedded in the set if there exists such that .

Theorem 2.1. Let in the domain the following conditions hold:

i) for any fixed the set-valued mapping is continuous;

ii) for any fixed the set-valued mapping is continuous;

iii) there exists a positive constant such that

for all ;

iv) there exists such that

for all ;

v)  and ;

vi) if the set can be embedded in the set , then the set can be embedded in the set for all ;

vii) .

Then equation (2.1) has a unique solution on the interval .

Proof. According to Definition 2.1. we associate with set integral equation (2.1) the following set integral equation

(2.2)

Let us prove the existence of a solution of equation (2.2) on interval .

a) As for all , then for all ^[28]. Therefore, as and , then there exists such that the Hukuhara difference exists for all ^[28].

b) Let us build the successive approximations of the solution

(2.3)

By a), exists and for all . Also by conditions i) ,ii) and iii) of the theorem is continuous on and the set can be embedded in the set for all , i.e. there exists such that , where for .

By condition vi), the set can be embedded in the set for all and . Then the set can be embedded in the set for all , i.e. there exists such that

Therefore, exists and for all . Also by conditions i), ii) and iii) of the theorem is continuous on and the set can be embedded in the set for all , i.e. there exists such that , where for .

By condition vi), the set can be embedded in the set for all and . Then the set can be embedded in the set for all , i.e. there exists such that

and so on.

Similarly, exists and for all and . Also by conditions i), ii) and iii) of the theorem is continuous on and the set can be embedded in the set for all for all .

Besides

and so on.

Therefore,

Then

Hence, it follows that the sequence of the set-valued mappings in uniformly bounded:

for all .

Let us show that the sequence of the set-valued mappings is a Cauchy sequence. For any we have

Hence,

Therefore, the sequence is a Cauchy sequence. Its limit is a continuous set-valued mapping that we will denote by . Owing to the theorem conditions in (2.3) it is possible to pass to the limit under the sign of the integral. We receive that the set-valued mapping satisfies equation (2.2), i.e. is the solution of (2.1) on the interval .

To prove the uniqueness, suppose that there exist at least two different solutions and of (2.1) on , then .

As and then

Similarly,

Then . Therefore, for any that contradicts . This concludes the proof.

Finally we consider example for case .

Example 3.1. Consider the following set-valued integral equation

(2.4)

where . It is obvious that

is the solution of equation (2.4) (see Figure 2.1)

Figure 2.1. The graph of a solution of system (2.4)

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3. Conclusion

Also it is possible to prove the similar results if instead of we consider a space of all nonempty -strongly convex closed sets of and all elements of ^[29], i.e. .

Example 3.1. Consider the example 2.1 if . It is obvious

is the solution (see Figure 3.1).

Figure 3.1. The graph of a solution of example 3.1

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