Integrals on time scales were considered, for example, by Liu and Zhao  , Mozyrska et al.  and by Peterson and Thompson  . Liu and Zhao  studied the McShane integral on time scales. On the other hand, Mozyrska et al.  studied the Riemann-Stieltjes integral on time scales. In turn, Peterson and Thompson  studied the Henstock-Kurzweil integral on time scales. Here we establish an extension of the Aumann integral. Thus, using the Lebesgue D-integral on time scales, see for example Guseinov  , we define the Aumann D-integral on time scales. To the best of our knowledge, the Aumann integral on time scales has not yet been considered in the literature. We get some basic properties for the Aumann D-integral on time scales in consonance with the basic properties of the Aumann integral considered by Aumann  . Furthermore, we established a formula that relates the Aumann D-integral on time scales and the Aumann integral, in analogy to the formula obtained by Cabada and Vivero  that relates the Lebesgue D-integral on time scales and the Lebesgue integral.
In this section we consider concepts and results necessary for the study of the Aumann D-integral on time scales.
2.1. Time Scales
A time scale is a nonempty closed subset of the real numbers. Here we use an arbitrary bounded time scale where and are such that .
Define the forward jump operator by
Here we assume that .
Lemma 1  There exist and such that
where stands for right scattered points of the time scale .
2.2. Lebesgue Integration on Time Scales
The definition of D-mensurable sets of , was considered, for example, by Guseinov  .
We denote the family of D-mensurable sets of by D. We remember that D is a s-algebra of subsets of the time scale .
It is said that a function is D-measurable if for each the set is D-measurable. The vector valued function is D-measurable if each component is D-measurable.
Consider a function and a set . We indicate by
the Lebesgue D-integral of f over E. If is a D-measurable function and , is integrable over E if each component is integrable over E. In this case
We denote by the set of functions D-integrable over E.
Cabada and Vivero  and Santos and Silva  consider a more complete approach to Lebesgue integration theory on time scales.
Given a function , define as
where and .
If , define
It follows from Cabada and Vivero  the next two results.
Proposition 1 Take a function . Then is D-measurable if and only if is Lebesgue measurable.
Theorem 1 Let be such that . Then if and only if . In this case
2.3. Measurable Multifunctions
Let be a measurable space. A multifunction is a set-valued function that takes points into subsets of . We say that the multifunction is -measurable if the set
is -measurable for all compact sets .
A function is a selection of the multifunction if for each .
A multifunction is said to be closed, compact, convex or nonempty when satisfies the required property, for each point .
We will use the following result due to Castaing and Valadier  .
Theorem 2 Let be a measurable space and a nonempty closed multifunction. If is -measurable then admits a measurable selection.
3. Aumann D-Integral on Time Scales
If , we denote the set by .
Consider a nonempty multifunction . Let be the set of all functions such that f is D-integrable over and for all . We define the Aumann D-integral of F over by
We note that the Aumann D-integral of F over coincides with the usual Aumann integral when . Hence the Aumann D-integral on time scales is a generalization of the usual Aumann integral.
From definition, if and is given by for
each , then . On the other hand, if and
is defined by for every , then
Below we establish properties for the Aumann D-integral on time scales.
Theorem 3 If is a convex nonempty multifunction, then
Proof. Let . If it follows that . Hence,
and thus is convex.
We say that the multifunction is D-integrably bounded if there is a function D-integrable over such that for all y and t such that .
Theorem 4 Let be a nonempty closed, D-integrably bounded
and D-measurable multifunction. Then is nonempty.
Proof. From Theorem 2 the multifunction F admits a D-measurable selection f. Since F is D-integrably bounded, it follows that f is D-integrable over .
Thus, and then is nonempty.
Given a multifunction , we define the multifunction by
Theorem 5 Let be a nonempty compact and convex multifunction. Then
Proof. Let be a selection of F. Suppose that f is D-integrable over . Hence the function is a selection of . Furthermore, it follows from Theorem 1 that
Consider a selection of . Suppose that g is Lebesgue integrable over .
Let . We have
Since F is a compact and convex multifunction, for each there exists such that
Define the function as
Hence the proof is complete.
Theorem 6 Let be a nonempty compact, convex and D-integrably bounded multifunction. Then is a compact set.
Proof. We know by Aumann  that the set
is compact. From Theorem 5 we may conclude that the set
By introducing the Aumann D-integral on time scales, the paper contributes to the theory of time scales, more specifically, for the integration on time scales. The Aumann integral on time scales is added to other extensions of integrals for the theory of time scales, namely, the McShane integral on time scales, the Riemann-Stieltjes integral on time scales and the Henstock-Kurzweil integral on time scales, among others. The paper also established properties for the Aumann D-integral on time scales. Moreover, a formula is also established that relates the Aumann D-integral on time scales and the Aumann integral. However, such a formula is restricted to multifunctions . Thus, future work might consider possibilities under which this formula remains valid for multifunctions .
 Cabada, A. and Vivero, D.R. (2006) Expression of the Lebesgue △-Integral on Time Scales as a Usual Lebesgue Integral: Application to the Calculus of △-Antiderivatives. Mathematical and Computer Modelling, 43, 194-207.
 Santos, I.L.D. and Silva, G.N. (2013) Absolute Continuity and Existence of Solutions to Dynamic Inclusions in Time Scales. Mathematische Annalen, 356, 373-399.