Existence and Approximation of Solutions to Dynamic Inclusions in Time Scales

This work contributes to the existence of solutions for dynamic inclusions in time scales. More specifically, it proves a result of existence and approximation of solutions for dynamic inclusions in time scales. The result obtained is a generalization of the existence and approximation of solutions for differential inclusions.


Introduction
Recently, considerable attention has been given to the study of the existence of solutions for dynamic inclusions in time scales.This can be witnessed by works [1,2,3,4,5,6,7].The work [1] proves the existence of solutions to first order dynamical inclusions in time scales with general boundary conditions.[2] investigates the existence of solutions and extrernal solutions for a first order impulsive dynamic inclusion on time scales.[3] proves the existence of solutions for second order dynamic inclusions in time scales with boundary conditions.The work [4] proves the existence of solutions for first order dynamic inclusions on time scales with nonlocal initial conditions.[5] studies existence results for systems of first order inclusions on time scales with an initial or a periodic boundary value condition.[6] studies the existence of solutions to nabla differential equations and nabla differential inclusions on time scales.[7] provides existence of solutions to a system of dynamical inclusions in time scales.
To the best of our knowledge, the approximation of solutions for dynamic inclusions in time scales has not been considered in the literature of time scales.In this work we have established a result of existence and approximation of solutions for dynamic inclusions in time scales.

Background and preliminaries
In this section, we gather basic concepts and results that will be useful in the development of the work.
We make the following conventions: (i) if x ∈ R n we denote the Euclidean norm of x by x ; (ii) B is the closed unit ball {x ∈ R n : x ≤ 1}; (iii) given a compact subset E ⊂ R and a function g : E → R n , we will indicate by g ∞ the supremum norm.

Time scales
A time scale is a nonempty closed subset T ⊂ R of the real numbers.An arbitrary bounded time scale T will be taken, such that a = min T and b = max T. We also suppose that a < b.
We define the forward jump operator σ : T → T by and the backward jump operator ρ : T → T by ρ(t) = sup{s ∈ T : s < t}.
Here we assume that inf ∅ = sup T and sup ∅ = inf T.
Lemma 1 (Cabada [8]).There exist I ⊂ N and {t i } i∈I ⊂ T such that where RS stands for right scattered points of the time scale T.
Define the function µ : T → [0, +∞) by for all s ∈ (t − δ, t + δ) T , we say that ξ is the delta derivative of f at t and we denote it by ξ := f ∆ (t).Now, consider a function f : T → R n and t ∈ T κ .We say that . The next result is proven in [9] for scalar valued functions.But the generalization for vector valued functions is straightforward.
Theorem 1 (Bohner [9]).Consider a function f : T → R n and t ∈ T κ .Then the following statements hold: (iii) If σ(t) = t, then f is ∆-differentiable at t if and only if there exists the limit

∆-measurable sets
Below, we recall the σ-algebra of subsets of the time scale T.
Denote by F the collection of all subintervals of T given by [ã, b) T = {t ∈ T : ã ≤ t < b}, where ã, b ∈ T. The interval [ã, ã) T is understood as an empty set.
Take an arbitrary subset E ⊂ T. If there exists at least one sequence of intervals The outer measure defined on R will be denoted by λ * .Properties of the outer measure m * can be founded in [8], [10], and [11].Below, we have considered some of these properties.
Using [11] one can prove the following lemma.
Thus we have the following result.
Theorem 2. The family of ∆-measurable sets is a σ-algebra of T.
We will indicate by ∆ the σ-algebra of ∆-measurable sets of T. We call the measure m * : ∆ → [0, +∞] of ∆-measure of Lebesgue and denoted it by m * ≡ µ ∆ .
Let E ⊂ T. We say that a statement P holds ∆-almost everywhere (∆-a.e.) on E, if the set N given by N = {t ∈ E : P does not hold at t} satisfies µ ∆ (N ) = 0.

∆-measurable functions and ∆-integrability
We say that a function f Proof.This result is stated in [8] for scalar valued functions f .However, it can be verified that it remains valid for vector valued functions.
For functions f : T → R the integration concept can be found, for example, in [11] and [14].The integral of a function f : We call this integral the Lebesgue ∆-integral of f over E and denote the set of functions Next, a result that relates the Lebesgue ∆-integral in time scales and the usual Lebesgue integral is presented.
If E ⊂ T we define the set Ẽ by where The following result is provided in [8] for scalar valued functions.However, one can see that it holds for vector valued functions as stated below.
We also have the following theorem.
We will use the following elementary result in the proof of lemma 6.

Lemma 6. Consider a function
for all j ≥ 1 and all t ∈ T, where s 0 = t.
k(s 1 )g(s 1 )ds 1 . If and then Suppose the lemma is valid for j ≥ 2. Below we find that the lemma is also valid for j + 1 and therefore by mathematical induction we conclude that the formula 1 is valid for all j.

Absolutely continuous functions in time scales
Theorem 6 given below is established in [13] for scalar valued functions.However, it is easy to see that it can be extended for vector valued functions as stated below.
Theorem 6 (Cabada [13]).A function f : T → R n is absolutely continuous if and only if the following assertions are valid: (ii) for each t ∈ T we have We say that the function f : T → R n is an arc if it is absolutely continuous.In the next lemma, we get a result on absolutely continuous functions in time scales.
We have the following consequence of the previous lemma.
R m given by Consider a function φ : T × R m → R n .We say that φ is a ∆-Carathéodory function if it satisfies the followings properties: Let B m denote the Borel σ-algebra of R m .We use the notation ∆ × B m for the product σ-algebra between ∆ and B m .We recall that the σ-algebra ∆ × B m is the least σ-algebra of T × R m that contains all products A × B, where A ∈ ∆ and B ∈ B m .Lemma 10 (Loewen [16]).
for some i ∈ I.
Proof.Denote by D the collection of subsets Thus ∆ × B m ⊂ D. Take arbitrarily a compact set V ⊂ R n .We have and then E ∈ D. Therefore for each t ∈ T. It follows from Lemma 13 that Continuing the sequence x j as previously, we get at each step x ∆ l (s) − x ∆ l+1 (s) .

Conclusion
This work contributes to the theory of time scales.More specifically, the Theorem 7 provides the existence and approximation of solutions for dynamic inclusions in time scales.Thus, we obtain a generalization of the result [[18], 3.1.6Theorem].In the spirit of the present study, [ [19], Theorem 1] can be cited as the first result of existence and approximation of solutions to differential inclusions.

Proposition 4 .
Let F : T × R m R n be a ∆ × B m -measurable set-valued function and u : T → R m a ∆-measurable function.Then the set-valued function G : T R n defined by