Dynamic equation on time scale with almost periodic coe cients

In this paper, we discuss a nonautonomous dynamical equation on time scale in a Banach space. The nonautonomous case is particularly important and needs to be studied because it is frequently met in the mathematical models of evolutionary processes. We give sufficient condition for equation to have an exponentially stable almost periodic solution in terms of the accretiveness of an operator. At the end, examples are given to illustrate the analytical findings.


Introduction
Time scale calculus was rst introduced by Stefan Hilger [1] in order to unify the theory of continuous and discrete calculus. Apart from these two calculus, the theory uni es several other calculus including quantum calculus and equation de ned over Cantor set etc. One advantage to work on di erential equation on time scale is that the results are more general and contains several results as a particular case. Also, it is useful in several situations when we need simultaneous modelling of both continuous and discrete calculus. For example, the insect "Paroha cicada" lives as larva for 17 years and then as adults for 7 days. In this case the time domain is the union of closed subsets of the real line. For the given species, we need the following timescale: where D := Days and Y :=Years. Recent decades have seen tremendous interest in the eld of di erential equations de ned on time scale. It covers di erential equations, di erence equations and several other kinds of evolutionary processes. For more details, we refer to [2][3][4][5][6][7][8][9] and references therein.
Almost periodicity is very important property of a dynamical system. It was introduced by Bohr [10] as a generalization of periodicity. The other de nition which in terms of sequence is given be Bochner [11]. There are several work on the almost periodic solutions of di erential equations, we refer to a nice monograph [12]. This concept is more natural especially in mathematical modeling as the growth rates may not be exactly periodic but periodic with certain error. Moreover, it may capture the dynamics which may not be possible using periodic functions.
In this work, we consider the following dynamical equation on time scale x ∆ = f (t, x), where t ∈ T and x belongs to a Banach space X. The set T denote the time scale which is any nonempty closed subset of real line. Here, we consider the problem over unbounded time scale only. The function f : T× X → X and time scale T are assumed to be almost periodic in t. If f does not depend on t, then it is well known that the equation x ∆ = f (x) generates a dynamical system on time scale under some additional hypothesis. These kind of systems are called autonomous systems. The theory for such systems is very rich and lots of analysis can be performed. But if f depends explicitly on t, which is nonautonomous case, the above mentioned fact is not true. As it is evident that most of the rates appearing in the mathematical models are time dependent, hence it is very important to study nonautonomous dynamical systems. There are several work when T = R, we refer to [13][14][15][16] and references therein, but there are not much work in this direction for general time scale

T.
Several authors have developed methods to study nonautonomous equations, for more details, we refer to [14,15,17,18] and the references therein. Another direction is to employ theory of semigroups of linear as well as nonlinear operators to study such systems. Some works in this direction are by [19,20] for linear equations, and [21][22][23] for nonlinear equations. In the current work, we adopt the approach given in [13], which is continuation of the approach studied in [20]. Of course, our equation is on time scale, so introduction of new term are required. We establish that the evolution semigroup associated with dynamical equation on time scale leaves the function space of almost periodic functions from T to X invariant. So, the smaller the space of "test functions" is the better the results are. We derive the su cient condition which ensure the existence of an exponentially stable almost periodic solution in terms of the accretiveness of the operator L = − d ∆ dt ∆ + f (t, ·) which acts on the set of almost periodic functions.

Preliminaries
In this section, we give important de nitions and results for time scales which are required for further work. A time scale, T, is subset of real line which is non empty and closed. We denote T+ = T∩R ≥ . Some important operators are backward, forward and graininess operators. The backward, forward and graininess operators are de ne by ρ(t) = sup{s ∈ T : s < t}, respectively.

De nition 2.1. A point t is a left dense point and left scattered point when ρ(t) = t and ρ(t) < t respectively with t > inf T. Also, t is right scattered point and right dense when σ(t) > t and σ(t) = t respectively with t < sup T.
De nition 2.5. [7] Let Λ : T → R and t ∈ T. ∆-derivative, Λ ∆ (t) is the number if exist, such that given any ε > , ∃ a neighbourhood U of t such that De nition 2.6. The exp function on T is de ned as eq(r, t) = exp For w > , Note that the function eq(t, s) is solution of x ∆ (t) = q(t)x(t), t ∈ T, x(s) = , where q is a regressive function.
One can see that for λ > , the following claim holds with the condition that µ is bounded. Moreover, for λ > we also have Lemma 2.9. [8] Let q ∈ R and c, d, a ∈ T, then d c q(ζ )eq(a, σ(ζ ))∆ζ = eq(a, c) − eq(a, d).

De nition 2.10. The time scale T is called periodic time scale if
Here R denotes the set of real numbers. Now, we give the de nition of almost periodic function in the sense of Bochner.

De nition 2.12. A function h : T × X → X is said to be almost periodic in t and uniformly for all x in any bounded subset K ⊂ X if for every given sequence {tn} there exists a subsequence {tn k } such that the sequence of functions h(· + tn k , x) is uniformly convergent on
The equation is said to have almost periodic coe cients if for every xed x ∈ X the function f (·, x) is almost periodic. The notation AP(X) is the collection of all almost periodic functions from T to X. One can note that this space is a Banach space under the supremum norm.
De nition 2. 13. A mapping f : T × X → X is said to be admissible if it satis es the following conditions: for all t ∈ T and x ∈ X.
We may replace the growth condition by the similar analysis can de performed without much change.
De nition 2.14. A mapping f : We have the following easy proposition.

Lemma 2.15. An admissible function f satis es condition H if and only if for every xed x ∈ X the function f (·, x) is almost periodic.
Let us denote by X(t, s)x the solution of the Cauchy problem The following de nition is given in [6].

De nition 2.16. Let S be a C semigroup, the liner operator A is generator of S if
The domain D(A) of A is the set of all x ∈ X for which the above limit exists uniformly in t.

De nition 2.17. The zero solution is called exponentially stable if there exists a positive constant d, a constant
C ∈ R + , and an M > such that for any solution We discuss now several versions of the Gronwall lemma, which is very important to prove several results [7].

Lemma 2.18.
Let y, f be rd-continuous functions and p ∈ R + , then Lemma 2. 19. Let y, f be rd-continuous functions and p ∈ R + , then Another version which we use is the following.
Lemma 2.20. Let y be rd-continuous functions and a, b, c ∈ R, with c > , then We make use of all the above versions in order to prove our results.

Nonlinear semigroup and almost periodicity
Let us consider the following di erential equation on a Banach space X : with initial data x(s) = x ∈ X. Let X(t, s) is the associate Cauchy operator. Now de ne the following operator Proof: Let u ∈ AP(X). It is easy to see that X(t, t − h)x is Lipschitz in x. Hence, for xed h, x, the operator X(·, · − h)x is almost periodic. Let us de ne Computing norm of S * , we obtain Now we show that (S * u)(t) is almost periodic. We use Bochner criteria to establish this result. Let {tn} n∈N be any sequence in T, then there exists a subsequence {tn k } k∈N such that {f (t + tn k , u(t + tn k ))} k∈N converges uniformly in t. It is due to the fact that f (t, u(t)) is almost periodic since u(t) is almost periodic. Hence, we obtain which implies that (S * u)(t) is almost periodic in t. Moreover, we can see that for each u, v ∈ AP(X). Hence if h is such that hL f < , the operator S * has a xed point in AP(X). In addition, it coincides with X(t, t − h)x, which implies that X(t, t − h) is almost periodic. Moreover, since the equation has almost periodic coe cients, one can see Let us now denote by A the in nitesimal generator of the semigroup {S h , h ∈ T+}. By de nition then v belongs to D(A) if and only if the limit as s → + . The above equation can be written as Now after taking limit s → , the above relation implies The functions f satis es the required condition and f (·, v(·)) is almost periodic. Hence, we can infer that the D(A) consists of all functions v which is almost periodic with the property that (3.8) The above calculation is valid for σ(t) > t. If σ(t) = t, then similar analysis will work by taking limit when t → . The later case covers R and any other set which is dense in R. This implies the delta derivative v ∆ is almost periodic. Hence D(A) = AP(X). Moreover, we have established that A = − d ∆ dt ∆ + f (t, ·). We can also deduce from the above analysis that if the function f (t, u) is uniform with respect to u ∈ X, then m(X(·, ·−h)u) ⊂ m(f ). Here the symbol m(u) denotes the module (see [12]) of an almost periodic function or a family of almost periodic functions. Now, we prove our next result. Proof: In order to prove the above result, we need to show that for every almost periodic g the following di erential equation has a unique solution in the set of almost periodic functions AP(X). We may consider time dependent t and get a more general result. So, let us consider the modi ed equation where p(t) := λ(t) is regressive function. Hence +p(t)µ(t) ≠ , which is equivalent to the condition λ(t)+µ(t) ≠ . The functions F (t, x) = − λ x σ + f (t, x) + λ g(t) satis es condition H as f satis es condition H. Let us assume P h : h ∈ T+ is the semigroup associate with the above equation. Also, let Y(t, s)x denotes the solution of above equation with the condition that Y(s, s) = I. Similar to last lemma, we can conclude that P h acts on the set of almost periodic functions for every h ∈ T+. Let us denote F(t, x) = f (t, x) + λ g(t). Since Y(t, s) is a solution, we have Multiplying both side by e pt + µ ∞ , we obtain Applying the Gronwall's lemma 2.19 for time scale, we obtain Here, we may assume that L f is time dependent. Rearranging the terms, we obtain We compute the following Thus for < e − ph) + µ ∞ e L f (t, t − h) < , the operator P h is a contraction. Hence, for every xed h ∈ T+ − { }, employing Banach contraction theorem, there exists a unique xed point v h of P h . Moreover, it is evident that this xed point is almost periodic. We can see that Moreover, v ∈ D(L) and Lv = , which implies that it is almost periodic. So if w(t) is any other almost periodic solution, then w(t) is a xed point of P h . Thus, we conclude that w(t) = v . Hence the di erential equation has at most one almost periodic solution.

Theorem 3.3. Let f satis es condition H and the operator αI − L is accretive for some α. Then there exists a unique almost periodic solution of equation (3.1) which is globally exponentially stable.
Proof: We show that the operator L = − d ∆ dt ∆ + f (t, ·) is closed. Let {xn} n∈N is a sequence in the space of almost periodic functions and xn → x, Lxn → y. Now, we have for t ∈ R. Moreover, d ∆ dt ∆ is closed in AP(X). (easy to check), which implies that the operator L is closed. If we pick y ∈ AP (X), then S h y is also in AP (X) due to fact that Hence d ∆ dt ∆ S t y = LS t y for all y ∈ AP (X)., Using Brezis-Pazy theorem [24], we obtain (i) limn→∞(I − h n L) −n u = S h u for u almost periodic and (ii) The inequality S h u−S h v ≤ eα(h, ) u−v . The mapping S h is a contraction if α < . (may depend on t.) So, one can infer the existence of unique almost periodic solution. Let us call this solution x * (t). Furthermore, for y ∈ X, let u(t) = y, we obtain The above relation implies that x * is globally exponentially stability.
We can also observe that m(x * ) ⊂ m(f ). Here the symbol m(u) denotes the module (see [12]) of an almost periodic function or a family of almost periodic functions Let us consider the space AP f (X) which consists of all almost periodic function v such that m(v) ⊂ m(f ). De ne the following operator The operator Q is contraction with xed point given by (S h v)(t). Hence x * ∈ AP f (X).
Next theorem is about perturbation of the equation (1).

Theorem 3.4.
If all condition of Theorem (3.3) are satis ed and let g satis es condition H with Lipschitz constant Lg such that β = α + Lg < , then the operator γI − A is accretive. The notation A = − d ∆ dt ∆ + f (t, ·) + g(t, ·) and the corresponding perturbed equation has a unique globally exponentially stable almost periodic solution.
Proof: It is enough to show that the operator γI − A is accretive and then we can apply Theorem (3.3). For every u, v ∈ AP(X) and λ > , computing the norm, we obtain Hence, we have our desired result. Next, we prove results regarding perturbed equation. Let us assume all the assumptions of previous theorem. may also note that, one can choose time dependent N and δ which are bounded. The proof will still work. Then, we have the following result. Proof: Let us assume that S h * is the evolution operator and X * (t, s) is the Cauchy operator associated with the perturbed equation. We can observe the following Denote δ = sup ξ g(ξ , ) , δ = max{δ , Lg}. We can compute Thus, we obtain Using the another version of Gronwall 2.20, we have Substituting the above relation, we obtain for |t − s| ≤ h. Again applying Gronwall 2.18, we obtain for t ∈ [s, s + h] T , we have also used the positivity of the exponential function since N + δ is positive. As we know Hence we need to compute So, we have x − xp ≤ δ e α+Lg (·, · − ) { + ( + Lg )(N + δ)( x (·) + )}.
Hence, we have the desired result.

Example
Let us consider the following equation x ∆ = A(t)x + b(t), t ∈ T, x ∈ X, (4.10) where A, b : T → R is rd-continuous and almost periodic functions. It is easy to see that f (t, x) = A(t)x + b(t), computing the norm, we get f (t, x) ≤ A(t) x + |b(t)|. Under the assumption that A and b are bounded, we obtain f (t, x) ≤ max{ A , b }( + x ). Hence, we can apply our results to ensure the existence of almost periodic solutions. One may consider the funntion b(t) = sin +cos t+cos √ t , t ∈ T. Another example, we may consider ∆ ∆t x(t, z) = c(t, z) ∆ ∆z x(t, z) + f (t, x(t, z)), t ∈ T, z ∈ [ , ] T Using the above formulations, the considered equation can be rewritten in the following abstract form X ∆ X(t) = A(t)X(t) + F(t, X(t)), X( ) = X t ∈ T, (4.12) in Y = L [ , ] T and X(t) = x(t, ·) which means X(t)z = x(t, z). Hence under the condition that F is almost periodic and satis es the conditions-A, the existence of almost periodic solution is ensured.