General Forms of Solutions for Linear Impulsive Fuzzy Dynamic Equations on Time Scales

A class of linear impulsive fuzzy dynamic equations on time scales is considered by using the generalized differentiability concept on time scales. Some novel criteria and general forms of solutions are established for such models whose significance lies in proposing the possibility to get unifying forms of solutions for discrete and continuous dynamical systems under uncertainty and to unify corresponding problems in the framework of fuzzy dynamic equations on time scales. Finally, some examples show the applicability of our results.


Introduction
In the real world, some processes vary continuously, while others vary discretely. ese processes can be modeled by differential and difference equations, respectively. ere are also some processes that vary both continuously and discretely. Usage of fuzzy differential and difference equations is a natural way to model dynamical systems under possibilistic uncertainty [1,2]. First-order linear fuzzy differential (difference) equations are one of the simplest fuzzy equations which are very basic, important, and may appear in many applications. us, it is reasonable to seek conditions under which the resulting fuzzy systems would have a solution with a general form. Much progress has been seen in the fuzzy differential (difference) equation direction, and many criteria are established based on different approaches (for instance, fuzzy differential equations [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and fuzzy difference equations [18][19][20][21][22][23]). Careful investigation reveals that it is similar to explore the existence of solutions for fuzzy differential equations and their discrete analogue in the approaches, methods, and the main results. For example, extensive research shows that many results concerning the existence of fuzzy differential equations can be carried over to their discrete analogues [24][25][26]. However, other results seem to be completely different [3]. It is natural to ask whether we can explore such an existence problem in a unified way and offer more general conclusions. For the certainty system, the theory of time scale calculus and dynamic equations on time scales provides us with a powerful tool for attacking such mixed processes [27]. e calculus on time scales (see [28][29][30][31]) was initiated by Hilger in [28] in order to unify continuous and discrete analysis under the certainty system, and it has a tremendous potential for applications and has recently received much attention.
e H-derivative of a fuzzy-number-valued function was introduced in [32], and it has its starting point in the Hukuhara derivative of set-valued functions. e first approach to modeling the uncertainty of dynamical systems uses the H-derivative or its generalized, and mainly the existence and uniqueness of the solution of a fuzzy differential equation are studied under this setting (see for example [11,14,[33][34][35]). Fuzzy differential equations have been studied under other approaches (see [12,36]). Furthermore, there are several works that have dealt with fuzzy-numbervalued functions on time scales and focused on a class of new derivative of such fuzzy functions, see [37][38][39][40], as well as the Hukuhara derivative of set-valued functions has been extended onto the time scales by Hong in [41][42][43] and fuzzy or set dynamic equations have afterward been discussed in cited above references and [44,45]. e aim of this paper is to establish a general form of solutions for linear fuzzy impulsive dynamic equations whose significance lies in proposing the possibility to get unifying forms of solutions for discrete and continuous dynamical systems under uncertainty and to build a unifying framework for the study of corresponding problems. As mentioned above, the notion of the H-derivative plays a fundamental role in the theory of fuzzy differential equations and the calculus on time scales has the features of unification and extension. In order to achieve our purpose, a derivative of fuzzy-number-valued functions on time scales, which is similar to the one in [37] and called the Δ H -derivative in this paper, will be developed to suit our study of fuzzy dynamic equations. e proposed approach forms the appropriate environment within which the study of fuzzy dynamic systems on time scales can be developed.
is paper contains four sections. In Section 2, we recall several basic definitions and properties of time scales and generalized differentiability of fuzzy-number-valued functions on time scales proposed by [38] which is the extension of that on the real axis R introduced in [24]. Moreover, it contains the Δ H -derivative introduced in [41]. In addition, some corresponding properties of the Δ H -derivative are explored which provide the necessary background for our further consideration. Subsequently, in Section 3, we consider first order linear fuzzy dynamic equations on account of Δ H -differentiability. e idea of the present section originates from the study of an analogous problem examined by Khastan et al. [3] for a variation of constant formula for the first-order linear fuzzy differential equations in R. As distinct from [3], we consider the impulsive problem on an infinite time scale interval instead of the initial value problem on a finite realnumber interval and present the solutions with general expressions in this setting. is study reveals that, when we deal with the existence of solutions with general expressions for linear fuzzy differential equations and the difference counterparts, it is unnecessary to prove results for fuzzy differential equations and separately again for their discrete analogues. In other words, one can get a unifying expression of solutions for such continuous and discrete uncertainty systems. Finally, several examples are given to illustrate the applicability of our results in Section 4.

Preliminaries
In this section, we first recall a notion of the time scale built by Hilger and Bernd Aulbach. For more details, we refer the reader to [28,29].
A closed nonempty subset T of real axis R is called a time scale or measure chain. For t ∈ T, we define the forward jump operator σ: In this definition, we put inf ∅ � sup T(i.e. σ(t) � t if T has a maximum t) and sup ∅ � inf T(i.e., ρ(t) � t if T has a minimum t), where ∅ denotes the empty set. t is said to be right (left) scattered if σ(t) > t(ρ(t) < t), and t is said to be A point is said to be isolated (dense) if it is right-scattered (right-dense) and leftscattered (left-sense) at the same time. In this paper, we stipulate that the time scale A function f is right-dense continuous (rd-continuous, for short) if f is continuous at each right-dense point in T and its left-sided limits exist at each left-dense points in T. For a function f: T ⟶ R and t ∈ T, S. Hilger defined the Δ-derivative of f at t, f Δ (t), to be the number (when it exists), with the property that, for each ε > 0, there exists a neighborhood We also recall the concept of the matrix-valued functions introduced by [29]. An m × n-matrix-valued function A: T ⟶ R mn (a collection of all m × n-real matrixes) is said to be Δ-differentiable on T provided each entry of A is Δ-differentiable on T. In this case, we put An n × n-matrix-valued function A on T is called regressive provided Here, I stands for an n × n-identity matrix (and so is it in what follows). Let R � A | A: T ⟶ R nn is a regressive and rd-continuous n × n matrix − valued function}, From now on, unless otherwise mentioned, the matrixvalued functions under consideration are always assumed to belong to R.
For A, B ∈ R, the "circle plus" and "circle minus" of matrix-valued functions are referred to as, respectively, A matrix exponential function e A (t, t 0 ) is defined as a unique matrix-valued solution of the following initial value problem: where A ∈ R and t 0 ∈ T. In [29], matrix exponential functions have been proved to possess the following properties: For an interval J ⊂ T, if a function g: J ⟶ R is Δ-differentiable and g Δ (t) � f(t); then, in [29], the authors defined the Cauchy integral by t a f(s)Δs � g(t) − g(a).
In this case, f is said to be Δ-integrable on J. In particular, by In the following, we introduce the necessary definitions and notation for fuzzy numbers on time scales which are the extension of the corresponding concepts in R (see, for example, [46]). Let us denote by T f the class of fuzzy subsets of T satisfying the following properties, that is, u ∈ T f , i.e. u: T ⟶ [0, 1] and (f1) u is normal, i.e., there exists s 0 ∈ T such that u(s 0 ) � 1 (f2) u is fuzzy convex on T, i.e., u(ta , it follows that the α-level set [u] α is a nonempty compact interval of T for all 0 ≤ α ≤ 1 if u belongs to T f (i.e., [u] α is an intersection of a nonempty compact interval of R and T). e notation denotes explicitly the α-level set of u on T. We refer to u and u as the lower and upper branches of u, respectively. For u ∈ T f , we define the length of u as For u, v ∈ T f and λ ∈ R, the sum u + v and the product λu are defined by α is defined as the same as the usual addition of two intervals (subsets) of T and λ[u] α means the usual product between a scalar and a subset of T.
e metric structure is given by the Hausdorff distance D: (T f , D) is a complete metric space [42,46], and the following properties are well known: Definition 1. Let x, y ∈ T f . If there exists z ∈ T f such that x � y + z, then z is called the H-difference of x, y and it is denoted by x− H y. Let us remark that, in general, x − H y ≠ x + (− 1)y. Usually, we denote x + (− 1)y by x − y, while x − H y stands for the H-difference. Similar to the analysis in [24,37,38], we have the following remark: e following lemma appears in references [37,38,47].

Lemma 1. If u − H v exists, it is unique and one has
In the sequel, we fix T + � T ∩ R + . e strongly generalized differentiability on the real axis R was introduced in [24] and studied in [1,26]. Motivated by these works, we introduce generalized differentiability on a time scale T which appears to [38] and can be regarded as a generalization of Δ H -differentiability introduced in [41].
Discrete Dynamics in Nature and Society (2) F is said to be Δ-left differentiable at t if there exists an element Δ − F(t) of T f and, for any given ε > 0, there exists a neighborhood U T of t such that either the H-differences F(t − h) − H F(σ(t)) exists and e principal properties of the Δ H -derivative in the sense of Definition 2 have been proposed in [37][38][39]41]. Next, we shall write some properties whose a majority of proofs are similar to the above mentioned references.
Proposition 1 (see [38]). Let F: T ⟶ T f . For t ∈ T, we have the following results:

(III) If t is right-dense, then F is Δ H -differentiable at t if and only if
exist as a finite number and satisfy any one of the following equations: Remark 1. Proposition 1 implies that, under the hypothesis that T is a discrete system, the fuzzy number-valued function F: T ⟶ T f is Δ H -differentiable if and only if F is continuous and the corresponding H-differences exist. However, if T is a continuous system, Δ H -derivative of F does not always exist even if the corresponding H-differences exist. (19) are identical to the relevant provisions in Definition 5 in [24] in which four cases for derivatives were considered in R. In addition, equation (16) is identical to (16)-differentiability and equation (19) to (17)-differentiability in [3].
If T � Z, then the previous definition expresses some generalized difference operators, for example, corresponding to the difference operator ΔF n � F n+1 − H F n given in [21].
For functions F, G: We have the following.
In the sequel, we say that a fuzzy function is Δ H -differentiable meaning that it is in two cases of (c 1 ) and (c 2 )-differentiability (denoted by (i)-differentiable) or (c 2 ) and (c 1 )-differentiability (denoted by (ii)-differentiable) on T.
Discrete Dynamics in Nature and Society Proof. For t ∈ T, if t is a right-dense point, it is the same as the proof of eorem 4 in [1]. If t is a right-scattered point, by means of Proposition 2-(II) and Lemma 1-(i) and (ii), we have is proof is complete. e following lemma roots in eorem 5 of [1].
Proof. We get into details regarding the discussion of the cases (b) and (c), while the proof of (a) is similar to (b).
(b) For any t ∈ T and ε > 0, there exists a neighborhood U T of t for some δ > 0 such that Note that a(σ(t)) has the same sign as a(t + h) for sufficiently small h > 0. In addition, a(σ(t))a Δ (t) < 0 and μ(t) − h ≤ 0 imply that a Δ (t)(μ(t) − h) has the same sign as a(σ(t)). Hence, It follows that In view of this and the continuity of G, together with the inequalities (26) and (27), we see that aG satisfies Discrete Dynamics in Nature and Society the first inequality of Definition 2-(c 2 ). We can similarly check the second inequality of Definition 2-(c 1 ). Consequently, the desired conclusion arrives.
(c) As in case (b), the inequalities (26) and (27) are valid. Moreover, − a Δ (t)(μ(t) − h) has the same sign as a(σ(t)) and a(t + h) under the hypothesis of (c). erefore, for 0 As in case (b), we arrive at the desired result. is proof is complete.
Proposition 4 for its proof is similar to eorem 5 in [11]. A fuzzy-number-valued function F: J ⊂ T ⟶ T f is called regulated provided its right-sided limit exists at any right-dense point in T and left-sided limit exists at any leftdense point in T.
F is called right dense continuous, denoted r d-continuous, provided F is continuous at each right dense point in T, its left-sided limits exist at each left dense points in T. Similarly, we can define l d-continuity.
e sets of all r d-continuous fuzzy-number-valued functions, and all such functions F: J ⟶ T f whose rd-continuous Δ H -derivative exist are denoted, respectively, by A function f: J ⊂ T ⟶ R is called an integrable selector of the fuzzy-number-valued function F: For the fuzzy version of the fundamental properties of calculus in the sense of (i)-differentiability, we refer to the analogue of set-valued functions in [41,43], and in the sense of (ii)-differentiability we present the following results which are similar to those proposed in [3,11]: As the authors pointed out in [3], in general, the function F(t) is not (ii)-differentiable. Indeed, suppose that is (ii)-differentiable, then the length of the support decreases in t, but the function F(t), if f is fuzzy non-realvalued, has increasing length of the support. A (ii)-differentiable function needs to have decreasing length of support which is a contradiction.

is such that the previous H-difference exists for t ∈ J. en, U is (ii)differentiable and Δ H U(t) � F(t); (ii) Let F be (ii)-differentiable and Δ H F be integrable on
T + . en, for each t ∈ T + with t ≥ t 0 ∈ T + , we have 6 Discrete Dynamics in Nature and Society

General Forms of Solutions for LIFDE
We emphasize the following notation: It is clear that (BC, D 0 ) is a complete metric space if it is endowed with the distance D 0 (U, V) � sup t∈T + D(U(t), V(t)).
Consider the first linear impulsive fuzzy dynamic equation (LIFDE): where r: T + ⟶ R, F ∈ PC and L k : PC 1 ⟶ PC 1 is a continuous linear operator, i.e., for any v, w ∈ T f and a, b ∈ R, one has L k (av ± g bw) � aL k (v) ± g bL k (w) whenever the H-difference exists. Let U ∈ PC 1 be a fuzzynumber-valued function such that Δ H U exists at every point t ∈ T + \J. By a (i)-solution of LIFDE (39), we mean U and Δ H U exist in the case of (i)-differential and satisfy problem (39). e definition that U is a (ii)-solution of (39) is similar.
To explore the existence of solutions to LIFDE (39), we need the following essential preliminaries. In virtue of eorem 5.24 in [29], the problem has a unique solution v: T ⟶ R n given by where A ∈ R, e A (t) � e A (t, 0) and w: T ⟶ R n is rd-continuous. Let A ∈ R, f: T + ⟶ R n be an rd-continuous function and lim t⟶t + k f(t) � f(t + k ) exist for k � 1, 2, . . ., and let L k be a continuous linear operator acting in R n for k � 1, 2, . . .. By an analogue of the proof of the above result, we can prove that the linear impulsive dynamic equation: for each t k ∈ J has a unique solution for k � 0, 1, 2, . . ., where u 0− 1 � v(t 0 ). us, we obtain that the linear impulsive dynamic equation: has a unique solution w � w(v 0 ) on T + which is left continuous on T + and defined by Similar to the formulation in [3], we study LIFDE (39) in three cases r(t) < 0, r(t) > 0 and r(t) � 0 for t ∈ T + , where r is a function given in LIPDE (39). We first observe that the hyperbolic functions proposed by Bohner and Peterson [31] can be extended to Discrete Dynamics in Nature and Society Let E r (t, s) � e r (t, s)e − r (t, s). Obviously, for all t, s ∈ T, the hyperbolic functions possess the following properties: We are now in a position to state and verify our main results. Theorem 1. If r ∈ R + 1 satisfies r(t) < 0 for all t ∈ T + , then (i) LIFDE (39) has a (i)-solution on T + given by where

. ., and (ii) e (ii)-solution of the LIFDE (39) on T + is given by
provided the H-differences exist and Proof. Proposition 4 shows us how to translate the LIFDE (39) into a system of ordinary dynamic equations (ODEs), that is, if r(t) < 0 with t ∈ T + and U is (i)-differentiable, then for all α ∈ [0, 1], and (39) is translated into the following impulsive system of ODEs: and (39) is translated into (44) with , v 0 given as in (51) and (i) Under the case of the (i)-differential, we check that A given by (51) belongs to R. From r ∈ R + 1 , it follows that the matrix is invertible for each t ∈ T + , that is, A ∈ R. Moreover, we easily check that with A given in (51). Now, by substituting this matrix exponential function for e A (t, t + k ) and e A (t, σ(τ)) of (43), we have erefore, for t ∈ J k with k � 0, 1, 2, . . ., we have Discrete Dynamics in Nature and Society en, by the property (p3), the solution of the linear dynamic equation system is us, we obtain the (i)-solution of LIFDE (39) on J k for k � 0, 1, 2, . . . when r(t) < 0 with t ∈ T + as follows: Similarly, if t ∈ J − , we have Let us remark that the H-difference always exists for s ∈ [0, σ(τ)] and r(t) < 0. As explicated in [3], the diameters of the α-level sets of F(τ)cosh r (σ(τ), s)/E r (σ(τ), s) and F(τ)sinh r (σ(τ), s)/E r (σ(τ), s) are, respectively, While the former is greater than the latter since we have the inequality: 10 Discrete Dynamics in Nature and Society from r ∈ R + 1 and r(t) < 0. Finally, note that cosh r (σ(t), s) · cosh Δ t r (t, s) � r cosh r (σ(t), s) · sinh r (t, s) > 0 and sinh r (σ(t), s) · sinh Δ t r (t, s) � r(t)sinh r (σ(t), s) · cosh r (t, s) > 0 for r(t) < 0, we see that U(t) is (i)-differentiable on T + \J in view of Lemma 2-(a). Consequently, LIFDE (39) has a (i)-solution and (47) holds.
(ii) For t ∈ J k , under the hypothesis of (ii)-differentiability, LIFDE (39) is translated into the corresponding system (42) with A and f given as (52).
Obviously, A ∈ R and e A (t, s) � e r (t, s)I.
By means of (43), we have Repeating the arguments of (i), we obtain that the solution of the corresponding ODEs system is for all α ∈ [0, 1]. We assert that the (ii)-solution of LIFDE (39) on J k is where k � 0, 1, 2, . . . and r(t) < 0. In fact, we observe that Now, the conditions in Lemma 2-(b) are met, so for k � 0, 1, 2 . . .. Similarly, for t ∈ J − , we have We obtain that LIFDE (39) has a (ii)-solution satisfying (49). e proof is complete. □ Theorem 2. If − r ∈ R + 1 and r(t) > 0 with t ∈ T + , then (i) LIFDE (39) has a (ii)-solution on T + given as in (47), where Discrete Dynamics in Nature and Society provided that the above H-differences exist and

(ii) e (i)-solution of LIFDE (39) on T + is given by
Proof. (i) LIFDE (39) with (ii)-differentiability is transformed into the linear impulsive dynamic equation system (44) with u(t), u t + k , A(t), v 0 given in (51) and f(t) as in (52). By (44), we obtain erefore, for t ∈ J k (k � 0, 1, 2, . . .), by r(t) > 0, we check that is (ii)-differentiable and U k is a (ii)-solution of LIFDE (39) on J k . e following argument is due to the proof of eorem 3.3 in [3]. First, by our hypothesis G k is well defined. Second, Lemma 3-(i) guarantees that G k is (ii)-differentiable and the diameter of G k is nonincreasing in the variable t for fixed α ∈ [0, 1]. Note that cosh r (t, t + k ) − sinh r (t, t + k ) � e − r (t, t + k ) is nonnegative and decreasing in t. us, diam[U k (t)] α is nonincreasing in t for fixed α ∈ [0, 1]. erefore, the H-differences U k (σ(t))− H U k (t + h) and U k (t − h)− right-dense point, in view of an analogous argument of eorem 3.3 in [3], we can check that In the light of Proposition 1-(III), we have en, From Proposition 1-(II) and the property (p2) it follows sinh r σ(t), t k − sinh r t + h, t k � r(t)cosh r t, t k μ(t), cosh r σ(t), t k − cosh r t + h, t k � r(t)sinh r t, t k μ(t). (80) Due to the above results, for each α ∈ [0, 1], we uniformly have Discrete Dynamics in Nature and Society Analogously, we can prove for all α ∈ [0, 1]. Hence, Proposition 1-(II) guarantees that Now, (71) holds, and (i) is proved.
(ii) For t ∈ J k , under the hypothesis of (i)-differentiability, LIFDE (39) is translated into the corresponding system (42) with A ∈ R given in (52) and f given in (51). Load e A (t, s) � e r (t, s)I in (43) and repeat the process of the proof of eorem 1, we have, for all α ∈ [0, 1], We now check that the (i)-solution of LIFDE (39) on J k for k � 0, 1, 2, . . . and r(t) > 0(t ∈ T + ) is U k (t) � e r t, t + k L k U k t k + t t k F(τ)e ⊖r σ(τ), t k Δτ .
If T � Z, then For t ∈ J k , we have erefore, in the discrete case, the phenomenon that the uncertainty asymptotically disappears on the fuzzy system arises only a > − 1.