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

Authors: Shihuang Hong; Xiaoyu Cao; Ji Chen; Haiyang Hou; Xinggang Luo
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

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.


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## Body

## 1. Introduction

In the real world, some processes vary continuously, while others vary discretely. These processes can be modeled by differential and difference equations, respectively. There 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. Thus, 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, 3–17] and fuzzy difference equations [18–23]). 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 [24–26]. 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]. The calculus on time scales (see [28–31]) 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.TheH-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. The 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, 33–35]). Fuzzy differential equations have been studied under other approaches (see [12, 36]). Furthermore, there are several works that have dealt with fuzzy-number-valued functions on time scales and focused on a class of new derivative of such fuzzy functions, see [37–40], as well as the Hukuhara derivative of set-valued functions has been extended onto the time scales by Hong in [41–43] and fuzzy or set dynamic equations have afterward been discussed in cited above references and [44, 45]. The 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 ΔH-derivative in this paper, will be developed to suit our study of fuzzy dynamic equations. The proposed approach forms the appropriate environment within which the study of fuzzy dynamic systems on time scales can be developed.This paper contains four sections. In Section2, 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 ΔH-derivative introduced in [41]. In addition, some corresponding properties of the ΔH-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 ΔH-differentiability. The 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 real-number interval and present the solutions with general expressions in this setting. This 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.


## 2. 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 subsetT of real axis ℝ is called a time scale or measure chain. For t∈T, we define the forward jump operator σ:T⟶T by σt=infτ∈T:τ>t, while the backward jump operator ρ:T⟶T is defined by ρt=supτ∈T:τ<t. The function μ:T⟶0,∞ called the graininess function is defined by μt=σt−tfort∈T. In this definition, we put inf∅=supT(i.e. σt=t if T has a maximum t) and sup∅=infT(i.e., ρt=t if T has a minimum t), where ∅ denotes the empty set. t is said to be right (left) scattered if σt>tρt<t, and t is said to be right (left) dense if σt=tρt=t. A point is said to be isolated (dense) if it is right-scattered (right-dense) and left-scattered (left-sense) at the same time. In this paper, we stipulate that the time scale T is T−M if T has a left-scattered maximum M.A functionf 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, fΔt, to be the number (when it exists), with the property that, for each ε>0, there exists a neighborhood UT of t (i.e. UT=t−δ,t+δ∩T) such that(1)fσt−fs−fΔtσt−s<εσt−s,for all t∈UT. A function f is said to be Δ-differentiable at t if its Δ-derivative exists at t and Δ-differentiable at T if its Δ-derivative exists at each t∈T.We also recall the concept of the matrix-valued functions introduced by [29]. An m×n-matrix-valued function A:T⟶ℝmn(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(2)AΔ=aijΔ1≤i≤m,1≤j≤n,whereA=aij.Ann×n-matrix-valued function A on T is called regressive provided(3)I+μtAtis invertible for allt∈T.Here,I stands for an n×n-identity matrix (and so is it in what follows). Let(4)ℛ=AA:T⟶ℝnnis a regressive andrd‐continuousn×nmatrix−valued function,ℛ1+=pp:T⟶ℝis ard‐continuous function and1+μtpt>0fort∈T.From now on, unless otherwise mentioned, the matrix-valued functions under consideration are always assumed to belong toℛ.ForA,B∈ℛ, the “circle plus” and “circle minus” of matrix-valued functions are referred to as, respectively,(5)A⊕Bt=At+Bt+μtAtBt,A⊖Bt=A⊕⊖Bt,with⊖At=−I+μtAt−1At=−AtI+μtAt−1.A matrix exponential functioneAt,t0 is defined as a unique matrix-valued solution of the following initial value problem:(6)YΔ=AtY,Yt0=I,where A∈ℛ and t0∈T. In [29], matrix exponential functions have been proved to possess the following properties:(a1)e0t,s≡I,eAt,t≡I,eAσt,s=I+μtAteAt,s(a2)eAs,t=eA−1t,s=e⊖A∗∗t,s, where A∗ stands for the conjugate transpose of the matrix A(a3)eAt,seAs,r=eAt,rFor an intervalJ⊂T, if a function g:J⟶ℝ is Δ-differentiable and gΔt=ft; then, in [29], the authors defined the Cauchy integral by(7)∫atfsΔs=gt−ga.In this case,f is said to be Δ-integrable on J. In particular, by ∫a∞ftΔt:=limb⟶∞∫abftΔt, we mean that f is Δ-integrable on J=a,∞∩T provided this limit exists.In the following, we introduce the necessary definitions and notation for fuzzy numbers on time scales which are the extension of the corresponding concepts inℝ (see, for example, [46]). Let us denote by Tf the class of fuzzy subsets of T satisfying the following properties, that is, u∈Tf, i.e. u:T⟶0,1 and(f1)u is normal, i.e., there exists s0∈T such that us0=1(f2)u is fuzzy convex on T, i.e., uta+1−tb≥minua,ub for all t∈0,1 with ta+1−tb∈T, where a,b∈T(f3)u is upper semicontinuous on T(f4)u0=s∈T:us>0¯∩T is compact, where A¯ denotes the closure of A in ℝThen,Tf is called the space of fuzzy numbers. Obviously, T⊂Tf. Here, T⊂Tf is understood as T=χxxis an element in the time scale. For 0<α≤1, set uα=s∈Tus≥α and u0=s∈Tus>0¯∩T. From (f1)–(f4), it follows that the α-level set uα is a nonempty compact interval of T for all 0≤α≤1 if u belongs to Tf (i.e., uα is an intersection of a nonempty compact interval of ℝ and T). The notation(8)uα=u¯α,u¯α∩T≕u¯α,u¯α,withu¯α,u¯α∈T,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∈Tf, we define the length of u as(9)diamu=u¯α−u¯α.Foru,v∈Tf and λ∈ℝ, the sum u+v and the product λu are defined by u+vα=uα+vα, λuα=λuα for any α∈0,1, where uα+vα is defined as the same as the usual addition of two intervals (subsets) of T and λuα means the usual product between a scalar and a subset of T. The metric structure is given by the Hausdorff distance D:Tf×Tf⟶ℝ+=0,∞⊂ℝ,(10)Du,v=supα∈0,1maxu¯α−u¯α,u¯α−v¯α.Tf,D is a complete metric space [42, 46], and the following properties are well known:(11)Du+w,v+w=Du,v,∀u,v,w∈Tf,Dku,kv=kDu,v,∀k∈ℝ,u,v∈Tf,Du+v,z+w≤Du,z+Dv,w,∀u,v,z,w∈Tf.Definition 1.
Letx,y∈Tf. If there exists z∈Tf such that x=y+z, then z is called the H-difference of x,y and it is denoted by x−Hy.
Let us remark that, in general,x−yH≠x+−1y. Usually, we denote x+−1y by x−y, while x−yH stands for the H-difference. Similar to the analysis in [24, 37, 38], we have the following remark:(1)
We denote0^∈Tf as a neutral element with respect to + if u+0^=0^+u=u for all u∈Tf. For instance, if 0∈T, then 0^=χ0(2)
λ+μu=λu+μu, with λμ>0(3)
λu+v=λu+λv
The following lemma appears in references [37, 38, 47].Lemma 1.
Ifu−vH exists, it is unique and one has(i)
u−uH=0^,u−0^H=u, 0^−uH=−u and u+v−vH=u(ii)
v−uH=−u−vH=−u−−vH provided v−uH existsIn the sequel, we fixT+=T∩ℝ+. The 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 ΔH-differentiability introduced in [41].Definition 2.
LetF:T⟶Tf, t∈T and a neighborhood UT of t be defined by UT=t−δ,t+δ∩T for some δ>0. Then,(1)
F is said to be Δ-right differentiable at t if there exists an element Δ+Ft of Tf and, for any given ε>0, there exists a neighborhood UT of t such that either the H-differences Ft+h−FHσt exists and(12)c1DFt+h−FHσt,Δ+Fth−μt≤εh−μtorFσt−FHt+hexists andc2DFσt−FHt+h,Δ+Ftμt−h≤εh−μt,for allt+h∈UT with 0≤h<δ. Moreover, Δ+Ft is called the Δ-right derivative of F at t.(2)
F is said to be Δ-left differentiable at t if there exists an element Δ−Ft of Tf and, for any given ε>0, there exists a neighborhood UT of t such that either the H-differences Ft−h−FHσt exists and(13)c1DFt−h−FHσt,Δ−Ft−h−μt≤εh+μtorFσt−FHt−hexists andc2DFσt−FHt−h,Δ−Fth+μt≤εμt+h,for allt−h∈UT with 0≤h<δ. Moreover, Δ−Ft is called the Δ-left derivative of F at t.(3)
F is said to be Δ-differentiable at t if F is both Δ-left and Δ-right differentiable at t and Δ−Ft=Δ+Ft. The element Δ+Ft or Δ−Ft is said to be the ΔH-derivative of F at t and is denoted by ΔHFt. We say that F is ΔH-differentiable at t if its ΔH-derivative exists at t. Moreover, we say F is ΔH-differentiable on T if its ΔH-derivative exists at each t∈T. The fuzzy set-valued function ΔHF:T⟶Tf is then called the ΔH-derivative of F on T.
The principal properties of theΔH-derivative in the sense of Definition 2 have been proposed in [37–39, 41]. Next, we shall write some properties whose a majority of proofs are similar to the above mentioned references.Proposition 1 (see [38]).
LetF:T⟶Tf. For t∈T, we have the following results:(I)
IfF is ΔH-differentiable at t, then F is continuous at t.(II)
IfF is continuous at t and t is right scattered, then F is ΔH-differentiable at t with(14)ΔHFt=Fσt−HFtμtorFσt=Ft+μtΔHFt.(III)
Ift is right-dense, then F is ΔH-differentiable at t if and only if(15)limh⟶0+Ft+h−FHthorlimh⟶0+Ft−FHt+h−h,limh⟶0+Ft−FHt−hhorlimh⟶0+Ft−h−FHt−h,exist as a finite number and satisfy any one of the following equations:(16)limh⟶0+Ft+h−FHth=limh⟶0+Ft−FHt−hh=ΔHFt,(17)limh⟶0+Ft+h−FHth=limh⟶0+Ft−h−FHt−h=ΔHFt,(18)limh⟶0+Ft−FHt+h−h=limh⟶0+Ft−FHt−hh=ΔHFt,(19)limh⟶0+Ft−FHt+h−h=limh⟶0+Ft−h−FHt−h=ΔHFt.Remark 1.
Proposition1 implies that, under the hypothesis that T is a discrete system, the fuzzy number-valued function F:T⟶Tf is ΔH-differentiable if and only if F is continuous and the corresponding H-differences exist. However, if T is a continuous system, ΔH-derivative of F does not always exist even if the corresponding H-differences exist.Remark 2.
IfT=ℝ, equations (16)–(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].
IfT=ℤ, then the previous definition expresses some generalized difference operators, for example, corresponding to the difference operator ΔFn=Fn+1−FnH given in [21].
For functionsF,G:T⟶Tf, we define the sum F+G by F+Gt=Ft+Gt and the H-difference F−GH by F−GHt=Ft−GHt for each t∈T. We have the following.Proposition 2.
Assume thatF,G:T⟶Tf are ΔH-differentiable at t∈T and λ is any constant. Then,(i)
The sum isΔH-differentiable at t∈T. Moreover,(20)ΔHF+Gt=ΔHFt+ΔHGt;(ii)
λF is ΔH-differentiable at t and ΔHλFt=λΔHFt.In the sequel, we say that a fuzzy function isΔH-differentiable meaning that it is in two cases of (c1) and (c2)-differentiability (denoted by (i)-differentiable) or (c2) and (c1)-differentiability (denoted by (ii)-differentiable) on T.Proposition 3.
LetF,G:T⟶Tf be ΔH-differentiable such that F is (i)-differentiable and G is (ii)-differentiable or F is (ii)-differentiable and G is (i)-differentiable on T. If the H-difference Ft−GHt exists for t∈T, then F−HG is ΔH-differentiable and(21)ΔHF−GHt=ΔFHt+−1⋅ΔGHt.Proof.
Fort∈T, if t is a right-dense point, it is the same as the proof of Theorem 4 in [1]. If t is a right-scattered point, by means of Proposition 2-(II) and Lemma 1-(i) and (ii), we have(22)ΔHF−Gt=F−GHσt−F−GHHtμt=Fσt−GHσt−Ft−GHtHμt=Fσt−FHtμt+Gt−GHσtμt=ΔHFt+−1⋅Gσt−GHtμt=ΔHFt+−1⋅ΔHGt.
This proof is complete.
The following lemma roots in Theorem 5 of [1].Lemma 2.
Leta:T⟶ℝ be Δ-differentiable and G:T⟶TfΔH-differentiable.(a)
IfaσtaΔt>0 and G is (i)-differentiable, then aG is i-differentiable and we have(23)ΔHaGt=aΔtGt+aσtΔHGt.(b)
IfaσtaΔt<0 and G is (ii)-differentiable, then aG is (ii)-differentiable and we have(24)ΔHaGt=aΔtGt+aσtΔHGt.(c)
IfaσtaΔt>0, G is (ii)-differentiable and the H-differences aGσt−aGHt+h and aGt−h−aGHσt exist, then aG is (ii)-differentiable, and we have(25)ΔHaGt=aσtΔHGt−−aΔtHGt.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 anyt∈T and ε>0, there exists a neighborhood UT of t for some δ>0 such that(26)DGσt−GHt+h,ΔHGtμt−h≤εh−μt,(27)aσt−at+h−aΔtμt−h≤εh−μt,for0≤h<δ with t+h∈UT. On the contrary, we have(28)DaσtGσt−aHt+hGt+h,aΔtGt+aσtΔHGtμt−h≤DaσtGσt−aHσtGt+h,aσtΔHGtμt−h+DaσtGt+h−aHt+hGt+h,aΔtGtμt−h≤DGσt−GHt+h,ΔHGtμt−haσt+DaσtGt+h−aHt+hGt+h,aΔtGtμt−h≤DGσt−GHt+h,ΔHGtμt−haσt+DaσtGt+h,at+hGt+h+aΔtGt+hμt−h+D0,aΔtGtμt−h−HaΔtGt+hμt−h.Note thataσt has the same sign as at+h for sufficiently small h>0. In addition, aσtaΔt<0 and μt−h≤0 imply that aΔtμt−h has the same sign as aσt. Hence, at+hGt+h+aΔtGt+hμt−h=at+h+aΔtμt−hGt+h and(29)DaσtGt+h,at+hGt+h+aΔtGt+hμt−h=aσt−at+h+aΔtμt−hGt+h.It follows that(30)DaσtGσt−aHt+hGt+h,aΔtGt+aσtΔHGtμt−h≤DGσt−GHt+h,ΔHGtμt−haσt+aσt−at+h+aΔtμt−hGt+h+DGt,Gt+haΔtμt−h.In view of this and the continuity ofG, together with the inequalities (26) and (27), we see that aG satisfies the first inequality of Definition 2-(c2). We can similarly check the second inequality of Definition 2-(c1). Consequently, the desired conclusion arrives.(c)
As in case (b), the inequalities (26) and (27) are valid. Moreover, −aΔtμt−h has the same sign as aσt and at+h under the hypothesis of (c). Therefore, for 0≤h<δ with t+h∈UT, we have(31)DaσtGσt−aHt+hGt+h,aσtΔHGt−−aΔtHGtμt−h≤DaσtGσt−aHσtGt+h,aσtΔHGtμt−h+DaσtGt+h−aHt+hGt+h,0^−−aΔtGtμt−hH≤DGσt−GHt+h,ΔHGtμt−haσt+DaσtGt+h,at+hGt+h+−aΔtGt+hμt−h+D0^,−−aΔtHGt+hμt−h−−aΔtGtμt−hH≤DGσt−GHt+h,ΔHGtμt−haσt+aσt−at+h+aΔtμt−hGt+h+DGt,Gt+haΔtμt−h.
As in case (b), we arrive at the desired result. This proof is complete.Proposition 4.
LetF:T⟶Tf and put Ftα=F¯αt,F¯αt for each α∈0,1.(i)
IfF is (i)-differentiable, then F¯α and F¯α are Δ-differentiable functions and(32)ΔHFtα=F¯αΔt,F¯αΔt.(ii)
IfF is (ii)-differentiable, then F¯α and F¯α are Δ-differentiable functions, and we have(33)ΔHFtα=F¯αΔt,F¯αΔt.Proposition4 for its proof is similar to Theorem 5 in [11].A fuzzy-number-valued functionF:J⊂T⟶Tf is called regulated provided its right-sided limit exists at any right-dense point in T and left-sided limit exists at any left-dense point in T.F is called right dense continuous, denoted rd-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 ld-continuity. The sets of all rd-continuous fuzzy-number-valued functions, and all such functions F:J⟶Tf whose rd-continuous ΔH-derivative exist are denoted, respectively, by(34)Crd=CrdJ=CrdJ,Tf,Crd1=Crd1J=Crd1J,Tf.A functionf:J⊂T⟶ℝ is called an integrable selector of the fuzzy-number-valued function F:J⟶Tf if f is Δ-integrable and ft∈Ftα for all t∈J with α∈0,1. F is called integrable on J if it has at least an integrable selector. The integral of F, denoted by ∫JFsΔs, is defined levelwise by(35)∫JFsΔsα=∫JFsαΔs=∫JfsΔs:fis an integrable selector ofFonJ.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]:The integralFFt=∫t0tFsΔs is (i)-differentiable and ΔHFt=Ft. If F∈CrdT,Tf and the function F is (ii)-differentiable, then ΔHFt=Ft.As the authors pointed out in [3], in general, the function Ft is not (ii)-differentiable. Indeed, suppose that is (ii)-differentiable, then the length of the support decreases in t, but the function Ft, if f is fuzzy non-real-valued, has increasing length of the support. A (ii)-differentiable function needs to have decreasing length of support which is a contradiction.Lemma 3 (see [42]).
LetJ=t0,T∩T with t0,T∈T. Then, we have(i)
LetF∈CrdJ,Tf and define Ut=γ−H∫t0t−HFsΔs for t∈J, where γ∈Tf is such that the previous H-difference exists for t∈J. Then, U is (ii)-differentiable and ΔHUt=Ft;(ii)
LetF be (ii)-differentiable and ΔHF be integrable on T+. Then, for each t∈T+ with t≥t0∈T+, we have(36)Ft=Ft0−∫t0t−ΔHHFτΔτH;(iii)
∫t0TFsΔs=∫t0tFsΔs+∫tTFssΔs. Specially, ∫ttFsΔs=0 for t∈J;(iv)
∫tσtFsΔs=μtFt, for t∈J;(v)
LetF is (i)-differentiable on T+. Then, Ft=γ+∫t0tFsΔs with γ∈Tf is (i)-differentiable and ΔHFt=Ft;(vi)
IfF,G∈CrdE, then DF⋅,G⋅:J⟶ℝ+ is Δ-integrable and(37)D∫t0TFsΔs,∫t0TGsΔs≤∫t0TDFs,GsΔs.


## 3. General Forms of Solutions for LIFDE

LetT+=t∈Tt≥0, J=tk∈T+0≤t0<t1<t2<⋯<tk<⋯,limk⟶∞tk=∞, J−=0,t0∩T+ and Jk=tk,tk+1∩T+ for k=0,1,…. Let xtk+=xtk+ represent the right limit of xt at tk if tk is right-dense and xtk+=xσtk if tk is right-scattered for k=1,2,…. We emphasize the following notation:(38)PC=U:T+⟶TfU∈Crdtk−1,tkandlimt⟶tk+Ut=Utk+exists fork=1,2,….BC=U∈PCUt=DUt,0^is bounded inT+,PC1=U∈BCUisΔH−differentiable in each intervaltk−1,tk.It is clear thatBC,D0 is a complete metric space if it is endowed with the distance D0U,V=supt∈T+DUt,Vt.Consider the first linear impulsive fuzzy dynamic equation (LIFDE):(39)ΔHUt=rtUt+Ft,t∈T+J,Utk+=LkUtk,tk∈Jk=0,1,2,…,U0=U0∈Tf,t0∈T+,where r:T+⟶ℝ,F∈PC and Lk:PC1⟶PC1 is a continuous linear operator, i.e., for any v,w∈Tf and a,b∈ℝ, one has Lkav±gbw=aLkv±gbLkw whenever the H-difference exists. Let U∈PC1 be a fuzzy-number-valued function such that ΔHU exists at every point t∈T+\J. By a i-solution of LIFDE (39), we mean U and ΔHU exist in the case of (i)-differential and satisfy problem (39). The 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 Theorem 5.24 in [29], the problem(40)uΔt=Atut+ft,t∈T,u0=v0,has a unique solution v:T⟶ℝn given by(41)vt=eAtv0+∫0teAt,σsfsΔs,where A∈ℛ, eAt=eAt,0 and w:T⟶ℝn is rd-continuous.LetA∈ℛ, f:T+⟶ℝn be an rd-continuous function and limt⟶tk+ft=ftk+ exist for k=1,2,…, and let Lk be a continuous linear operator acting in ℝn for k=1,2,…. By an analogue of the proof of the above result, we can prove that the linear impulsive dynamic equation:(42)uΔt=Atut+ft,t∈Jk,utk+=Φk=Lkutk,utk=uk−1tk,for each tk∈J has a unique solution(43)ukt=eAt,tk+Φk+∫tkteAt,στfτΔτ,t∈Jk,uktk=uk−1tk,for k=0,1,2,…, where u0−1=vt0. Thus, we obtain that the linear impulsive dynamic equation:(44)uΔt=Atut+ft,t∈T+\J,utk+=Φk=Lkutk,tk∈Jk=0,1,2,…,u0=v0,has a unique solution w=wv0 on T+ which is left continuous on T+ and defined by(45)wt=vt,t∈J−,u0t,t∈J0,⋯⋯ukt,t∈Jk;⋯⋯.Similar to the formulation in [3], we study LIFDE (39) in three cases rt<0,rt>0 and rt=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(46)coshrt,s=ert,s+e−rt,s2,sinhrt,s=ert,s−e−rt,s2.LetErt,s=ert,se−rt,s. Obviously, for all t,s∈T, the hyperbolic functions possess the following properties:(p1)
coshrs,s=1,sinhrs,s=0(p2)
coshrΔtt,s=rsinhrt,s,sinhrΔtt,s=rcoshrt,s, where αΔtt,s means the Δ-derivative of α with respect to the variable t(p3)
coshrs,t=1/Ert,scoshrt,s,sinhrs,t=−1/Ert,ssinhrt,s(p4)
sinhrt,s+coshrt,s=ert,s,coshrt,s−sinhrt,s=e−rt,s and coshr2t,s−sinhr2t,s=Ert,sWe are now in a position to state and verify our main results.Theorem 1.
Ifr∈ℛ1+ satisfies rt<0 for all t∈T+, then(i)
LIFDE (39) has a (i)-solution on T+ given by(47)Ut=Vt,t∈J−,Ukt,t∈Jkk=0,1,2,…,where(48)Vt=coshrtU0+∫0tFτcoshrστ,0Erστ,0−FHτsinhrστ,0Erστ,0Δτ+sinhrtU0+∫0tFτcoshrστ,0Erστ,0−FHτsinhrστ,0Erστ,0Δτ,Ukt=coshrt,tk+LkUktk+∫tktFτcoshrστ,tkErστ,tk−FHτsinhrστ,tkErστ,tkΔτ+sinhrt,tk+LkUktk+∫tktFτcoshrστ,tkErστ,tk−FHτsinhrστ,tkErστ,tkΔτ,withU0t0=Vt0,Uktk=Uk−1tk for k=0,1,2,…, and(ii)
The (ii)-solution of the LIFDE (39) on T+is given by(49)Ut=Vt=ertU0−H∫0t−FHτe⊖rστΔτ,t∈J−,Ukt=ert,tk+LkUktk−∫tktFτe⊖rστ,tkΔτH,t∈Jk,provided the H-differences exist and U0t0=Vt0,Uktk=Uk−1tk for k=0,1,2,….Proof.
Proposition4 shows us how to translate the LIFDE (39) into a system of ordinary dynamic equations (ODEs), that is, if rt<0 with t∈T+ and U is (i)-differentiable, then ΔHUtα=U¯αΔt,U¯αΔt with Uα=U¯α,U¯α for all α∈0,1, and (39) is translated into the following impulsive system of ODEs:(50)U¯αΔt=rtU¯αt+F¯αt,t∈T+\J,U¯αΔt=rtU¯αt+F¯αt,t∈T+\J,U¯αtk+=LkU¯αtk,U¯αtk+=LkU¯αtk,tk∈Jk=0,1,2,…,U¯α0=U¯α0,U¯α0=U¯α0,where Ftα=F¯αt,F¯αt for all α∈0,1. For solving system (50), we translate it into system (44) with(51)ut=U¯αtU¯αt,utk+=U¯αtk+U¯αtk+,ft=F¯αtF¯αt,v0=U¯α0U¯α0,α∈0,1,At=0rtrt0.
Similarly, ifU is (ii)-differentiable then ΔHUtα=U¯αΔt,U¯αΔt and (39) is translated into (44) with u,utk+,v0 given as in (51) and(52)At=rt00rt,ft=F¯αtF¯αt.(i)
Under the case of the (i)-differential, we check thatA given by (51) belongs to ℛ. From r∈ℛ1+, it follows that the matrix(53)I+μtAt=1μtrtμtrt1,is invertible for eacht∈T+, that is, A∈ℛ. Moreover, we easily check that(54)eAt,s=coshrt,ssinhrt,ssinhrt,scoshrt,s,withA given in (51).Now, by substituting this matrix exponential function foreAt,tk+ and eAt,στ of (43), we have(55)ukt=coshrt,tk+sinhrt,tk+sinhrt,tk+coshrt,tk+Φk+∫tktcoshrt,στsinhrt,στsinhrt,στcoshrt,στfτΔτ,t∈Jk,uktk=uk−1tk,whereu0−1t0=vt0. Since eAt,στ=eAt,tkeAtk,στ, we have(56)ukt=coshrt,tk+sinhrt,tk+sinhrt,tk+coshrt,tk+Φk+∫tktcoshrtk,στsinhrtk,στsinhrtk,στcoshrtk,στfτΔτ.Therefore, fort∈Jk with k=0,1,2,…, we have(57)U¯αtU¯αt=Uk¯αtUk¯αt=coshrt,tk+sinhrt,tk+sinhrt,tk+coshrt,tk+×LkUk¯αtkLkUk¯αtk+∫tktcoshrtk,στsinhrtk,στsinhrtk,στcoshrtk,στF¯ατF¯ατΔτ=coshrt,tk+sinhrt,tk+sinhrt,tk+coshrt,tk+×LkUk¯αtk+∫tktF¯ατcoshrtk,στ+F¯ατsinhrtk,στΔτLkUk¯αtk+∫tktF¯ατsinhrtk,στ+F¯ατcoshrtk,στΔτ.Then, by the property (p3), the solution of the linear dynamic equation system is(58)u¯αt=Uk¯αt=coshrt,tk+LkUk¯αtk+∫tktF¯ατcoshrστ,tkErστ,tk−F¯ατsinhrστ,tkErστ,tkΔτ+sinhrt,tk+LkUk¯αtk+∫tkt−F¯ατsinhrστ,tkErστ,tk+F¯ατcoshrστ,tkErστ,tkΔτ,U¯αt=Uk¯αt=sinhrt,tk+LkUk¯αtk+∫tktF¯ατcoshrστ,tkErστ,tk−F¯ατsinhrστ,tkErστ,tkΔτ+coshrt,tk+LkUk¯αtk+∫tkt−F¯ατsinhrστ,tkErστ,tk+F¯ατcoshrστ,tkErστ,tkΔτ.Thus, we obtain the (i)-solution of LIFDE (39) on Jk for k=0,1,2,… when rt<0 with t∈T+ as follows:(59)Ut=Ukt=coshrt,tk+LkUktk+∫tktFτcoshrστ,tkErστ,tk−FHτsinhrστ,tkErστ,tkΔτ+sinhrt,tk+LkUktk+∫tktFτcoshrστ,tkErστ,tk−FHτsinhrστ,tkErστ,tkΔτ.Similarly, ift∈J−, we have(60)Ut=Vt=coshrtU0+∫0tFτcoshrστ,0Erστ,0−FHτsinhrστ,0Erστ,0Δτ+sinhrtU0+∫0tFτcoshrστ,0Erστ,0−FHτsinhrστ,0Erστ,0Δτ.Let us remark that theH-difference(61)Fτcoshrστ,sErστ,s−FHτsinhrστ,sErστ,s,always exists fors∈0,στ and rt<0. As explicated in [3], the diameters of the α-level sets of Fτcoshrστ,s/Erστ,s and Fτsinhrστ,s/Erστ,s are, respectively,(62)diamFταcoshrστ,sErστ,s,diamFτα−sinhrστ,sErστ,s.While the former is greater than the latter since we have the inequality:(63)sinhrστ,sErστ,s+coshrστ,sErστ,s=erστ,sErστ,s=1e−rστ,s>0,fromr∈ℛ1+ and rt<0.Finally, note thatcoshrσt,s⋅coshrΔtt,s=rcoshrσt,s⋅sinhrt,s>0 and sinhrσt,s⋅sinhrΔtt,s=rtsinhrσt,s⋅coshrt,s>0 for rt<0, we see that Ut is (i)-differentiable on T+\J in view of Lemma 2-(a). Consequently, LIFDE (39) has a (i)-solution and (47) holds.(ii)
Fort∈Jk, 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(64)eAt,s=ert,sI.
By means of (43), we have(65)ukt=ert,tk+IΦk+∫tkte⊖rστ,tkIfτΔτ,t∈Jk,uktk=uk−1tk.
Repeating the arguments of (i), we obtain that the solution of the corresponding ODEs system is(66)Uk¯αt=ert,tk+LkUk¯αtk+∫tktF¯ατe⊖rστ,tkΔτ,Uk¯αt=ert,tk+LkUk¯αtk+∫tktF¯ατe⊖rστ,tkΔτ,for all α∈0,1. We assert that the (ii)-solution of LIFDE (39) on Jk is (67)Ut=Ukt=ert,tk+LkUktk−∫tkt−Fτe⊖rστ,tkΔτH,where k=0,1,2,… and rt<0. In fact, we observe that erσt,tkerΔtt,tk=rterσt,tkert,tk<0. If we denote Gkt=LkUktk−∫tkt−Fτe⊖rστ,tkΔτH, then Lemma 3-(i) guarantees that Gk is (ii)-differentiable and(68)ΔHGkt=Fte⊖rσt,tk.
Now, the conditions in Lemma2-(b) are met, so(69)ΔHUkt=rtert,tk+Gkt+erσt,tkΔHGkt=rtUkt+Ft,for k=0,1,2…. Similarly, for t∈J−, we have(70)Ut=Vt=ert,0U0−∫0t−Fgτe⊖rστ,0Δτg.
We obtain that LIFDE (39) has a (ii)-solution satisfying (49). The proof is complete.Theorem 2.
If−r∈ℛ1+ and rt>0 with t∈T+, then(i)
LIFDE (39) has a (ii)-solution on T+ given as in (47), where(71)Vt=coshrtU0−∫0tFτsinhrστ,0Erστ,0−FHτcoshrστ,0Erστ,0HΔτ−−sinhrtHU0−∫0tFτsinhrστ,0Erστ,0−FHτcoshrστ,0Erστ,0HΔτ,t∈J−,Ukt=coshrt,tk+LkUktk−∫tktFτsinhrστ,tkErστ,tk−HFτcoshrστ,tkErστ,tkHΔτ−−sinhrt,tk+HLkUktk−∫tktFτsinhrστ,tkErστ,tk−HFτcoshrστ,tkErστ,tkHΔτ,t∈Jk,provided that the above H-differences exist and U0t0=Vt0,Uktk=Uk−1tk for k=0,1,2,….(ii)
The (i)-solution of LIFDE (39) on T+ is given by(72)Ut=Vt=ertU0+∫0tFτe⊖rστΔτ,t∈J−,Ukt=ert,tk+LkUktk+∫tktFτe⊖rστ,tkΔτ,t∈Jk,where U0t0=Vt0,Uktk=Uk−1tk for k=0,1,2,….Proof.
(i) LIFDE (39) with (ii)-differentiability is transformed into the linear impulsive dynamic equation system (44) with ut,utk+,At, v0 given in (51) and ft as in (52). By (44), we obtain(73)ukt=At,tk+Φk+∫tktAtk,στfτΔτ,t∈Jk,uktk=uk−1tk,where u0−1t0=vt0. Therefore, for each α∈0,1, we have(74)u¯αt=Uk¯αt=coshrt,tk+LkUk¯αtk+∫tktF¯ατcoshrστ,tkErστ,tk−F¯ατsinhrστ,tkErστ,tkΔτ+sinhrt,tk+LkUk¯αtk+∫tkt−F¯ατsinhrστ,tkErστ,tk+F¯ατcoshrστ,tkErστ,tkΔτ,U¯αt=Uk¯αt=sinhrt,tk+LkUk¯αtk+∫tktF¯ατcoshrστ,tkErστ,tk−F¯ατsinhrστ,tkErστ,tkΔτ+coshrt,tk+LkUk¯αtk+∫tkt−F¯ατsinhrστ,tkErστ,tk+F¯ατcoshrστ,tkErστ,tkΔτ.
Therefore, fort∈Jkk=0,1,2,…, by rt>0, we check that(75)Ukt=coshrt,tk+Gkt−H−sinhrt,tk+Gkt,with(76)Gkt=LkUktk−H∫tktFτsinhrστ,tkErστ,tk−FHτcoshrστ,tkErστ,tkΔτ,is (ii)-differentiable and Uk is a (ii)-solution of LIFDE (39) on Jk.
The following argument is due to the proof of Theorem 3.3 in [3]. First, by our hypothesis Gk is well defined. Second, Lemma 3-(i) guarantees that Gk is (ii)-differentiable and the diameter of Gk is nonincreasing in the variable t for fixed α∈0,1. Note that coshrt,tk+−sinhrt,tk+=e−rt,tk+ is nonnegative and decreasing in t. Thus, diamUktαis nonincreasing in t for fixed α∈0,1. Therefore, the H-differences Ukσt−HUkt+h and Ukt−h−HUkσt exist. Third, we check that the (ii)-derivative of Uk is rtUkt+Ft for t∈Ekk=0,1,2,…. If t is a right-dense point, in view of an analogous argument of Theorem 3.3 in [3], we can check that(77)limh⟶0+Ukt−HUkt+h−h=limh⟶0+Ukt−h−HUkt−h=rtUkt+Ft.
In the light of Proposition1-(III), we have ΔHUkt=rtUkt+Ft. If t is a right-scattered point, we denote(78)ξkt=∫tktF¯ατcoshrστ,tkErστ,tk−F¯ατsinhrστ,tkErστ,tkΔτ,ηkt=∫tkt−F¯ατsinhrστ,tkErστ,tk+F¯ατcoshrστ,tkErστ,tkΔτ.
Then,(79)ξkΔt=F¯αtcoshrσt,tkErσt,tk−F¯αtsinhrσt,tkErσt,tk,ηkΔt=−F¯αtsinhrσt,tkErσt,tk+F¯αtcoshrσt,tkErσt,tk,ξkσt−ξkt=∫tσtF¯ατcoshrστ,tkErστ,tk−F¯ατsinhrστ,tkErστ,tkΔτ=μtF¯αtcoshrσt,tkErσt,tk−μtF¯αtsinhrσt,tkErσt,tk,ηkσt−ηkt=−μtF¯αtsinhrσt,tkErσt,tk+μtF¯αtcoshrσt,tkErσt,tk.
From Proposition1-(II) and the property (p2) it follows(80)sinhrσt,tk−sinhrt+h,tk=rtcoshrt,tkμt,coshrσt,tk−coshrt+h,tk=rtsinhrt,tkμt.
Due to the above results, for eachα∈0,1, we uniformly have (81)Uk¯ασt−Uk¯αt−rtUk¯αt+F¯αtμt=coshrσt,tk+LkUk¯αtk+ξkσt+sinhrσt,tk+LkUk¯αtk+ηkσt−coshrt,tk+LkUk¯αtk+ξkt−sinhrt,tk+LkUk¯αtk+ηkt−rtμtsinhrt,tk+LkUk¯αtk+ξkt+coshrt,tk+LkUk¯αtk+ηkt−F¯αtμt=coshrσt,tk−coshrt,tk−rtμtsinhrt,tkLkUk¯αtk+ξkσt+sinhrσt,tk−sinhrt,tk−rtμtcoshrt,tkLkUk¯αtk+ηkσt+coshrt,tkξkσt−ξkt+rtμtsinhrt,tkξkσt−ξkt+sinhrt,tkηkσt−ηkt+rtμtcoshrt,tkηkσt−ηkt−F¯αtμt=μtcoshrt,tk+rtsinhrt,tkF¯αtcoshrσt,tkErσt,tk−F¯αtsinhrσt,tkErσt,tk+μtsinhrt,tk+rtcoshrt,tk−F¯αtsinhrσt,tkErσt,tk+F¯αtcoshrσt,tkErσt,tk−F¯αtμt=μtcoshrσt,tkF¯αtcoshrσt,tkErσt,tk−F¯αtsinhrσt,tkErσt,tk+μtsinhrσt,tk−F¯αtsinhrσt,tkErσt,tk+F¯αtcoshrσt,tkErσt,tk−F¯αtμt=0,that is, Uk¯ασt−Uk¯αt=rtUk¯αt+F¯αtμt. Analogously, we can prove(82)Uk¯ασt−Uk¯αt=rtUk¯αt+F¯αtμt,for all α∈0,1. Hence, Proposition 1-(II) guarantees that ΔHUkt=rtUkt+Ftfor all right-scattered point t∈Jk. In consequence, Uk is a (ii)-solution of LIFDE (39) on Jk for k=0,1,2…. Similarly, for t∈J−, we have(83)Ut=Vt=coshrtU0−H∫0tFτsinhrστ,0Erστ,0−HFτcoshrστ,0Erστ,0Δτ−H−sinhrtU0−H∫0tFτsinhrστ,0Erστ,0−HFτcoshrστ,0Erστ,0Δτ.
Now, (71) holds, and (i) is proved.(ii)
Fort∈Jk, 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 eAt,s=ert,sI in (43) and repeat the process of the proof of Theorem 1, we have, for all α∈0,1,(84)Uk¯αt=ert,tk+LkUk¯αtk−∫tkt−F¯ατe⊖rστ,tkΔτ,Uk¯αt=ert,tk+LkUk¯αtk−∫tkt−F¯ατe⊖rστ,tkΔτ.
We now check that the (i)-solution of LIFDE (39) on Jk for k=0,1,2,… and rt>0(t∈T+) is(85)Ukt=ert,tk+LkUktk+∫tktFτe⊖rστ,tkΔτ.
Note thaterσt,tkerΔt,tk=rterσt,tkert,tk>0 for rt>0. From Lemma 3-(v), it follows that LkUktk+∫tktFτe⊖rστ,tkΔτ is (i)-differentiable on T+, and from Lemma 2-(a) it follows that(86)ΔHUkt=rtert,tk+LkUktk+∫tktFτe⊖rστ,tkΔτ+erσt,tk+Fte⊖rσt,tk=rtUkt+Ft,for k=0,1,2,…. Similarly, we have for t∈J−(87)Ut=Vt=ertU0+∫0tFτe⊖rστΔτ.
This shows that (71) holds, and LIFDE (39) has a (i)-solution on T+. This proof is complete.Theorem 3.
Ifr≡0, then LIFDE (39) has a (i)-solution(88)Ut=Vt=U0+∫0tFτΔτ,t∈J−,Ukt=LkUktk+∫tktFτΔτ,t∈Jkk=0,1,2,…,and a (ii)-solution(89)Ut=Vt=U0−H∫0t−FτΔτ,t∈J−,Ukt=LkUktk−H∫tkt−FτΔτ,t∈Jkk=0,1,2,…,provided the above H-differences exist and U0t0=Vt0,Uktk=Uk−1tk for k=0,1,2,….Remark 3.
IfT=ℝ, then Ert,s≡1, sinhrt,s=sinh∫strτdτ, and coshrt,s=cosh∫strτdτ. Thus, our results are the extension and improvement of the corresponding results in [3]. In particular, the present results are identical to those in [3] in case that we consider t restricted to J− and under H-difference.
IfT=ℤ, then Ert,s=∏τ=st−11−r2τ, sinhrt,s=1/2∏τ=st−11+rτ−∏τ=st−11−rτ and coshrt,s=1/2∏τ=st−11+rτ+∏τ=st−11−rτ provided that r is never ±1 and s<t. Thus, we obtain the general form of solutions to linear impulsive fuzzy difference equation:(90)Un+1=rnUn+Un+Fn,n∈ℤ+\J,Utk+=LkUtk,tk∈Jk=0,1,2,…,U0=U0∈Tf,t0∈T+.
In addition, we also extend these “classical cases” to cases “in between”, for instance,T=∑k=1n1/k:n∈ℕ. In this case, (91)Ert,s=n−s+rn−s⋅n−s−rn−s,sinhrt,s=12n−s+rn−s−n−s−rn−s,coshrt,s=12n−s+rn−s+n−s−rn−s,where t=∑k=1n1/k. We can also consider the so-called q-difference problems.Remark 4.
Although problem (44) has a unique solution on T+, the solution of LIFDE (39) is not unique in general.


## 4. Examples

In this section, we present several examples to further illustrate the applicability of the results involved in the above sections.Example 1.
Consider the impulsive fuzzy dynamic equation:(92)ΔHUt=aUt+t,t∈T+\J,Utk+=LkUtk,k=0,1,2,…,Ut0=U0,where a<0,a∈ℛ+1, and Lk is linear bounded. It is not difficult to infer the existence of the H-differences U0−H∫0tτe⊖aστΔτ and LkUtk−∫tktτe⊖aστ,tkΔτH. Thus, by Theorem 1, (ii)-solution of LIFDE (92) on T+ is given by(93)Ut=Vt=eatU0−H∫0tτe⊖aστΔτ,t∈J−,Ukt=eat,tk+LkUktk−H∫tktτe⊖aστ,tkΔτ,t∈Jk,where U0t0=Vt0,Uktk=Uk−1tk for k=0,1,2,….Remark 5.
IfT=ℝ, then we get(94)Vt=eatU0−H∫0t−τe−aτdτ,t∈J−,Ukt=eat−tkLkUktk−H∫tktτe−aτ−tkdτ,t∈Jkk=0,1,2….
We obtain by calculating(95)Ukt=1at+1a1˜+LkUktk+ta+1a2eat−tk,with 1˜=χ1,t∈Jk. We observe that in this case DUt,1/at+1/a1˜≤LkUktk+t/a+t/a2eat, and this implies that limt⟶∞DUt,1/at+1/a1˜=0. An interpretation in the light of [1] is that the uncertainty asymptotically disappears on the fuzzy system.
IfT=ℤ, then(96)Vt=1+atU0−H∑τ=0t−1−τ1+a−τ+1,t∈J−,Ukt=1+at−tkLkUktk−H∑τ=tkt−1−τ1+a−τ+1−tk,t∈Jkk=0,1,2,….
Fort∈Jk, we have(97)Ut=Ukt=LkUktk1+at−tk−H1+at∑τ=tkt−1−τ1+a−τ+1=1+a1−tka1˜+LkUtk−1+aa1+at−tk.
Therefore, in the discrete case, the phenomenon that the uncertainty asymptotically disappears on the fuzzy system arises onlya>−1.Example 2.
Let us consider the LIFDE(98)ΔHUt=2tUt+tγ,t∈T+\J,Utk+=4Utk,k=0,1,2,…,U0=γ,where γα=α−1,1−α with α∈0,1.


## 5. Conclusion

LetT=ℝ and tk=k+1ln2 for k=0,1,2,…. As a result of Theorem 2, the (i)-solution of (98) is(99)Ut=Vt=123et2−1γ,t∈J−=0,ln2∩T+,Ukt=4Ukk+1ln2et2−k+1ln2+12et2−k+1ln2−1γ,t∈Jkk=0,1,2,….To seek the (ii)-solution, we note thatcoshrt,tk=cosht2−tk2andsinhrt,tk=sinht2−tk2 when rt=2t. The solution of the ODEs system corresponding to (43) is(100)Uk¯αt=4Uk¯αtkcosht2−tk2+4Uk¯αtksinht2−tk2−α−12cosht2−tk2+sinht2−tk2−1cosht2−tk2−sinht2−tk2,Uk¯αt=4Uk¯αtksinht2−tk2+4Uk¯αtkcosht2−tk2−1−α2cosht2−tk2+sinht2−tk2−1cosht2−tk2−sinht2−tk2.It is easy to see that theH-difference(101)γ−H∫0tτγsinhτ2τγcoshτ2dτ,exists on 0,ln2, and the (ii)-solution Vt of (98) on 0,ln2 can be written as(102)Vt=123e−t2−1γ.In particular,Vln2=1/4γ. From L0U0t0=4×1/4γ=γ, it follows that the H-difference(103)L0U0t0−H∫t0tτγsinhτ2−t02τγcoshτ2−t02dτ,=γ−H∫ln2tτγsinhτ2−ln2τγcoshτ2−ln2dτ,exists on ln2,2ln2, and the (ii)-solution U0t of (98) on ln2,2ln2 can be written as(104)U0t=cosht2−ln2γ−H∫ln2tτγsinhτ2−ln2−Hτγcoshτ2−ln2dτ−H−sinht2−ln2γ−H∫ln2tτγsinhτ2−ln2−τγcoshτ2−ln2dτ=1−12sinht2−ln2+cosht2−ln2−1cosht2−ln2−sinht2−ln2γ=123e−t2−ln2−1.In particular,U02ln2=1/4γ. On the analogy of this process, we obtain that the (ii)-solution of (98) on Jk=k+1ln2,k+2ln2 can be expressed by(105)Ukt=123e−t2−k+1ln2−1,fort∈Jk,Ukk+2ln2=14γ,k=1,2,…LetT=ℤ and tk=2k+1k=0,1,2,…. rt=2t implies that ert,s=∏τ=st−11+2τ. Therefore, the (i)-solution of (98) is(106)Ut=Vt=γ,t∈J−,Ukt=∏τ=2k+1t−11+2τ⋅4Uk−1tk+∑τ=2k+1t−1τ∏s=2k+1τ1+2sγ,t∈Jkk=0,1,2,….Here,J−=0,1∩ℤ+,Jk=2k+1,2k+3∩ℤ+, U0−1t0=γ,andU0t1=3⋅54γ+1/5γ+2γ,…Similarly, we can present the expression of the (ii)-solutions of (98) in the discrete case.



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*Source: 4894921-2020-03-12.xml*