On conformable delta fractional calculus on time scales

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1 Avilble online t J. Mth. Computer Sci. 6 (06), Reserch Article On conformble delt frctionl clculus on time scles Dfng Zho, Tongxing Li b,c, School of Mthemtics nd Sttistics, Hubei Norml University, Hungshi, Hubei 43500, P. R. Chin. b LinD Institute of Shndong Provincil Key Lbortory of Network Bsed Intelligent Computing, Linyi University, Linyi, Shndong 76005, P. R. Chin. c School of Informtics, Linyi University, Linyi, Shndong 76005, P. R. Chin. Abstrct In this pper, we introduce nd investigte the concepts of conformble delt frctionl derivtive nd conformble delt frctionl integrl on time scles. Bsic properties of the theory re proved. c 06 All rights reserved. Keywords: Conformble delt frctionl derivtive, conformble delt frctionl integrl, time scle. 00 MSC: 6A33, 6E70.. Introduction Frctionl clculus is generliztion of ordinry differentition nd integrtion to rbitrry (non-integer) order. This subject is s old s the clculus of differentition nd goes bck to times when Leibniz, Guss, nd Newton invented this kind of clcultion. The frctionl clculus lwys ttrcted interest of reserchers due to its numerous pplictions in engineering, economics nd finnce, signl processing, dynmics of erthqukes, geology, probbility nd sttistics, chemicl engineering, physics, splines, thermodynmics, neurl networks, nd so on; see, for instnce, the monogrphs by Crpinteri nd Minrdi [9], Herrmnn [0], Miller nd Ross [6], Oldhm nd Spnier [7], Ortigueir [8], Podlubny [30], Sbtier et l. [3], Smko et l. [3], nd the references cited therein. Severl definitions of frctionl derivtive hve been proposed. These definitions include Riemnn Liouville, Grunwld Letnikov, Weyl, Cputo, Mrchud, nd Riesz frctionl derivtives. Corresponding uthor Emil ddresses: dfngzho@63.com (Dfng Zho), litongx007@63.com (Tongxing Li) Received

2 D. Zho, T. Li, J. Mth. Computer Sci. 6 (06), However, nerly ll frctionl derivtives do not stisfy the well-known formul of the derivtive of the product (the quotient) of two functions nd the chin rule, etc. Recently, Khlil et l. [3] defined new well-behved simple frctionl derivtive which is clled the conformble frctionl derivtive depending just on the bsic limit definition of the derivtive s follows: Let f : [0, ] R be given function. Then for ll t > 0 nd α (0, ), define T α (f) by f(t + εt α ) f(t) T α (f)(t) := lim, ε 0 ε T α (f) is termed the conformble frctionl derivtive of f of order α. By virtue of this definition, ll the clssicl properties of the derivtive hold; see [8, ]. The notion of conformble frctionl derivtive ws developed; we refer the reder to the ppers [ 6, 9, 0, 3] nd the references cited there. In prticulr, Benkhettou et l. [3] introduced conformble frctionl clculus on n rbitrry time scle, which is nturl extension of the conformble frctionl clculus. A time scle T is n rbitrry nonempty closed subset of rel numbers R with the subspce topology inherited from the stndrd topology of R. The theory of time scles ws born in 988 with the Ph.D. thesis of Hilger []. The im of this theory is to unify vrious definitions nd results from the theories of discrete nd continuous dynmicl systems, nd to extend such theories to more generl clsses of dynmicl systems. It hs been extensively studied on vrious spects by severl uthors; see, for instnce, the ppers by [7,,, 4 8, 4, 5, 9]. In this pper, we introduce nd investigte the concepts of conformble delt frctionl derivtive nd conformble delt frctionl integrl on time scles which re different from those of [3]. This pper is orgnized s follows: Section contins bsic concepts of time scles. In Section 3, the definition of conformble delt frctionl derivtive is introduced, nd the bsic properties of frctionl derivtive re investigted. In Section 4, we introduce nd develop the notion of conformble delt frctionl integrl on time scles. We end Section 5 with conclusions nd future reserch.. Preliminries Let T be time scle. For, b T we define the closed intervl [, b] T by [, b] T := {t T : t b}. The open nd hlf-open intervls re defined in similr wy. For t T we define the forwrd jump opertor σ : T T nd bckwrd jump opertor ρ : T T by σ(t) := inf{s T : s > t} nd ρ(t) := sup{s T : s < t}, respectively, where inf := sup T nd sup := inf T, denotes the empty set. Assume t T. If σ(t) > t, then t is right-scttered, nd if ρ(t) < t, then t is left-scttered. If σ(t) = t nd t < sup T, then t is right-dense, while if ρ(t) = t nd t > inf T, then t is left-dense. A point t T is dense if it is right-dense nd left-dense t the sme time; isolted if it is rightscttered nd left-scttered t the sme time. The forwrd grininess function µ : T [0, ) nd the bckwrd grininess function η : T [0, ) re defined by µ(t) := σ(t) t nd η(t) := t ρ(t) for ll t T, respectively. If sup T is finite nd left-scttered, then T k := T\{sup T}; otherwise, T k := T. If inf T is finite nd right-scttered, then T k := T\{inf T}; otherwise, T k := T. We set T k k := Tk T k. A function f : T R is clled rd-continuous provided it is continuous t ll right-dense points in T nd its left-sided limits exist (finite) t ll left-dense points in T. A function f : T R is clled regulted provided its right-sided limits exist (finite) t ll right-dense points in T nd its left-sided limits exist (finite) t ll left-dense points in T. Assume f : T R is function nd let t T k. Then we define f (t) to be the number (provided it exists) with the property tht given ny ε > 0, there exists neighborhood U of t such tht f(σ(t)) f(s) f (t)(σ(t) s) ε σ(t) s,

3 D. Zho, T. Li, J. Mth. Computer Sci. 6 (06), for ll s U. We cll f (t) the delt derivtive of f t t nd we sy tht f is delt differentible on T k provided f (t) exists for ll t T k. In wht follows, let α (0, ]. 3. Conformble delt frctionl derivtive Definition 3.. Assume f : T R is function nd let t T k. Then we define T α (f )(t) to be the number (provided it exists) with the property tht given ny ε > 0, there exists neighborhood U of t such tht (f(σ(t)) f(s))σ α (t) T α (f )(t)(σ(t) s) ε σ(t) s for ll s U. We cll T α (f )(t) the conformble delt ( ) frctionl derivtive of f of order α t t. Moreover, we sy tht f is conformble frctionl differentible of order α on t T k provided T α (f )(t) exists for ll t T k. The function T α (f ) : T k R is then clled the conformble frctionl derivtive of f of order α on T k. We define the conformble frctionl derivtive t 0 s T α (f )(0) = lim t 0 T α (f )(t). If t is right-dense, then we obtin the conformble frctionl derivtive reported in [3]. Some useful properties of conformble frctionl derivtive of f of order α re given in the following theorem. Theorem 3.. Let T be time scle, t T k, nd α (0, ]. Then we hve the following. (i) If f is conformble frctionl differentible of order α t t, then f is continuous t t. (ii) If f is continuous t t nd t is right-scttered, then f is conformble frctionl differentible of order α t t with T α (f f(σ(t)) f(t) )(t) = σ α (t). µ(t) (iii) If t is right-dense, then f is conformble frctionl differentible of order α t t if nd only if the limit f(t) f(s) lim t α, exists s finite number. In this cse, T α (f f(t) f(s) )(t) = lim t α. (iv) If f is conformble frctionl differentible of order α t t, then f(σ(t)) = f(t) + µ(t)t α (f )(t)σ α (t). Proof. Prt (i). Assume tht f is conformble frctionl differentible of order α t t. Then for ech ε > 0, there exists neighborhood U of t such tht (f(σ(t)) f(s))σ α (t) T α (f )(t)(σ(t) s) ε σ(t) s

4 D. Zho, T. Li, J. Mth. Computer Sci. 6 (06), for ll s U, where ε = ε + T α (f )(t) + µ(t) σ α (t). Therefore, we hve, for ll s U (t ε, t + ε ), f(t) f(s) = f(σ(t)) f(s) T α (f )(t)(σ(t) s)σ α (t) [ f(σ(t)) f(t) T α (f )(t)(σ(t) t)σ α (t) ] + T α (f )(t)(t s)σ α (t) ε (σ(t) s)σ α (t) + ε (σ(t) t)σ α (t) + T α (f )(t)(t s)σ α (t) ε σ α (t) ( µ(t) + s t + µ(t) + T α (f )(t) ) < ε σ α (t) ( + T α (f )(t) + µ(t) ) = ε. It follows tht f is continuous t t. Prt (ii). Assume tht f is continuous t t nd t is right-scttered. By continuity, f(σ(t)) f(s) lim σ α (t) = s t σ(t) s f(σ(t)) f(t) σ α (t) = σ(t) t f(σ(t)) f(t) σ α (t). µ(t) Hence, given ε > 0, there exists neighborhood U of t such tht f(σ(t)) f(s) σ α f(σ(t)) f(t) (t) σ α (t) σ(t) s µ(t) ε for ll s U. It follows tht (f(σ(t)) f(s)) σ α (t) for ll s U. Hence, we get the desired result f(σ(t)) f(t) µ(t) (σ(t) s) σ α (t) ε σ(t) s T α (f )(t) = f(σ(t)) f(t) σ α (t). µ(t) Prt (iii). Assume tht f is conformble frctionl differentible of order α t t nd t is right-dense. Then for ech ε > 0, there exists neighborhood U of t such tht (f(σ(t)) f(s))σ α (t) T α (f )(t)(σ(t) s) ε σ(t) s for ll s U. Since σ(t) = t, we hve (f(t) f(s))t α T α (f )(t)(t s) ε t s for ll s U. It follows tht f(t) f(s) t α T α (f )(t) t s ε for ll s U nd s t. Thus, we conclude tht T α (f f(t) f(s) )(t) = lim t α.

5 D. Zho, T. Li, J. Mth. Computer Sci. 6 (06), On the other hnd, if the limit f(t) f(s) lim t α, exists s finite number nd is equl to J, then for ech ε > 0, there exists neighborhood U of t such tht (f(t) f(s))t α J(t s) ε t s for ll s U. Since t is right-dense, we obtin (f(σ(t)) f(s))σ α (t) J(σ(t) s) ε σ(t) s. Therefore, f is conformble frctionl differentible of order α t t nd T α (f f(t) f(s) )(t) = lim t α. Prt (iv). If t is right-dense, then µ(t) = 0 nd f(σ(t)) = f(t) = f(t) + µ(t)t α (f )(t)σ α (t). If t is right-scttered, then σ(t) > t, nd by virtue of (ii), The proof is complete. f(σ(t)) f(t) f(σ(t)) = f(t) + µ(t) µ(t) Exmple 3.3. We consider the two cses T = R nd T = Z. = f(t) + µ(t)t α (f )(t)σ α (t). (i) If T = R, then f : R R is conformble frctionl differentible of order α t t R if nd only if the limit f(t) f(s) lim t α, exists s finite number. In this cse, If α =, then T α (f f(t) f(s) )(t) = lim t α. T α (f )(t) = f (t) = f (t). (ii) If T = Z, then f : Z R is conformble frctionl differentible of order α t t Z with If α =, then T α (f )(t) = f(t + ) f(t) f α (t + ) = f α (t + ) (f(t + ) f(t)). T α (f )(t) = f(t + ) f(t) = f(t), where is the usul forwrd difference opertor. Exmple 3.4. (i) If f : T R is defined by f(t) = C for ll t T, where C R is constnt, then T α (f )(t) 0. This is cler becuse for ny ε > 0, (f(σ(t)) f(s))σ α (t) 0 (σ(t) s) = (C C)σ α (t) = 0 ε σ(t) s holds for ll s T.

6 D. Zho, T. Li, J. Mth. Computer Sci. 6 (06), (ii) If f : T R is defined by f(t) = t for ll t T, then T α (f )(t) = σ α (t). This is vlid becuse for ny ε > 0, (f(σ(t)) f(s))σ α (t) σ α (t)(σ(t) s) = (σ(t) s)σ α (t) σ α (t)(σ(t) s) holds for ll s T. If α =, then T α (f )(t). = 0 ε σ(t) s Exmple 3.5. Suppose tht f : T R is defined by f(t) = t for ll t T := {n/ : n N 0 }. By virtue of Theorem 3. (ii), we hve tht f is conformble frctionl differentible of order α t t T with ( T α (f )(t) = t + ) ( t + α. ) Theorem 3.6. Assume f, g : T R re conformble frctionl differentible of order α t t T k. Then we hve the following. (i) For ll constnts λ nd λ, the sum λ f + λ g : T R is conformble frctionl differentible of order α t t T k with T α ((λ f + λ g) )(t) = λ T α (f )(t) + λ T α (g )(t). (ii) The product fg : T R is conformble frctionl differentible of order α t t with T α ((fg) )(t) = T α (f )(t)g(t) + f(σ(t))t α (g )(t) = f(t)t α (g )(t) + T α (f )(t)g(σ(t)). (iii) If f(t)f(σ(t)) 0, then /f is conformble frctionl differentible of order α t t with ( ( ) ) T α (t) = T α(f )(t) f f(t)f(σ(t)). (iv) If g(t)g(σ(t)) 0, then f/g is conformble frctionl differentible of order α t t with ( (f ) ) T α (t) = T α(f )(t)g(t) f(t)t α (g )(t). g g(t)g(σ(t)) Proof. Prt (i). Let ε > 0. Then there exist neighborhoods U nd U of t such tht for ll s U nd (λ f(σ(t)) λ f(s))σ α (t) λ T α (f )(t)(σ(t) s) ε λ (σ(t) s) (λ g(σ(t)) λ g(s))σ α (t) λ T α (g )(t)(σ(t) s) ε λ (σ(t) s) for ll s U. Let U := U U nd λ := mx{λ, λ }. Then, we hve, for ll s U, ((λ f + λ g)(σ(t)) (λ f + λ g)(s))σ α (t) (λ T α (f )(t) + λ T α (g )(t))(σ(t) s)

7 D. Zho, T. Li, J. Mth. Computer Sci. 6 (06), (λ f(σ(t)) λ f(s))σ α (t) λ T α (f )(t)(σ(t) s) + (λ g(σ(t)) λ g(s))σ α (t) λ T α (g )(t)(σ(t) s) ε λ (σ(t) s) + ε λ (σ(t) s) ε λ(σ(t) s). Therefore, λ f + λ g is conformble frctionl differentible of order α t t T k with Prt (ii). Let 0 < ε <. Define T α ((λ f + λ g) )(t) = λ T α (f )(t) + λ T α (g )(t). ε := ε + g(σ(t)) + f(t) + T α (g )(t). Then 0 < ε <. Since f, g : T R re conformble frctionl differentible of order α t t T k, there exist neighborhoods U nd U of t such tht for ll s U nd (f(σ(t)) f(s))σ α (t) T α (f )(t)(σ(t) s) ε σ(t) s (g(σ(t)) g(s))σ α (t) T α (g )(t)(σ(t) s) ε σ(t) s for ll s U. From Theorem 3. (i), there exists neighborhood U 3 of t such tht f(t) f(s) ε for ll s U 3. Let U := U U U 3. Then, we hve, for ll s U, [f(σ(t))g(σ(t)) f(s)g(s)]σ α (t) [T α (f )(t)g(σ(t)) + f(t)t α (g )(t)](σ(t) s) [(f(σ(t)) f(s))σ α (t) T α (f )(t)(σ(t) s)]g(σ(t)) + [(g(σ(t)) g(s))σ α (t) T α (g )(t)(σ(t) s)]f(t) + [(g(σ(t)) g(s))σ α (t) T α (g )(t)(σ(t) s)](f(s) f(t)) + T α (g )(t)(σ(t) s)(f(s) f(t)) ε σ(t) s ( g(σ(t)) + f(t) + ε + T α (g )(t) ) ε σ(t) s. Thus, we deduce tht T α ((fg) )(t) = f(t)t α (g )(t) + T α (f )(t)g(σ(t)). One cn esily obtin nother product rule by interchnging the role of f nd g. Prt (iii). From Exmple 3.4, ( ( T α f ) ) (t) = T α (() )(t) = 0. f Therefore, T α ( ( f ) ) (t)f(σ(t)) + T α (f )(t) f(t) = 0,

8 D. Zho, T. Li, J. Mth. Computer Sci. 6 (06), nd consequently T α ( ( f ) ) (t) = T α(f )(t) f(t)f(σ(t)). Prt (iv). Applictions of (ii) nd (iii) imply tht ( (f ) ) ( ( ) ) T α (t) = f(t)t α (t) + T α (f )(t) g g g(σ(t)) The proof is complete. = f(t) T α(g )(t) g(t)g(σ(t)) + T α(f )(t) g(σ(t)) = T α(f )(t)g(t) f(t)t α (g )(t). g(t)g(σ(t)) Theorem 3.7. Let c be constnt nd m N + := {,,...}. (i) If f(t) = (t c) m, then (ii) If g(t) = /f(t) = /(t c) m, then provided (σ(t) c)(t c) 0. m T α (f )(t) = σ α (t) (σ(t) c) i (t c) m i. m T α (g )(t) = σ α (t) (σ(t) c) m i (t c), i+ Proof. Prt (i). We prove the first formul by induction. If m =, then f(t) = t c, nd T α (f )(t) = σ α (t) holds when using Exmple 3.4 nd Theorem 3.6 (i). We ssume now tht m T α (f )(t) = σ α (t) (σ(t) c) i (t c) m i, holds for f(t) = (t c) m nd let F (t) := (t c) m+ = (t c)f(t). An ppliction of Theorem 3.6 (ii) yields T α (F ) (t) = σ α (t)f(σ(t)) + (t c)t α (f )(t) Hence, prt (i) is intct. m = σ α (t)(σ(t) c) m + (t c)σ α (t) (σ(t) c) i (t c) m i = σ α (t) m (σ(t) c) i (t c) m i.

9 D. Zho, T. Li, J. Mth. Computer Sci. 6 (06), Prt (ii). For g(t) = /(t c) m = /f(t), we use Theorem 3.6 (iii) to rrive t T α (g) (t) = T α(f )(t) f(t)f(σ(t)) = σ α (t) m (σ(t) c)i (t c) m i (σ(t) c) m (t c) m m = σ α (t) (σ(t) c) m i (t c), i+ provided (σ(t) c)(t c) 0. This completes the proof. Exmple 3.8. If f : T R is defined by f(t) = /t for ll t T := { n : n N 0 }, then we hve tht f is conformble frctionl differentible of order α t t T with ( T α (f )(t) = σ α (t) tσ (t) + ) = ( ( t t σ(t) + ) α t t + + ). t 4. Conformble delt frctionl integrtion Definition 4.. Assume f : T R is regulted function. We define the indefinite α-conformble frctionl integrl of f by I α (f )(t) + C = f(t) α t = f(t)σ α (t) t, where C is n rbitrry constnt. I α (f )(t) is clled pre-ntiderivtive of f. We define the Cuchy α-conformble frctionl integrl by f(t) α t = I α (f )(b) I α (f )() for ll, b T. A function I α (f ) : T R is clled n ntiderivtive of f : T R provided for ll t T k. (T α I α (f ))(t) = f(t) Theorem 4. (Existence of ntiderivtives). For every rd-continuous function f : T R, there exists function I α (f ) such tht (T α I α (f ))(t) = f(t). Proof. Suppose f : T R is rd-continuous. By [6, Theorem.60], f is regulted. Similr to the proof of [6, Theorem 8.3], we conclude tht I α (f ) is conformble frctionl differentible of order α t t T k. Then This completes the proof. We cn esily get the following theorem. (T α I α (f ))(t) = σ α (t)σ α (t)f(t) = f(t).

10 D. Zho, T. Li, J. Mth. Computer Sci. 6 (06), Theorem 4.3. Let, b, c T, λ, λ R, nd f, g : T R be rd-continuous functions. Then (i) (λ f(t) + λ g(t)) α t = λ f(t) αt + λ g(t) αt; (ii) f(t) αt = b f(t) αt; (iii) f(t) αt = c f(t) αt + c f(t) αt; (iv) f(σ(t))t α(g )(t) α t = f(b)g(b) f()g() T α(f )(t)g(t) α t; (v) f(t)t α(g )(t) α t = f(b)g(b) f()g() T α(f )(t)g(σ(t)) α t; (vi) f(t) αt = 0; (vii) if f(t) g(t) on [, b) T, then f(t) αt g(t) αt; (viii) if f(t) 0 for ll t [, b) T, then f(t) αt 0. Theorem 4.4. Let, b T, nd let f : T R be n rd-continuous function. Then we hve the following. (i) If T = R, then f(t) αt = f(t)/t α dt, where the integrl on the right is the conformble frctionl integrl given in [3]. If α =, then it reduces to the usul Riemnn integrl. (ii) If [, b] T consists of only isolted points, then t [,b) T σ α (t)µ(t)f(t) if < b, f(t) α t = 0 if = b, t [b,) T σ α (t)µ(t)f(t) if > b. (iii) If T = hz = {hk : k Z}, where h > 0, then b h k= (hk + h) α hf(hk) if < b, h f(t) α t = 0 if = b, h (hk + h) α hf(hk) if > b. k= b h (4.) (4.) (iv) If T = Z, then b t= (t + )α f(t) if < b, f(t) α t = 0 if = b, t=b (t + )α f(t) if > b. (4.3) Proof. Prt (i). The proof is not difficult nd so is omitted. Prt (ii). First, note tht [, b] T consists of only finitely isolted points. Assume tht < b nd [, b] T = {t 0, t,..., t n }, where = t 0 < t < t <... < t n = b.

11 D. Zho, T. Li, J. Mth. Computer Sci. 6 (06), By virtue of Theorem 4.3 (iii), Consequently, n f(t) α t = ti+ n f(t) α t = t i f(t) α t = σ(ti ) n f(t) α t = µ(t i )σ α (t i )f(t i ). t i t [,b) T σ α (t)µ(t)f(t). If > b, then the result follows from wht we just proved nd Theorem 4.3 (ii). If = b, then the result follows from Theorem 4.3 (vi). Prt (iii) nd (iv) re specil cses of Prt (ii). The proof is complete. Exmple 4.5. If f : T R is defined by f(t) = t 3 for ll T = R nd α = /, then 4 f(t) α t = 4 t 3 t = 4 t 3 t = 4 t 5 dt = Exmple 4.6. If f : T R is defined by f(t) = t for ll t T := {n/ : n N 0 } nd α = /, then 3 5. Conclusions 3 f(t) α t = t t = = ( ) ( ) ( ) 5 This pper investigted the conformble delt frctionl clculus on time scles. The results of the pper give common generliztion of the conformble frctionl derivtive nd the usul delt derivtive. Another interesting line of reserch is to investigte conformble frctionl derivtive on time scles in other different directions rther thn the one considered here. For instnce, insted of following the delt pproch tht we hve dopted, one cn develop nbl, dimond, or symmetric time scle frctionl clculus. These problems will be subject of future reserch. Acknowledgements This reserch ws supported by NNSF of P. R. Chin (Grnt Nos , , nd ), CPSF (Grnt No. 05M5809), NSF of Shndong Province (Grnt No. ZR0FL06), Eductionl Commission of Hubei Province (Grnt No. Q05505), DSRF of Linyi University (Grnt No. LYDX05BS00), nd the AMEP of Linyi University, P. R. Chin. References [] T. Abdeljwd, On conformble frctionl clculus, J. Comput. Appl. Mth., 79 (05), [] T. Abdeljwd, M. Al Horni, R. Khlil, Conformble frctionl semigroups of opertors, J. Semigroup Theory Appl., 05 (05), 9 pges. [3] I. Abu Hmmd, R. Khlil, Frctionl Fourier series with pplictions, Amer. J. Comput. Appl. Mth., 4 (04), 87 9.

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