The fuzzy C-delta integral on time scales

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1 Avilble online t J. Nonliner Sci. Appl., 11 (2018, Reserch Article Journl Homepge: The fuzzy C-delt integrl on time scles Xuexio You,b, Dfng Zhob,c, Jin Chengb, Tongxing Lid,e, College b School of Computer nd Informtion, Hohi University, Nnjing, Jingsu , P. R. Chin. of Mthemtics nd Sttistics, Hubei Norml University, Hungshi, Hubei , P. R. Chin. c College of Science, Hohi University, Nnjing, Jingsu , P. R. Chin. d LinD Institute of Shndong Provincil Key Lbortory of Network Bsed Intelligent Computing, Linyi University, Linyi, Shndong , P. R. Chin. e School of Informtion Science nd Engineering, Linyi University, Linyi, Shndong , P. R. Chin. Communicted by C. Vetro Abstrct In this pper, we introduce nd study the C-delt integrl of intervl-vlued functions nd fuzzy-vlued functions on time scles. Also, some bsic properties of the fuzzy C-delt integrl re proved. Finlly, we give two necessry nd sufficient conditions of integrbility. Keywords: C-Delt integrl, fuzzy-vlued function, time scle MSC: 26A42, 26E50, 26E70. c 2018 All rights reserved. 1. Introduction It is well known tht the Henstock-Kurzweil integrl integrtes highly oscillting functions nd encompsses Newton, Riemnn, nd Lebesgue integrls 22, 28. As n importnt brnch in the HenstockKurzweil integrtion theory, the theory of fuzzy Henstock-Kurzweil integrl hs been studied extensively 7, 12, 15, 16, 24, 29, 31, 32. In 1986, Bruckner et l. 9 considered the function x sin x12, if 0 < x 6 1, F(x = (1.1 0, if x = 0. It is primitive for the Henstock integrl, but it is neither Lebesgue primitive, differentible function, nor sum of Lebesgue primitive nd differentible function. The nturl question is: is there miniml Corresponding uthor Emil ddresses: youxuexio@126.com (Xuexio You, dfngzho@163.com (Dfng Zho, jinchenghs@163.com (Jin Cheng, litongx2007@163.com (Tongxing Li doi: /jns Received: Revised: Accepted:

2 X. You, D. Zho, J. Cheng, T. Li, J. Nonliner Sci. Appl., 11 (2018, integrl including the Lebesgue integrl nd derivtives? To solve this question, Bongiorno 4 provided miniml constructive integrtion process of Riemnn type, i.e., C-integrl, which includes the Lebesgue integrl nd lso integrtes the derivtives of differentible functions. The theory of C-integrtion hs developed rther intensively in the pst few yers; see, for instnce, the ppers 5, 6, 8, 11, 13, 20, 21, 26, 30, 33, 34, 36, 38 nd the references cited there. 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 1988 with the Ph.D. thesis of Hilger 18. 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, e.g., 1 3, 17, 23, 25, 27, 35, 37. To the best of our knowledge, the C-delt integrl of fuzzy-vlued functions hs not received ttention in the literture of time scles. The min gol of this pper is to generlize the results bove by constructing the C-delt integrl of fuzzy-vlued functions on time scles. The pper is orgnized s follows. Section 2 contins bsic concepts of fuzzy sets, time scles, nd C-integrl. In Section 3, we give the definition of C-delt integrl of intervl-vlued functions, nd discuss some of its bsic properties. In Section 4, the definition of fuzzy C-delt integrl is introduced, nd two necessry nd sufficient conditions of integrbility re presented. We end with Section 5 of conclusions nd future work. 2. Preliminries In this section, we recll some bsic definitions, nottion, properties, nd results on fuzzy sets nd the time scle clculus, which re used throughout the pper. Let us denote by R I the set of ll nonempty compct intervls of R, tht is, R I = {u, u u, u R nd u < u}. u nd u re clled the lower nd the upper brnches of u, u, respectively. The usul intervl opertions, i.e., Minkowski ddition nd sclr multipliction, re defined by u, u + v, v = u + v, u + v nd λu, u λu, λu, if λ > 0, = {0}, if λ = 0, λu, λu, if λ < 0. The distnce between intervls u, u nd v, v is defined by d ( u, u, v, v { } = mx u v, u v. Let us denote by R F the clss of fuzzy subsets of the rel xis. Assume u : R 0, 1 stisfies the following properties: (1 u is norml, i.e., there exists n x 0 R with u(x 0 = 1; (2 u is convex fuzzy set, i.e., for ll x 1, x 2 R, λ (0, 1, we hve (3 u is upper semi-continuous; u(λx 1 + (1 λx 2 min{u(x 1, u(x 2 }; (4 u 0 = {x R : u(x > 0} is compct, where A denotes the closure of the set A. Then R F is clled the spce of fuzzy numbers. For 0 < α 1, denote u α = {x R : u(x α}. From the conditions (1-(4, it follows tht the α-level set u α is nonempty compct intervl for ll α 0, 1. We

3 X. You, D. Zho, J. Cheng, T. Li, J. Nonliner Sci. Appl., 11 (2018, write u α = u α, u α. As distnce between fuzzy numbers we use the Husdorff metric defined by D(u, v = sup d ( u α, v α { } = sup mx u α v α, u α v α α 0,1 α 0,1 for u, v R F. Then (R F, D is complete metric spce. The following properties re well-known: (1 D(u w, v w = D(u, v; (2 D(λ u, λ v = λ D(u, v; (3 D(u v, w e D(u, w + D(v, e for ll u, v, w, e, R F nd λ R. Let T be time scle, i.e., nonempty closed subset of R. 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 by σ(t = inf{s > t : s T}, where inf = sup T, while the bckwrd jump opertor ρ is defined by ρ(t = sup{s < t : s T}, where sup = inf T. If σ(t > t, then we sy tht t is right-scttered, while if ρ(t < t, then we sy tht t is left-scttered. If σ(t = t nd t < sup T, then we sy tht t is right-dense, while if ρ(t = t nd t > inf T, then we sy tht 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 right-scttered 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 we define 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. Throughout this pper, ll considered intervls will be intervls in T. A prtition D of, b T is finite collection of intervl-point pirs {(t i 1, t i T, ξ i } n, where { = t 0 < t 1 < < t n 1 < t n = b} nd ξ i, b T for i = 1, 2,..., n. By t i = t i t i 1 we denote the length of the ith subintervl in the prtition D. δ(ξ = (δ L (ξ, δ R (ξ is -guge for, b T provided tht δ L (ξ > 0 on (, b T, δ R (ξ > 0 on, b T, δ L ( 0, δ R (b 0, nd δ R (ξ µ(ξ for ll ξ, b T. Let δ 1 (ξ nd δ 2 (ξ be -guges for, b T such tht 0 < δ 1 L (ξ < δ2 L (ξ for ll ξ (, b T nd 0 < δ 1 R (ξ < δ2 R (ξ for ll ξ, b T. Then δ 1 (ξ is finer thn δ 2 (ξ nd we write δ 1 (ξ < δ 2 (ξ. We sy tht D = {(t i 1, t i T, ξ i } n is (1 prtil prtition of, b T if n t i 1, t i T, b T ; (2 prtition of, b T if n t i 1, t i T =, b T ; (3 δ-fine McShne prtition of, b T if t i 1, t i T (ξ i δ L (ξ i, ξ i + δ R (ξ i T nd ξ i, b T for ll i = 1, 2,..., n; (4 δ-fine C-prtition of, b T if it is δ-fine McShne prtition of, b T stisfying the condition n dist(ξ i, t i 1, t i T < 1 ε for the given rbitrry ε > 0, where dist(ξ i, t i 1, t i T denotes the distnce of ξ i from t i 1, t i T. Given δ-fine C-prtition (McShne prtition D = {(t i 1, t i T, ξ i } n of, b T, we write S(f, D, δ = n f(ξ i (t i t i 1 for integrl sums over D, whenever f :, b T R or f :, b T R F.

4 X. You, D. Zho, J. Cheng, T. Li, J. Nonliner Sci. Appl., 11 (2018, Definition 2.1. A function f :, b T R is clled McShne delt integrble on, b T if there exists n A R such tht for ech ɛ > 0 there exists -guge, δ, for, b T such tht S(f, D, δ, A < ɛ for ech δ-fine McShne prtition D = {(t i 1, t i T, ξ i } n of, b T. In this cse, A is clled the McShne delt integrl of f on, b T nd is denoted by A = (M f(t t. Definition 2.2. A function f :, b T R is clled C-delt integrble on, b T if there exists n A R such tht for ech ɛ > 0 there exists -guge, δ, for, b T such tht S(f, D, δ, A < ɛ for ech δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T. In this cse, A is clled the C-delt integrl of f on, b T nd is denoted by A = (C f(t t. The collection of ll functions tht re C-delt integrble on, b T will be denoted by C (,,bt. Lemm 2.3 (19. If u R F, then (1 u α is closed intervl, α 0, 1; (2 u α 1 u α 2 whenever 0 α 1 α 2 1; (3 for ny α n converging incresingly to α (0, 1, n=1 uα n = u α. Conversely, if {Ã α : α 0, 1} is fmily of subsets of R stisfying (1-(3, then there exists u R F such tht u α = Ã α for α (0, 1 nd u 0 = Ã α Ã 0. Lemm 2.4 (14. If u R F, then (1 u α is bounded nondecresing function on 0, 1; (2 u α is bounded nonincresing function on 0, 1; 0<α 1 (3 u 1 u 1 ; (4 for c (0, 1, lim α c u α = u c, lim α c u α = u c ; (5 lim α 0 + u α = u 0, lim α 0 + u α = u 0. Conversely, if u α nd u α stisfy (1-(5, then there exists v R F such tht v α = v α, v α = u α, u α. 3. C-Delt integrl of intervl-vlued functions Definition 3.1. An intervl-vlued function f :, b T R I is clled intervl C-delt integrble (IC - integrble on, b T if there exists n A R I such tht for ech ɛ > 0 there exists -guge, δ, for, b T such tht d ( S(f, D, δ, A < ɛ for ech δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T. In this cse, A is clled the IC -integrl of f on, b T nd is denoted by A = (IC f(t t. The collection of ll functions tht re IC -integrble on, b T will be denoted by IC (,,bt. For the IC -integrl, we hve the following properties. Theorem 3.2. An intervl-vlued function f(t IC (,,bt if nd only if f(t, f(t C (,,bt nd (IC f(t t = (C f(t t, (C f(t t.

5 X. You, D. Zho, J. Cheng, T. Li, J. Nonliner Sci. Appl., 11 (2018, Proof. Let f(t IC (,,bt. Then there exists n A R I such tht for ech ɛ > 0 there exists -guge, δ, for, b T such tht d ( S(f, D, δ, A < ɛ for ech δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T. It follows tht ( S(f, mx D, δ A, S(f, D, δ A < ɛ. Then we hve S(f, D, δ A < ɛ, S(f, D, δ A < ɛ. By Definition 2.2, f(t, f(t C (,,bt nd (IC f(t t = (C f(t t, (C f(t t Conversely, if f(t C (,,bt, then there exists n A R such tht for ech ɛ > 0 there exists -guge, δ 1, for, b T such tht S(f, D1, δ 1 A < ɛ for ech δ 1 -fine C-prtition D 1 of, b T. Similrly, there exists -guge δ 2 such tht S(f, D2, δ 2 A < ɛ for ech δ 2 -fine C-prtition D 2 of, b T. Let δ 2 = min{δ 1, δ 2 } nd A = A, A. Then d ( S(f, D, δ, A < ɛ for ech δ-fine C-prtition D of, b T nd the proof is complete. The following Theorems 3.3 nd 3.4 re obvious, becuse their proofs re similr to those of 27. Theorem 3.3. If f(t, g(t IC (,,bt nd α, β R, then αf(t + βg(t IC (,,bt nd (IC (αf(t + βg(t t = α(ic f(t t + β(ic. g(t t. Theorem 3.4. Let < c < b. If f(t IC (,,ct nd f(t IC (, c,bt, then so it is on, b T nd (IC f(t t = (IC 4. C-Delt integrl of fuzzy-vlued functions c f(t t + (IC c f(t t. Definition 4.1. A fuzzy-vlued function f :, b T R F is clled fuzzy C-delt integrble (FC - integrble on, b T if there exists fuzzy number à R F such tht for ech ɛ > 0 there exists -guge, δ, for, b T such tht D ( S(f, D, δ, à < ɛ for ech δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T. In this cse, à is clled the FC -integrl of f on, b T nd is denoted by à = (FC f(t t. The collection of ll functions tht re FC -integrble on, b T will be denoted by FC (,,bt.

6 X. You, D. Zho, J. Cheng, T. Li, J. Nonliner Sci. Appl., 11 (2018, Remrk 4.2. It is cler tht if f is rel-vlued function, then Definition 4.1 yields the definition of C-delt integrl introduced by 13. For the FC -integrl, we hve the following properties. Theorem 4.3. If f(t FC (,,ct, then the integrl of f(t is determined uniquely. Theorem 4.4. If f(t, g(t FC (,,ct nd α, β R, then αf(t + βg(t FC (,,ct nd (FC (αf(t + βg(t t = α(fc f(t t + β(fc g(t t. Theorem 4.5 (Cuchy-Bolzno condition. A function f(t FC (,,ct if nd only if for ech ɛ > 0 there exists -guge, δ, for, b T such tht for ech pir of δ-fine C-prtitions D 1, D 2 of, b T. D ( S(f, D 1, δ, S(f, D 2, δ < ɛ Theorem 4.6. Let < c < b. If f(t FC (,,ct nd f(t FC (, c,bt, then so it is on, b T nd (FC f(t t = (FC c f(t t + (FC c f(t t. Theorem 4.7. If f(t FC (,,bt, then f(t FC (, c,dt for ny c, d T, b T. Proof. We only prove tht Theorem 4.5 holds, the others re obvious. (Necessity. Suppose tht f(t FC (,,ct nd ɛ > 0. Then, there exists -guge, δ, for, b T such tht ( D S(f, D, δ, (FC f(t t for ech δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T. Let D 1 nd D 2 be two δ-fine C-prtitions of, b T. Then, ( D ( S(f, D 1, δ, S(f, D 2, δ D S(f, D 1, δ, (FC f(t t + D ( < ɛ 2 S(f, D 2, δ, (FC f(t t < ɛ 2 + ɛ 2 = ɛ. (Sufficiency. For ech n N, choose n -guge, δ n, for, b T such tht for ny two δ n -fine C-prtitions D 1 nd D 2 of, b T we hve D ( S(f, D 1, δ n, S(f, D 2, δ n < 1 n. Replcing δ n with n j=1 δ j = δ n, we my ssume tht δ n+1 δ n. For ech n, fix δ n -fine C-prtitions D n. Note tht for j > n, since δ j δ n, D j is δ n -fine C-prtitions of, b T. Thus, D ( S(f, D n, δ n, S(f, D j, δ n < 1 n, which implies tht the sequence {S(f, D n, δ n } is Cuchy sequence, nd hence converges. Let à be the limit of this sequence. It follows from the previous inequlity tht D ( S(f, D n, δ n, à < 1 n. It remins to show tht à stisfies Definition 4.1. Fix ɛ > 0 nd choose N > 2/ɛ. Let D be δ N -fine C-prtitions of, b T. Then, D ( S(f, D, δ N, à D ( S(f, D 1, δ N, S(f, D, δ N + D ( S(f, D, δ N, à < 1 N + 1 N < ɛ. It follows now tht f(t FC (,,ct. The proof is complete.

7 X. You, D. Zho, J. Cheng, T. Li, J. Nonliner Sci. Appl., 11 (2018, Definition 4.8 (27. We sy tht subset E of time scle T hs delt mesure zero provided tht E contins no right-scttered points nd E hs Lebesgue mesure zero. We sy tht property A holds delt lmost everywhere (delt.e. on T provided tht there is subset E of T such tht the property A holds for ll t E nd E hs delt mesure zero. Theorem 4.9. If f(t = g(t holds delt.e. on, b T nd f(t FC (,,bt, then g(t FC (,,bt nd (FC f(t t = (FC g(t t. Proof. Let à denote the integrl vlue of f(t on, b T. Given ɛ > 0 there exists -guge, δ, for, b T such tht D ( S(f, D, δ, à < ɛ for ech δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T. Set E = j=1 E j, where E j = {t : j 1 < D(f(t, g(t j, j = 1, 2,..., t, b T }. For ech j, there is n F j which is the union of countble number of open intervls with the totl length less thn ɛ 2 j j 1 nd such tht E j F j. Then define δ(ξ = { (δ 0 L (ξ, δ0 R (ξ, if ξ, b T\E, (δ 1 L (ξ, δ1 R (ξ, such tht (ξ δ1 L (ξ, ξ + δ1 R (ξ T F j, ξ E j. Then, for ny δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T, we hve D ( S(g, D, δ, à ( = D g(ξ i (t i t i 1, à ξ i,b T ( = D g(ξ i (t i t i 1 + The proof is complete. ξ i E ( = D g(ξ i (t i t i 1 + ξ i,b T \E g(ξ i (t i t i 1, à f(ξ i (t i t i 1 + ξ i E ξ i,b T \E ξ i E à + f(ξ i (t i t i 1 ξ i E ( D f(ξ i (t i t i 1, à ξ i,b T ɛ + D j=1 ξ i E j ( + D ξ i E ( g(ξi, f(ξ i (t i t i 1 2ɛ. Theorem 4.10 (Dominted convergence theorem. Assume (1 lim n f n (t = f(t holds delt.e. on, b T ; (2 g(t f n h(t holds delt.e. on, b T nd f n, g, h C (,,bt. g(ξ i (t i t i 1, f(ξ i (t i t i 1, ξ i E f(ξ i (t i t i 1 Then f(t C (,,bt nd lim (C f n (t t = (C n f(t t.

8 X. You, D. Zho, J. Cheng, T. Li, J. Nonliner Sci. Appl., 11 (2018, Proof. By hypotheses, the function φ(t = h(t g(t is McShne delt integrble on, b T nd f n (t f m (t φ(t on, b T for ll n nd m. By the dominted convergence theorem for McShne integrl (see 35, f(t f 1 (t is McShne delt integrble on, b T nd lim (C (f n (t f 1 (t t = (C n (f(t f 1 (t t. In prticulr, the sequence { (C f n(t t } converges. Let ɛ > 0. Since the function Φ(x = { x φ(t t} is bsolutely continuous on, b T (see 10, there exists δ > 0 such tht n Φ(t i Φ(t i 1 < ɛ whenever t i 1 t i nd {t i 1, t i T } n is finite collection of non-overlpping intervls in, b T stisfying n (t i t i 1 < δ. By Egorov s theorem, there exists n open set G with m(g < δ such tht lim n f n (t = f(t uniformly for t, b T \G. Choose positive integer N such tht (C f n (t t (C f m (t t < ɛ nd f n f m < ɛ for ll m, n > N nd for ll t, b T \G. Let δ Φ (ξ = (δ L (ξ, δ R (ξ be -guge for, b T such tht S(Φ, D, δ Φ (M Φ(t t < ɛ nd S(f n, D, δ Φ (C f n (t t < ɛ for 1 n N whenever D is δ Φ -fine C-prtition of, b T. Define -guge, δ, for, b T by { δ Φ (ξ, if ξ, b T \G, δ(ξ = min{δ Φ (ξ, ρ(ξ, G}, if ξ G, where ρ(ξ, G = inf{ ξ ξ : ξ G}. Suppose tht D is δ-fine C-prtition of, b T nd fix n > N. Then S(f n, D, δ (C f n (t t S(f n, D, δ S(f N, D, δ + S(f N, D, δ f N (t t + (C f N (t t (C f n (t t f n (ξ i (t i t i 1 f N (ξ i (t i t i 1 ξ i,b T \G ξ i,b T \G + f n (ξ i (t i t i 1 f N (ξ i (t i t i 1 + ɛ + ɛ ξ i G ξ i G (b + 2ɛ + f n (ξ i f N (ξ i (t i t i 1 ξ i G (b + 2ɛ + (b + 4ɛ. ξ i G φ(ξ i (t i t i 1 (M G φ(t t + (M G φ(t t

9 X. You, D. Zho, J. Cheng, T. Li, J. Nonliner Sci. Appl., 11 (2018, It follows tht { (C f n(t t } is Cuchy sequence. Consequently, we hve f(t C (,,bt nd The proof is complete. lim (C f n (t t = (C n f(t t. Now, we hve the necessry mchinery to prove the following theorem. Theorem Let f(t be fuzzy-vlued function. Then f(t FC (,,bt if nd only if u α, u α C (,,bt for ny α 0, 1 uniformly, i.e., where -guge in Definition 4.1 is independent of α 0, 1. Proof. (Necessity. Let à denote the integrl vlue of f(t on, b T. Given ɛ > 0 there exists -guge, δ, for, b T such tht D ( S(f, D, δ, à < ɛ for ech δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T. Then { n α n α } sup mx f(ξ α 0,1 i (t i t i 1 à α, f(ξ i (t i t i 1 à α { } n = sup mx f(ξ α 0,1 i α (t i t i 1 à α n, f(ξ i α (t i t i 1 à α < ɛ. Hence, for ny α 0, 1 nd for ny δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T, we hve n f(ξ i α (t i t i 1 à α n < ɛ, f(ξ i α (t i t i 1 à α < ɛ. This implies tht u α, u α C (,,bt for ny α 0, 1 uniformly. (Sufficiency. Since u α, u α C (,,bt for ny α 0, 1 uniformly, given ɛ > 0 there exists -guge, δ, for, b T such tht n f(ξ i α (t i t i 1 à α n < ɛ, f(ξ i α (t i t i 1 à α < ɛ for ny δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T nd for ny α 0, 1, where à α nd à α re the integrl vlues of f(ξ i α nd f(ξ i α, respectively. {Ãα } We cn prove tht the clss of closed intervls, à α, α 0, 1 determines fuzzy number. In Ãα fct,, à α stisfies ll conditions of Lemm 2.3. Ãα (1. Since f(t α f(t α, α 0, 1, we hve à α à α, i.e.,, à α is closed intervl, α 0, 1. (2. Since f(t α is nondecresing function on 0, 1 nd f(t α is nonincresing function on 0, 1, for ny 0 α 1 α 2 1, we get (FC which yields Ãα 1, à 1 α f(t α 1 t (FC Ãα 2, à α 2. f(t α 2 t (FC f(t α 2 t (FC f(t α 1 t,

10 X. You, D. Zho, J. Cheng, T. Li, J. Nonliner Sci. Appl., 11 (2018, (3. For ny α n converging incresingly to α (0, 1, αn n=1 f(t = f(t α, i.e., n=1 f(t α n, f(t α n = f(t α, f(t α. Tht is, We lso hve lim n f(tα n = f(t α, lim n f(tα n = f(t α. f(t 0 f(t α n f(t 1, f(t 1 f(t α n f(t 0. Thnks to Theorem 4.10, we infer tht f(t α, f(t α C (,,bt nd Consequently, we obtin lim (C f(t α n t = (C n f(t α t, Ãα n, à n α = n=1 lim (C f(t α n t = (C f(t α t. n Ãα, à α. Define à s fuzzy number which is determined by the closed intervls clss Then, for ny δ-fine C-prtition D = {(t i 1, t i T, ξ i } n of, b T, we hve The proof is complete. D ( S(f, D, δ, à < ɛ. {Ãα }, à α, α 0, Conclusions This pper investigted the C-delt integrl of intervl-vlued functions nd fuzzy-vlued functions on time scles. we gve generliztions of some results on the C-delt integrl on time scles. The next steps in the reserch direction proposed here is to study the chrcteriztions of fuzzy C-delt integrble functions. Acknowledgment The uthors re grteful to the editors nd nonymous referees for very thorough reding of the mnuscript nd for kindly prompting improvements in presenttion. This reserch is supported by the Fundmentl Reserch Funds for the Centrl Universities (Grnt No. 2017B19714, Eductionl Commission of Hubei Province (Grnt No. B , Nturl Science Foundtion of Shndong Province (Grnt No. ZR2016JL021, nd Doctorl Scientific Reserch Foundtion of Linyi University (Grnt No. LYDX2015BS001. References 1 N. Benkhettou, A. M. C. Brito d Cruz, D. F. M. Torres, A frctionl clculus on rbitrry time scles: Frctionl differentition nd frctionl integrtion, Signl Process., 107 (2015, M. Bohner, A. Peterson, Dynmic equtions on time scles, An introduction with pplictions, Birkhäuser, Boston, ( M. Bohner, A. Peterson, Advnces in dynmic equtions on time scles, Birkhäuser, Boston, ( B. Bongiorno, Un nuovo integrle per il problem delle primitive, Le Mtemtiche, 51 (1997, B. Bongiorno, On the miniml solution of the problem of primitives, J. Mth. Anl. Appl., 251 (2000, D. Bongiorno, On the problem of nerly derivtives, Sci. Mth. Jpn., 61 (2005,

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