ON CONVERGENCE THEOREMS FOR FUZZY HENSTOCK INTEGRALS

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1 Irnin Journl of Fuzzy Systems Vol. 14, No. 6, 2017 pp ON CONVERGENCE THEOREMS FOR FUZZY HENSTOCK INTEGRALS B. M. UZZAL AFSAN Abstrct. The min purpose of this pper is to estblish different types of convergence theorems for fuzzy Henstock integrble functions, introduced by Wu nd Gong [12]. In fct, we hve proved fuzzy uniform convergence theorem, convergence theorem for fuzzy uniform Henstock integrble functions nd fuzzy monotone convergence theorem. Finlly, necessry nd sufficient condition under which the point-wise limit of sequence of fuzzy Henstock integrble functions is fuzzy Henstock integrble hs been estblished. 1. Introduction The concept of Henstock integrtion =guge integrtion for rel-vlued functions ws introduced by Henstock [8] nd Kurzweil [9] independently nd is considered s one of the powerful integrtion theory in modern dys. It not only generlizes the concepts of Riemnn integrtion s well s Lebesgue integrtion but lso is equivlent to the Denjoy integrtion nd Perron integrtion of rel vlued functions. In ddition, this integrtion theory stisfies most of the desired properties of integrl. Becuse of growing importnce, generliztion of such concept in fuzzy setting is lmost inevitble; in fct, in 2001, the concept of Henstock integrl ws fuzzyfied by Wu nd Gong [12]. Some recent works relted to fuzzy Henstock integrls re found in literture in the form of published ppers of Bongiorno nd Pizz [2], Gong nd Wng [6] nd Musi l [11]. As convergence theory is one of the fundmentl concepts in mesure theory nd hs vrious pplictions in integrtion theory s well, we re tempted to estblish some convergence theorems for the fuzzy Henstock integrble functions. In fct, we hve proved fuzzy uniform convergence theorem, convergence theorem for fuzzy uniform Henstock integrble functions nd fuzzy monotone convergence theorem. At the end, we hve given necessry nd sufficient condition under which the point-wise limit of sequence of fuzzy Henstock integrble functions is fuzzy Henstock integrble. It is to be mentioned tht, if {ψ k } is sequence of fuzzy Henstock integrble functions on [, b] which pointwise converges to ψ : [, b] E 1 in the metric spce Received: December 2015; Revised: December 2016; Accepted: April 2017 Key words nd phrses: Fuzzy number, Fuzzy number function, Fuzzy Henstock integrl, Fuzzy monotone sequence.

2 88 B. M. Uzzl Afsn E 1,, then limh is not true in generl see Exmple 3.1. ψ k = H ψ. 2. Preliminries Throughout this pper, symbols R, R + nd N stnd for the rel line with usul topology, the set of ll positive rel numbers nd the set of ll positive integers respectively. For ny two given sets A nd B, B A denotes the set of ll mppings with domin A nd codomin B. Definition 2.1. A mpping α [0, 1] R is clled fuzzy number if i α is norml, i.e. αr = 1 for some r R, ii α is convex, i.e. αλr λr 2 min{αr 1, αr 2 } for ll r 1, r 2 R nd λ [0, 1], iii α is semi-continuous, i.e. for every λ [0, 1], the set {x R : αx λ} is closed, iv cl[α] 0 = cl{x R : αx > 0} is compct, where cla is the closure of A R. The set of ll fuzzy numbers is denoted by E 1. Let α E 1. Then Wu nd Ming [13] showed tht for ech λ 0, 1], [α] λ = {x R : αx λ} is closed nd bounded intervl nd [α] 1. For ech λ 0, 1], let [α] λ = [α λ 1, α λ 2 ]. Goetschel nd Voxmn [3] estblished the following lemm. Lemm 2.2. [3] Let α 1, α 2 R [0,1] be two mpping sending ech λ [0, 1] to α 1 λ = α λ 1 nd α 2 λ = α λ 2 respectively with the properties: i α 1 is bounded incresing function, ii α 2 is bounded decresing function, iii α 1 1 α 2 1 nd iv α 1 nd α 2 re both left continuous on 0, 1] nd right continuous t 0. Then there exists unique fuzzy number α E 1 such tht [α] λ = [α λ 1, α λ 2 ] for ech λ [0, 1]. Let Ω = { = [, ā] :, ā R, ā} be the fmily of ll bounded closed intervls. Let, b Ω. We define i = b if nd only if = b, ā = b, ii b if nd only if b, ā b, iii + b = [, ā] + [b, b] = [ + b, ā + b], iv.b = {st : s, t b}, v.b = min{.b,. b, ā.b, ā. b}, vi.b = mx{.b,. b, ā.b, ā. b}.

3 On Convergence Theorems for Fuzzy Henstock Integrls 89 Here we observe tht is prtil order in Ω nd the mpping ρ : Ω Ω R defined by ρ, b = mx{ b, ā b } for ll, b Ω is metric clled Husdorff metric on Ω. Now it is esy to verify tht the mpping : E 1 E 1 R defined by α, β = sup{ρ[α] λ, [β] λ : λ [0, 1]} for ll α, β E 1 is metric on E 1. The results of the following theorem hve been used frequently in this pper. Theorem 2.3. [7, 10, 13] i E 1, is complete metric spce. ii α + γ, β + γ = α, β for ll α, β, γ E 1. iii λα, λβ = λ α, β for ll α, β E 1 nd λ R. iv α + γ, β + η α, β + γ, η for ll α, β, γ, η E 1. v α + β, θ α, θ + β, θ for ll α, β E 1 nd θ is the chrcteristic function of zero. vi α + β, γ α, γ + β, θ for ll α, β, γ E 1. vii If α, β, γ E 1, α, β α, γ nd β, γ α, γ. Definition 2.4. [8] A tgged prtition of [, b] consist of prtition Σ = {x 0, x 1, x 2,..., x i 1, x i,..., x n}, where = x 0 < x 1 < x 2 <... < x i 1 < x i,... < x n = b of [, b] nd ξ = {ξ i : i = 1, 2,..., n}, where ξ i [x i 1, x i ], i = 1, 2,..., n nd it is denoted by Σ, ξ. Also let σ i = ξ i ξ i 1, i = 1, 2,..., n. Let δ R [,b] +, Σ = {x 0, x 1, x 2,..., x i 1, x i,..., x n } [, b] nd ξ = {ξ i : i = 1, 2,..., n}, where ξ i [x i 1, x i ] for ech i = 1, 2,..., n. i The pir Σ, ξ is clled δ-fine division of [, b] if = x 0 < x 1 < x 2 <... < x i 1 < x i,... < x n = b nd [x i 1, x i ] ξ i δξ i, ξ i + δξ i. ii The pir Σ, ξ is clled δ-fine subdivision of [, b] if x 0 x 1 x 2... x i 1 x i,... < x n b nd [x i 1, x i ] ξ i δξ i, ξ i + δξ i. Now we recll the definition of Henstock integrl [8] for function ψ R [,b]. Definition 2.5. [8] A mpping ψ R [,b] is clled Henstock integrble on [, b] with rel vlue l if for ech ε > 0, there exists δ R [,b] + such tht n ψξ iσ i l < ε for every δ-fine division Σ, ξ of [, b]. Any function ψ E 1[,b] is clled fuzzy function defined on [, b]. Wu nd Gong introduced the notion of fuzzy Henstock integrl of fuzzy function defined on closed intervl [, b]. Definition 2.6. A fuzzy function ψ defined on [, b] is clled fuzzy Henstock integrble [12] on [, b] with vlue α E 1 if for ech ε > 0, there exists δ R [,b] + such tht n ψξ iσ i, α < ε for every δ-fine division Σ, ξ of [, b]. In symbol, we write or H ψ = α H ψ = α [,b]

4 90 B. M. Uzzl Afsn if it exists. Here α is clled Henstock integrl vlue of ψ on [, b]. The set of ll fuzzy Henstock integrble fuzzy functions defined on [, b] is denoted by F H[, b]. Wu nd Gong [12] hve chieved the following bsic results of fuzzy Henstock integrble function. Theorem 2.7. [12] Let ψ, ψ 1, ψ 2 E 1[,b]. Then i If H ψ exists, then its vlue is unique. ii ψ F H[, b] if nd only if for ech ε > 0, there exists δ R [,b] + such tht ll δ-fine divisions Σ, ξ nd Σ, ξ of [, b] stisfy n m ψξ i σ i, ψξ iσ i < ε. iii If ψ 1, ψ 2 F H[, b], then H iv If ψ F H[, b], then for ny λ R. ψ 1 + ψ 2 = H H λψ = λh ψ 1 + H v If ψ F H[, b] nd [c, d] [, b], then ψ F H[c, d]. vi If c [, b], ψ F H[, c] nd ψ F H[c, b], then ψ F H[, b] with H ψ = H c vii If ψ = θ lmost everywhere on [,b], then H ψ + H ψ = θ. viii If ψ = φ lmost everywhere on [,b], then H ψ = H Let µ be rel constnt, i.e. µ R. Then define µ : R [0, 1] by { 1 if x = µ µx = 0 if x µ Clerly, µ E 1. Thus rel number cn be viewed s fuzzy number in this wy. For α, β E 1, we define the reltion α β if nd only if αx βx for ll x R. Zhng Gung-Qun [7] introduced the concepts of bounds of set of fuzzy numbers. φ. ψ d c ψ. ψ 2.

5 On Convergence Theorems for Fuzzy Henstock Integrls 91 Definition 2.8. [7] A fuzzy number α 0 E 1 is clled the lest upper bound or suprimum of A E 1 if i α α 0 for ll α A i.e. α is n upper bound of A nd ii for ny ε > 0, there exists t lest one β A such tht α 0 < β + ε. We write α 0 = sup A. Similrly, the gretest lower bound or infimum [7] of A E 1 hs been defined nd is denoted by inf A. A sequence {α k }, α k E 1 is sid to be monotoniclly incresing resp. monotoniclly decresing [7] if α k α k+1 resp. α k+1 α k for ll k N. Zhng Gung-Qun [7] estblished the following simple but importnt theorem. Theorem 2.9. [7] Every monotoniclly incresing resp. monotoniclly decresing sequence {α k }, α k E 1 with n upper bound resp. lower bound converges to sup{α k : k N} resp. inf{α k : k N} in the metric spce E 1,. 3. Convergence Theorems Let {ψ k } be sequence of fuzzy Henstock integrble functions in E 1[,b] tht fuzzy converges to the fuzzy function ψ E 1[,b] in the metric spce E 1,. It is quite nturl to expect tht ψ F H[, b] nd H ψ = limh ψ k. But the following exmple shows tht this is not true in generl. Exmple 3.1. For ech k N, let A k = 0, 1 k nd define φ k : [0, 1] E 1 by { λ if t Ak φ k t = θ if t A k where λ E 1 is defined by { 1 if x = 1 λx = 0 if x 1. Now consider for ech k N, ψ k = k φ k. Since ech φ k is step function hving only two discontinuities 0 nd 1 k, clerly φ k F H[0, 1] nd so by Theorem 2.7, for ech k N, ψ k F H[0, 1]. Now by Theorem 2.7, H 1 0 ψ k = H 1 k 0 = k H ψ k + H 1 k Using Riemnn type sum, it is esy to verify tht 1 k ψ k 1 k φ k + θ = k H λ = λ 1 k. 1 k 0 λ.

6 92 B. M. Uzzl Afsn So H 1 0 ψ k = λ. Now consider the fuzzy function ψ : [0, 1] E 1 defined by ψt = θ for ll t [0, 1]. Then {ψ k } fuzzy converges to the fuzzy function ψ E 1[,b] in the metric spce E 1, nd by vii of Theorem 2.7, Thus H H 1 0 ψ = θ. ψ limh The min purpose of this pper is to estblish some sufficient conditions such tht the limit ψ E 1[,b] of sequence {ψ k } of fuzzy Henstock integrble functions in E 1[,b] is fuzzy Henstock integrble on [, b] nd H ψ = limh Definition 3.2. A sequence {ψ k } in E 1[,b] is sid to be fuzzy uniformly converge to ψ E 1[,b] on [, b] if for ech ε > 0, there exists k 0 N such tht ψ k x, ψx < ε for ll k k 0 nd for ll x [, b]. Theorem 3.3. Fuzzy uniform convergence theorem. Let {ψ k } be sequence of fuzzy Henstock integrble functions in E 1[,b] tht fuzzy uniformly converges to the fuzzy function ψ E 1[,b]. Then i ψ is Henstock integrble on [, b] nd ii H ψ = limh Proof. First we shll show tht {H ψ k} is Cuchy sequence in E 1,. Let ε > 0. Since for ech k N, there exists δ k R [,b] + such tht n b ψ k ξ iσ i, H ψ k < ε 3 for every δ k -fine division Σ, ξ of [, b]. Agin since {ψ k } in E 1[,b] fuzzy uniformly converges to ψ E 1[,b], there exists k 0 N such tht ε ψ k ξ i, ψξ i < 3b for ll k k 0 nd for ll i = 1, 2,..., n. ψ k. ψ k. ψ k.

7 On Convergence Theorems for Fuzzy Henstock Integrls 93 So, for ll k, l k 0, tking n rbitrry prtition Σ, ξ simultneously δ k - nd δ l -fine, we hve H ψ k, H ψ l n b n n n b ψ k ξ i σ i, H ψ k + ψ k ξ i σ i, ψ l ξ i σ i + ψ l ξ i σ i, H ψ l < ε n 3 + ψ k ξ i, ψ l ξ i σ i + ε 3 < ε 3 + ε 3b b + ε 3 = ε. Thus {H ψ k} is Cuchy sequence in E 1,. Since by Theorem 2.3, E 1, is complete, {H ψ k} converges in the metric spce E 1,. Suppose limh ψ k = α. Let ε > 0 be given. By the condition of the theorem there exists k 0 N such tht ε ψ k x, ψx < 3b for ll k k 0 nd for ll x [, b]. Now for ny tgged prtition Σ, ξ of [, b] nd k k 0, by Theorem 2.3, we get n n n ψ k ξ iσ i, ψξ iσ i σ i ψ k ξ i, ψξ i b n ψ k ξ i, ψξ i < ε 3. Now since limh ψ k = α, there exists p k 0 N such tht H ψ p, α < ε 3. Since H ψ p exists, there exists δ p R [,b] + such tht every δ p -fine division Σ, ξ of [, b] stisfies n b ψ pξ i σ i, H ψ p < ε 3. Thus n ψξ i σ i, α n n n b ψξ i σ i, ψ pξ i σ i + ψ pξ i σ i, H ψ p + H for every δ p -fine division Σ, ξ of [, b]. ψ p, α < ε

8 94 B. M. Uzzl Afsn Hence H ψ exists nd H ψ = α. Definition 3.4. A sequence {ψ k } of fuzzy Henstock integrble functions in E 1[,b] is clled fuzzy uniform Henstock integrble on [, b] if for ech ε > 0, there exists δ R [,b] + such tht every δ-fine division Σ, ξ of [, b] stisfies n b ψ k ξ i σ i, H ψ k < ε for ll k N. Z. Gong nd Y. Sho [5] proved convergence theorem see Theorem 5.1 of [5] for the strongly fuzzy Henstock integrl [4] which showed tht the controlled convergence [5] of sequence of strongly fuzzy Henstock integrble functions implies the equi-integrbility [5] of subsequence of the sequence. In the next theorem, we shll prove convergence theorem for fuzzy uniform Henstock integrble functions following the rgument used in Theorem 5.1 of [5]. Theorem 3.5. Convergence theorem for fuzzy uniform Henstock integrble functions. Let {ψ k } be fuzzy uniform Henstock integrble sequence of fuzzy Henstock integrble functions in E 1[,b] nd ψ E 1[,b] be such tht for ech x [, b], {ψ k x} converges to ψx in the metric spce E 1[,b],. Then i ψ is Henstock integrble on [, b] nd ii H ψ = limh ψ k. Proof. Let ε > 0 be given. Since {ψ k } is fuzzy uniform Henstock integrble sequence on [, b], there exists δ-fine division Σ, ξ of [, b] tht stisfies n b ψ k ξ i σ i, H ψ k < ε 3 for ll k N. Agin by the condition of the theorem there exists k 0 N such tht ε ψ k ξ i, ψ l ξ i < 3b for ll k, l > k 0 nd so n n n ψ k ξ i σ i, ψ l ξ i σ i σ i ψ k ξ i, ψ l ξ i for ll k, l > k 0. Then for ll k, l > k 0. n b ψ k ξ i, ψ l ξ i < ε 3 H ψ k, H ψ l b n n n H ψ k, ψ k ξ i σ i + ψ k ξ i σ i, ψ l ξ i σ i n b + ψ l ξ i σ i, H ψ l < ε

9 On Convergence Theorems for Fuzzy Henstock Integrls 95 So {H ψ k} is Cuchy sequence in the complete metric spce E 1,. Therefore {H ψ k} converges in the metric spce E 1,. Suppose We clim tht limh H ψ k = α. ψ = α. To show this, let ε > 0 be given. Since {ψ k } is fuzzy uniformly Henstock integrble sequence on [, b], there exists δ R [,b] + such tht every δ-fine division Σ, ξ of [, b] stisfies n b ψ k ξ i σ i, H ψ k < ε 3 for ll k N. Since limh ψ k = α, there exists k 1 N such tht H ψ k, α for ll k k 1. Agin by the given condition of the theorem, there exists k 2 k 1 N such tht n n n ψ k ξ i σ i, ψξ i σ i b ψ k ξ i, ψξ i < ε 3 Thus n ψξ iσ i, α < ε 3 n n n ψξ iσ i, ψ k ξ iσ i + ψ k ξ iσ i, H ψ k + H for every δ-fine division Σ, ξ of [, b]. Hence H ψ exists nd H ψ = α. ψ k, α < ε Sks-Henstock lemm plys n importnt role in Henstock integrtion theory. Now we shll estblish the fuzzy version of this lemm. Lemm 3.6. Let ψ E 1[,b] be fuzzy Henstock integrble function in E 1[,b], let φx = H x ψ

10 96 B. M. Uzzl Afsn for ll x [, b] nd let ε > 0. Further suppose i δ R [,b] + is positive rel-vlued function such tht n n ψξ i σ i, φξ i σ i < ε for every δ-fine division Σ, ξ of [, b]. If Σ, ξ = {ξ i, [ i, b i ] : i = 1, 2,..., n} is δ-fine subdivision of [, b], then m m ψξ iσ i, φξ iσ i ε. Proof. Let ε 0 > 0 nd {F i : i = 1, 2,..., r} be the fmily of closed intervls in [, b] such tht {F i : i = 1, 2,..., r} Σ is prtition of [, b]. Here ψ E 1[,b] is fuzzy Henstock integrble on ech of the intervls F 1, F 2,..., F r nd hence for ech k {1, 2,..., r}, there exists δ k R [,b] + such tht every δ k -fine division Σ k, ξ k, Σ k = {x k 0, x k 1,..., x k i 1, xk i,..., xk n k }, ξ k = {ξ1 k,..., ξi k,..., ξk n k } of [, b] stisfies nk ψξi k σi k, H F t ψ < ε 0 r. Without loss of generlity, we cn ssume tht δ k x δx for ll x F k, k {1, 2,..., r}. If we tke Σ = Σ 1 Σ 2... Σ r Σ nd ξ = ξ 1 ξ 2... ξ r ξ, then Σ, ξ is δ-fine division of [, b]. Thus using condition i nd Theorem 2.3, we get r nk m m nk m ψξi k σi k + ψξ iσ i, φξi k σi k + φξ iσ i < ε. k=1 Now m m ψξ iσ i, φξ iσ i k=1 n k n r k m = ψξi k σi k + ψξ iσ i k=1 k=1 n r k ψξi k σi k, k=1 n m m m k φξi k σi k + φξ iσ i φξi k σi k k=1 r k=1 nk k=1 m m ψξi k σi k + ψξ iσ, n r k n k ψξi k σi k, φξi k σi k < ε + r ε0 r = ε + ε0 nd consequently, k=1 nk m m ψξ iσ i, φξ iσ i ε. m φξi k σi k + φξ iσ i +

11 On Convergence Theorems for Fuzzy Henstock Integrls 97 A sequence {ψ k }, ψ k E 1[,b] is clled fuzzy incresing resp. fuzzy decresing in [, b] if ψ k x ψ k+1 x resp. ψ k+1 x ψ k x for ll x [, b] nd k N. A sequence {ψ k } is clled fuzzy monotone on [, b] if it is either fuzzy incresing or fuzzy decresing in [, b]. Theorem 3.7. Fuzzy monotone convergence theorem. Let {ψ k } be fuzzy monotone sequence of fuzzy Henstock integrble functions in E 1[,b], {H ψ k} be fuzzy bounded nd ψ E 1[,b] be such tht for ech x [, b], {ψ k x} converges to ψx in the metric spce E 1[,b],. Then i ψ is Henstock integrble on [, b] nd ii H ψ = limh Proof. Let {ψ k } be fuzzy incresing sequence of fuzzy Henstock integrble functions in E 1[,b]. Then {H ψ k} is fuzzy incresing nd bounded. Then by Theorem 2.9, {H ψ k} must be fuzzy converges to α = sup{h ψ k}. 1 Let ε > 0 be given. Then we cn choose n r N such tht 2 < ε r 2 3 nd H ψ r, α < ε 3. Agin since {ψ k } is sequence of fuzzy Henstock integrble functions on [, b], for ech k N there exists δ k R [,b] + such tht every δ k -fine division Σ k, ξ k of [, b] stisfies nk b ψ k ξi k σi k, H ψ k < 1 2. k Agin by the condition, for ech x [, b], we cn select k x r N such tht ε ψ kx x, ψx < 3b. Consider the function δ = δ kx nd let Σ, ξ = {, ξ i : i = 1, 2,..., n} be ny δ-fine division of [, b]. Here n n n ψξ iσ i, α ψξ iσ i, ψ kξi ξ iσ i + n n ψ kξi ξ iσ i, H ψ kξi ψ k. n + H ψ kξi, α. Now we estimte the three vlues in the right-hnded sum of the lst inequlity. Estimtion of n n ψξ iσ i, ψ kξi ξ iσ i : By Theorem 2.3, n n ψξ i σ i, ψ kξi ξ i σ i n ε ψξ i, ψ kξi ξ i σ i < 3b b = ε 3.

12 98 B. M. Uzzl Afsn b Estimtion of n n ψ kξi ξ i σ i, H Suppose p = mx{k ξi : i = 1, 2,..., n}. Then n n p ψ kξi ξ iσ i, H ψ kξi t=r ψ kξi i {1,2,...,n:k ξi =t} : ψ kξi ξ i, H ψ kξi. Now pplying Lemm 3.6, nd hence c Estimtion of i {1,2,...,n:k ξi =t} n n ψ kξi ξ i σ i, H ψ kξi ξ i, H ψ kξi 1 2 t 1 ψ kξi < p t=r n H ψ kξi, α : 1 2 t 1 < 1 2 r 2 < ε 3. Here r k ξi p implies ψ r x ψ kξi x ψ p x for ll x [, b] nd so ψ r ψ kξi ψ p. Hence ψ r n ψ kξi ψ p α. Therefore by Theorem 2.3, n b H ψ kξi, α ψ r, α < ε 3. Thus n ψξ i σ i, α < ε. So ψ is Henstock integrble on [, b] nd H ψ = α = limh ψ k.

13 On Convergence Theorems for Fuzzy Henstock Integrls 99 Brtle [1] found necessry nd sufficient conditions for Henstock integrl convergence theorem of rel functions. This pper of Brtle [1] inspires to estblish the finl theorem of this section. Actully, this theorem provides us necessry nd sufficient condition such tht the point-wise limit ψ E 1[,b] of sequence {ψ k } of fuzzy Henstock integrble functions is to be fuzzy Henstock integrble on [, b] nd the equlity holds. H ψ = limh Theorem 3.8. Let {ψ k } be sequence of fuzzy Henstock integrble functions in E 1[,b] nd ψ E 1[,b] be such tht for ech x [, b], {ψ k x} converges to ψx in the metric spce E 1[,b],. Then the following conditions re equivlent: i ψ is fuzzy Henstock integrble on [, b] nd H ψ = limh ii for ech ε > 0, there exists m N such tht for ech k m, there exists δ R [,b] + such tht every δ-fine division Σ k, ξ k of [, b] stisfies nk n k ψ k ξi k σi k, ψξi k σi k < ε. Proof. i ii. Let ε > 0. Since limh there exists n m N such tht H ψ k = H ψ k, H for ll k m. Agin since {ψ k } is sequence of fuzzy Henstock integrble functions, for ech k m, we cn find δ k R [,b] + such tht every δ k -fine division Σ k, ξ k of [, b] stisfies nk b ψ k ξi k σi k, H ψ k < ε 3. Agin since ψ is fuzzy Henstock integrble on [, b], we cn find δ 0 R [,b] + such tht every δ 0 -fine division Σ 0, ξ 0 of [, b] stisfies n0 b ψξi 0 σi 0, H ψ < ε 3. ψ ψ k, ψ k, ψ, < ε 3

14 100 B. M. Uzzl Afsn We tke δ R [,b] + defined by δx = min{δ 0 x, δ k x}. Then nk n k ψ k ξi k σk i, nk b ψξi k σk i ψ k ξi k σk i, H ψ k + H n k + H ψ, ψξi k σk i < ε ψ k, H for every δ-fine division Σ k, ξ k of [, b] nd k m. ii i. Let ε > 0 nd ii holds. Then we clim tht {H ψ k} is Cuchy sequence. By ii, there exists m N such tht for ech k, l m, there exist δ k, δ l R [,b] + such tht nk n k ψ k ξi k σi k, ψξi k σi k < ε 4 for every δ k -fine division Σ k, ξ k of [, b] nd nl n l ψ l ξiσ l i, l ψξiσ l i l < ε 4 every δ l -fine division Σ l, ξ l of [, b]. Since ψ k nd ψ l re fuzzy Henstock integrble functions on [, b], we cn find ς k, ς l R [,b] + such tht sk b ψ k τi k ϱ k i, H ψ k < ε 4 for every ς k -fine division k, τ k of [, b] nd sl ψ k τi l ϱ l i, H for every ς l -fine division l, τ l of [, b]. Now define δ R [,b] + by δx = min{δ k x, δ l x, ς k x, ς l x}. Then for ll k, l m, H ψ k, H ψ l < ε 4 ψ l < ε nd so {H ψ k} is Cuchy sequence. Completeness of metric spce E 1, ensures tht exists in E 1. Suppose limh limh ψ k ψ k = α. ψ

15 On Convergence Theorems for Fuzzy Henstock Integrls 101 Then we cn choose p m N such tht H ψ k, α < ε 3. Also by ii, there exists δ 1 R [,b] + such tht every δ 1 -fine division Σ 1, ξ 1 of [, b] stisfies n1 n 1 ψ p ξi 1 σi 1, ψξi 1 σi 1 < ε 3. Since ψ p is fuzzy Henstock integrble function, there exists n δ 2 R +[,b] such tht every δ 2 -fine division Σ 2, ξ 2 of [, b] stisfies n2 ψ p ξi 2 σ i, H ψ p < ε 3. We define δ R [,b] + by δx = min{δ 1 x, δ 2 x}. Then α, n ψξ i σ i α, H ψ p + H n n ψ p ξ i σ i, ψξ i σ i < ε. So ψ is Henstock integrble on [, b] nd H ψ = α = limh 4. Conclusions ψ k. ψ k, n ψ p ξ i σ i + Let {ψ k } be sequence of fuzzy Henstock integrble functions in E 1[,b] which pointwise converges to ψ E 1[,b] in the metric spce E 1,. In Exmple 3.1, we hve shown tht limh ψ k = H is not true in generl. As result, finding vrious sufficient conditions s when the bove equlity will hold, re very much desired for fuzzy Henstock integrble functions. Being tempted, we hve estblished, in this pper, three coveted convergence theorems for fuzzy Henstock integrble functions: fuzzy uniform convergence theorem, convergence theorem for fuzzy uniform Henstock integrble functions nd fuzzy monotone convergence theorem ; we hve lso chieved in finding necessry nd sufficient condition under which the point-wise limit of sequence of fuzzy Henstock integrble functions is fuzzy Henstock integrble. In this pper, ttempts hve been mde in estblishing some bsic convergence theorems, but more nd ψ.

16 102 B. M. Uzzl Afsn more subsequent venture in this ren will emerge mny non trivil results tht will definitely enrich the Henstock integrtion theory in fuzzy setting. Acknowledgements. The uthor is grteful to the lerned reviewers for their constructive comments nd vluble suggestions, which improved the pper to gret extent. References [1] R. G. Brtle, A convergence theorem for generlized Riemnn integrls, Rel Anl. Exchnge, , [2] B. Bongiorno, L. Di Pizz nd K. Musi l, A decomposition theorem for the fuzzy Henstock integrl I, Fuzzy Sets nd Systems, , [3] R. Goetschel nd W. Voxmn, Elementry fuzzy clculus, Fuzzy Sets nd Systems, , [4] Z. Gong, On the problem of chrcterizing derivtives for the fuzzy-vlued functions II: lmost everywhere differentibility nd strong Henstock integrl, Fuzzy Sets nd Systems, , [5] Z. Gong nd Y. Sho, The controlled convergence theorems for the strong Henstock integrls of fuzzy-number-vlued functions, Fuzzy Sets nd Systems, , [6] Z. Gong nd L. Wng, The Henstock-Stieltjes integrl for fuzzy-number-vlued functions, Inform. Sci., , [7] Z. Gung-Qun, Fuzzy continuous function nd its properties, Fuzzy Sets nd Systems, , [8] R. Henstock, Theory of Integrtion, Butterworths, London, [9] J. Kurzweil, Generlized ordinry differentil equtions nd continuous dependence on prmeter, Czechoslovk Mth. J., , [10] M Ming, On embedding problem of fuzzy number spce: Prt 4, Fuzzy Sets nd Systems, , [11] K. Musi l, A decomposition theorem for Bnch spce vlued fuzzy Henstock integrl, Fuzzy Sets nd Systems, , [12] C. Wu nd Z. Gong, On Henstock integrl of fuzzy-number-vlued functions, Fuzzy Sets nd Systems, , [13] C. Wu nd M Ming, On embedding problem of fuzzy number spce: Prt 1, Fuzzy Sets nd Systems, , B. M. Uzzl Afsn, Deprtment of Mthemtics, Sript Singh College, Jignj , Murshidbd, West Bengl, Indi E-mil ddress: uzlfsn@gmil.com