Hindawi Publishing Corporation Journal of Spaces Volume 2015, Article ID 870179, 10 pages http://dx.doi.org/10.1155/2015/870179

Research Article On of Sequences and of Fuzzy-Valued Functions

ULur Kadak1,2 and Hakan Efe1

1 Department of Mathematics, Faculty of Sciences, Gazi University, 06500 Ankara, Turkey 2Department of Mathematics, Faculty of Sciences and Arts, Bozok University, 66100 Yozgat, Turkey

Correspondence should be addressed to Ugur˘ Kadak; [email protected]

Received 18 June 2014; Revised 26 August 2014; Accepted 10 September 2014

Academic Editor: Mahmut Is¸ik

Copyright © 2015 U. Kadak and H. Efe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The class of membership functions is restricted to trapezoidal one, as it is general enough and widely used. In the present paper since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct for a fuzzy-valued function via related trapezoidal membership function. We derive uniform convergence of fuzzy-valued function sequences and series with some illustrated examples. Also we study Hukuhara differentiation and Henstock integration of a fuzzy- valued function with some necessary inclusions. Furthermore, we introduce the with fuzzy coefficients and define the radius of convergence of power series. Finally, by using the notions of H-differentiation and radius of convergence we examine the relationship between term by term H-differentiation and uniform convergence of fuzzy-valued function series.

1. Introduction The effectiveness of level sets is based on not only their required storage capacity but also their two-valued nature. The term uniform convergence was probably first used by Also the definition of these sets offers some advantages over Christoph Gudermann, in an 1838 paper on elliptic functions, the related membership functions. where he employed the phrase “convergence in a uniform Many authors have developed the different cases of way” when the “mode of convergence” of a series is inde- sequence sets with fuzzy metric on a large scale. Bas¸arir pendent of two variables. While he thought it a “remarkable [3] has recently promoted some new sets of sequences of fact” when a series converged in this way, he did not give fuzzy numbers generated by a nonnegative regular matrix 𝐴, a formal definition or use the property in any of his proofs some of which reduced to Maddox’s spaces ℓ∞(𝑝; 𝐹), 𝑐(𝑝; 𝐹), [1]. Later Karl Weierstrass, who attended his course on 𝑐0(𝑝; 𝐹),andℓ(𝑝; 𝐹) for the special cases of that matrix 𝐴. elliptic functions in 1839-1840, coined the term uniformly Quiterecently,TaloandBas¸ar [4]havedevelopedthemain convergentwhichheusedinhis1841paperZurTheorie results of Bas¸ar and Altay [5] to fuzzy numbers and defined der Potenzreihen, published in 1894. Independently a similar the alpha-, beta-, and gamma-duals and introduced the duals concept was used by Imre [2] and G. Stokes but without of these sets together with the classes of infinite matrices of having any major impact on further development. fuzzy numbers mapping one of the classical set into another Duetotherapiddevelopmentofthefuzzylogictheory, one. Also, Kadak and Ozluk [6]haveintroducedsomenew however, some of these basic concepts have been modified sets of sequences of fuzzy numbers with respect to the partial andimproved.Oneofthemisintheformofintervalvalued metric. fuzzy sets. To achieve this we need to promote the idea of The rest of this paper is organized as follows. In Section 2, the level sets of fuzzy numbers and the related formulation we give some necessary definitions and propositions related of a representation of an interval valued fuzzy set in terms to the fuzzy numbers, sequences, and series of fuzzy numbers. of its level sets. Once having the structure we then can We also report the most relevant and recent literature in supply the required extension to interval valued fuzzy sets. this section. In Section 3, first the definition of fuzzy-valued 2 Journal of Function Spaces function is given which will be used in the proof of our main whose membership function 𝜇(𝑥) can generally be defined as results. In this section, generalized Hukuhara differentiation [9] and Henstock integration are presented according to fuzzy- 𝑥−𝑢 valued functions depending on real values 𝑥 and 𝑡.Thefinal { 1 ,𝑢≤𝑥≤𝑢, {𝑢 −𝑢 1 2 section is completed with the concentration of the results on { 2 1 { uniform convergence of fuzzy-valued sequences and series. 1, 𝑢2 ≤𝑥≤𝑢3, 𝜇 (𝑥) := { 𝑢 −𝑥 (2) Also we examine the relationship between the radius of { 4 ,𝑢≤𝑥≤𝑢, { 3 4 convergence of power series and the notion of uniform {𝑢4 −𝑢3 convergence with respect to fuzzy-valued function. {0, 𝑥<𝑢1, 𝑥>𝑢4,

𝑢− :[𝑢,𝑢 ]→[0,1] 𝑢+ :[𝑢,𝑢 ]→[0,1] 2. Preliminaries, Background, and Notation where 1 2 and 3 4 are two strictly monotonical and continuous mappings from A fuzzy number is a fuzzy set on the real axis, that is, a R to the closed interval [0, 1].If𝜇(𝑥) is piecewise linear, mapping 𝑢:R → [0, 1] which satisfies the following four then 𝑢 is referred to as a trapezoidal fuzzy number and is conditions. usually denoted by 𝑢=(𝑢1,𝑢2,𝑢3,𝑢4). In particular, when 𝑢2 ≡𝑢3, the trapezoidal form is reduced to a triangular (i) 𝑢 is normal; that is, there exists an 𝑥0 ∈ R such that form; that is, 𝑢=(𝑢1,𝑢2,𝑢4). So, triangular forms are special − + 𝑢(𝑥0)=1. cases of trapezoidal forms. Since 𝑢 and 𝑢 are both strictly monotonical and continuous functions, their inverse exists (ii) 𝑢 is fuzzy convex; that is, 𝑢[𝜆𝑥 + (1 − 𝜆)𝑦] ≥ and should also be continuous and strictly monotonical. min{𝑢(𝑥), 𝑢(𝑦)} for all 𝑥, 𝑦 ∈ R and for all 𝜆∈[0,1]. 1 Let 𝑢, V,𝑤 ∈ 𝐸 and 𝛼∈R.Thensomealgebraic (iii) 𝑢 is upper semicontinuous. operations, that is, level set addition, scalar multiplication, 1 and product, are defined on 𝐸 by (iv) The set [𝑢]0 = {𝑥 ∈ R :𝑢(𝑥)>0} is compact (cf. {𝑥 ∈ R :𝑢(𝑥)>0} Zadeh [7]), where denotes the 𝑢⊕V =𝑤⇐⇒[𝑤]𝜆 = [𝑢]𝜆 ⊕ [V]𝜆 closure of the set {𝑥 ∈ R :𝑢(𝑥)>0}in the usual − − − topology of R. ⇐⇒ 𝑤 (𝜆) =𝑢 (𝜆) + V (𝜆) , (3)

1 + + + We denote the set of all fuzzy numbers on R by 𝐸 and called 𝑤 (𝜆) =𝑢 (𝜆) + V (𝜆) , 1 it the space of fuzzy numbers and the 𝜆-level set [𝑢]𝜆 of 𝑢∈𝐸 is defined by [𝛼𝑢]𝜆 =𝛼[𝑢]𝜆 and 𝑢V =𝑤⇔[𝑤]𝜆 =[𝑢]𝜆[V]𝜆 for all 𝜆∈[0,1],where

{{𝑡∈R :𝑢(𝑡) ≥𝜆} , 0<𝜆≤1, − − − − + 𝑤 (𝜆) = min {𝑢 (𝜆) V (𝜆) ,𝑢 (𝜆) V (𝜆) , [𝑢]𝜆 := { (1) {{𝑡 ∈ R : 𝑢(𝑡) > 𝜆}, 𝜆=0. 𝑢+ (𝜆) V− (𝜆) ,𝑢+ (𝜆) V+ (𝜆)},

+ − − − + (4) The set [𝑢]𝜆 is closed, bounded, and nonempty interval for 𝑤 (𝜆) = {𝑢 (𝜆) V (𝜆) ,𝑢 (𝜆) V (𝜆) , − + max each 𝜆∈[0,1]which is defined by [𝑢]𝜆 := [𝑢 (𝜆), 𝑢 (𝜆)]. 𝑢+ (𝜆) V− (𝜆) ,𝑢+ (𝜆) V+ (𝜆)}. Theorem 1 (representation theorem [8]). Let [𝑢]𝜆 = − + 1 [𝑢 (𝜆), 𝑢 (𝜆)] for 𝑢∈𝐸 and for each 𝜆∈[0,1].Thenthe Let 𝑊 be the set of all closed bounded intervals 𝐴 of R with following statements hold. endpoints 𝐴 and 𝐴;thatis,𝐴:=[𝐴, 𝐴].Definetherelation 𝑑 on 𝑊 by 𝑑(𝐴, 𝐵) := max{|𝐴 −𝐵|, |𝐴−𝐵|}.Thenitcan 𝑢− (i) is a bounded and nondecreasing left continuous easily be observed that 𝑑 is a metric on 𝑊 (cf. Diamond ]0, 1] function on . and Kloeden [10]) and (𝑊,𝑑) is a complete metric space (cf. + 𝐷 𝐸1 (ii) 𝑢 is a bounded and nonincreasing left continuous Nanda [11]).Now,wecangivethemetric on by means 𝑑 function on ]0, 1]. of the Hausdorff metric as − + (iii) The functions 𝑢 and 𝑢 are right continuous at the 𝐷 (𝑢, V) := sup 𝑑([𝑢]𝜆, [V]𝜆) point 𝜆=0. 𝜆∈[0,1] (5) − + 󵄨 − − 󵄨 󵄨 + + 󵄨 (iv) 𝑢 (1) ≤ 𝑢 (1). := sup max {󵄨𝑢 (𝜆) − V (𝜆)󵄨 , 󵄨𝑢 (𝜆) − V (𝜆)󵄨}. 𝜆∈[0,1] − + Otherwise, if the pair of functions 𝑢 and 𝑢 holds the 1 It is trivial that conditions (i)–(iv), then there exists a unique element 𝑢∈𝐸 − + such that [𝑢]𝜆 := [𝑢 (𝜆), 𝑢 (𝜆)] for each 𝜆∈[0,1]. 󵄨 − 󵄨 󵄨 + 󵄨 𝐷(𝑢,0) = sup max {󵄨𝑢 (𝜆)󵄨 , 󵄨𝑢 (𝜆)󵄨} 𝜆∈[0,1] A fuzzy number is a convex fuzzy subset of R and is (6) 󵄨 − 󵄨 󵄨 + 󵄨 defined by its membership function. Let 𝑢 be a fuzzy number, = max {󵄨𝑢 (0)󵄨 , 󵄨𝑢 (0)󵄨}. Journal of Function Spaces 3

1 Proposition 2 (see [12]). Let 𝑢, V,𝑤,𝑧∈𝐸 and 𝛼∈R.Then, where the limits are in the Hausdorff metric 𝑑 for 1 intervals. (i) (𝐸 ,𝐷)is a complete metric space (cf. Puri and Ralescu [13]); (ii) 𝐷(𝛼𝑢, 𝛼V) = |𝛼|𝐷(𝑢, V); 3. Fuzzy-Valued Functions with the Level Sets (iii) 𝐷(𝑢 ⊕ V,𝑤⊕V)=𝐷(𝑢,𝑤); Definition 7 (see [17]). Consider a fuzzy-valued function 𝑢𝑡(𝑥) R 𝐸1 (iv) 𝐷(𝑢 ⊕ V,𝑤⊕𝑧)≤𝐷(𝑢,𝑤)+𝐷(V,𝑧); from into with respect to a membership function 𝜇(𝑥) which is called trapezoidal fuzzy number and is inter- (v) |𝐷(𝑢, 0) − 𝐷(V, 0)| ≤ 𝐷(𝑢, V)≤𝐷(𝑢,0) + 𝐷(V, 0). preted as follows: Remark 3 (cf. [14]). Then the following remarks can be given. 𝑥−𝑢 (𝑡) { 1 ,𝑢(𝑡) ≤𝑥≤𝑢 (𝑡) , {𝑢 (𝑡) −𝑢 (𝑡) 1 2 (a) Obviously the sequence (𝑢𝑛)∈𝜔(𝐹)converges to a { 2 1 − + { fuzzy number 𝑢 if and only if {𝑢𝑛 (𝜆)} and {𝑢𝑛 (𝜆)} 1, 𝑢2 (𝑡) ≤𝑥≤𝑢3 (𝑡) , − + 𝜇 (𝑥) = { 𝑢 (𝑡) −𝑥 (11) converge uniformly to 𝑢 (𝜆) and 𝑢 (𝜆) on [0, 1], { 4 { ,𝑢3 (𝑡) ≤𝑥≤𝑢4 (𝑡) , respectively. {𝑢4 (𝑡) −𝑢3 (𝑡) 0, 𝑥 < 𝑢 (𝑡) , 𝑥>𝑢 (𝑡) . (b) The boundedness of the sequence (𝑢𝑛)∈𝜔(𝐹)is { 1 4 equivalent to the fact that 𝑢−,𝑢+ 𝑡∈R 󵄨 󵄨 󵄨 󵄨 Then, the pair of depending on can be written as 𝐷(𝑢 , 0) = {󵄨𝑢− (𝜆)󵄨 , 󵄨𝑢+ (𝜆)󵄨}<∞. 𝑡 1 sup 𝑛 sup sup max 󵄨 𝑛 󵄨 󵄨 𝑛 󵄨 𝑢 (𝑥) = [(𝑢2(𝑡) −1 𝑢 (𝑡))𝜆1 +𝑢 (𝑡),4 𝑢 (𝑡) − 4(𝑢 (𝑡) −3 𝑢 (𝑡))𝜆] ∈𝐸 𝑛∈N 𝑛∈N𝜆∈[0,1] (7) 𝑡 for all 𝜆∈[0,1].Then,thefunction𝑢 is said to be a fuzzy- valued function on R. If the sequence (𝑢𝑘)∈𝜔(𝐹)is bounded then the sequences − + of functions {𝑢𝑘 (𝜆)} and {𝑢𝑘 (𝜆)} areuniformlyboundedin Remark 8. The functions 𝑢𝑖 with 𝑖 ∈ {1,2,3,4} given in [0, 1]. Definition 7 arealsodefinedby𝑓𝑖(𝑡) = 𝑘 for all 𝑡∈R and the constant 𝑘. Definition 4 (see [14]). Let (𝑢𝑘)∈𝜔(𝐹). Then the expression ∑ 𝑢 ⊕ 𝑘 𝑘 is called a series of fuzzy numbers with the level Now, following Kadak [18]wegivetheclassicalsets summation ⊕∑. Define the sequence (𝑠𝑛) via 𝑛th partial level 𝐶 [𝑎, 𝑏] 𝐵 [𝑎, 𝑏] 𝑠 =𝑢 ⊕𝑢 ⊕𝑢 ⊕⋅⋅⋅⊕𝑢 𝑛∈N 𝐹 and 𝐹 consisting of the continuous and sum of the series by 𝑛 0 1 2 𝑛 for all .If bounded fuzzy-valued functions; that is, the sequence (𝑠𝑛) converges to a fuzzy number 𝑢,thenwesay ∑ 𝑢 𝑢 that the series ⊕ 𝑘 𝑘 of fuzzy numbers converges to and 𝐶𝐹 [𝑎,] 𝑏 write ⊕∑𝑘 𝑢𝑘 =𝑢which implies that := {𝑓𝑡 |𝑓𝑡 : [𝑎,] 𝑏 󳨀→ 𝐸 1 ∋𝑓𝑡 𝑛 𝑛 ∑𝑢− (𝜆) =𝑢− (𝜆) , ∑𝑢+ (𝜆) =𝑢+ (𝜆) , 𝑛→∞lim 𝑘 𝑛→∞lim 𝑘 (8) 𝑘=0 𝑘=0 continuous fuzzy-valued function where the summation is in the sense of classical summation ∀𝑥 ∈ [𝑎,] 𝑏 ,𝑡∈R}, (12) and converges uniformly in 𝜆∈[0,1]. 𝐵𝐹 [𝑎,] 𝑏 2.1. Generalized Hukuhara Difference. A generalization of the := {𝑓𝑡 |𝑓𝑡 : [𝑎,] 𝑏 󳨀→ 𝐸 1 ∋𝑓𝑡 Hukuhara difference proposed in [15]aimstoovercomethis situation. bounded fuzzy-valued function ∀𝑥, 𝑡 ∈ [𝑎,] 𝑏 }.

Definition 5 (see [15, Definition 1]). The generalized − + 𝐴⊖𝐵 𝐴, 𝐵 ∈ K Obviously, from Theorem 1, each function 𝑓 ,𝑓 is left Hukuhara difference of two sets is defined 𝜆 ∈ (0, 1] 𝜆=0 as follows: continuous on and right continuous at .It was shown that 𝐶𝐹[𝑎, 𝑏] and 𝐵𝐹[𝑎, 𝑏] are complete with the 𝐴=𝐵+𝐶, metric 𝐷𝐹 as 𝐴⊖𝐵=𝐶⇐⇒{ ∞ 𝐵=𝐴+(−1) 𝐶. (9) 𝐷 (𝑓𝑡,𝑔𝑡):= {𝐷 (𝑓𝑡 (𝑥) ,𝑔𝑡 (𝑥))} 𝐹∞ sup Proposition 6 (see [16]). The following statements hold. 𝑥∈[𝑎,𝑏] 𝐴 𝐵 𝐷(𝐴, 𝐵) = (a) If and are two closed intervals, then = { 𝑑([𝑓 𝑡 (𝑥)] ,[𝑔𝑡 (𝑥)] )} 𝐷(𝐴 ⊖ 𝐵, {0}) sup sup 𝜆 𝜆 . 𝑥∈[𝑎,𝑏] 𝜆∈[0,1] − + (b) Let 𝑢:[𝑎,𝑏]→be 𝐼 such that 𝑢(𝑥) = [𝑢 (𝑥), 𝑢 (𝑥)]. (13) Then, we have 󵄨 − − 󵄨 := max { sup sup 󵄨𝑓𝜆 (𝑡) −𝑔𝜆 (𝑡)󵄨 , 𝜆∈[0,1]𝑡∈[𝑎,𝑏] lim 𝑢 (𝑥) =ℓ⇐⇒ lim (𝑢 (𝑥) ⊖ℓ) = {0} , 𝑥→𝑥0 𝑥→𝑥0 󵄨 󵄨 (10) 󵄨𝑓+ (𝑡) −𝑔+ (𝑡)󵄨}, lim 𝑢 (𝑥) =𝑢(𝑥0)⇐⇒ lim (𝑢 (𝑥) ⊖𝑢(𝑥0)) = {0} , sup sup 󵄨 𝜆 𝜆 󵄨 𝑥→𝑥0 𝑥→𝑥0 𝜆∈[0,1]𝑡∈[𝑎,𝑏] 4 Journal of Function Spaces

𝑡 𝑡 𝑡 𝑡 where 𝑓 =𝑓(𝑥) and 𝑔 =𝑔(𝑥) are the elements of the sets Remark 12. We remark that the integrals on the right hand 𝐶𝐹[𝑎, 𝑏] or 𝐵𝐹[𝑎, 𝑏] with 𝑥, 𝑡 ∈ [𝑎,. 𝑏] side of (16) exist in the usual sense for all 𝜆∈[0,1].Itiseasyto ± see that the pair of functions 𝑓𝜆 : [𝑎, 𝑏] → R are continuous. 3.1. Generalized Hukuhara Differentiation and Henstock Inte- gration. The notion of fuzzy differentiability comes from 4. Uniform Convergence of a generalization of the Hukuhara difference for compact Fuzzy-Valued Functions convex sets. We prove several properties of the derivative of 𝑡 fuzzy-valued functions considered here. As a continuation of Definition 13. Let {𝑓𝑛(𝑥)} be a sequence of fuzzy-valued Hukuhara derivatives for real fuzzy-valued functions [19], we functions defined on a set 𝐴⊆R with respect to the 𝑡 can define H-differentiation of 𝑓 with respect to level sets. sequence 𝑡=(𝑡𝑛) with real or complex terms. We say that 𝑡 {𝑓𝑛(𝑥)} converges pointwise on 𝐴 if for each 𝑥∈𝐴the 𝑓𝑡 : R →𝐸1 𝑡 Definition 9 (cf. [20]). A fuzzy-valued function sequence {𝑓𝑛(𝑥)} converges for all 𝑥∈𝐴.Ifasequence 𝑡 is said to be generalized H-differentiable with respect to the {𝑓𝑛(𝑥)} converges pointwise, then we can define a fuzzy- 𝑡 󸀠 1 level sets at the point 𝑥:if(𝑓 ) (𝑥) ∈ 𝐸 exists such that, for all 𝑢𝑡 :𝐴 →1 𝐸 𝑡 𝑡 valued function by ℎ>0sufficiently near to 0,theH-difference𝑓 (𝑥+ℎ)⊖𝑓 (𝑥) (𝑓𝑡)󸀠(𝑥) 𝑓𝑡 (𝑥) =𝑢𝑡 (𝑥) 𝑥∈𝐴,𝑛∈N. exists then the H-derivative is given as follows: 𝑛→∞lim 𝑛 for each (17)

𝑡 󸀠 𝑡 𝑡 (𝑓 ) (𝑥) On the other hand, {𝑓𝑛(𝑥)} converges to 𝑢 on 𝐴 if and only if, for each 𝑥∈𝐴and for an arbitrary 𝜖>0, there exists 𝑡 𝑡 𝑡 𝑡 𝑓 (𝑥+ℎ) ⊖𝑓 (𝑥) an integer 𝑁 = 𝑁(𝜖, 𝑥) such that 𝐷(𝑓𝑛(𝑥), 𝑢 (𝑥)) < 𝜖 = lim [ ] ℎ→0+ ℎ 𝑛>𝑁 𝑁 𝜆 whenever . The integer in the definition of pointwise convergence may, in general, depend on both 𝜖>0and − − + + (14) 𝑓𝜆 (𝑡+ℎ) −𝑓𝜆 (𝑡) 𝑓𝜆 (𝑡+ℎ) −𝑓𝜆 (𝑡) 𝑥∈𝐴. If, however, one integer can be found that works for all =[lim , lim ] ℎ→0+ ℎ ℎ→0+ ℎ points in 𝐴, then the convergence is said to be uniform. That 𝑡 is, a sequence of fuzzy-valued functions {𝑓𝑛(𝑥)} converges − 󸀠 + 󸀠 𝑡 =[(𝑓𝜆 (𝑡)) ,(𝑓𝜆 (𝑡)) ]. uniformly to 𝑢 if, for each 𝜖>0, there exists an integer 𝑁(𝜖) such that 𝑡 From here, we remind that the H-derivative of 𝑓 at 𝑥∈R 𝑡 𝑡 𝑡 𝐷(𝑓𝑛 (𝑥) ,𝑢 (𝑥))<𝜖 whenever 𝑛>𝑁(𝜖) ,∀𝑥∈𝐴. (18) depends on 𝑡 and 𝜆. Therefore, 𝑓 is H-differentiable if and − + only if 𝑓𝜆 and 𝑓𝜆 are classical differentiable functions. 𝑡 Obviously the sequence (𝑓𝑛) of fuzzy-valued functions con- 𝑡 Definition 10 (see [21, Definition 8.7]). A fuzzy valued func- verges (uniformly) to a fuzzy valued-function 𝑢 if and only 𝑡 𝑡 − 𝑡 + 𝑡 − 𝑓 if {(𝑓𝑛) (𝜆)} and {(𝑓𝑛) (𝜆)} converge uniformly to {(𝑢 ) (𝜆)} tion is said to be fuzzy Henstock, in short FH-integrable if, 𝑡 + 𝜖>0 𝛿>0 and {(𝑢 ) (𝜆)} in 𝜆∈[0,1], respectively. On the other hand for any , there exists such that 𝑡 𝑡 (𝑓𝑛) converges (uniformly) to 𝑢 if and only if the sequence 𝑡𝑛 converges to a real (complex) number 𝑡. 𝐷( ∑ (V −𝑢) 𝑓𝑡 (𝜉) ,𝐼) ⊕ 𝑡 𝑃 Example 14. Consider a sequence 𝑓𝑛 =(𝑡𝑛,𝑢1,𝑢2,𝑢3) of 1 󵄨 󵄨 fuzzy-valued functions from [𝑡𝑛,𝑢3] into 𝐸 where (𝑡𝑛)∈𝜔, 󵄨 󵄨 = {󵄨∑ (V −𝑢) 𝑓− (𝑡) −𝐼−󵄨 , whose membership function 𝜇(𝑥) is defined as sup max 󵄨 𝜆 𝜆 󵄨 (15) 𝜆∈[0,1] 󵄨 𝑃 󵄨 𝑥−𝑡 󵄨 󵄨 { 𝑛 ,𝑡≤𝑥≤𝑢, 󵄨 󵄨 {𝑢 −𝑡 𝑛 1 󵄨∑ (V −𝑢) 𝑓+ (𝑡) −𝐼+󵄨}<𝜖 { 1 𝑛 󵄨 𝜆 𝜆 󵄨 {1, 𝑢 ≤𝑥≤𝑢, 󵄨 𝑃 󵄨 𝜇 (𝑥) = 1 2 { 𝑢3 −𝑥 (19) { ,𝑢≤𝑥≤𝑢, {𝑢 −𝑢 2 3 for any division 𝑃={[𝑢,V]; 𝜉} of [𝑎, 𝑏] with the norms Δ(𝑃) < { 3 2 𝑏 0, 𝑥<𝑡,𝑥>𝑢, 𝛿 𝐼:=( )∫ 𝑓𝑡(𝑥)𝑑𝑥 𝑓𝑡 { 𝑛 3 ,where FH 𝑎 and is also called FH- ∑ integrable. One can conclude that 𝑃 in (15) denotes the usual where max{𝑡𝑛}≤𝑢1 for all 𝑛∈N. Then, the membership 𝑃 [𝑎, 𝑏] Riemann sum for any division of . function can be written as Theorem 11 𝑓𝑡 ∈𝐶 [𝑎, 𝑏] − + (see [21,Theorem8.8]). Let 𝐹 and let [𝑓𝑡] =[(𝑓𝑡) (𝜆) ,(𝑓𝑡) (𝜆)] 𝑡 𝑛 𝜆 𝑛 𝑛 𝑓 be a FH-integrable function. If there exists 𝑥0 ∈ [𝑎, 𝑏] such − + (20) that 𝑓𝜆 (𝑥0)=𝑓𝜆 (𝑥0)=1,then =[(𝑢1 −𝑡𝑛)𝜆+𝑡𝑛,𝑢3 −(𝑢3 −𝑢2)𝜆]

𝑏 𝑥0 𝑏 − + [( ) ∫ 𝑓𝑡(𝑥)𝑑𝑥] =[∫ 𝑓− (𝑡) 𝑑𝑡, ∫ 𝑓+ (𝑡) 𝑑𝑡] . consisting of each function 𝑓 ,𝑓 depending on 𝑡 and 𝜆∈ FH 𝜆 𝜆 (16) [0, 1] (𝑡 ) {𝑓𝑡} 𝑎 𝜆 𝑎 𝑥0 . Suppose that the sequence 𝑛 converges; then 𝑛 Journal of Function Spaces 5

𝑡 𝑡 converges uniformly to the fuzzy-valued function 𝑢 which Example 18. Consider a sequence 𝑓𝑛 =(𝑢1,𝑡𝑛,𝑡𝑛 +𝑘,𝑡𝑛 +𝐿) is given by of fuzzy-valued functions for the constants 𝑘, 𝐿 with 𝐿≥𝑘, 𝜇(𝑥) 𝑥−𝑡 whose membership function is defined: { , 𝑡≤𝑥≤𝑢, { 1 {𝑢1 −𝑡 𝑥−𝑢 { { 1 ,𝑢≤𝑥≤𝑡, 1, 𝑢1 ≤𝑥≤𝑢2, { 1 𝑛 𝜇 (𝑥) = { 𝑢 −𝑥 (21) {𝑡𝑛 −𝑢1 { 3 ,𝑢≤𝑥≤𝑢, { { 2 3 1, 𝑡𝑛 ≤𝑥≤𝑡𝑛 +𝑘, {𝑢3 −𝑢2 𝜇 𝑡 (𝑥) = { 𝑓𝑛 {𝑡 +𝐿−𝑥 (25) { 𝑛 {0, 𝑥<𝑡,𝑥>𝑢3. { ,𝑡𝑛 +𝑘≤𝑥≤𝑡𝑛 +𝐿, { 𝐿−𝑘 0, 𝑥<𝑢1,𝑥>𝑡𝑛 +𝐿, In this form one can easily conclude that 𝑡𝑛 →𝑡if and only { 𝑡 − 𝑡 − if (𝑓𝑛) (𝜆) → (𝑢 ) (𝜆) uniformly in 𝜆∈[0,1]. 𝑡 where 𝑢1 ≤ min{𝑡𝑛} for all 𝑛∈N.Then,itisclearthat[𝑓 ]𝜆 = (𝑡 ) 𝑛 Remark 15. If the sequence 𝑛 is, respectively, replaced by [(𝑡𝑛 −𝑢1)𝜆+𝑢1,𝑡𝑛 +𝐿−𝜆(𝐿−𝑘)]for all 𝜆∈[0,1].Then𝑡𝑛 →𝑡 𝑡 + 𝑡 + 𝑡 − 𝑡 − each of the fix numbers 𝑢1,𝑢2,and𝑢3 in Example 14,thatis, (𝑓 ) (𝜆) → (𝑢 ) (𝜆) (𝑓 ) (𝜆) → (𝑢 ) (𝜆) 𝑡 if and only if 𝑛 and 𝑛 𝑓𝑛 =(𝑢1,𝑡𝑛,𝑢2,𝑢3) where 𝑡𝑛 ∈[𝑢1,𝑢2] for all 𝑛∈N,then 𝜆∈[0,1] 𝑡 𝑡 𝑡 − uniformly in where 𝑓𝑛 →𝑢uniformly if and only if 𝑡𝑛 →𝑡or (𝑓𝑛) (𝜆) → (𝑢𝑡)−(𝜆) 𝜆∈[0,1] 𝑢𝑡 =(𝑢,𝑡,𝑢 ,𝑢 ) uniformly in where 1 2 3 . [𝑢𝑡] =[(𝑡−𝑢)𝜆+𝑢 , (𝑡+𝐿) −𝜆(𝐿−𝑘)], (𝑢 ≤𝑡). 𝑡 𝜆 1 1 1 (26) Similarly we take 𝑓𝑛 =(𝑢1,𝑢2,𝑡𝑛,𝑢3) where 𝑡𝑛 ∈[𝑢2,𝑢3] for 𝑡 𝑡 all 𝑛∈N and then 𝑓𝑛 →𝑢uniformly if and only if 𝑡𝑛 →𝑡 𝑡 + 𝑡 + 𝑡∈R (𝑓 ) (𝜆) → (𝑢 ) (𝜆) Remark 19. If we take 𝑢1 as a function based on 𝑡 in for all ,or 𝑛 uniformly for each 𝑡 𝜆∈[0,1] 𝑢 =(𝑢,𝑢 ,𝑡,𝑢 ) Example 18,thatis,𝑓𝑛 =(𝑡𝑛,𝑡𝑛 +𝑘1,𝑡𝑛 +𝑘2,𝑡𝑛 +𝑘3) where the where 𝑡 1 2 3 . 𝑡 𝑡 constants 𝑘1 ≤𝑘2 ≤𝑘3,then𝑓𝑛 →𝑢uniformly if and only 𝑡 + 𝑡 + Now, in the following example, we give some cases of if 𝑡𝑛 →𝑡or (𝑓𝑛) (𝜆)→(𝑢) (𝜆) uniformly in 𝜆∈[0,1] membership functions with two constants. where 𝑢𝑡 =(𝑡,𝑡+𝑘1,𝑡+𝑘2,𝑡+𝑘3). 𝑡 Example 16. Consider a sequence 𝑓𝑛 =(𝑢1,𝑡𝑛,𝑡𝑛 +𝑘,𝑢4) of Theorem 20 (see [17]). Then, the following statements are 1 fuzzy-valued functions from [𝑢1,𝑢4] into 𝐸 for the constant valid. 𝑘, whose membership function 𝜇(𝑥) is defined as (i) A sequence of fuzzy-valued functions {(𝑓𝑡)𝑛(𝑥)} defined 𝑥−𝑢1 { ,𝑢1 ≤𝑥≤𝑡𝑛, on a set 𝐴⊆R converges uniformly to a fuzzy-valued {𝑡 −𝑢 𝑡 { 𝑛 1 function 𝑓 on 𝐴 if and only if {1, 𝑡 ≤𝑥≤𝑡 +𝑘, 𝜇 (𝑥) = 𝑛 𝑛 { 𝑢4 −𝑥 (22) { ,𝑡𝑛 +𝑘≤𝑥≤𝑢4, 𝛿 = 𝐷[(𝑓) (𝑥) ,𝑢 (𝑥)] {𝑢 −𝑡 −𝑘 𝑛 sup 𝑡 𝑛 𝑡 { 4 𝑛 𝑥∈𝐴 {0, 𝑥<𝑢1,𝑥>𝑢4, = { 𝑑([(𝑓 ) (𝑥)] ,[𝑢 (𝑥)] )} (27) where 𝑢1 ≤ min{𝑡𝑛} and 𝑢4 ≥ max{𝑡𝑛 +𝑘}for all 𝑛∈N and sup sup 𝑡 𝑛 𝜆 𝑡 𝜆 𝑥∈𝐴 𝜆∈[0,1] the constant 𝑘.Then,itisobviousthat 𝑡 𝛿 = 0. [𝑓 ] = [(𝑡 −𝑢 ) 𝜆+𝑢 ,𝑢 − (𝑢 −𝑡 −𝑘) 𝜆] with 𝑛→∞lim 𝑛 𝑛 𝜆 𝑛 1 1 4 4 𝑛 (23) for all 𝜆∈[0,1].Hence,aswehaveseenabove𝑡𝑛 →𝑡if 𝑡 + 𝑡 + 𝑡 − 𝑡 − and only if (𝑓𝑛) (𝜆)→(𝑢) (𝜆) and (𝑓𝑛) (𝜆)→(𝑢) (𝜆) (ii) The limit of a uniformly convergent sequence of con- uniformly in 𝜆∈[0,1]where tinuous fuzzy-valued functions {(𝑓𝑡)𝑛} on a set 𝐴 is 𝑎∈𝐴 − + continuous.Thatis,foreach , [𝑢𝑡] =[(𝑢𝑡) (𝜆) ,(𝑢𝑡) (𝜆)] 𝜆 (24) 𝑡 lim [ lim 𝑓𝑛 (𝑥)] = lim [ lim (𝑓𝑡) (𝑥)] . (28) =[(𝑡−𝑢1)𝜆+𝑢1,𝑢4 −𝜆(𝑢4 −𝑡−𝑘)], 𝑥→𝑎 𝑛→∞ 𝑛→∞ 𝑥→𝑎 𝑛

𝑢1 ≤𝑡and 𝑢4 ≥𝑡+𝑘for all 𝑡∈R and the constant 𝑘. 𝑓𝑡 →𝑢𝑡 Therefore, we derive that 𝑛 converges (uniformly). Theorem 21. A sequence of fuzzy-valued functions {(𝑓𝑡)𝑘} 𝐴 𝑡 𝑡 +𝑘 defined on a set convergesuniformlyifandonlyifitis Remark 17. If 𝑛 and 𝑛 are,respectively,replacedbyeachof uniformly Cauchy; that is, for an arbitrary 𝜀>0there is a the fixed numbers 𝑢𝑖 with 𝑖∈{1,2,3,4}given in Example 16, 𝑡 number 𝑁=𝑁(𝜀)such that that is, 𝑓𝑛 =(𝑢1,𝑢2,𝑡𝑛,𝑡𝑛 +𝑘)where 𝑢2 ≤ min{𝑡𝑛} for all 𝑡 𝑡 𝑛∈N and a constant 𝑘,then𝑓𝑛 →𝑢uniformly if and only 𝑡 𝑡 𝑡 + 𝑡 + 𝐷(𝑓𝑚 (𝑥) ,𝑓𝑛 (𝑥))<𝜀 whenever 𝑚>𝑛>𝑁(𝜀) ,∀𝑥∈𝐴, (29) if 𝑡𝑛 →𝑡or (𝑓𝑛) (𝜆)→(𝑢) (𝜆) uniformly in 𝜆∈[0,1] where 𝑢𝑡 =(𝑢1,𝑢2,𝑡,𝑡+𝑘). Other cases with two constants 𝑡 𝑡 canbeobtainedsimilarly. or equivalently, sup𝑥∈𝐴𝐷(𝑓𝑚(𝑥),𝑛 𝑓 (𝑥)) < 𝜀. 6 Journal of Function Spaces

𝑡 Theorem 22 (cf. [17]). Suppose that 𝑓𝑛(𝑥) ∈𝐹 𝐶 [𝑎, 𝑏] for all and therefore, 𝑡 𝑡 𝑛∈N such that {𝑓𝑛(𝑥)} converges uniformly to 𝑓 (𝑥).By 𝜀 𝜀 combining this and inclusion (28),theequalities 𝐿(𝑃,𝑓𝑡 )− <𝐿(𝑃,𝑓𝑡)≤𝑈(𝑃,𝑓𝑡)<𝑈(𝑃,𝑓𝑡 )+ . 𝑁 3 𝑁 3 (37) 𝑏 [( ) ∫ 𝑓𝑡 (𝑥) 𝑑𝑥] 𝑛→∞lim FH 𝑛 𝑎 𝜆 Consequently,

𝑏 𝑡 =[( ) ∫ 𝑓 (𝑥) 𝑑𝑥] 𝑡 𝑡 𝑡 𝑡 FH 𝑛→∞lim 𝑛 (30) 𝑈(𝑃,𝑓)−𝐿(𝑃,𝑓)<𝑈(𝑃,𝑓 )−𝐿(𝑃,𝑓 ) 𝑎 𝜆 𝑁 𝑁 𝑏 𝑝 𝑏 2𝜀 𝜀 2𝜀 (38) =[( ) ∫ 𝑓𝑡 (𝑥) 𝑑𝑥] =[∫ 𝑓− (𝑡) 𝑑𝑡, ∫ 𝑓+ (𝑡) 𝑑𝑡] + < + =𝜀, FH 𝜆 𝜆 3 3 3 𝑎 𝜆 𝑎 𝑝

− + 𝑏 𝑡 𝑓𝑡 [𝑎, 𝑏] 𝑛≥𝑁 hold for 𝑓 =𝑓 =𝑝where the integral (FH)∫ 𝑓 (𝑥)𝑑𝑥 showing that is FH-integrable on .Finally,for 1 1 𝑎 𝑟∈[𝑎,𝑏] exists for all 𝑥, 𝑡 ∈ [𝑎, 𝑏] and 𝜆∈[0,1]. and for each ,(34)impliesthat

𝑡 𝑟 𝑟 Theorem 23. Suppose that {𝑓𝑛} is a sequence of FH-integrable 𝑡 𝑡 𝑡 𝑡 𝐷∞ ((FH) ∫ 𝑓𝑛 (𝑥) 𝑑𝑥, (FH) ∫ 𝑓 (𝑥) 𝑑𝑥) functions defined on a closed interval [𝑎, 𝑏].If𝑓𝑛 →𝑓 𝑎 𝑎 𝑡 uniformly on [𝑎, 𝑏],then𝑓 is FH-integrable on [𝑎, 𝑏],and 𝑟 𝑟 𝑡 𝑡 = sup 𝑑([ (FH) ∫ 𝑓𝑛 (𝑥) 𝑑𝑥] ,[(FH) ∫ 𝑓 (𝑥) 𝑑𝑥] ) 𝑏 𝑏 𝑥∈[𝑎,𝑏] 𝑎 𝜆 𝑎 𝜆 𝑡 𝑡 lim [(FH) ∫ 𝑓𝑛(𝑥)𝑑𝑥] =[(FH) ∫ 𝑓 (𝑥)𝑑𝑥] . (31) 𝑟 𝑛→∞ 󵄨 𝑡 − 𝑡 −󵄨 𝑎 𝜆 𝑎 𝜆 ≤ ∫ { {󵄨(𝑓 ) −(𝑓) 󵄨 , (39) sup sup max 󵄨 𝑛 𝜆 𝜆󵄨 𝑎 𝑡∈[𝑎,𝑏]𝜆∈[0,1] 󵄨 󵄨 𝑟∈[𝑎,𝑏] Also, for each ,then 󵄨 + 󵄨 󵄨(𝑓𝑡) −𝑓(𝑡)+󵄨}} 𝑑𝑡 󵄨 𝑛 𝜆 𝜆󵄨 𝑟 𝑟 [( ) ∫ 𝑓𝑡(𝑥)𝑑𝑥] 󳨀→ [ ( ) ∫ 𝑓𝑡(𝑥)𝑑𝑥] 𝜀 (𝑟−𝑎) 𝜀 (𝑏−𝑎) 𝜀 FH 𝑛 FH (32) < ≤ = 𝑎 𝜆 𝑎 𝜆 3 (𝑏−𝑎) 3 (𝑏−𝑎) 3 uniformly in 𝜆∈[0,1]. and the proof is complete. Proof. By taking into account Theorem 22, we need only to 𝑡 Theorem 24. {𝑓𝑡} show that the limit function 𝑓 is FH-integrable on [𝑎, 𝑏].We Consider 𝑛 is a sequence of fuzzy-valued 𝑡 𝑡 see that {𝑓𝑛} is bounded, because each {𝑓𝑛} is FH-integrable functions such that 𝑡 on [𝑎, 𝑏].Also,thelimitfunction𝑓 is bounded, since, using 𝑡 1 triangle inequality and Theorem 20,wehave (i) {𝑓𝑛}∈𝐶𝐹[𝑎, 𝑏];

𝑡 𝑡 𝑡 𝑡 𝐷(𝑓 (𝑥) , 0) ≤ 𝐷 𝑛(𝑓 (𝑥) ,𝑓 (𝑥)) (ii) there exists a point 𝑥0 ∈ [𝑎, 𝑏] such that {𝑓𝑛(𝑥0)} (33) converges; 𝑡 𝑡 +𝐷(𝑓𝑛 (𝑥) , 0) ≤ 𝛿𝑛 +𝐷(𝑓𝑛 (𝑥) , 0) , 𝑡 󸀠 𝑡 (iii) (𝑓𝑛) →𝑔uniformly on [𝑎, 𝑏]. 𝑡 𝑡 where 𝛿𝑛 ∈𝑐0(𝐹). Besides this, 𝑓𝑛 →𝑓uniformly on [𝑎, 𝑏], 𝜀>0 𝑁(𝜀) 𝑡 𝑡 given any , there exists an integer such that Then {𝑓𝑛} converges uniformly to some 𝑓 on [𝑎, 𝑏] such that 𝑡 󸀠 𝑡 𝜀 (𝑓 ) (𝑥) = 𝑔 (𝑥) on [𝑎, 𝑏]. 𝐷(𝑓𝑡 (𝑥) ,𝑓𝑡 (𝑥))< 𝑛 3 (𝑏−𝑎) (34) 𝑡 󸀠 𝑡 Proof. By (iii), {(𝑓𝑛) } is uniformly convergent to 𝑔 on any ∀𝑥, 𝑡 ∈ [𝑎,] 𝑏 ,𝜆∈[0, 1] , closed interval contained in [𝑎, 𝑏], say in an interval with endpoints 𝑥0 and 𝑥∈[𝑎,𝑏].ByusingTheorem 22,wehave 𝑡 for 𝑛≥𝑁(𝜀).Since{𝑓𝑁} is FH-integrable, there exists a partition 𝑃 of [𝑎, 𝑏] such that 𝑥 [(FH) ∫ 𝑔 (𝑦) 𝑑𝑦] 𝜀 𝑥0 𝜆 𝑈 (𝑃, 𝑓𝑡 ) −𝐿(𝑃, 𝑓𝑡 ) < . 𝑁 𝑁 3 (35) 𝑥 𝑡 󸀠 = lim [(FH) ∫ (𝑓𝑛) (𝑦) 𝑑𝑦] 𝑛→∞ 𝑥 For each 𝑥∈[𝑎,𝑏], using the inclusion (34)with𝑛=𝑁 0 𝜆 (40) implies that = [𝑓𝑡 (𝑥) −𝑓𝑡 (𝑥 )] 𝑛→∞lim 𝑛 𝑛 0 𝜆 ± 𝜀 ± ± 𝜀 𝑓 𝑡 − <𝑓𝑡 <𝑓 𝑡 + 𝑡 − 𝑡 − 𝑡 + 𝑡 + 𝑁( )𝜆 ( )𝜆 𝑁( )𝜆 (36) =[ ((𝑓 ) −(𝑓0 ) ), ((𝑓 ) −(𝑓0 ) )] , 3 (𝑏−𝑎) 3 (𝑏−𝑎) 𝑛→∞lim 𝑛 𝜆 𝑛 𝜆 𝑛→∞lim 𝑛 𝜆 𝑛 𝜆 Journal of Function Spaces 7

∞ 𝑡 for all 𝑡 and 𝑡0 belonging to [𝑎, 𝑏] by the fundamental theorem that ⊕∑𝑘=1 𝑓𝑘 converges (pointwise) on 𝐴 and write the level of , and the convergence is uniform on [𝑎, 𝑏].Since sum function as 𝑡 lim𝑛𝑓𝑛(𝑥0) exists by (ii), we can add this term to both sides 𝑛 𝑓𝑡 (𝑥) : ∑𝑓𝑡 (𝑥) . and obtain 𝑛→∞lim ⊕ 𝑘 (45) 𝑘=1 𝑥 𝑓𝑡 (𝑥) = [( ) ∫ 𝑔(𝑦)𝑑𝑦] + [ 𝑓𝑡(𝑥 )] ∑∞ 𝑓𝑡(𝑥) 𝑛→∞lim 𝑛 FH 𝑛→∞lim 𝑛 0 (41) Definition 26. The series ⊕ 𝑘=1 𝑘 is said to be uniformly 𝑥 𝜆 𝑡 0 𝜆 convergent to a fuzzy-valued function 𝑓 (𝑥) on 𝐴 if the {𝑆𝑡 (𝑥)} 𝑓𝑡(𝑥) 𝐴 and the convergence is uniform on [𝑎, 𝑏].Wemaynowtake partial level sum 𝑛 converges uniformly to on . 𝑡 𝑡 𝑓𝑡(𝑥) 𝑓 (𝑥) = lim𝑛→∞𝑓𝑛(𝑥),andthelastequationholds That is, the series converges uniformly to if given any 𝜀>0, there exists an integer 𝑛0(𝜀) such that 𝑥 𝑡 𝑡 𝑛 𝑓 (𝑥) = [(FH) ∫ 𝑔(𝑦)𝑑𝑦] + [ lim 𝑓𝑛(𝑥0)] . (42) 𝑡 𝑡 𝑥 𝑛→∞ 𝜆 𝐷( ∑𝑓 (𝑥) ,𝑓 (𝑥)) 0 𝜆 ⊕ 𝑘 𝑘=1 𝑔𝑡 [𝑎, 𝑏] 󵄨 𝑛 󵄨 Now, , being the fuzzy limit on ,iscontinuouson 󵄨 󵄨 𝑡 𝑥 󵄨 − −󵄨 [𝑎, 𝑏] 𝐺 (𝑥) = [( )∫ 𝑔(𝑦)𝑑𝑦] = max { sup sup 󵄨∑𝑓𝑘(𝑡)𝜆 −𝑓(𝑡)𝜆󵄨 , (46) ,andso FH 𝑥 𝜆 is H-differentiable 󵄨 󵄨 0 𝜆∈[0,1] 𝑡∈𝐴 󵄨𝑘=1 󵄨 𝑡 󸀠 𝑡 (𝐺 ) (𝑥) = 𝑔 (𝑥) [𝑎, 𝑏] 󵄨 𝑛 󵄨 and 𝑛 on . Therefore, the last inequality 󵄨 󵄨 󵄨∑𝑓 (𝑡)+ −𝑓(𝑡)+󵄨}<𝜀 and Definition 9 imply that sup sup 󵄨 𝑘 𝜆 𝜆󵄨 𝜆∈[0,1] 𝑡∈𝐴 󵄨𝑘=1 󵄨 󸀠 󸀠 󸀠 (𝑓𝑡) (𝑥) =𝑔𝑡 (𝑥) , (𝑓𝑡) (𝑥) = (𝑓𝑡) (𝑥) 𝑛→∞lim 𝑛 (43) whenever 𝑛≥𝑛0(𝜀). 𝑡 for all 𝑥, 𝑡 ∈ [𝑎, 𝑏] and 𝜆∈[0,1].Thiscompletestheproof. Corollary 27. If each {𝑓𝑘} is a continuous fuzzy-valued func- 𝑡 tion on 𝐴⊆R for each 𝑘≥1and if ⊕∑𝑘≥1 𝑓𝑘(𝑥) is uniformly 𝑡 𝑡 convergent to 𝑓 (𝑥) on 𝐴,then𝑓 must be continuous on 𝐴 for Theorem 24 continues to hold under a weaker hypothesis. all 𝑥, 𝑡. ∈𝐴 However, we cannot replace the third condition, namely, (𝑓𝑡)󸀠 ∞ 𝑡 theuniformconvergenceofthesequence 𝑛 ,withpoint- Corollary 28. Aseries⊕∑𝑘=1 𝑓𝑘(𝑥) converges uniformly on wise convergence. Now we state an improved version of aset𝐴 if and only if the sequence of partial level sums is Theorem 24 without its proof. uniformly Cauchy on 𝐴;thatis,foranarbitrary𝜀>0there 𝑁=𝑁(𝜀) 𝑡 is a number such that Theorem 25. Suppose that {𝑓 } is a sequence of fuzzy-valued 𝑛 𝐷(𝑆𝑡 (𝑥) ,𝑆𝑡 (𝑥)) functions such that 𝑚 𝑛 𝑡 (i) each {𝑓 } is H-differentiable on [𝑎, 𝑏]; 𝑚 𝑛 =𝐷( ∑ 𝑓𝑡 (𝑥) , 0) 𝑡 𝑘 (ii) there exists a point 𝑥0 ∈ [𝑎, 𝑏] such that {𝑓𝑛(𝑥0)} 𝑘=𝑛+1 converges; 󵄨 𝑚 󵄨 󸀠 󵄨 󵄨 (47) (𝑓𝑡) →𝑔𝑡 [𝑎, 𝑏] = { 󵄨 ∑ 𝑓 (𝑡)−󵄨 , (iii) 𝑛 uniformly on . max sup sup 󵄨 𝑘 𝜆󵄨 𝑡∈𝐴 𝜆∈[0,1] 󵄨𝑘=𝑛+1 󵄨 𝑡 𝑡 Then {𝑓 } converges uniformly to some 𝑓 on [𝑎, 𝑏] such that 𝑛 󵄨 𝑚 󵄨 𝑡 󸀠 𝑡 󵄨 󵄨 (𝑓 ) (𝑥) = 𝑔 (𝑥) on [𝑎, 𝑏]. 󵄨 ∑ 𝑓 (𝑡)+󵄨}<𝜀, sup sup 󵄨 𝑘 𝜆󵄨 𝑡∈𝐴 𝜆∈[0,1] 󵄨𝑘=𝑛+1 󵄨

5. Uniform Convergence of Fuzzy-Valued whenever 𝑚>𝑛≥𝑛0(𝜀).

Function Series 𝑡 Corollary 29. Suppose that {𝑓𝑘(𝑥)} is a sequence in 𝐶𝐹[𝑎, 𝑏] ∞ 𝑡 𝑡 Definition 13 suggests that we continue our discussion from and that ⊕∑𝑘=0 𝑓𝑘(𝑥) converges uniformly to 𝑓 (𝑥) on [𝑎, 𝑏]. sequences of fuzzy-valued functions to series of fuzzy-valued Then, functions with the level sets. Consider a sequence of functions ∞ 𝑏 𝑡 ∞ 𝑡 𝑡 {𝑓𝑛(𝑥)} defined on a set 𝐴.Recallthatthelevelsum⊕∑𝑘=1 𝑓𝑘 [( ) ∑ ∫ 𝑓 (𝑥)𝑑𝑥] FH ⊕ 𝑘 is called a series of fuzzy-valued functions. Form a new 𝑘=0 𝑎 𝑡 𝜆 sequence of partial level sums of functions {𝑆𝑛(𝑥)} defined by 𝑏 ∞ 𝑡 𝑛 =[(FH) ∫ ∑𝑓𝑘 (𝑥) 𝑑𝑥] (48) 𝑡 𝑡 𝑎 ⊕ 𝑆 (𝑥) = ∑𝑓 (𝑥) , (𝑥∈𝐴,𝑡∈R) . 𝑘=0 𝜆 𝑛 ⊕ 𝑘 (44) 𝑘=1 𝑏 =[( ) ∫ 𝑓𝑡 (𝑥) 𝑑𝑥] , {𝑆𝑡 (𝑥)} 𝑥∈𝐴 FH If the sequence 𝑛 converges at a point ,thenwe 𝑎 𝜆 𝑥 say that the series of fuzzy-valued functions converges at .If 𝑏 {𝑆𝑡 (𝑥)} 𝐴 ( )∫ 𝑓𝑡(𝑥)𝑑𝑥 𝑥, 𝑡 ∈ [𝑎, 𝑏] 𝜆∈[0,1] the sequence 𝑛 converges at all points of ,thenwesay where FH 𝑎 exists for all and . 8 Journal of Function Spaces

Corollary 28 showsthatananalogueofthecomparison which is also given provided that the limit on the right hand test, namely, a sufficient condition for the uniform conver- side exists, where 0≤𝑅≤∞. gence of a fuzzy-valued function series. Indeed, the following ∞ 𝑛 result is a simple and direct test for the uniform convergence Definition 32. Suppose a power series ⊕∑𝑛=1 𝑎𝑛(𝑥−𝑥0) with of these series. radius of convergence 𝑅.Then,thesetofthepointsfroman interval at which the series is convergent is called the interval ∞ 𝑡 Theorem 30. Let ⊕∑𝑘=0 𝑓𝑘(𝑥) be a series of fuzzy-valued of convergence such that |𝑥 −0 𝑥 |<𝑅⇔𝑥∈(𝑥0 −𝑅,𝑥0 +𝑅) functions on a subset 𝐴 of R. Consider there exists a convergent which must be either ]𝑥0 −𝑅,𝑥0 +𝑅[, ]𝑥0 −𝑅,𝑥0 +𝑅], [𝑥0 − ∞ series ∑𝑘=1 𝑀𝑘 of nonnegative real numbers such that for all 𝑅,0 𝑥 +𝑅[,or[𝑥0 −𝑅,𝑥0 +𝑅]. 𝑘∈N 𝑥∈𝐴 and we have 𝑛 Theorem 33. A power series with fuzzy coefficients ⊕∑𝑛≥0 𝑎𝑛𝑥 󵄨 󵄨 󵄨 󵄨 𝑛 󵄨 𝑡 − 󵄨 󵄨 𝑡 + 󵄨 and the -fold derived series defined by max {sup sup 󵄨(𝑓𝑘) (𝜆)󵄨 , sup sup 󵄨(𝑓𝑘) (𝜆)󵄨}≤𝑀𝑘. (49) 𝑡∈𝐴 𝜆∈[0,1] 󵄨 󵄨 𝑡∈𝐴 𝜆∈[0,1] 󵄨 󵄨 ∑𝑛 (𝑛−1) ⋅⋅⋅(𝑛−𝑘+1) 𝑎 𝑥𝑛−𝑘 ⊕ 𝑛 (53) ∞ 𝑡 𝑛≥𝑘 Then ⊕∑𝑘=0 𝑓𝑘(𝑥) converges uniformly.

∞ 𝑡 Proof. Suppose that ⊕∑𝑘=0 𝑓𝑘(𝑥) is dominated by a conver- ∞ ∞ 𝑡 ± have the same radius of convergence. gent series ∑𝑘=1 𝑀𝑘.Thateachoftheseries∑𝑘=0 |(𝑓𝑘) (𝜆)| 𝑛 converges on 𝐴 comes from the comparison test for real Theorem 34 (cf. [23]). If ⊕∑𝑛≥0 𝑎𝑛𝑥 hasradiusofconvergence 𝑡 𝑡 𝑛 series. To verify the uniform convergence of ⊕∑𝑘 𝑓𝑘(𝑥),take 𝑅>0 𝑓 (𝑥) = ∑ 𝑎 𝑥 |𝑥| < 𝑅 ∞ ,then ⊕ 𝑛≥0 𝑛 is H-differentiable in 󸀠 into account Cauchy criterion for the series ∑𝑘=1 𝑀𝑘.Hence, 𝑡 𝑛−1 𝑡 (𝑛) and (𝑓 ) (𝑥) = ⊕∑𝑛≥1 𝑛𝑎𝑛𝑥 .Consequently,(𝑓 ) (𝑥) exists given 𝜖>0, there exists an integer 𝑛0 =𝑛0(𝜖) such that, for 𝑚 for every 𝑛≥1and every 𝑥 with |𝑥| <,and 𝑅 𝑚>𝑛>𝑛0,wehave∑𝑘=𝑛+1 𝑀𝑘 <𝜖.Forall𝑚>𝑛>𝑛0,we also obtain ∞ 𝑡 (𝑛) 𝑛−𝑘 𝑡 𝑡 (𝑓 ) (𝑥) = ∑𝑛 (𝑛−1) ⋅⋅⋅(𝑛−𝑘+1) 𝑎 𝑥 , 𝐷(𝑆 (𝑥) ,𝑆 (𝑥)) ⊕ 𝑛 𝑚 𝑛 𝑛=𝑘 (54) 󵄨 𝑚 󵄨 󵄨 𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 = { 󵄨 ∑ 𝑓 (𝑡)−󵄨 , 󵄨 ∑ 𝑓 (𝑡)+󵄨} (|𝑥| <𝑅) . max sup sup 󵄨 𝑘 𝜆󵄨 sup sup 󵄨 𝑘 𝜆󵄨 𝑡∈𝐴 𝜆∈[0,1] 󵄨𝑘=𝑛+1 󵄨 𝑡∈𝐴 𝜆∈[0,1] 󵄨𝑘=𝑛+1 󵄨 𝑚 󵄨 𝑡 − 󵄨 󵄨 𝑡 + 󵄨 (50) The fuzzy coefficients 𝑎𝑛 are uniquely determined, and ≤ ∑ { 󵄨(𝑓 ) (𝜆)󵄨 , 󵄨(𝑓 ) (𝜆)󵄨} max sup sup 󵄨 𝑘 󵄨 sup sup 󵄨 𝑘 󵄨 𝑘=𝑛+1 𝑡∈𝐴 𝜆∈[0,1] 𝑡∈𝐴 𝜆∈[0,1] − + 𝑚 𝑎𝑛 =[(𝑎𝑛)𝜆,(𝑎𝑛)𝜆] ≤ ∑ 𝑀𝑘 <𝜖. 𝑘=𝑛+1 − + 0 (𝑛) 0 (𝑛) (𝑓𝑡)(𝑛)(0) [((𝑓 ) ) ((𝑓 ) ) ] (55) =[ ] = [ 𝜆 , 𝜆 ] . Therefore, by the Cauchy criterion in Corollary 28,theseries 𝑛! [ 𝑛! 𝑛! ] ∑ 𝑓𝑡(𝑥) 𝐴 𝜆 ⊕ 𝑘 𝑘 converges uniformly on . [ ]

Wemaynowgivethefollowing,whichisthemainfor Proof. Let some applications of power series of fuzzy numbers (see [22]). We may state a condition under which term-by-term H- ∞ 𝑓𝑡 (𝑥) = ∑𝑎 𝑥𝑛 differentiation of an infinite series is allowable. ⊕ 𝑛 𝑛=0 (56) Definition 31. Let 𝑥 be any element and 𝑥0 the fixed element. ∞ ∞ 𝑎 − 𝑛 + 𝑛 1 Then, the power series with fuzzy coefficients 𝑛 is in the form =[∑(𝑎𝑛)𝜆𝑡 , ∑(𝑎𝑛)𝜆𝑡 ]∈𝐸 𝑛=0 𝑛=0 ∞ ∑𝑎 (𝑥 − 𝑥 )𝑛 =𝑎 ⊕𝑎 (𝑥−𝑥 )⊕𝑎(𝑥 − 𝑥 )2 ⊕ 𝑛 0 0 1 0 2 0 have radius of convergence 𝑅>0.Wehavetoshowthe 𝑛=0 (51) 󸀠 (𝑓𝑡) (𝑥) |𝑡| < 𝑅 𝑛 existence of H-differentiable in and that ⊕⋅⋅⋅⊕𝑎 (𝑥−𝑥 ) ⊕⋅⋅⋅ 𝑡 󸀠 𝑛 0 (𝑓 ) is of the stated form. By Theorem 33 with 𝑘=1,the 𝑛−1 derived series ⊕∑𝑛≥1 𝑛𝑎𝑛𝑥 converges for |𝑡| < 𝑅 and defines 𝑅 𝑡 and the radius of convergence is defined by a fuzzy-valued function, say 𝑔 (𝑥),in|𝑡| < 𝑅.Weshowthat 𝑡 󸀠 𝑡 (𝑓 ) (𝑥) = 𝑔 (𝑥) for all 𝑥, 𝑡 ∈ (−𝑅,. 𝑅) 𝑎+ (0) 𝑡∈(−𝑅,𝑅) 𝑟<𝑅 𝑅:= 𝑛 Let be fixed. Then take a positive such 𝑛→∞lim + (52) |𝑡| < 𝑟 𝑟 = (𝑅 + |𝑡|)/2 ℎ∈R 𝑎𝑛+1 (0) that ;thatis, .Also,let with Journal of Function Spaces 9

0 < |ℎ| < (𝑅−|𝑡|)/2.Wehave|𝑡+ℎ| ≤ |𝑡|+|ℎ| < |𝑡|+(𝑅−|𝑡|)/2 = Acknowledgment (𝑅 + |𝑡|)/2. =𝑟 We consider The authors record their pleasure to the anonymous referee 𝑓𝑡 (𝑥+ℎ) ⊖𝑓𝑡 (𝑥) 𝐷( ,𝑔𝑡 (𝑥)) for his/her constructive report and many helpful suggestions ℎ onthemainresults. 𝑓− (𝑡+ℎ) −𝑓− (𝑡) =[ 𝜆 𝜆 −𝑔− (𝑡) , ℎ 𝜆 References (57) [1] H. N. Jahnke, The Foundation of Analysis in the 19th Century: 𝑓+ (𝑡+ℎ) −𝑓+ (𝑡) 𝜆 𝜆 −𝑔+ (𝑡)] Weierstrass. A History of Analysis,AMSBookstore,2003. ℎ 𝜆 𝜆 [2] L. Imre, Proofs and Refutations, Cambridge University Press, 1976. (𝑡+ℎ)𝑛 −𝑡𝑛 = ∑𝑎 ( −𝑛𝑡𝑛−1), [3] M. Bas¸arir, “On some new sequence spaces of fuzzy numbers,” ⊕ 𝑛 𝑛≥2 ℎ Indian Journal of Pure and Applied Mathematics,vol.34,no.9, pp. 1351–1357, 2003. {|𝑡|, |𝑡+ℎ|} ≤ 𝑟<𝑅 where max . By taking into account Taylors [4] O. Talo and F. Bas¸ar, “On the space bv𝑝(F) of sequences of p- theorem on the interval with endpoints, we get bounded variation of fuzzy numbers,” Acta Mathematica Sinica, 󵄨 𝑛 𝑛 󵄨 English Series,vol.24,no.7,pp.965–972,2008. 󵄨(𝑡+ℎ) −𝑡 𝑛−1󵄨 𝑛 (𝑛−1) 𝑛−2 󵄨 −𝑛𝑡 󵄨 ≤ |ℎ| 𝑟 . [5] F. Bas¸ar and B. Altay, “On the space of sequences of p- 󵄨 ℎ 󵄨 2 (58) 󵄨 󵄨 bounded variation and related matrix mappings,” Ukrainian Therefore we have ℎ→0, Mathematical Journal,vol.24,no.1,pp.136–147,2003. [6] U. Kadak and M. Ozluk, “Some new sets of sequences of fuzzy 𝑡 󸀠 𝑡 𝐷((𝑓) (𝑥) ,𝑔 (𝑥)) numbers with respect to the partial metric,” The Scientific World Journal,vol.2014,ArticleID735703,11pages,2014. (𝑡+ℎ)𝑛 −𝑡𝑛 [7] L. A. Zadeh, “Fuzzy sets,” Information and Computation,vol.8, =𝐷( ∑𝑎 ( −𝑛𝑡𝑛−1),0) ⊕ 𝑛 ℎ pp. 338–353, 1965. 𝑛≥2 [8]R.GoetschelandW.Voxman,“Elementaryfuzzycalculus,” 󵄨 󵄨 (59) Fuzzy Sets and Systems,vol.18,no.1,pp.31–43,1986. 󵄨ℎ − 󵄨 ≤ {󵄨 ∑(𝑎 ) 𝑛 (𝑛−1) 𝑟𝑛−2󵄨 , [9] K. H. Lee, First Course on Fuzzy Theory and Applications, sup max 󵄨 2 𝑛 𝜆 󵄨 𝜆∈[0,1] 󵄨 𝑛≥2 󵄨 Springer, Berlin, Germany, 2005. 󵄨 󵄨 [10] P.Diamond and P.Kloeden, “Metric spaces of fuzzy sets,” Fuzzy 󵄨ℎ + 󵄨 󵄨 ∑(𝑎 ) 𝑛 (𝑛−1) 𝑟𝑛−2󵄨}󳨀→0. Sets and Systems,vol.35,no.2,pp.241–249,1990. 󵄨 2 𝑛 𝜆 󵄨 [11] S. Nanda, “On sequences of fuzzy numbers,” Fuzzy Sets and 󵄨 𝑛≥2 󵄨 Systems,vol.33,no.1,pp.123–126,1989. 𝑡 󸀠 Consequently, by (57), it follows that (𝑓 ) (𝑥) exists and [12] B. Bede and S. G. Gal, “Almost periodic fuzzy-number-valued 𝑡 equals 𝑔 (𝑥).Since𝑥 is arbitrary, this holds at any interior functions,” Fuzzy Sets and Systems,vol.147,no.3,pp.385–403, point in |𝑥| <. 𝑅 On the other hand, this argument gives that 2004. all the H-derivatives exist in |𝑥| <. 𝑅 [13] M. L. Puri and D. A. Ralescu, “Differentials of fuzzy functions,” Journal of Mathematical Analysis and Applications,vol.91,no.2, 𝑡 pp. 552–558, 1983. Corollary 35. Consider {𝑓𝑛} is a sequence of fuzzy-valued functions. Then the following hold: [14] O. Talo and F.Bas¸ar, “Determination of the duals of classical sets 𝑡 1 of sequences of fuzzy numbers and related matrix transforma- (i) {𝑓𝑛}∈𝐶𝐹[𝑎, 𝑏] (H-differentiable continuous fuzzy- tions,” Computers & Mathematics with Applications,vol.58,no. valued function); 4, pp. 717–733, 2009. 𝑡 [15] L. Stefanini, “A generalization of Hukuhara difference for (ii) there exists a point 𝑥0 ∈ [𝑎, 𝑏] such that ⊕∑𝑘 𝑓𝑘(𝑥0) converges; interval and fuzzy arithmetic,” in Soft Methods for Handling Variability and Imprecision,vol.48ofSeries on Advances in Soft 𝑡 󸀠 (iii) ⊕∑𝑘 (𝑓𝑘) converges uniformly on [𝑎, 𝑏]. Computing,2008. 𝑡 [16] L. Stefanini and B. Bede, “Generalized Hukuhara differen- Then ⊕∑𝑘 𝑓𝑘 converges uniformly on [𝑎, 𝑏] to a H-differentiable 𝐹 tiability of interval-valued functions and interval differential fuzzy-valued function , equations,” Nonlinear Analysis. Theory, Methods & Applications, 󸀠 vol.71,no.3-4,pp.1311–1328,2009. 󸀠 𝑡 𝑡 󸀠 𝐹 (𝑥) =( ∑𝑓 (𝑥)) = ∑(𝑓 (𝑥)) (60) [17] U. Kadak and F. Bas¸ar, “On some sets of fuzzy-valued sequences ⊕ 𝑘 ⊕ 𝑘 𝑘 𝑘 with the level sets,” Contemporary Analysis and Applied Mathe- 𝑥, 𝑡 ∈ [𝑎, 𝑏] 𝜆∈[0,1] matics,vol.1,no.2,pp.70–90,2013. for all and . [18] U. Kadak, “On the sets of fuzzy-valued function with the level sets,” Journal of Fuzzy Set Valued Analysis,vol.2013,ArticleID Conflict of Interests 00171, 13 pages, 2013. [19] M. Hukuhara, “Integration´ des applications mesurables dont la The authors declare that there is no conflict of interests valeur est un compact convex,” Funkcialaj Ekvacioj,vol.10,pp. regarding the publication of this paper. 205–229, 1967. 10 Journal of Function Spaces

[20] U. Kadak and F. Bas¸ar, “On of fuzzy-valued functions,” The Scientific World Journal,vol.2014,ArticleID 782652, 13 pages, 2014. [21] G. A. Anastassiou, Fuzzy Mathematics: Approximation Theory, Studies in Fuzziness and Soft Computing, 251, 2010. [22] U. Kadak and F. Bas¸ar, “Power series of fuzzy numbers,” AIP Conference Proceedings, vol. 1309, pp. 538–550, 2010. [23] S. Ponnusamy, Foundations of Mathematical Analysis,Springer, 2010. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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