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On Uniform Convergence of Sequences and Series of Fuzzy-Valued Functions Hindawi Publishing Corporation Journal of Function Spaces Volume 2015, Article ID 870179, 10 pages http://dx.doi.org/10.1155/2015/870179 Research Article On Uniform Convergence of Sequences and Series 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 power series 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 formal definition or use the property in any of his proofs some of which reduced to Maddox’s spaces ℓ∞(; ), (; ), [1]. Later Karl Weierstrass, who attended his course on 0(; ),andℓ(; ) for the special cases of that matrix . 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 () can generally be defined as results. In this section, generalized Hukuhara differentiation [9] and Henstock integration are presented according to fuzzy- − valued functions depending on real values and .Thefinal { 1 ,≤≤, { − 1 2 section is completed with the concentration of the results on { 2 1 { uniform convergence of fuzzy-valued sequences and series. 1, 2 ≤≤3, () := { − (2) Also we examine the relationship between the radius of { 4 ,≤≤, { 3 4 convergence of power series and the notion of uniform {4 −3 convergence with respect to fuzzy-valued function. {0, <1, >4, − :[, ]→[0,1] + :[, ]→[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].If() is piecewise linear, mapping :R → [0, 1] which satisfies the following four then is referred to as a trapezoidal fuzzy number and is conditions. usually denoted by =(1,2,3,4). In particular, when 2 ≡3, the trapezoidal form is reduced to a triangular (i) is normal; that is, there exists an 0 ∈ R such that form; that is, =(1,2,4). So, triangular forms are special − + (0)=1. cases of trapezoidal forms. Since and are both strictly monotonical and continuous functions, their inverse exists (ii) is fuzzy convex; that is, [ + (1 − )] ≥ and should also be continuous and strictly monotonical. min{(), ()} for all , ∈ R and for all ∈[0,1]. 1 Let , V, ∈ and ∈R.Thensomealgebraic (iii) is upper semicontinuous. operations, that is, level set addition, scalar multiplication, 1 and product, are defined on by (iv) The set []0 = { ∈ R :()>0} is compact (cf. { ∈ R :()>0} Zadeh [7]), where denotes the ⊕V =⇐⇒[] = [] ⊕ [V] closure of the set { ∈ R :()>0}in the usual − − − topology of R. ⇐⇒ () = () + V () , (3) 1 + + + We denote the set of all fuzzy numbers on R by and called () = () + V () , 1 it the space of fuzzy numbers and the -level set [] of ∈ is defined by [] =[] and V =⇔[] =[][V] for all ∈[0,1],where {{∈R :() ≥} , 0<≤1, − − − − + () = min { () V () , () V () , [] := { (1) {{ ∈ R : () > }, =0. + () V− () ,+ () V+ ()}, + − − − + (4) The set [] is closed, bounded, and nonempty interval for () = { () V () , () V () , − + max each ∈[0,1]which is defined by [] := [ (), ()]. + () V− () ,+ () V+ ()}. Theorem 1 (representation theorem [8]). Let [] = − + 1 [ (), ()] for ∈ and for each ∈[0,1].Thenthe Let be the set of all closed bounded intervals of R with following statements hold. endpoints and ;thatis,:=[, ].Definetherelation on by (, ) := max{| −|, |−|}.Thenitcan − (i) is a bounded and nondecreasing left continuous easily be observed that is a metric on (cf. Diamond ]0, 1] function on . and Kloeden [10]) and (,) is a complete metric space (cf. + 1 (ii) is a bounded and nonincreasing left continuous Nanda [11]).Now,wecangivethemetric on by means function on ]0, 1]. of the Hausdorff metric as − + (iii) The functions and are right continuous at the (, V) := sup ([], [V]) point =0. ∈[0,1] (5) − + − − + + (iv) (1) ≤ (1). := sup max { () − V () , () − V ()}. ∈[0,1] − + Otherwise, if the pair of functions and holds the 1 It is trivial that conditions (i)–(iv), then there exists a unique element ∈ − + such that [] := [ (), ()] for each ∈[0,1]. − + (,0) = sup max { () , ()} ∈[0,1] A fuzzy number is a convex fuzzy subset of R and is (6) − + defined by its membership function. Let be a fuzzy number, = max { (0) , (0)}. Journal of Function Spaces 3 1 Proposition 2 (see [12]). Let , V,,∈ and ∈R.Then, where the limits are in the Hausdorff metric for 1 intervals. (i) ( ,)is a complete metric space (cf. Puri and Ralescu [13]); (ii) (, V) = ||(, V); 3. Fuzzy-Valued Functions with the Level Sets (iii) ( ⊕ V,⊕V)=(,); Definition 7 (see [17]). Consider a fuzzy-valued function () R 1 (iv) ( ⊕ V,⊕)≤(,)+(V,); from into with respect to a membership function () which is called trapezoidal fuzzy number and is inter- (v) |(, 0) − (V, 0)| ≤ (, V)≤(,0) + (V, 0). preted as follows: Remark 3 (cf. [14]). Then the following remarks can be given. − () { 1 ,() ≤≤ () , { () − () 1 2 (a) Obviously the sequence ()∈()converges to a { 2 1 − + { fuzzy number if and only if { ()} and { ()} 1, 2 () ≤≤3 () , − + () = { () − (11) converge uniformly to () and () on [0, 1], { 4 { ,3 () ≤≤4 () , respectively. {4 () −3 () 0, < () , > () . (b) The boundedness of the sequence ()∈()is { 1 4 equivalent to the fact that −,+ ∈R Then, the pair of depending on can be written as ( , 0) = {− () , + ()}<∞. 1 sup sup sup max () = [(2() −1 ())1 + (),4 () − 4( () −3 ())] ∈ ∈N ∈N∈[0,1] (7) for all ∈[0,1].Then,thefunction is said to be a fuzzy- valued function on R. If the sequence ()∈()is bounded then the sequences − + of functions { ()} and { ()} areuniformlyboundedin Remark 8. The functions with ∈ {1,2,3,4} given in [0, 1]. Definition 7 arealsodefinedby() = for all ∈R and the constant . Definition 4 (see [14]). Let ()∈(). Then the expression ∑ ⊕ is called a series of fuzzy numbers with the level Now, following Kadak [18]wegivetheclassicalsets summation ⊕∑. Define the sequence () via th partial level [, ] [, ] = ⊕ ⊕ ⊕⋅⋅⋅⊕ ∈N and consisting
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