On Uniform Convergence of Sequences and Series of Fuzzy-Valued Functions

The


Introduction
The term uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series is independent of two variables.While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition or use the property in any of his proofs [1].Later Karl Weierstrass, who attended his course on elliptic functions in 1839-1840, coined the term uniformly convergent which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894.Independently a similar concept was used by Imre [2] and G. Stokes but without having any major impact on further development.
Due to the rapid development of the fuzzy logic theory, however, some of these basic concepts have been modified and improved.One of them is in the form of interval valued fuzzy sets.To achieve this we need to promote the idea of the level sets of fuzzy numbers and the related formulation of a representation of an interval valued fuzzy set in terms of its level sets.Once having the structure we then can supply the required extension to interval valued fuzzy sets.
The effectiveness of level sets is based on not only their required storage capacity but also their two-valued nature.Also the definition of these sets offers some advantages over the related membership functions.
Many authors have developed the different cases of sequence sets with fuzzy metric on a large scale.Bas ¸arir [3] has recently promoted some new sets of sequences of fuzzy numbers generated by a nonnegative regular matrix , some of which reduced to Maddox's spaces ℓ ∞ (; ), (; ),  0 (; ), and ℓ(; ) for the special cases of that matrix .Quite recently, Talo and Bas ¸ar [4] have developed the main results of Bas ¸ar and Altay [5] to fuzzy numbers and defined the alpha-, beta-, and gamma-duals and introduced the duals of these sets together with the classes of infinite matrices of fuzzy numbers mapping one of the classical set into another one.Also, Kadak and Ozluk [6] have introduced some new sets of sequences of fuzzy numbers with respect to the partial metric.
The rest of this paper is organized as follows.In Section 2, we give some necessary definitions and propositions related to the fuzzy numbers, sequences, and series of fuzzy numbers.We also report the most relevant and recent literature in this section.In Section 3, first the definition of fuzzy-valued function is given which will be used in the proof of our main results.In this section, generalized Hukuhara differentiation and Henstock integration are presented according to fuzzyvalued functions depending on real values  and .The final section is completed with the concentration of the results on uniform convergence of fuzzy-valued sequences and series.Also we examine the relationship between the radius of convergence of power series and the notion of uniform convergence with respect to fuzzy-valued function.

Preliminaries, Background, and Notation
A fuzzy number is a fuzzy set on the real axis, that is, a mapping  : R → [0, 1] which satisfies the following four conditions.
We denote the set of all fuzzy numbers on R by  1 and called it the space of fuzzy numbers and the -level set []  of  ∈  1 is defined by The set []  is closed, bounded, and nonempty interval for each  ∈ [0, 1] which is defined by Theorem 1 (representation theorem [8]).(iii) The functions  − and  + are right continuous at the point  = 0.
Otherwise, if the pair of functions  − and  + holds the conditions (i)-(iv), then there exists a unique element  ∈  1 such that [𝑢] A fuzzy number is a convex fuzzy subset of R and is defined by its membership function.Let  be a fuzzy number, whose membership function () can generally be defined as [9]  ( where  − : [ Let  be the set of all closed bounded intervals  of R with endpoints  and ; that is,  := [, ].Define the relation  on  by (, ) := max{| − |, | − |}.Then it can easily be observed that  is a metric on  (cf.Diamond and Kloeden [10]) and (, ) is a complete metric space (cf.Nanda [11]).Now, we can give the metric  on  1 by means of the Hausdorff metric  as It is trivial that Proposition 2 (see [12]).Let , V, ,  ∈  1 and  ∈ R.Then, (i) ( 1 , ) is a complete metric space (cf.Puri and Ralescu [13]); Remark 3 (cf.[14]).Then the following remarks can be given.
If the sequence (  ) ∈ () is bounded then the sequences of functions { −  ()} and { +  ()} are uniformly bounded in [0, 1].Definition 4 (see [14]).Let (  ) ∈ ().Then the expression ⊕ ∑    is called a series of fuzzy numbers with the level summation ⊕ ∑.Define the sequence (  ) via th partial level sum of the series by If the sequence (  ) converges to a fuzzy number , then we say that the series ⊕ ∑    of fuzzy numbers converges to  and write ⊕ ∑    =  which implies that lim where the summation is in the sense of classical summation and converges uniformly in  ∈ [0, 1].

Generalized Hukuhara Difference.
A generalization of the Hukuhara difference proposed in [15] aims to overcome this situation.
Definition 5 (see [15,Definition 1]).The generalized Hukuhara difference  ⊖  of two sets ,  ∈ K is defined as follows: Proposition 6 (see [16]).The following statements hold.Then, we have where the limits are in the Hausdorff metric  for intervals.

Fuzzy-Valued Functions with the Level Sets
Definition 7 (see [17]).Consider a fuzzy-valued function   () from R into  1 with respect to a membership function () which is called trapezoidal fuzzy number and is interpreted as follows: Then, the pair of  − ,  + depending on  ∈ R can be written as Then, the function   is said to be a fuzzyvalued function on R.

Generalized Hukuhara Differentiation and Henstock Integration.
The notion of fuzzy differentiability comes from a generalization of the Hukuhara difference for compact convex sets.We prove several properties of the derivative of fuzzy-valued functions considered here.As a continuation of Hukuhara derivatives for real fuzzy-valued functions [19], we can define H-differentiation of   with respect to level sets.
Definition 9 (cf.[20]).A fuzzy-valued function   : R →  1 is said to be generalized H-differentiable with respect to the level sets at the point : if (  )  () ∈  1 exists such that, for all ℎ > 0 sufficiently near to 0, the H-difference   ( + ℎ) ⊖   () exists then the H-derivative (  )  () is given as follows: From here, we remind that the H-derivative of   at  ∈ R depends on  and .
Remark 12.We remark that the integrals on the right hand side of ( 16) exist in the usual sense for all  ∈ [0, 1].It is easy to see that the pair of functions  ±  : [, ] → R are continuous.

Uniform Convergence of Fuzzy-Valued Functions
Definition 13.Let {   ()} be a sequence of fuzzy-valued functions defined on a set  ⊆ R with respect to the sequence  = (  ) with real or complex terms.We say that {   ()} converges pointwise on  if for each  ∈  the sequence {   ()} converges for all  ∈ .If a sequence {   ()} converges pointwise, then we can define a fuzzyvalued function On the other hand, {   ()} converges to   on  if and only if, for each  ∈  and for an arbitrary  > 0, there exists an integer  = (, ) such that (   (),   ()) <  whenever  > .The integer  in the definition of pointwise convergence may, in general, depend on both  > 0 and  ∈ .If, however, one integer can be found that works for all points in , then the convergence is said to be uniform.That is, a sequence of fuzzy-valued functions {   ()} converges uniformly to   if, for each  > 0, there exists an integer () such that  (   () ,   ()) <  whenever  >  () , ∀ ∈ .(18) Obviously the sequence ( where max{  } ≤  1 for all  ∈ N.Then, the membership function can be written as consisting of each function  − ,  + depending on  and  ∈ [0, 1].Suppose that the sequence (  ) converges; then {   } converges uniformly to the fuzzy-valued function   which is given by In where  1 ≤ min{  } and  4 ≥ max{  + } for all  ∈ N and the constant .Then, it is obvious that for all  ∈ [0, 1].Hence, as we have seen above   →  if and only if where where Theorem 20 (see [17]).Then, the following statements are valid.
(i) A sequence of fuzzy-valued functions {(  )  ()} defined on a set  ⊆ R converges uniformly to a fuzzy-valued function   on  if and only if (ii) The limit of a uniformly convergent sequence of continuous fuzzy-valued functions {(  )  } on a set  is continuous.That is, for each  ∈ , Theorem 21.A sequence of fuzzy-valued functions {(  )  } defined on a set  converges uniformly if and only if it is uniformly Cauchy; that is, for an arbitrary  > 0 there is a number  = () such that  (   () ,    ()) <  whenever  >  >  () , ∀ ∈ , (29) or equivalently, sup ∈ (   (),    ()) < .

Uniform Convergence of Fuzzy-Valued Function Series
Definition 13 suggests that we continue our discussion from sequences of fuzzy-valued functions to series of fuzzy-valued functions with the level sets.
If the sequence {   ()} converges at a point  ∈ , then we say that the series of fuzzy-valued functions converges at .If the sequence {   ()} converges at all points of , then we say that ⊕ ∑ ∞ =1    converges (pointwise) on  and write the level sum function as   () : lim Definition 26.The series ⊕ ∑ ∞ =1    () is said to be uniformly convergent to a fuzzy-valued function   () on  if the partial level sum {   ()} converges uniformly to   () on .That is, the series converges uniformly to   () if given any  > 0, there exists an integer  0 () such that If each {   } is a continuous fuzzy-valued function on  ⊆ R for each  ≥ 1 and if ⊕ ∑ ≥1    () is uniformly convergent to   () on , then   must be continuous on  for all ,  ∈ .Corollary 28.A series ⊕ ∑ ∞ =1    () converges uniformly on a set  if and only if the sequence of partial level sums is uniformly Cauchy on ; that is, for an arbitrary  > 0 there is a number  = () such that Suppose that {   ()} is a sequence in   [, ] and that ⊕ ∑ ∞ =0    () converges uniformly to   () on [, ].