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On Uniform Convergence of Sequences and Series of Fuzzy-Valued Functions
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.
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 . 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  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. Başarir  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 , , , and for the special cases of that matrix . Quite recently, Talo and Başar  have developed the main results of Başar and Altay  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  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 fuzzy-valued 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.
2. Preliminaries, Background, and Notation
A fuzzy number is a fuzzy set on the real axis, that is, a mapping which satisfies the following four conditions.(i)is normal; that is, there exists an such that .(ii) is fuzzy convex; that is, for all and for all .(iii) is upper semicontinuous.(iv)The set is compact (cf. Zadeh ), where denotes the closure of the set in the usual topology of .
We denote the set of all fuzzy numbers on by and called it the space of fuzzy numbers and the -level set of is defined by The set is closed, bounded, and nonempty interval for each which is defined by .
Theorem 1 (representation theorem ). Let for and for each . Then the following statements hold. (i) is a bounded and nondecreasing left continuous function on .(ii) is a bounded and nonincreasing left continuous function on .(iii)The functions and are right continuous at the point .(iv).Otherwise, if the pair of functions and holds the conditions (i)–(iv), then there exists a unique element such that for each .
A fuzzy number is a convex fuzzy subset of and is defined by its membership function. Let be a fuzzy number, whose membership function can generally be defined as  where and are two strictly monotonical and continuous mappings from to the closed interval . If is piecewise linear, then is referred to as a trapezoidal fuzzy number and is usually denoted by . In particular, when , the trapezoidal form is reduced to a triangular form; that is, . So, triangular forms are special cases of trapezoidal forms. Since and are both strictly monotonical and continuous functions, their inverse exists and should also be continuous and strictly monotonical.
Let and . Then some algebraic operations, that is, level set addition, scalar multiplication, and product, are defined on by
and for all , where Let be the set of all closed bounded intervals of with endpoints and ; that is, . Define the relation on by . Then it can easily be observed that is a metric on (cf. Diamond and Kloeden ) and is a complete metric space (cf. Nanda ). Now, we can give the metric on by means of the Hausdorff metric as It is trivial that
Remark 3 (cf. ). Then the following remarks can be given. (a)Obviously the sequence converges to a fuzzy number if and only if and converge uniformly to and on , respectively.(b)The boundedness of the sequence is equivalent to the fact that If the sequence is bounded then the sequences of functions and are uniformly bounded in .
Definition 4 (see ). 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 for all . If the sequence converges to a fuzzy number , then we say that the series of fuzzy numbers converges to and write which implies that where the summation is in the sense of classical summation and converges uniformly in .
2.1. Generalized Hukuhara Difference
A generalization of the Hukuhara difference proposed in  aims to overcome this situation.
Definition 5 (see [15, Definition 1]). The generalized Hukuhara difference of two sets is defined as follows:
Proposition 6 (see ). The following statements hold. (a)If and are two closed intervals, then .(b) Let be such that . Then, we have where the limits are in the Hausdorff metric for intervals.
3. Fuzzy-Valued Functions with the Level Sets
Definition 7 (see ). Consider a fuzzy-valued function from into with respect to a membership function which is called trapezoidal fuzzy number and is interpreted as follows: Then, the pair of depending on can be written as for all . Then, the function is said to be a fuzzy-valued function on .
Remark 8. The functions with given in Definition 7 are also defined by for all and the constant .
Now, following Kadak  we give the classical sets and consisting of the continuous and bounded fuzzy-valued functions; that is, Obviously, from Theorem 1, each function is left continuous on and right continuous at . It was shown that and are complete with the metric as where and are the elements of the sets or with .
3.1. 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 , we can define H-differentiation of with respect to level sets.
Definition 9 (cf. ). A fuzzy-valued function is said to be generalized H-differentiable with respect to the level sets at the point : if exists such that, for all sufficiently near to , the H-difference exists then the H-derivative is given as follows: From here, we remind that the H-derivative of at depends on and . Therefore, is H-differentiable if and only if and are classical differentiable functions.
Definition 10 (see [21, Definition 8.7]). A fuzzy valued function is said to be fuzzy Henstock, in short FH-integrable if, for any , there exists such that for any division of with the norms , where and is also called FH-integrable. One can conclude that in 15 denotes the usual Riemann sum for any division of .
Theorem 11 (see [21, Theorem 8.8]). Let and let be a -integrable function. If there exists such that , then
Remark 12. We remark that the integrals on the right hand side of 16 exist in the usual sense for all . It is easy to see that the pair of functions are continuous.
4. Uniform Convergence of Fuzzy-Valued Functions
Definition 13. Let be a sequence of fuzzy-valued functions defined on a set 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 fuzzy-valued function by
On the other hand, converges to on if and only if, for each and for an arbitrary , there exists an integer such that whenever . The integer in the definition of pointwise convergence may, in general, depend on both 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 , there exists an integer such that Obviously the sequence of fuzzy-valued functions converges (uniformly) to a fuzzy valued-function if and only if and converge uniformly to and in , respectively. On the other hand converges (uniformly) to if and only if the sequence converges to a real (complex) number .
Example 14. Consider a sequence of fuzzy-valued functions from into where , whose membership function is defined as where for all . Then, the membership function can be written as consisting of each function depending on and . Suppose that the sequence converges; then converges uniformly to the fuzzy-valued function which is given by In this form one can easily conclude that if and only if uniformly in .
Remark 15. If the sequence is, respectively, replaced by each of the fix numbers , and in Example 14, that is, where for all , then uniformly if and only if or uniformly in where . Similarly we take where for all and then uniformly if and only if for all , or uniformly for each where .
Now, in the following example, we give some cases of membership functions with two constants.
Example 16. Consider a sequence of fuzzy-valued functions from into for the constant , whose membership function is defined as where and for all and the constant . Then, it is obvious that for all . Hence, as we have seen above if and only if and uniformly in where and for all and the constant . Therefore, we derive that converges (uniformly).
Remark 17. If and are, respectively, replaced by each of the fixed numbers with given in Example 16, that is, where for all and a constant , then uniformly if and only if or uniformly in where . Other cases with two constants can be obtained similarly.
Example 18. Consider a sequence of fuzzy-valued functions for the constants with , whose membership function is defined: where for all . Then, it is clear that for all . Then if and only if and uniformly in where
Remark 19. If we take as a function based on in Example 18, that is, where the constants , then uniformly if and only if or uniformly in where .
Theorem 20 (see ). Then, the following statements are valid. (i)A sequence of fuzzy-valued functions defined on a set 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 there is a number such that or equivalently, .
Theorem 23. Suppose that is a sequence of -integrable functions defined on a closed interval . If uniformly on , then is -integrable on , and Also, for each , then uniformly in .
Proof. By taking into account Theorem 22, we need only to show that the limit function is FH-integrable on . We see that is bounded, because each is FH-integrable on . Also, the limit function is bounded, since, using triangle inequality and Theorem 20, we have where . Besides this, uniformly on , given any , there exists an integer such that for . Since is FH-integrable, there exists a partition of such that For each , using the inclusion 34 with implies that and therefore, Consequently, showing that is FH-integrable on . Finally, for and for each , 34 implies that and the proof is complete.
Theorem 24. Consider is a sequence of fuzzy-valued functions such that (i);(ii)there exists a point such that converges;(iii) uniformly on .Then converges uniformly to some on such that on .
Proof. By (iii), is uniformly convergent to on any closed interval contained in , say in an interval with endpoints and . By using Theorem 22, we have for all and belonging to by the fundamental theorem of calculus, and the convergence is uniform on . Since exists by (ii), we can add this term to both sides and obtain and the convergence is uniform on . We may now take , and the last equation holds Now, , being the fuzzy limit on , is continuous on , and so is H-differentiable and on . Therefore, the last inequality and Definition 9 imply that for all and . This completes the proof.
Theorem 24 continues to hold under a weaker hypothesis. However, we cannot replace the third condition, namely, the uniform convergence of the sequence , with pointwise convergence. Now we state an improved version of Theorem 24 without its proof.
Theorem 25. Suppose that is a sequence of fuzzy-valued functions such that (i)each is H-differentiable on ;(ii)there exists a point such that converges;(iii) uniformly on .Then converges uniformly to some on such that on .
5. 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. Consider a sequence of functions defined on a set . Recall that the level sum is called a series of fuzzy-valued functions. Form a new sequence of partial level sums of functions defined by 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 converges (pointwise) on and write the level sum function as
Definition 26. The series 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 , there exists an integer such that whenever .
Corollary 27. If each is a continuous fuzzy-valued function on for each and if is uniformly convergent to on , then must be continuous on for all .
Corollary 28. A series converges uniformly on a set if and only if the sequence of partial level sums is uniformly Cauchy on ; that is, for an arbitrary there is a number such that whenever .
Corollary 29. Suppose that is a sequence in and that converges uniformly to on . Then, where exists for all and .
Corollary 28 shows that an analogue of the comparison test, namely, a sufficient condition for the uniform convergence of a fuzzy-valued function series. Indeed, the following result is a simple and direct test for the uniform convergence of these series.
Theorem 30. Let be a series of fuzzy-valued functions on a subset of . Consider there exists a convergent series of nonnegative real numbers such that for all and we have Then converges uniformly.
Proof. Suppose that is dominated by a convergent series . That each of the series converges on comes from the comparison test for real series. To verify the uniform convergence of , take into account Cauchy criterion for the series . Hence, given , there exists an integer such that, for , we have . For all , we also obtain Therefore, by the Cauchy criterion in Corollary 28, the series converges uniformly on .
We may now give the following, which is the main for some applications of power series of fuzzy numbers (see ). We may state a condition under which term-by-term H-differentiation of an infinite series is allowable.
Definition 31. Let be any element and the fixed element. Then, the power series with fuzzy coefficients is in the form and the radius of convergence is defined by which is also given provided that the limit on the right hand side exists, where .
Definition 32. Suppose a power series with radius of convergence . Then, the set of the points from an interval at which the series is convergent is called the interval of convergence such that which must be either , , , or .
Theorem 33. A power series with fuzzy coefficients and the -fold derived series defined by have the same radius of convergence.
Theorem 34 (cf. ). If has radius of convergence , then is H-differentiable in and . Consequently, exists for every and every with , and The fuzzy coefficients are uniquely determined, and
have radius of convergence . We have to show the existence of H-differentiable in and that is of the stated form. By Theorem 33 with , the derived series converges for and defines a fuzzy-valued function, say , in . We show that for all .
Let be fixed. Then take a positive such that ; that is, . Also, let with . We have . We consider where . By taking into account Taylors theorem on the interval with endpoints, we get Therefore we have , Consequently, by 57, it follows that exists and equals . Since is arbitrary, this holds at any interior point in . On the other hand, this argument gives that all the H-derivatives exist in .
Corollary 35. Consider is a sequence of fuzzy-valued functions. Then the following hold: (i) (H-differentiable continuous fuzzy-valued function);(ii) there exists a point such that converges;(iii) converges uniformly on . Then converges uniformly on to a H-differentiable fuzzy-valued function , for all and .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors record their pleasure to the anonymous referee for his/her constructive report and many helpful suggestions on the main results.
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