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In the present study, we have introduced the concept of Abel summability for sequences and series of fuzzy numbers. Also, some tauberian results in classical analysis have been generalized to fuzzy analysis. This is a preview of subscription content, log in to check access.

Rent this article via DeepDyve. J Intell Fuzzy Syst — Soft Comput 16 6 — J Fuzzy Math 9 3 — Fuzzy Sets Syst — J Intell Fuzzy Syst 26 6 — J Intell Fuzzy Syst 27 2 — Soft Comput 15 4 — Diamond P, Kloeden P Metric spaces of fuzzy sets.

Goetschel R, Voxman W Elementary fuzzy calculus. Filomat 26 3 — Matloka M Sequences of fuzzy numbers. Busefal — Nanda S On sequence of fuzzy numbers. Soft Comput 16 4 — J Anal — Comput Math Appl 58 4 — Appl Math Lett — J Intell Fuzzy Syst 24 1 — J Intell Fuzzy Syst 25 3 — Download references. The authors would like to express their pleasure to referees for many helpful suggestions on the main results of the earlier version of the paper which improved the presentation of the paper.

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

Yavuz, E. Abel summability of sequences of fuzzy numbers. Soft Comput 20, — Download citation. Published : 21 December Issue Date : March Search SpringerLink Search. Abstract In the present study, we have introduced the concept of Abel summability for sequences and series of fuzzy numbers.In mathematicsfuzzy sets a. Fuzzy sets were introduced independently by Lotfi A.

In classical set theorythe membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. Fuzzy sets generalize classical sets, since the indicator functions aka characteristic functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics.

Although the complement of a fuzzy set has a single most common definition, the other main operations, union and intersection, do have some ambiguity. By the definition of the t-norm, we see that the union and intersection are commutativemonotonicassociativeand have both a null and an identity element. Since the intersection and union are associative, it is natural to define the intersection and union of a finite family of fuzzy sets by recursion.

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Fuzzy sets are disjoint, iff their supports are disjoint according to the standard definition for crisp sets. This family is pairwise disjoint iff. Because fuzzy sets are unambiguously defined by their membership function, this metric can be used to measure distances between fuzzy sets on the same universe:.

Other distances like the canonical 2-norm may diverge, if infinite fuzzy sets are too different, e. These are usually called L -fuzzy setsto distinguish them from those valued over the unit interval. These kinds of generalizations were first considered in by Joseph Goguenwho was a student of Zadeh. An extension of fuzzy sets has been provided by Atanassov and Baruah. After all, we have a percentage of approvals, a percentage of denials, and a percentage of abstentions.

For this situation, special 'intuitive fuzzy' negators, t- and s-norms can be provided.

Fuzzy Set Field and Fuzzy Metric

The concept of IFS has been extended into two major models. The two extensions of IFS are neutrosophic fuzzy sets and Pythagorean fuzzy sets. Neutrosophic fuzzy sets were introduced by Smarandache in This value indicates that the degree of undecidedness that the entity x belongs to the set. The other extension of IFS is what is known as Pythagorean fuzzy sets. Pythagorean fuzzy sets are more flexible than IFS. This is why Yager proposed the concept of Pythagorean fuzzy sets.

With these valuations, many-valued logic can be extended to allow for fuzzy premises from which graded conclusions may be drawn.

Some new results on sequence spaces with respect to non-Newtonian calculus

This extension is sometimes called "fuzzy logic in the narrow sense" as opposed to "fuzzy logic in the wider sense," which originated in the engineering fields of automated control and knowledge engineeringand which encompasses many topics involving fuzzy sets and "approximated reasoning. Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic.

Fuzzy numbers can be likened to the funfair game "guess your weight," where someone guesses the contestant's weight, with closer guesses being more correct, and where the guesser "wins" if he or she guesses near enough to the contestant's weight, with the actual weight being completely correct mapping to 1 by the membership function.

The latter means that fuzzy intervals are normalized fuzzy sets. As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous. However, there are other concepts of fuzzy numbers and intervals as some authors do not insist on convexity.Since the utilization of Zadeh's extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct some classical notions.

In this paper, we present the sets of bounded, convergent, and null series and the set of sequences of bounded variation of fuzzy level sets, based on the partial metric. We examine the relationships between these sets and their classical forms and give some properties including definitions, propositions, and various kinds of partial metric spaces of fuzzy level sets.

Furthermore, we study some of their properties like completeness and duality. We define the classical sets bs Hcs Hand cs 0 H consisting of the sets of all bounded, convergent, and null series, respectively; that is.

We can show that bs Hcs Hand cs 0 H are complete metric spaces with the partial metric H s defined by.

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Many authors have extensively developed the theory of the different sets of sequences and its matrix transformations [ 12 ]. The rest of this paper is organized as follows. In Section 2some required definitions and consequences related with the partial metric and fuzzy level sets, sequences, and convergence are given.

Motivated by experience from computer science, nonzero self-distance seen to be plausible for the subject of finite and infinite sequences.

Thus d s xy is defined to be 1 over 2 to the power of the length of the longest initial sequence common to both x and y. Each partial metric space thus gives rise to a metric space with the additional notion of nonzero self-distance introduced. Thus, a metric space can be defined to be a partial metric space in which each self-distance is zero.

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Clearly, a limit of a sequence in a partial metric space need not be unique. If p is a partial metric on Xthen the function p s defined by. Let Xp be a partial metric space and x n a sequence in Xp. Then, we say the following:. It is easy to see that every closed subset of a complete partial metric space is complete.

We denote the set of all fuzzy numbers on R by E 1 and call it as the space of fuzzy numbers. Representation Theorem 1 see [ 22 ]. Then the following statements hold. Then it can easily be observed that d is a metric on W cf. Diamond and Kloeden [ 24 ] and Wd is a complete metric space, cf. Nanda [ 25 ].

on the classical sets of sequences with fuzzy b-metric 1

Now, we can define the metric D on E 1 by means of the Hausdorff metric d as. Then, the following statements hold. Also, the space l p H is complete metric space with the partial metric H p defined by. Since the proof is similar for the spaces cs H and cs 0 Hwe prove the theorem only for the space bs H.

It remains to prove the completeness of the space bs H. We must show that. Since u m is an arbitrary Cauchy sequence, bs H is complete.The class of membership functions is restricted to trapezoidal one, as it is general enough and widely used. 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.

Later Karl Weierstrass, who attended his course on elliptic functions incoined the term uniformly convergent which he used in his paper Zur Theorie der Potenzreihen, published in 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.

on the classical sets of sequences with fuzzy b-metric 1

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. 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 2we 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 3first 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. A fuzzy number is a fuzzy set on the real axis, that is, a mapping which satisfies the following four conditions.

Zadeh [ 7 ]where denotes the closure of the set in the usual topology of.

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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 [ 8 ]. Let for and for each. Then the following statements hold. Otherwise, if the pair of functions and holds the conditions i — ivthen 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.Contractive sequences in fuzzy metric spaces.

Download PDF. Recommend Documents. On fuzzy contractive mappings in fuzzy metric spaces. Completable fuzzy metric spaces. On fuzzy metric spaces. Intuitionistic fuzzy metric spaces. Fuzzy pseudo-metric spaces. Fuzzy metric neighbourhood spaces. Metric spaces of fuzzy variables.

On completable fuzzy metric spaces. Fixed point theorems in fuzzy metric spaces. Curve arclength in fuzzy metric spaces. Mihet, which is not Cauchy in a fuzzy metric space in the sense of George and Veeramani. To overcome this drawback we introduce and study a concept of strictly fuzzy contractive sequence.

Then, we also make an appropriate correction to Lemma 3. All rights reserved. Keywords: Fuzzy metric space; Fuzzy contractive mapping; Fixed point 1. Introduction In this paper we deal with the concept of fuzzy metric due to George and Veeramani [1] which is a modification of the one given by Kramosil and Michalek [9,3].

This topology is metrizable [2,7] and consequently, topics related to metrics have been systematically extended and studied in this fuzzy setting. In particular, fuzzy fixed point theory is a field of high activity. This sequence converges in X since contractive sequences are Cauchy.

Some New Sets of Sequences of Fuzzy Numbers with Respect to the Partial Metric

But, what about this statement in fuzzy setting? We notice that in [8] the authors introduced a concept of fuzzy contractive sequence and they posed the following question: Is every fuzzy contractive sequence a Cauchy sequence? So far, there is no answer to this question D. Mihet [11] gave a negative answer, but for fuzzy metrics in the sense of Kramosil and Michalek.

The purpose of this article is to make a new contribution to this field and, at the same time, to correct an error appeared in [5]. For it, we will introduce and study a concept of strictly fuzzy contractive sequence. Regarding the last paragraph, on the one hand, we notice that there are several concepts of Cauchy sequence in the literature [6].

Here we focus our attention in the two concepts used in fuzzy fixed point theory. Gregori et al. Grabiec in [3] and it will be denoted by G-Cauchy see Definition 2. The second one will be called, simply, Cauchy see Definition 2.Metrics details. As an alternative to classical calculus, Grossman and Katz Non-Newtonian Calculus, introduced the non-Newtonian calculus consisting of the branches of geometric, anageometric and bigeometric calculus etc. Following Grossman and Katz, we construct the field R N of non-Newtonian real numbers and the concept of non-Newtonian metric.

During the Renaissance many scholars, including Galileo, discussed the following problem: Two estimates, 10 and 1, are proposed as the value of a horse. Which estimate, if any, deviates more from the true value of ? The scholars who maintained that deviations should be measured by differences concluded that the estimate of 10 was closer to the true value.

However, Galileo eventually maintained that the deviations should be measured by ratios and concluded that two estimates deviated equally from the true value cf. From the story, the question comes out this way: What if we measure by ratios?

The answer is the main idea of non-Newtonian calculus which consists of many calculi such as classical, geometric, anageometric, bigeometric calculus etc. Even you can create your own calculus by choosing a different function as a generator. Although all arithmetics are structurally equivalent, only by distinguishing among them do we obtain suitable tools for constructing all the non-Newtonian calculi. But the usefulness of arithmetics is not limited to the construction of calculus; we believe there is a more fundamental reason for considering alternative arithmetics.

They may also be helpful in developing and understanding the new systems of measurement that could yield simpler physical laws cf.

Bashirov et al. Quite recently, Uzer [ 3 ] has extended the multiplicative calculus to the complex valued functions and was interested in the statements of some fundamental theorems and concepts of multiplicative complex calculus, and demonstrated some analogies between the multiplicative complex calculus and classical calculus by theoretical and numerical examples. Non-Newtonian calculus is an alternative to the usual calculus of Newton and Leibniz. It provides differentiation and integration tools based on non-Newtonian operations instead of classical operations.

Every property in classical calculus has an analogue in non-Newtonian calculus. Generally speaking, non-Newtonian calculus is a methodology that allows one to have a different look at problems which can be investigated via calculus.

In some cases, for example, for wage-rate in dollars, euro etc. Throughout this document, non-Newtonian calculus is denoted by NCand classical calculus is denoted by CC. Each generator generates exactly one arithmetic and, conversely, each arithmetic is generated by exactly one generator.

Following Bashirov et al. To do this, we need some preparatory knowledge about non-Newtonian calculus. There are infinitely many arithmetics, all of which are isomorphic, that is, structurally equivalent.

on the classical sets of sequences with fuzzy b-metric 1

Nevertheless, the fact that two systems are isomorphic does not preclude their separate uses. In [ 1 ], it is shown that each ordered pair of arithmetics gives rise to a calculus by a judicious use of the first arithmetic for function arguments and the second arithmetic for function values. The rest of the paper is organized, as follows. Further, the non-Newtonian exponent, surd and absolute value are defined and some of their properties are given. Additionally, some required inequalities are presented in the sense of non-Newtonian calculus, and the concepts of non-Newtonian metric and non-Newtonian norm are introduced.

In the final section of the paper, we note the significance of non-Newtonian calculus and record some further suggestions. In the present section, we construct the non-Newtonian real field R N and give some related properties. Theorem 2. Therefore, we have. The non-Newtonian distance between two numbers x 1 and x 2 is defined by. The following proposition shows that this also holds in non-Newtonian calculus.

Proposition 2. In this section, we deal with the vector spaces over the non-Newtonian real field R N. First, we present the required inequalities in the sense of NC.In this paper, we present the sets of bounded, convergent, and null series and the set of sequences of bounded variation of fuzzy level sets, based on the partial metric.

We examine the relationships between these sets and their classical forms and give some properties including definitions, propositions, and various kinds of partial metric spaces of fuzzy level sets. Furthermore, we study some of their properties like completeness and duality. Bywe denote the set of all sequences of fuzzy numbers. We define the classical sets, and consisting of the sets of all bounded, convergent, and null series, respectively; that is We can show that, and are complete metric spaces with the partial metric defined by where and are the elements of the sets, or.

Secondly, we introduce the sets, and consisting of sequences of -bounded variation by using the partial metric with respect to the partial orderingas follows: where the distance function denotes the partial metric of fuzzy level sets defined by for any with the partial ordering.

One can conclude that the sets, and are complete metric spaces with the following partial metrics: respectively, where and are the elements of the sets, or and for all. Many authors have extensively developed the theory of the different sets of sequences and its matrix transformations [ 12 ]. Finally, Kadak and Ozluk [ 11 — 13 ] have introduced the sets,and of classical sequences of fuzzy level sets and sufficient conditions for partial completeness of these are established by means of fuzzy level sets.

The rest of this paper is organized as follows.

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In Section 2some required definitions and consequences related with the partial metric and fuzzy level sets, sequences, and convergence are given. Section 3 is devoted to the completeness of the sets of sequences, and, of fuzzy level sets and some related notions. Motivated by experience from computer science, nonzero self-distance seen to be plausible for the subject of finite and infinite sequences.

Definition 1 see [ 14 ]. Let be a nonempty set and be a function from to the set of nonnegative real numbers. Then the pair is called a partial metric space and is a partial metric forif the following partial metric axioms are satisfied for all : P1 if and only ifP2P3P4. Proposition 2 Nonzero self-distance [ 15 ].

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Let be the set of all infinite sequences over a set. For all such sequences and letwhere is the largest number possibly such that for each. Thus is defined to be over to the power of the length of the longest initial sequence common to both and. It can be shown that is a metric space. Each partial metric space thus gives rise to a metric space with the additional notion of nonzero self-distance introduced. Also, a partial metric space is a generalization of a metric space; indeed, if an axiom is imposed, then the above axioms reduce to their metric counterparts.


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On the classical sets of sequences with fuzzy b-metric 1
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