Fuzzy stability of multi-additive mappings

The main purpose of this paper is to consider the strong law of large numbers for random sets in fuzzy metric space. Since many years ago, limited theorems have been expressed and proved for fuzzy random variables, but despite the uncertainty in fuzzy discussions, the nonfuzzy metric space has been used. Given that the fuzzy random variable is defined on the basis of random sets, in this paper, we generalize the strong law of large numbers for random sets in the fuzzy metric space. The embedded theorem for compact convex sets in the fuzzy normed space is the most important tool to prove this generalization. Also, as a result and by application, we use the strong law of large numbers for random sets in the fuzzy metric space for the bootstrap mean.

Regularly open sets in fuzzy topological spaces

In this paper,we investigate the generalized Hyers-Ulam-Rassias stability of Jensen functional equation in Felbin’s type fuzzy normed linear spaces . 1.Introduction In 1940, Ulam[1] proposed the general Ulam stability problem.Next year, Hyers[2] solved this problem.In 1978, Rassias[3] took account of the unbounded Cauchy difference in Hyers’ theorem and obtained the results for linear mappings. The stability problems of several functional equations have been extensively investigated by a number of authors (see [4,5] and references therein).In 1989,Kominek[6] proved the stability of Jensen functional equation on a restricted domain.In1998, Jung[7] proved the Hyers-Ulam-Rassias stability of Jensen functional equation. In 2014, Eskandani and Rassias[8] investigated the stability of a general cubic functional equation in Felbin’s type fuzzy normed linear spaces. In this paper,we investigate the generalized Hyers-Ulam-Rassias stability of Jensen functional equation in Felbin’s type fuzzy normed linear spaces . We consider some basic concepts concerning in the theory of fuzzy real numbers. Let be a fuzzy subset on R, i.e., a mapping associating with each real number t its grade of membership . Definition1.1 ] 9 [ A fuzzy subset on R is called a fuzzy real number,whose -level set is denoted by ,i.e., ,if it satisfies two axioms: (1)There exists such that . (2)For each ; where . The set of all fuzzy real numbers denoted by .If and whenever ,then is called a nonnegative fuzzy real number and denotes the set of all non-negative fuzzy real numbers. Definition1.2 ] 9 [ Let X be a real linear space, L and R (respectively, left norm and right norm) be 3rd International Conference on Mechatronics, Robotics and Automation (ICMRA 2015) © 2015. The authors Published by Atlantis Press 563 symmetric and non-decreasing mappings in both arguments from into satisfying and .The mapping from X into is called a fuzzy norm if for and : (1) if and only if , (2) for all and (3)For all , (a)if , and then     t s y x    ), , ( t y s x L (b)if , and then   t s y x       ) , ( t y s x R . The quaternary (X, ,L,R) is called a fuzzy normed linear space. Definition1.3 ] 9 [ Let (X, ,L,R) be a fuzzy normed linear space and .A sequence in X is said to converge to ,denoted by ,if for every and is called a Cauchy sequence if for every .A subset in X is said to be complete if every Cauchy sequence in A converges in A.The fuzzy normed space (X, ,L,R) is said to be a fuzzy Banach space if it is complete. Theorem1.4 ] 10 [ Let (X,‖.‖,L,R) be a fuzzy normed linear space, if ,then for any , for all A mapping is called a Jensen function if f satisfies the functional equation for .For a given mapping we define the difference operator for Then f is a Jensen function if for all

We consider a space of continuous set-valued mappings defined on a locally compact space T with a countable base. The values of these mappings are closed (not necessarily bounded) sets in a metric space (X, d(·)) whose closed balls are compact. The space (X, d(·)) is locally compact and separable. Let Y be a countable dense set in X. The distance ρ(A,B) between sets A and B belonging to the family CL(X) of all nonempty closed subsets of X is defined as follows: $$\rho (A, B) = \sum_{i=1}^\infty\frac{1}{2^i}\frac{|d(y_i, A) - d(y_i, B)|}{1+|d(y_i, A) - d(y_i, B)|},$$ where d(yi,A) is the distance from the point yi ∈ Y to the set A. This distance is independent of the choice of the set Y, and the function ρ(A,B) is a metric on the space CL(X). The convergence of a sequence of sets An, n ≥ 1, in the metric space (CL(X), ρ(·)) is equivalent to the Kuratowski convergence of this sequence. We prove the completeness and separability of the space (CL(X), ρ(·)) and give necessary and sufficient conditions for the compactness of sets in this space. The space C(T,CL(X)) of all continuous mappings from T to (CL(X), ρ(·)) is endowed with the topology of uniform convergence on compact sets in T. We prove the completeness and separability of the space C(T,CL(X)) and give necessary and sufficient conditions for the compactness of sets in this space. These results are reformulated for the space C(T,CCL(X)), where T = [0, 1], X is a finite-dimensional Euclidean space, and CCL(X) is the space of all nonempty closed convex sets in X with metric ρ(·). This space plays a crucial role in the study of sweeping processes. We give a counterexample showing the significance of the assumption of compactness of closed balls in X.

This article discusses a variety of important notions, including ideal convergence and ideal Cauchyness of topological sequences produced by fuzzy normed spaces. Furthermore, the connections between the concepts of the ideal limit and ideal cluster points of a sequence in a fuzzy normed linear space are investigated. In a fuzzy normed space, we investigated additional effects, such as describing compactness in terms of ideal cluster points and other relevant but previously unresearched ideal convergence and adjoint ideal convergence aspects of sequences and nets. The countable compactness of a fuzzy normed space and its link to it were also defined. The terms ideal and its adjoint divergent sequences are then introduced, and specific aspects of them are explored in a fuzzy normed space. Our study supports the importance of condition (AP) in examining summability via ideals. It is suggested to use a fuzzy point symmetry-based genetic clustering method to automatically count the number of clusters in a data set and determine how well the data are fuzzy partitioned. As long as the clusters have the attribute of symmetry, they can be any size, form, or convexity. One of the crucial ways that symmetry is used in fuzzy systems is in the solution of the linear Fuzzy Fredholm Integral Equation (FFIE), which has symmetric triangular (Fuzzy Interval) output and any fuzzy function input.

In various articles, fuzzy <img src=image/13427754_06.gif>-normed space concept for <img src=image/13427754_02.gif> is constructed from fuzzy normed space which uses intuitionistic approach or <img src=image/13427754_03.gif>-norm approach concept. However, fuzzy normed space can be approached using fuzzy point too. This paper shows that fuzzy <img src=image/13427754_06.gif>-normed space for <img src=image/13427754_02.gif> can be constructed from fuzzy normed space using fuzzy point approach of fuzzy set. Furthermore, for <img src=image/13427754_07.gif>, it is also discussed how to construct fuzzy (<img src=image/13427754_05.gif>)-normed space from fuzzy <img src=image/13427754_06.gif>-normed space using fuzzy point approach. The method that can be used is as follows. From fuzzy normed space, we construct a norm function that satisfies properties of fuzzy <img src=image/13427754_06.gif>-normed, so that fuzzy <img src=image/13427754_06.gif>-normed space is derived. Conversely, from fuzzy <img src=image/13427754_06.gif>-normed space, we construct a normed function that satisfies properties of fuzzy (<img src=image/13427754_05.gif>)-normed, so that fuzzy (<img src=image/13427754_05.gif>)-normed space is obtained. Finally, we get two new theorems that state that a fuzzy <img src=image/13427754_06.gif>-normed space from any fuzzy normed space and fuzzy (<img src=image/13427754_05.gif>)-normed space for <img src=image/13427754_07.gif> from fuzzy <img src=image/13427754_06.gif>-normed space using fuzzy point of fuzzy set always can be constructed.

Betweenness relations and gated sets in fuzzy metric spaces

In this paper we will highlight some properties of fuzzy bounded sets in fuzzy normed linear spaces. More precisely, we will prove that any finite union or sum of fuzzy bounded sets is fuzzy bounded. We show that the closure of a fuzzy bounded set is also fuzzy bounded. We give a characterization for fuzzy bounded sets and finally we obtain that the Cartesian product of two fuzzy bounded sets is also fuzzy bounded.

In this paper , the concept of the Cartesian Product of two fuzzy&#x0D; normed spaces is presented. Some basic properties and theorems on this&#x0D; concept are proved. The main goal of this paper is to prove that the&#x0D; Cartesian product of two complete fuzzy normed spaces is a complete&#x0D; fuzzy normed space.&#x0D; Key words:Fuzzy normed space , Cartesian product , Cauchy sequence ,&#x0D; complete fuzzy normed space.&#x0D; 1- Introduction&#x0D; &#x0D; The fuzzy set concepts was introduced in mathematics by K.Menger&#x0D; in 1942 and reintroduced in the system theory by L.A.Zadeh in 1965.&#x0D; In 1984, Katsaras [ 1 ] , first introduced the notation of fuzzy norm on&#x0D; linear space, in the same year Wu and Fang [ 4 ] also introduced a notion of&#x0D; fuzzy normed space . Later on many other mathematicians like Felbin [ 2 ]&#x0D; , Cheng and Mordeson [ 10] , Bag and Samanta [12], J.Xiao and X.Zhu&#x0D; [8,9] , Krishna and Sarma [11] , Balopoulos and Papadopoulos [ 13] etc,&#x0D; have given different definitions of fuzzy normed spaces .&#x0D; J.Kider introduced the definition of fuzzy normed space[ 7 ] , we use this&#x0D; definition to prove that the Cartesian product of two fuzzy normed spaces&#x0D; is also fuzzy normed space.&#x0D; &#x0D; The structure of the paper is as follow : In section 2 we&#x0D; present some fundamental concepts . In section 3, the definition of fuzzy&#x0D; normed space appeared [7] is used to prove that the cartesain product of&#x0D; two fuzzy normed spaces is also fuzzy normed space, then we prove that&#x0D; the cartesain product of two complete fuzzy normed spaces is complete&#x0D; fuzzy normed space.&#x0D; &#x0D; 2. Preliminaries&#x0D; In this section, we briefly recall some definitions and preliminary&#x0D; results which are used in this paper.

The main purpose of this paper is to study the approximate best proximity pair of cyclic maps and their diameter in fuzzy normed spaces defined by Bag and Samanta. First, approximate best point proximity points on fuzzy normed linear spaces are defined and four general lemmas are given regarding approximate fixed point and approximate best proximity pair of cyclic maps on fuzzy normed spaces. Using these results, we prove theorems for various types of well-known generalized contractions in fuzzy normed spaces. Also, we apply our results to get an application of approximate fixed point and approximate best proximity pair theorem of their diameter.

Fuzzy sets and functions on fuzzy spaces

Rassias(2001) introduced the pioneering cubic functional equation in the history of mathematical analysis: and solved the pertinent famous Ulam stability problem for this inspiring equation. This Rassias cubic functional equation was the historic transition from the following famous Euler-Lagrange-Rassias quadratic functional equation: to the cubic functional equations. In this paper, we prove the Ulam-Hyers stability of the cubic functional equation: in fuzzy normed linear spaces. We use the definition of fuzzy normed linear spaces to establish a fuzzy version of a generalized Hyers-Ulam-Rassias stability for above equation in the fuzzy normed linear space setting. The fuzzy sequentially continuity of the cubic mappings is discussed.

Fuzzy Closure Spaces vs. Fuzzy Rough Sets

A new I-vector topology generated by a fuzzy norm

In this paper some interesting relationships known between continuous, sequentially continuous, strongly continuous maps, bounded and weakly bounded linear operators on the crisp normed spaces are generalized to the case of fuzzy normed spaces.