Fuzzy Number Space Research Articles

( 2003-5778) Stability problem for Pexiderized Cauchy-Jensen type functional equations of fuzzy number-valued mappings

We investigate the stability problems of the n-dimensional Cauchy-Jensen type and the n-dimensional Pexiderized Cauchy-Jensen type fuzzy number-valued functional equations in Banach spaces by using the metric defined on a fuzzynumber space. Under some suitable conditions, some properties of the solutions for these equations such as existence and uniqueness are discussed. Our results can be regarded as important extensions of stability results corresponding tosingle-valued functional equations and set-valued functional equations, respectively.

(2008-6104) Calculation of centroid of high dimensional fuzzy number and application

In this paper, the conception of centroid of n-dimensional fuzzy number is introduced viaregarding its membership function as the density function on its support set, and some properties of it are obtained. Compared with the mean of the multi dimensional fuzzy number, the centroid takes into account the overall relationship between the edge membership functions of the membership function of the multi dimensional fuzzy number. Therefore, it can approximate (characterize) the fuzzy number more objectively and reasonably than using the mean of the multi dimensional fuzzy number. The most important work of this paper is that for two special kinds of multi dimensional fuzzy numbers (fuzzy n-cell numbers and fuzzy n-ellipsoid numbers), we respectively give calculation formulas, which can be used conveniently in application since the formulas are based on a definite integral of the level set functions of the multi dimensional fuzzy number on the unit interval [0,1], rather than the multiple integral of the membership function of the multi dimensional fuzzy number itself on its support set. Then, by using the calculation formulas, we obtain another special property of the centroid for fuzzy n-cell number and fuzzy n-ellipsoid number. Finally, as an example of application, by using the centroid of multi dimensional fuzzy number, we define a fuzzy order on n-dimensional fuzzy number space, which can be used to rank uncertain or imprecise multichannel digital information.

Fractional calculus of linear correlated fuzzy-valued functions related to Fréchet differentiability

In this paper, we will introduce some types of Fréchet fractional derivative defined on the class of linear correlated fuzzy-valued functions. Firstly, we study Fréchet derivative and R-derivative of integer order and investigate their relationship with the well-known generalized Hukuhara derivatives in fuzzy metric space. Secondly, the Riemann-Liouville fractional integral of linear correlated fuzzy-valued functions is well-defined via an isomorphism between R2 and subspace of fuzzy numbers space RF. That allows us to introduce three types of Fréchet fractional derivatives, which are Fréchet Caputo derivative, Fréchet Riemann-Liouville derivative and Fréchet Caputo-Fabrizio derivative. Moreover, some common properties of fuzzy Laplace transform for linear correlated fuzzy-valued function are investigated. Finally, some applications to fuzzy fractional differential equations are presented to demonstrate the usefulness of theoretical results.

On n-polygonal interval-valued fuzzy sets

The structure of interval-valued fuzzy sets is complex in regard to their arithmetic operations. To reduce the computational complexity of arithmetic operations on interval-valued fuzzy sets, we propose the notion of an n-polygonal interval-valued fuzzy set considering two 2n+2-tuple ordered real numbers. Complete representation of n-polygonal interval-valued fuzzy sets and numbers is provided. Moreover, we demonstrate that the n-polygonal interval-valued fuzzy numbers can approximate the general interval-value fuzzy numbers with any precision. Next, we introduce arithmetic operations on n-polygonal interval-valued fuzzy numbers by capitalizing on the good characteristics of n-polygonal interval-valued fuzzy numbers. The properties of the introduced arithmetic operations are also addressed. In addition, with the aid of a concrete example, we verify the effectiveness of the approximation ability of the n-polygonal interval-valued fuzzy set. Furthermore, we study the properties of the topological space of n-polygonal interval-valued fuzzy numbers. This proves that this space is a complete, separable and local compact metric space when endowed with the newly defined distance between two n-polygonal interval-valued fuzzy numbers. By product, it shows that arithmetic operations introduced here on n-polygonal interval-valued fuzzy numbers are continuous. Finally, the practicability of n-polygonal interval-valued fuzzy numbers is verified by an example.

A New Fuzzy PID Control System Based on Fuzzy PID Controller and Fuzzy Control Process

In this paper, we present a fuzzy PID control system as a combination of a fuzzy PID controller and a fuzzy control process, which is represented by a fuzzy control differential equation in linear form. We use the concepts of the generalized Hukuhara differentiability and the fuzzy integral of fuzzy-valued functions to study some qualitative properties for this system in the space of fuzzy numbers. We also study the existence and uniqueness result for solutions of fuzzy PID control differential equations under some suitable conditions. A number of examples are also provided to illustrate the results of the theory.

Applications of contractive-like mapping principles to fuzzy fractional integral equations with the kernel $$\psi $$-functions

In this work, we present a new class of generalized fractional integral equations with respect to the kernel $$\psi $$ -function under the fuzzy concept. The results of this problem can be used to recover a wide class of fuzzy fractional integral equations by the choice of the kernel $$\psi $$ -function. Without the Lipschitzian right-hand side, we investigate the existence and uniqueness of the fuzzy solutions by employing the fixed point theorem of weakly contractive mappings in the partially ordered space of fuzzy numbers. The proposed approach is based on the concept of a fuzzy metric space endowed with a partial order and the altering distance functions. In addition, the continuous dependence of solutions on the order and the initial condition of the given problem is also shown. Some concrete examples are presented in order to consolidate the obtained result.

Fixed Point Theorems in Normed Linear Space of Triangular Fuzzy Numbers

Fixed Point Theorems in Normed Linear Space of Triangular Fuzzy Numbers

Tauberian theorems for statistical logarithmic summability of strongly measurable fuzzy valued functions

We define statistical logarithmic summability of strongly measurable fuzzy valued functions and we give slowly decreasing type Tauberian conditions under which statistical limit at infinity and statistical logarithmic summability of strongly measurable fuzzy valued functions imply ordinary limit at infinity in one dimensional fuzzy number space $E^1$. Besides, we give slowly oscillating type Tauberian conditions for statistical limit and statistical logarithmic summability of strongly measurable fuzzy valued functions in $n-$dimensional fuzzy number space $E^n$.

Fixed Point Theorems in Normed Linear Space of Triangular Fuzzy Numbers

Fixed Point Theorems in Normed Linear Space of Triangular Fuzzy Numbers

L-metric on the space of L-fuzzy numbers

In this article, a natural L-topology is constructed on the set of L-fuzzy numbers, which is called the standard L-topology. It is proved that the standard L-topology can be induced by an L-metric. The space of trapezoidal fuzzy numbers can be regarded as a subspace of the space of L-fuzzy numbers. An L-metric on the set of trapezoidal fuzzy numbers is presented.

A special fuzzy star-shaped numbers space with endograph metric

In this paper, for a non-degenerate convex set Y in Rn containing 0, two special function spaces S0 (Y) and E0 (Y) which consist of all fuzzy star-shaped numbers and of all fuzzy numbers in Rn with respect to 0 and their supports being included in Y with the endograph metric D are investigated. So me conclusions and methods in topology are used to discuss the topological structure of (S0 (Y) , D) and the pair ((S0 (Y) , D) , (E0 (Y) , D)). The main results are as follows: 1. The space (S0 (Y) , D) is homeomorphic to the Hilbert cube Q = [-1, 1] N if and only if S0 (Y) is compact if and only if Y is compact. 2. There exists a homeomorphism h : (S0 (Y) , D) → Q such that h (E0 (Y)) = {1} × [-1, 1] N\{1} if Y is compact but not a segment. 3. The space (S0 (Y) , D) homeomorphic to the pseudoboundary of the Hilbert cube if and only if S0 (Y) is non-compact and σ-compact if and only if Y is non-compact and locally compact.

The structure stability of periodic solutions for first-order uncertain dynamical systems

This paper studies the structural stability of periodic solutions for first-order fuzzy differential equations (FDEs) understood as differential inclusions, i.e., first-order uncertain dynamical systems. The existence and uniqueness of periodic solutions for this first-order fuzzy problems have been obtained on general fuzzy number space. When the forcing function has specific perturbations, the structural stability of the periodic solutions are discussed by using the support function, the Dini Theorem and the Convergence Theorem in the differential inclusion theory.

Ulam stability of two fuzzy number-valued functional equations

In this paper, the Ulam stability of two fuzzy number-valued functional equations in Banach spaces is investigated by using the metric defined on a fuzzy number space. Under some suitable conditions, some properties of the solutions for these equations such as existence and uniqueness are discussed.

An interval-valued fuzzy distance measure between two interval-valued fuzzy numbers

Interval-valued fuzzy numbers have found remarkable importance in various real-world applications. Such cases often require an evaluation of the distance measure between two interval-valued fuzzy numbers to solve the problems in the intended application. In this regard, an interval-valued fuzzy distance measure is provided in this study. For this purpose, an exact distance measure was first suggested and some of its main properties were investigated via some theorems and lemmas and compared with the results obtained by other studies. Then the proposed exact distance measure was extended in space of interval-valued fuzzy numbers and its axiomatic properties, such as triangular and robustness properties, were also investigated. Two numerical evaluations were presented to illustrate the calculation of the distance between two interval-valued fuzzy numbers as an interval-valued fuzzy number, as well.

Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization

Abstract The existing results on the variational inequality problems for fuzzy mappings and their applications were based on Zadeh’s decomposition theorem and were formally characterized by the precise sets which are the fuzzy mappings’ cut sets directly. That is, the existence of the fuzzy variational inequality problems in essence has not been solved. In this paper, the fuzzy variational-like inequality problems is incorporated into the framework of n-dimensional fuzzy number space by means of the new ordering of two n-dimensional fuzzy-number-valued functions we proposed [Fuzzy Sets and Systems 295 (2016) 19-36]. As a theoretical basis, the existence and the basic properties of the fuzzy variational inequality problems are discussed. Furthermore, the relationship between the variational-like inequality problems and the fuzzy optimization problems is discussed. Finally, we investigate the optimality conditions for the fuzzy multiobjective optimization problems.

Narrow Metric Semi-linear Space of Intuitionistic Fuzzy Numbers: Application to AIDS Model

In this paper, we study two new binary operations, which are addition and scalar multiplication, for intuitionistic fuzzy numbers (IFNs). Thereafter, we introduce a semi-linear space for IFNs that it is called the narrow metric semi-linear space— $$L^{*}$$ . At the same time, we present a new type of intuitionistic fuzzy functions with a real domain and propose a number of concepts and properties for these functions such as geometric difference, geometric differentiability, derivative and integral. In addition, we give a model of initial value problem for intuitionistic fuzzy differential equations and present its application to an AIDS model. Some examples are given to illustrate the theoretical results.

Dynamic ranking of multi-dimensional uncertain information based on fuzzy order on the fuzzy ellipsoid number space

Dynamic ranking of multi-dimensional uncertain information based on fuzzy order on the fuzzy ellipsoid number space

A process capability index for normal random variable with intuitionistic fuzzy information

In this study, a process control criterion was extended based on intuitionistic fuzzy information in cases where the underlying population is normal with intuitionistic fuzzy mean and exact variance. The proposed process control criterion was constructed based on the arithmetic operations and a common distance measure in the space of intuitionistic fuzzy numbers. For this purpose, one of the most popular process capability indices and its corresponding estimator were extended based on intuitionistic fuzzy specific limits and intuitionistic fuzzy target when the intuitionistic fuzzy mean and/or variance are unknown. A criterion was also proposed to investigate the level of process condition. The effectiveness of the proposed method was also examined by a practical example.

Asymptotic estimates for n-width of fuzzy numbers

n-widths in approximation theory characterize how well one can approximate a subset by some “good” subsets of a normed linear space. Especially, n-widths of sets of mathbb{R}^{N} have been studied deeply. Now the following problem is posed: we know that mathbb{R} ^{N} can be embedded in the fuzzy number space E^{N}. Is it then possible to define n-widths of set A in E^{N} and obtain asymptotic estimates for these n-widths?In this paper, we shall introduce four n-widths of A in E^{N} and determine these n-widths of Zadeh’s extension of diagonal matrices.

Some solving methods for a fuzzy multi-point boundary value problem

In this paper, we consider a fuzzy multi-point boundary value problem-FMBVP [or a multi-point boundary value problem (MBVP) for fuzzy second-order differential equations (FSDEs) under generalized Hukuhara differentiability]. We present solving methods for a FMBVP in the space of fuzzy numbers $$E^{1}$$, such that we have shown the ability to and methods to find solution of the MBVP for FSDEs in the form of $$(FH^{gi}-FH^{gj})$$-solutions. In addition, we provide with a new idea to develop the real Green’s function method and give two examples being simple illustration of this FMBVP.