Fuzzy Henstock Research Articles

The strong fuzzy variational Henstock multiple integral on n-dimensional fuzzy number space

In this paper, we define strongly fuzzy variational Henstock multiple integral and fuzzy Henstock integral by support functions on n-dimensional fuzzy number space. And we prove these two types of integrals are equivalent. After that, we propose the support-based derivative for strongly fuzzy variational Henstock integral functions and obtain the necessary and sufficient conditions for this differential. By the properties of ACGδ⁎⁎, we use the concepts of inner small variation and singular point covering to derive some characterization theorems of the primitives for strongly fuzzy variational Henstock integral. Finally, the Dominated Convergence Theorem for SFVH integral is obtained in the sense of UACGδ⁎⁎.

On retarded fuzzy functional differential equations and nonabsolute fuzzy integrals

We prove a generalized convergence theorem for fuzzy Henstock integrals based on the sequence of fuzzy-number-valued functions which are weak generalized uniformly absolute continuous (UACG⁎⁎). To widen the applications of this convergence theorem, we provide some existence theorems for the generalized solution for retarded fuzzy functional differential equations and the dependence of the solutions on a parameter under the assumption of the strong generalized derivative of the solution. Our results generalize existence results in a Kaleva integral setting to a fuzzy Henstock integral setting.

Fuzzy Laplace transform based on the Henstock integral and its applications in discontinuous fuzzy systems

The existing results on the fuzzy Laplace transform and their applications were based on Zaheh's decomposition theorem and were formally characterized by the integrals of real-valued functions directly. That is, the existence of the fuzzy Laplace transform in essence has not been solved. In this article, the fuzzy Laplace transform is incorporated into the framework of the Henstock integral and proposed by use of fuzzy Henstock integrals on infinite intervals. In addition, as a theoretical basis, the existence and the basic properties of the fuzzy Laplace transform are investigated, the convolution of fuzzy-valued functions and real-valued functions is defined, and the convolution theorem of the fuzzy Laplace transform is given. Finally, discontinuous fuzzy initial value problems and two kinds of fuzzy Volterra integral equations are discussed with use of the fuzzy Laplace transform presented in this article.

ON CONVERGENCE THEOREMS FOR FUZZY HENSTOCK INTEGRALS

The main purpose of this paper is to establish different types of convergence theorems for fuzzy Henstock integrable functions, introduced by Wu and Gong cite{wu:hiff}. In fact, we have proved fuzzy uniform convergence theorem, convergence theorem for fuzzy uniform Henstock integrable functions and fuzzy monotone convergence theorem. Finally, a necessary and sufficient condition under which the point-wise limit of a sequence of fuzzy Henstock integrable functions is fuzzy Henstock integrable has been established.

Controlled convergence theorems for infinite dimension Henstock integrals of fuzzy valued functions based on weak equi-integrability

In this paper, we give the concept of weak Henstock equi-integrability for a sequence of fuzzy-number-valued functions. Under this notion, we investigate a new version of the Henstock's Lemma of fuzzy-number-valued functions. Thus, it is possible to discuss the controlled convergence theorems of fuzzy Henstock integral in sense of Vitali covering. Moreover, we prove that a uniform version of Sklyarenko's and Lusin's integrability condition of fuzzy Henstock integrals together with pointwise convergence of a sequence of integrable functions is sufficient for a convergence theorem of fuzzy Henstock integrals. As the applications of the controlled convergence theorem, we discuss the existence theorems of generalized solution for a class of discontinuous fuzzy differential equations.

On generalized solutions for discontinuous fuzzy differential equations and strong fuzzy Henstock integrals

On generalized solutions for discontinuous fuzzy differential equations and strong fuzzy Henstock integrals

A decomposition theorem for Banach space valued fuzzy Henstock integral

We establish the following decomposition theorem for fuzzy mappings with values in a Banach space: a fuzzy mapping is fuzzy Henstock integrable if and only if it can be represented as a sum of a fuzzy McShane integrable fuzzy mapping and of a fuzzy Henstock integrable fuzzy mapping generated by a Henstock integrable function.

Fuzzy Integral Equations and Strong Fuzzy Henstock Integrals

By using the strong fuzzy Henstock integral and its controlled convergence theorem, we generalized the existence theorems of solution for initial problems of fuzzy discontinuous integral equation.

Fuzzy spline interpolation with optimal property in parametric form

In this paper we propose a new fuzzy interpolation procedure that uses cubic splines with an optimal property obtained in parametric form. This fuzzy interpolation operator is also a linear operator of uniform approximation that preserves the monotonicity. Moreover, by integrating this fuzzy cubic spline we obtain the trapezoidal quadrature rule for fuzzy Henstock integrable functions.

Discontinuous fuzzy Fredholm integral equations and strong fuzzy Henstock integrals

By using the properties of strong fuzzy Henstock integrals, the existence theorems of solution for a kind of the discontinuous fuzzy Fredholm integral equations are established. The results are generalizations of earlier investigation for fuzzy continuous systems.

The Strong Fuzzy Henstock Integrals and Discontinuous Fuzzy Differential Equations

We generalized the existence theorems and the continuous dependence of a solution on parameters for initial problems of fuzzy discontinuous differential equation by the strong fuzzy Henstock integral and its controlled convergence theorem.

A decomposition theorem for the fuzzy Henstock integral

We study the fuzzy Henstock and the fuzzy McShane integrals for fuzzy-number valued functions. The main purpose of this paper is to establish the following decomposition theorem: a fuzzy-number valued function is fuzzy Henstock integrable if and only if it can be represented as a sum of a fuzzy McShane integrable fuzzy-number valued function and of a fuzzy Henstock integrable fuzzy number valued function generated by a Henstock integrable function.

The Henstock–Stieltjes integral for fuzzy-number-valued functions

In this paper, we firstly define and discuss the Henstock–Stieltjes integral for fuzzy-number-valued functions which is an extension of the usual fuzzy Riemann–Stieltjes integral. In addition, several necessary and sufficient conditions of the integrability for fuzzy-number-valued functions are given by means of the Henstock–Stieltjes integral of real-valued functions and Henstock integral of fuzzy-number-valued functions. Secondly, the continuity and the differentiability of the primitive for the fuzzy Henstock–Stieltjes integral are discussed. We find that there exists a fuzzy-number-valued function which is fuzzy Henstock–Stieltjes integrable, but whose primitive is not α-differentiable almost everywhere. Thirdly, we introduce some quadrature rules for the fuzzy Henstock–Stieltjes integral by giving error bounds for the mappings of bounded variation and of Lipschitz type. We also consider the generalization of classical quadrature rules, such as midpoint-type, trapezoidal and Simpson’s quadrature. Finally, we propose the concept of weak equi-integrability for sequences of fuzzy Henstock–Stieltjes integrable functions. Under this concept, we prove two convergence theorems for sequences of the fuzzy Henstock–Stieltjes integrable functions. At the same time, the formula of integration by parts is also studied.

The numerical calculus of expectations of fuzzy random variables

In this paper, the concept of the fuzzy Henstock integral on infinite interval is proposed in order to calculate the expectation of fuzzy random variables. Two calculating methods are proposed: one is to calculate directly by the fuzzy Henstock integral proposed in this paper, including quadrature rules and the error estimates such as the midpoint-type rule, trapezoidal-type rule, Simpson's formula, δ -fine quadrature rules and their error estimates; another is to calculate by using the equivalent characteristic of fuzzy Henstock integrability, whose membership function could be obtained by solving a nonlinear programming problem.

On the problem of characterizing derivatives for the fuzzy-valued functions (II): almost everywhere differentiability and strong Henstock integral

In this paper, the concept of strong fuzzy Henstock integrability for the fuzzy-valued functions is presented; a necessary and sufficient condition of almost everywhere differentiability for the fuzzy-valued functions is given by means of this concept.

Bounded variation, absolute continuity and absolute integrability for fuzzy-number-valued functions

In this paper, the concepts of bounded variation and absolute continuity for the fuzzy-number-valued functions are presented and discussed by means of the representation of the absolute value for fuzzy numbers. The relations among bounded variation, absolute continuity, Kaleva's integral and fuzzy Henstock integral are characterized. Especially, the representation of absolute continuous fuzzy valued functions is given by Henstock (Kaleva) integral of absolutely Henstock (Kaleva) integrable fuzzy-number-valued functions.

On the problem of characterizing derivatives for the fuzzy-valued functions

In this paper, the concept of locally mean fuzzy Henstock integrability for the fuzzy-valued functions is presented; several necessary and sufficient conditions of differentiability for the fuzzy-valued functions are given by means of this concept.