Wu Congxin

Embedding problem of fuzzy number space: Part I

Abstract Using a concrete structure into which we embed the fuzzy number space E1, several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory.

Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions

Kaleva, Fuzzy sets and Systems 35 (1990) 389–396, has concluded that the Peano theorem does not hold for fuzzy differential equations on (En, D). This paper deals with the Cauchy problem of fuzzy differential equations, the existence theorems under compactness-type conditions are obtained.

On the regularity of the fuzzy measure on metric fuzzy measure spaces

Abstract In this paper, the concept of the regular fuzzy measure on metric spaces is introduced, some of its properties are discussed and Lusins theorem is shown. In addition, a condition for which a uniformly autocontinuous set function is a fuzzy measure is given.

Pseudo-atoms of fuzzy and non-fuzzy measures

In this paper, we introduce the concept of pseudo-atoms of set functions, and establish some fundamental conclusions about pseudo-atoms on the assumption of null-null-additivity. We give the necessary and sufficient conditions which insure that the set of non-null-measure contains no pseudo-atoms. Further, we obtain several decomposition theorems about pseudo-atoms and atoms of set functions.

Fundamental convergence of sequences of measurable functions on fuzzy measure space

Abstract The concepts of “fundamental almost everywhere” and “fundamental pseudo-almost everywhere” on fuzzy measure space are introduced, the relations among convergence of sequences of measurable functions are further discussed, the corresponding results on classical measure space are generalized and some of these results are improved in essence.

On the extension of the fuzzy number measures in Banach spaces: Part I. Representation of the fuzzy number measures

Abstract In this paper, the definition of the fuzzy number measures which extends the concepts of the set-valued measures in Banach spaces is introduced. The relations between the fuzzy number measures and the set-valued measures are studied. The definition of the indefinite integral of the random fuzzy numbers is also introduced and its properties are discussed.

Some properties of fuzzy integrable function space L 1 (μ)

Abstract We introduce definitions of fuzzy Orlicz class LΦ(μ) and fuzzy Φ-mean convergence. We show that fuzzy Φ-mean convergence is equivalent to convergence in fuzzy measure and LΦ(μ)=L1(μ) for any Orlicz function Φ. This conclusion refines a result of D. Ralescu (Toward a general theory of fuzzy variable, J. Math. Anal. Appl. 86 (1982) 176–193). Moreover a necessary and sufficient condition for L1(μ)=S(μ) is given. It also improves a result of D. Ralescu and G. Adams (The fuzzy integral, J. Math. Anal. Appl. 75 (1980) 562–570.)

An extension of Sugeno integral

Abstract In this paper, we extend the concept of Sugeno fuzzy integral from nonnegative fuzzy measurable functions to extended real-valued fuzzy measurable functions and discuss the lost genuine properties for this extension; several necessary and sufficient conditions of absolute (S)-integrability for extended real-valued fuzzy measurable functions are given. Moreover, the space (S(μ),ρ(.,.)) of all fuzzy measurable functions will be proved to be a pseudo-metric space under a necessary and sufficient condition. Finally, as an application of this extension the Pettis integral will be established for this kind of fuzzy integral.

A note on the extension of the null-additive set function on the algebra of subsets

Abstract In this work, we point out that the proof of Theorem 2 in [E. Pap, Extension of null-additive set functions on algebra of subsets, Novi Sad J. Math. 31 (2) (2001) 9–13] is incorrect and give a correct proof. Moreover, we also get a corresponding theorem on extension of the weakly null-additive set function.

On the basic solutions to the generalized fuzzy integral equation

Abstract For the fuzzy measure space ( X , A , μ ), the structure of the basic solutions to the generalized fuzzy integral equation >. esh ; x ( φ ( x ) ∧ h ( x )) dμ = β is considered, where h ( x ) is taken as a nonnegative measurable kernel function, β ϵ(0, ∞) is a constant, ∧ denotes the logic multiplication (minimum operator). At first, the existence of the constant basic solution to Eq. (1) is investigated, and the solvable sufficient and necessary conditions to the Eq. (1) are characterized, then the characteristic functional basic solution to Eq. (1) is shown. In addition, two convergent theorems corresponding to the sequences of approximate basic solutions each of which is constituted by the series with the logic addition (maximum operator denoted by ⋁) for the characteristic functions are obtained.For the fuzzy measure space (X, &,p), the structure of the basic solutions to the generalized fuzzy integral equation s x (f(x) A W)dh = P (1) is considered, where h(x) is taken as a nonnegative measurable kernel function, j E (0, co) is a constant, A denotes the logic multiplication (minimum operator). At first, the existence of the constant basic solution to Eq. (1) is investigated, and the solvable sufficient and necessary conditions to the Eq. (1) are characterized, then the characteristic functional basic solution to Eq. (1) is shown. In addition, two convergent theorems corresponding to the sequences of approximate basic solutions each of which is constituted by the series with the logic addition (maximum operator denoted by V) for the characteristic functions are obtained.