Dual K-quasi-additive Fuzzy Number Valued Integral And Fuzzy Number Valued Integral Based On Fuzzy Structuring Element

The paper mainly has two parts.The first part: at first, on the K- quasi-additive fuzzy measures space, aiming at a kind of(?)-integrable fuzzy number valued functions, the dual K-quasi-additive fuzzy number valued integral is established by using Darboux upper-sum, and the new integral transformation theorem is obtained by introducing inductive operator K. Applying this theorem, some important basic properties are discussed with respect to the K-quasi-additive fuzzy number valued integral. On the other hand, on the condition of quasi-minus operator K, function's integrability is discussed on finite measure spaces. At the same time, a series of convergence theorems are obtained.And then, regarding the whole kind of integrals as a set functions taking valued fuzzy numbers on the measurable spaces, the autocontinuity and converautocontinuity of this kind of fuzzy integrals are discussed by using the integral transformation theorem and some properties of inductive operator. At last, illustrate the application of this kind of integral in the productive practice throuth a specfic example.The second part: by introducing the concept of fuzzy structured element, aiming at the integral of fuzzy-valued function based on the fuzzy structured element,proofing that the integral of the functions which are generated by the same fuzzy structured element is functional additive, positive homogeneous and interval additive at first. The basic theorem of integrable-differential calculus of this kind of new fuzzy valued integral is presented. According to the proof of the Newton-Leibniz formula based on the standard fuzzy arithmetic condition of this new integral,the Newton-Leibniz formula based on the constrain fuzzy arithmetic condition of this new integral is proofed. The Newton-Leibniz formula based on the constrain fuzzy arithmetic condition of this new integral have more in common with the classical Newton-Leibniz formula. Consequently, the theory of the fuzzy integral with respect to fuzzy mathematics are riched.