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

  • Authors:

  • Zengtai Gong;Liangliang Wang

  • Affiliations:

  • College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, PR China;College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, PR China

  • Venue:
  • Information Sciences: an International Journal
  • Year:
  • 2012

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Abstract

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 @a-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.