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

Volume 10, Issue 4, pp 2181--2195 http://dx.doi.org/10.22436/jnsa.010.04.70

Publication Date: April 20, 2017 Submission Date: July 11, 2016

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Authors

Ya-Bin Shao - School of Science, Chongqing University of Posts and Telecommunications, Nanan, 400065, Chongqing, P. R. China. Zeng-Tai Gong - College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, P. R. China. Zi-Zhong Chen - College of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Nanan, 400065, Chongqing, P. R. China.

Abstract

In this paper, under the notion of strong uniformly \(AC^\nabla\) of fuzzy-number-valued functions, we prove a generalized controlled convergence theorem of strong fuzzy Henstock integral. As the applications of this convergence theorem, we provide sufficient conditions which guarantee the existence of generalized solutions to initial value problems for the fuzzy differential equations by using properties of strong fuzzy Henstock integrals under strong GH-differentiability. In comparison with some previous works, we consider equations whose right-hand side functions are not integrable in the sense of Kaleva on certain intervals and their solutions are not absolute continuous functions.

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Keywords

• Fuzzy number
• strong fuzzy Henstock integral
• generalized controlled convergence theorem
• fuzzy differential equations
• generalized solution.

MSC

•  03E72
•  26A39
•  46G05

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