# ON CONVERGENCE THEOREMS FOR FUZZY HENSTOCK INTEGRALS

Document Type: Research Paper

Author

Department of Mathematics, Sripat Singh College, Jiaganj-742123, Murshidabad, West Bengal, India

Abstract

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.

Keywords

### References

[1] R. G. Bartle, A convergence theorem for generalized Riemann integrals, Real Anal. Exchange,
20(2) (1994-95), 119{124.
[2] B. Bongiorno, L. Di Piazza and K. Musia l, A decomposition theorem for the fuzzy Henstock
integral (I), Fuzzy Sets and Systems, 200 (2012), 36{47.
[3] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986),
31{43.
[4] Z. Gong, On the problem of characterizing derivatives for the fuzzy-valued functions (II):
almost everywhere diff erentiability and strong Henstock integral, Fuzzy Sets and Systems,
145 (2004), 381{393.
[5] Z. Gong and Y. Shao, The controlled convergence theorems for the strong Henstock integrals
of fuzzy-number-valued functions, Fuzzy Sets and Systems, 160 (2009), 1528{1546.
[6] Z. Gong and L. Wang, The Henstock-Stieltjes integral for fuzzy-number-valued functions,
Inform. Sci., 188 (2012), 276{297.
[7] Z. Guang-Quan, Fuzzy continuous function and its properties, Fuzzy Sets and Systems, 43
(1991), 159{171.
[8] R. Henstock, Theory of Integration, Butterworths, London, 1963.
[9] J. Kurzweil, Generalized ordinary di erential equations and continuous dependence on a
parameter, Czechoslovak Math. J., 7(82) (1957), 418{446.
[10] Ma Ming, On embedding problem of fuzzy number space: Part 4, Fuzzy Sets and Systems,
58 (1993), 185{193.
[11] K. Musia l, A decomposition theorem for Banach space valued fuzzy Henstock integral, Fuzzy
Sets and Systems, 259 (2015), 21{28.
[12] C. Wu and Z. Gong, On Henstock integral of fuzzy-number-valued functions, Fuzzy Sets and
Systems, 120 (2001), 523{532.
[13] C. Wu and Ma Ming, On embedding problem of fuzzy number space: Part 1, Fuzzy Sets and
Systems, 44 (1991), 33{38.