# THE STRONG PERRON INTEGRAL IN THE n-DIMENSIONAL SPACE ℝn

• PARK, JAE-MYUNG (Department of Mathematics Chungnam National University) ;
• KIM, BYUNG-MOO (Department of Mathematics Chungju National University) ;
• LEE, DEUK-HO (Department of Mathematics Education Kongju National University)
• Published : 2005.04.01

#### Abstract

In this paper, we introduce the SP-integral and the $SP_\alpha-integral$ defined on an interval in the n-dimensional Euclidean space $\mathbb{R}^n$. We also investigate the relationship between these two integrals.

#### References

1. B. Bongiorno, L. Di Piazza, and V. Skvortsov, On continuous major and mi- nor functions for the n-dimensional Perron integral, Real Anal. Exchange 22 (1996/1997), no. 1, 318-327
2. B. Bongiorno, On the n-dimensional Perron integral defined by ordinary derivatives, Real Anal. Exchange 26 (2000/2001), no. 1, 371-380
3. R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Amer. Math. Soc., 1994
4. J. Kurzweil and J. Jarnik, Equivalent definitions of regular generalized Perron integral, Czechoslovak Math. J. 42 (1992), 365-378
5. J. Kurzweil, Differentiability and integrability in n dimensions with respect to $\alpha$-regular intervals, Results Math. 21 (1992), no. 1-2, 138-151 https://doi.org/10.1007/BF03323075
6. M. P. Navarro and V. A. Skvortsov, On n-dimensional Perron integral, South- east Asian Bull. Math. 20 (1997), no. 2, 111-116
7. K. M. Ostaszewski, Henstock Integration in the Plane, vol. 353, Mem. Amer. Math. Soc., 1986
8. Jae Myung Park, The Denjoy extension of the Riemann and McShane integrals, Czechoslovak Math. J. 50 (2000), no. 125, 615-625 https://doi.org/10.1023/A:1022845929564
9. S. Saks, Theory of the Integral, Dover, New York, 1964
10. V. A. Skvortsov, Continuity of $\delta$-variation and construction of continuous major and minor functions for the Perron integral, Real Anal. Exchange 21 (1995/1996), no. 1, 270-277