### Journals Information

**
Mathematics and Statistics Vol. 10(5), pp. 1038 - 1049 DOI: 10.13189/ms.2022.100515 Reprint (PDF) (431Kb) **

## Henstock - Kurzweil Integral for Banach Valued Function

**T. G. Thange ^{1}^{,*}, S. S. Gangane ^{2}**

^{1}Department of Mathematics,Yogeshwari Mahavidyalaya, Ambajogai,Dist.Beed, Maharashtra state, 431517, India

^{2}Department of Mathematics,Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, Maharashtra State - 431004, India

**ABSTRACT**

In this paper, we have studied the Henstock - Kurzweil integral which is a generalized Riemann integral means. Hen-stock - Kurzweil integral is the natural extension of Riemann integral. We defined Henstock - Kurzweil integral of Banach space valued function with respect to a function of bounded variation which is an extension of real valued Henstock - Kurzweil integral with respect to an increasing function. We investigated elementary properties of the Henstock - Kurzweil integral of Banach space valued function with respect to a function of bounded variation. We proved the convergence theorems and Saks - Henstock lemma of the Henstock - Kurzweil integral of Banach valued functions with respect to a function of bounded vari-ation. Equi-integrability with respect to Banach space valued function is defined and equi-integrable theorem of Henstock - Kurzweil integral of Banach space valued function with respect to a function of bounded variation is proved. Finally Bochner Henstock - Kurzweil integral of Banach valued function with respect to a function of bounded variation is defined and the relation between Bochner Henstock - Kurzweil integral and Henstock - Kurzweil integral is exhibited.

**KEYWORDS**

λ Henstock - Kurzweil Integral, Banach Space, Bounded Variation Function, Bochner Integral

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] T. G. Thange , S. S. Gangane , "Henstock - Kurzweil Integral for Banach Valued Function," Mathematics and Statistics, Vol. 10, No. 5, pp. 1038 - 1049, 2022. DOI: 10.13189/ms.2022.100515.

(b). APA Format:

T. G. Thange , S. S. Gangane (2022). Henstock - Kurzweil Integral for Banach Valued Function. Mathematics and Statistics, 10(5), 1038 - 1049. DOI: 10.13189/ms.2022.100515.