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This article studies some convergence results for the McShane integral of functions mapping the interval [0, 1] into a Banach space X from the point of view of an open problem proposed by D.H. Fremlin and J. Mendoza in [2], also the anthors give a negative answer to this open problem.

In this paper, we prove two integration by parts formulas for the Denjoy-Bochner integral, and give a representation theorem for the space of Denjoy-Bochner integrable functions.

We give the Riemann-type extensions of Dunford integral and Pettis integral, Henstock-Dunford integral and Henstock-Pettis integral. We discuss the relationships be- tween the Henstock-Dunford integral and Dunford integral, Henstock-Pettis integral and Pettis integral. We prove the Harnack extension theorems and the convergence theorems for Henstock-Dunford and Henstock-Pettis integrals. 2000 Mathematics Subject Classification. Primary 26A39, 28B05; Secondary 28B20, 46G10, 46G12. 1. Introduction. During 1957-1958, R. Henstock and J. Kurzweil, independently, gave a Riemann-type integral called the Henstock-Kurzweil integral (or Henstock inte- gral) (see (7)). It is a kind of nonabsolute integral and contains the Lebesgue integral. It has been proved that this integral is equivalent to the special Denjoy integral (7). The Dunford, Pettis integrals are generalizations of the Lebesgue integral to Banach- valued functions. In (5), R. A. Gordon gave two Denjoy-type extensions of the Dunford, Pettis integrals, the Denjoy-Dunford and Denjoy-Pettis integrals, and discussed their properties. In this paper, we give the Riemann-type extensions of Dunford, Pettis integrals, the Henstock-Dunford, Henstock-Pettis integrals, and discuss the relationships between the Henstock-Dunford integral and Dunford integral, Henstock-Pettis integral and Pet- tis integral. We prove the Harnack extension theorems and the convergence theorems for Henstock-Dunford and Henstock-Pettis integrals. Throughout this paper, X denotes a real Banach space and X ∗ its dual. B(X ∗ ) =