By H. Majima

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The proof of this theorem is done by the same argument as above. The assertion (i) was proven in [M]. 5. (vanishing theorem of global version of non-commutative case) The following statements are equivalent. If HI(M,GL(m,~)) is trivial~ then all are valid: (i) the image of the mapping from HI(M-,GL(m,~-)) or to HI(M-,GL(m,~-)) (ii) t_9_oHI(M-,GL(m,~'-)) is the neutral element, HI(M-,GL(m,~-)I ) is isomorphic to HO(M,GL(m,~MIH))/H0(M,GL(m,~)) m as a set, (iii) the kernel of the mapping from HI(M-,GL(m,~'-)) ^ HI(M-,GL~m,-Jr '--)) t__9oH 1 (M- ,pr~ (GL(m,~IH))) (or from (or to HI~M-,GL~m,~ _ .

A~-... ))) is the neutral element. 3 except for the difference between the additive operation and the (non-commutative) multi plicative operation. 15), because we can obtain them by replacing~-, ~'-, 0 etc. , respectively. 4. ASYMPTOTIC d-POINCARE'S LEMMA AND FURTHER PROPERTIES OF STRONGLY ASYMPTOTIC EXPANSIONS. 3. We denote by d the exterior derivative on M and use the same notation for the exterior derivative on M-. ~'-~q(~H) ~. pr~q(~H) pr @ = ~'-~) , ~ pr ~q(~H) pr ~ pr~ q = AO- ~ ~ pr~q(~H) .

As we recently noticed, a work of Sibuya in 1968 [70] also suggested research in this direction. In the following sections, in the notation of Part I, we prove existence theorems of integrable systems of partial differential equations of the first order under certain general conditions by developing Hukuhara's method [28]. In order to construct formal power series solutions of systems of differential equations, we provide analogues for systems of algebraic equations. ,x n as parameters. Xn,, e i ~ u = ai(x,u) i=l .....