Some Inequalities on a System of Solutions of Linear Simultaneous Differential Equations,
by
Taiiti KITAMURA, Sendai.
1. Let us consider a system of linear simultaneous differential equations
(1)
in a≦t≦b, where aik(t) and bi(t) are all continuous. For the sake of brevity, we put
X
t)=(xi(t)),A(t)=(aik(t)),B(t)=(bi(t)),
and similarly for B(t), and
then we can write the equation (1) in the form (1')
In this paper, we study some inequalities on a system of con-tinuous solutions of (1), which has at mast a finite number of zero points in a<t<b.
Now let us consider a relation of such X(t). Since
and for
(2)
SOME INEQUALITIES ON A SYSTEM OF SOLUTIONS ETO. 309
we get
(4)
consequegtly by Holde's inequality,
(5)
Therefore from (2), for a system of continuous solution of (1) which
has at most a finite number of zero points, are
continuous and (6)
except these points.
2. Now from (1') and the definition of A(t) (7)
Therefore we have except at most a finite number of points in
a<t<b
that is (8)
Now from the second inequality of (8), we get (8')
3 10 TAIITI KITAMURA;
and by integrating both sides from to (α≦t0<t≦b
that is
(9)
Similarly, from the first enequality of (8), we get (10)
Therefore from (9) and (10), we obtain (11)
for a≦t0<t≦b.
Now we consider the case where the equal signs of (11) occur. From (2), (3), (4) and (5) it is seen that the equal sign of (6) occurs when and only when
xi
(t)=cix(t) (ci being any constants and x(t)>0.) and
and moreover the first equal sign of (7) occurs when and only when
Σn v-1 aiv(t)xv(t)/bi(t) are all equal to a fixed positive function, whence
(1)becomes (1")
k(t) being a positive function, consequently, since bi(t)/ci are all equal
to a fixed function β(t)(say),(1") become
(12)
SOME INEQUALITIES ON A SYSTEM OF SOLUTIONS ETC. 311
(13)
And we see that the second equal sign of (7) occurs when and only when
whence we get
Therefore from x(t)>0, k(t)>0 and (12) it follows that according to the case x(t) x'(t)>0 or<0 we have β(t)>0 or<0, consequently
k(t) β(t)=A(t) or_A(t), and also in this case from (3) we
hav e
Therefore from (13) we see that the second or the first equal sign of (11) occurs when and only when the solution of (1) is
(14)
Especially, when B(t)=0, (11) and (14) become
and
which is Mr. Toyama's result(2).
Remark. For p=2, it can be seen that by defining A(t) as
instead of (11) does hold(3).
(Received the 5th March, 1948.)
(1) For this result Prof. Y. Okada kindly adviced me.
(2) H. TByama: TBhoku Math. Jour.. Vol. 47 (1940), pp. 210-216.
(3) See J. H. M. Wedderburn: 'Bull. Amer. Math. Soc., vol. 31 (1926), pp. 304-3 08; especially p. 305.
