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1 Tohoku Math. Journ. Vol. 19, No. 1, 1967 ON THE EXISTENCE OF O-CURVES*) JUNJI KATO (Received September 20, 1966) Introduction. In early this century, Perron [5] (also, cf. [1]) has shown the followings : There exists a k-parameter family of solutions, which tend to zero as time increases infinitely, of the system (E) x-ax+f(t,x), under the conditions (c1) k characteristic roots of the constant matrix A have negative real parts and (c2) f(t, x) is continuous and of the class o( x ) for x small and t large, that is, for any ƒã>0 there exist T 0 and ƒâ> 0 such that f(t,x) ƒã x if t T and x ƒâ, where x denotes a Euclidean norm. Moreover, if the condition (c3) for given ƒã>0, there exist ƒâ>0 and T 0 such that f(t,x)-f(t, y) ƒã x-y if t T, x ƒâ, y 6 is satisfied, then for any given solution x*, (t) of the system x=ax remaining near the trivial solution, we can find a solution x(t) of (E) satisfying *) This work was partly supported by the Sakko-kai Foundations.

2 J. KATO x(t)-x*(t) 0 as t. In this article, a solution which tends to zero as t will be called an 0-curve, and we shall discuss a similar problem to the above, that is, the existence of O-curves of the system (E) by replacing the conditions (c2) and (c3) by the condition (c4) there exists a continuous function ƒé(t, ƒ ) such that f(t, x) ƒé(t, a),if x ƒ and that (L) for any ƒ 0, where p is some positive integer which will be given in theorems. Here, we shall apply similar arguments to those used in the previous paper [3]. As was stated in [1], under the conditions (c1), (c2) and (c3), such k- parameter family of O-curves consists a k-manifold, and moreover, it is obvious that if A has a characteristic root with the positive real part, then the trivial solution of (E) is unstable under the condition (c2). However, in our case, even if A has a characteristic root with the positive real part, and even if f(t,0) = 0 for all t 0, the trivial solution of (E) is not necessarily unstable (see Example). We can discuss the same problem for more general systems by assuming the existence of Liapunov functions. However, for simplicity, we shall only consider a linear system and its perturbed system. 1. We shall consider a linear system of differential equations with constant coefficients (1) where x is an n-vector and A is a real (n, n)-matrix. First of all, we shall prove the following lemma. LEMMA 1. There exists a real non-singular matrix P(t) such that both of P(t) and P(t)-t are continuous and bounded on [0, ) and that by the transformation x=p(t)y, the system (1) is transformed into a system

3 ON THE EXISTENCE OF O-CURVES where A' = diag(a1,, Am), A; is an (kj, kj)-matrix of the form (if kj = 1, Aj = ƒ j), k1 + +km = n, and ƒ j is the real part of a characteristic root of A. PROOF. By well-known normalization, there exists a real non-singular constant matrix P1 such that P1-1 1 AP1= diag (A1,..., As, A's+1,, A'r). Here, Aj is an (kj, kj)-matrix of the form (2), where ƒ j is a real characteristic root of A (j= 1,, s), and A'j is an (k'j, k'j)-matrix of the form where k'j is even (if k'j = 2, A'j = Sj), E and O are the unit and the zero, respectively, (2, 2)-matrix, and ƒ j } iƒàj, (ƒàj 0) are characteristic roots of A. Let Q'j(t) be defined by and let Qj(t) be the (k'j, k'j,)-matrix of the form

4 J. KATO Q(t)=diag(Q3(t),, Q'j.(t)) (j=s+1,,r). Putting P2(t) = diag(e1,..., Es, Qs+1(t),, Qr (t)) where Ej is the unite (kj, kj)-matrix, the transformation x = P1P2(t)z transforms the syetem (1) into a system where C = diag(a1,, As, A's+1,, A''r) and A''j is the (k'j, k'j)-matrix of the form Finally, we can find a non-singular matrix R, such that Rj-1 A''j Rj = diag(a*j, A*j), j = s+1,.', r, where A*j is of the form (2) and the order of A; is k'j/2. Let P3 = diag(e1, one. Es,Rs+1,, Rr). Then, the matrix P(t) = P1P2(t) P3 is the required, By Lemma 1, we can find a non-singular real (n, n)-matrix P(t) such that both of P(t) and P(t)-1 are continuous and bounded on [o, oo) and that by the transformation (3) x = P(t) y the system (1) is transformed into a system (4) where B is a constant real (n, n)-matrix of a form

5 ON THE EXISTENCE OF O-CURVES 53 B = diag(b1,, Bl, C, D). Here, By is an (no, n,)-matrix of the form (if nƒë=1, B,=0), C is a (q, q)-matrix and D is a (k, k)-matrix, n1+ +nl+q + k = n. All characteristic roots of C have positive real parts and those of D have negative real parts. Let y be decomposed in the following way and yv= y = (yi,, y1, z, u) (y1v,,ynvv) (v = 1,,l) where yƒë is an nv-vector (ƒë = 1,, l), z is a q-vector, u is a k-vector and yjv is a scalar (j =1,, nƒë ; v=1,, l), Now, assuming that n1 n2 > n1, we shall put (5) where m, is the largest integer v such that l ƒë 1 and nƒë j + 1, Morover, we put m0 =l+ q and ƒöo = ƒë. Obviously, this transformation is non-singular and tranforms the system (4) into a system (6) Here, Ass =D, B!1 is the (m0, m0)-matrix diag (O*, C) and C*j is the (mj, mj-1)- matrix (Ej, Oj) (j =1,, n1-1), where O* is the zero (l, l)-matrix, Ej is the unit (mj, mj)-matrix and 0j is the zero (mj, mj-1 \mj)-matrix (it must be noted that m0 l m1 mn1-1). In the case where l = 0 (or n1 = 1), the system (6) becomes

6 54 J. KATO where A* = D and B* = C (or B* = diag(o*, C)) and ƒë = z (or ƒë = (y11,, y1l, z)). 2. We consider a perturbed system (7) of the system (1). It is the purpose of this article to prove the following theorem. THEOREM 1. We shall denote by p the maximum degrees of the elementary divisors o f characteristic roots of A with zero real parts if such a root exists, and otherwise we shall put p =1. Suppose that f (t, x) is continuous on [0, ) x Rn (Rn represents the Euclidean n-space) and satisfies the condition (c4). Then, there exists at least one O-curve of (7). Furthermore, if A satisfies the condition (c1), then there exists a k- parameter family of O-curves of (7). In the proof of this theorem, we shall apply the following lemma (for the proof, see [4]). LEMMA 2. Suppose that f(t, x, y) and g(t, x, y) are continuous and bounded on [a, b] x Rn x Rm. Then, for given (x0, y0) Rn ~ Rm there exists a solution (x(t), y(t)) of the system such that x(a) = xo and y(b) = y0. 3. Now, we shall prove Theorem 1. As was stated in 1, the system (1) can be transformed into a system of the form (6) by transformations of the forms (3) and (5). By these transformations, the system (7) will be transformed into a system (8)

7 ON THE EXISTENCE OF O-CURVES 55 where w=(ƒö1,,ƒöp-1). In the above, if p=1, the system (8) is considered to be By the properties of these transformations, the condition (c4) implies that F(t, u, v, w), G(t, u, v, w) and H,(t, u, v, w) (j =1,, p -1) are continuous on the domain,d = [0, ) ~ Rk ~ Rm ~ Rn-k-m m = m0, and that there exists a continous function ƒé *(t, ƒ ) satisfying and a F(t,u,v,w) a, ag(t,u,v,w) a, ahj(t,u,v,w) a ƒéx (t, ƒ ) (j= 1,, p-1) far any (t, u, v, w) such that t [0, ) and max { au a, av a, aw a} ƒ. Since all characteristic roots of A* have negative real parts, we can find a Liapunov function V(t, u) which is continuous in (t, u) on [0, ) ~ Rk and which satisfies the conditions i) au a V(t,u) K au a ii) V(t, u) - V(t, u') K au-u' a iii) where K and c are positive constants. On the other hand, the real part of each characteristic root of is positive or zero and in the latter case, the corresponding elementary divisor is linear, and hence there exists a continuous Liapunov function W(t, v) defined on [0, ) ~ Rm and satisfying the similar conditions to (i) and (ii) with the constant K and the condition iv)

8 56 J. KATO For any given ƒ > 0 we can find a constant T = T(ƒ ) 0 such that where Let us define F* (t, u, v, w), G* (t, u, v, w) and H*j (t, u, v, w) by replacing (t, u, v, w) in F(t, u, v, w), G(t, u, v, w) and Hj(t, u, v, w), respectively, by (t, ƒó( au a) u, ƒó( av a) v, ƒó ( aw a) w), where Obviously, F*(t, u, v, w), G*(t, u, v, w) and H*j (t, u, v, w) (j= 1,, p-1) are continuous, and the norms of those are bounded by ƒé*(t, 2Ka) on D. If we set f(t,u,v,w) =ƒó( au a)a*u + F*(t,u,v,w) g(t,u,v,w) = ƒó( av a) B*v + G*(t,u,v,w) and hj(t,u,v,w) = ƒó( aw a)c*j wj-1 + H*j(t,u,v,w) (j=1,...,p-1), these functions are continuous and bounded on D if t is restrained in a finite interval. Therefore, by applying Lemma 2, for any given constants a, b (b> a 0) and u0 Rk, we can find a solution (u(t), v(t), w(t)) of the system such that If we can see that u(a) = u0, v(b) = 0 and w(b) = 0.

9 ON THE EXISTENCE UP O-CURVES 57 ( 9) a u(t) a, av(t) a, a w(t) a 2Kƒ on [a, b], then (u(t), v(t), w(t)) is obviously a solution of the system (8) on [a, b]. Assuming that au0 a < ƒ and a T(ƒ ), (9) will be easily shown : So long as u(t) av(t) a 2Kƒ, we have a (10) d/dt V (t, u (t)) - cv (t, u (t)) + KƒÉ* (t, 2 Kƒ ), d/dt W (t, v(t)) - KƒÉ*(t, 2Kƒ ) for almost everywhere, and hence, these inequalities and the assumptions u(a) a <ƒ, a T(ƒ ), v(b) = 0 a imply that a u(t) a, a v(t) a 2Kƒ on [a, b] and that (11) Further, from the inequalities (11) and it follows that (12) and hence we have aw(t) a 2Ka on [a, b]. For given a and u0 such that a T(ƒ ), u0 Rk and au0 a <ƒ, let (u(t ; s), v(t ; s), w(t ; s)) be a continuous function defined on [a, ) such that it is a solution of the system (8) on [a, s] satisfying the conditions u(a ; s) = u0, v(s ; s) = 0 and w(s ; s) = 0 and that u(t ; s) = u(s ; s), v(s ; s) = 0 and w(t ; s) = 0 for all t s for any given s a. Since we have

10 58 J. ICATO u(t; s) a, av(t; s) a, aw(t a ; s) a 2Kƒ on [a, ) by (9), the family {(u(t ; s), v(t ; s), w(t ; s)) ; s a} are uniformly bounded and equicontinuous on any bounded subinterval of [a, ). From this, it follows that there exists a divergent sequence {SƒÊ}, for which the sequence {(u(t ; SƒÊ), v(t ; SƒÊ), w(t ; SƒÊ))} converges a solution (u(t), v(t), w(t)) of the system (8). Obviously, it satisfies the conditions u(a) = u0 and u(t) a, av(t) a, a w(t) a 2Ka on [ƒ, a ). Moreover, since each (v(t ; s), w(t ; s)) satisfies the conditions (11) and (12) on [a, ), so does (v(t), w(t)). Hence, we obtain v(t) 0, w(t) 0 as t, because Aj(t) 0 as t (j= 1, E E E, p-1). Referring the inequality (10), w e can easily see that u(t) 0 as t. Thus, (u(t), v(t), w(t)) is an O-curve of the system (8). Hence, by the properties of transformations of the forms (3) and (5), we completely prove Theorem Let a (t) be any bounded solution of the system (1) (or (7)), and consider the transformation x = x*(t) + y in the system (7) (or (1), respectively). Then, we have dy/dt = Ay + g(t, y), where g(t, y) = f (t, x*(t) + y) (or -f (t, x* (t))). If f (t, x) in (7) satisfies the condition (c4), then g(t, y) also satisfies the condition (c4), where X(t, a) will be replaced by ƒé(t, ƒ +M) (or X(t, M), respectively) and M is a bound of x*(t). Thus, the following theorem is an immediate consequence of Theorem 1. THEOREM 2, Under the same assumptions in Theorem 1, for any bounded solution x*(t) of one o f the system (1) and (7) there exists a solution x(t) o f the other system such that

11 ON THE EXISTENCE OF O-CURVES 59 x(t) \ x*(t) a 0 as t. a Furthermore, if A satisfies the condition (c1), then there exists a k- parameter family of such solutions. 5. The results above can be extended for functional differential equations. For a fixed constant r > 0, let Cn be the space of all continuous Rn-valued functions defined on [-r, 0], and let = sup{ a aƒó a ƒó(ƒæ) a ; ƒæ [-r,0]} for a ƒó Cn. For any continuous Rn-valued function x(t), we shall denote by xt the function of Cn such that xt(ƒæ)= x(t + ƒæ), ƒæ [-r,0], and by x(t) the right-hand derivative of x(t). In the system (13) x(t) = f (xt) + X(t, xt), we assume that f (ƒó) is a continuous linear function of Cn into Rn and that X(t, q.) is a completely continuous Rn-valued function defined on [0, ) x Cn. Let be a linear operator such that for ƒó Cn, for which the right-hand derivative ƒó(ƒæ) exists for each ƒæ [- r, 0). By referring Hale's result in [2], we can prove the following theorem by using the arguments similar to those in the proof of Theorem 1. The proof is omitted (refer [3]). THEOREM 3. Let p be the maximum dimensions of the eigen spaces of the spectra of t with zero real parts if such a spectrum exists, and let p =1 otherwise. Suppose that there exists a continuous function ƒé(t, ƒ ) satisfying the inequality (L) in the condition (c4) and (t, ƒó) a ƒé(t, ƒ ), a X if aƒó a r ƒ

12 60 J. KATO for any t [0, ) and ƒ 0 and that k spectra of have negative real parts. Then, there exists a k-parameter family of O-curves of the system (13). Moreover, for any given bounded solution x*(t) o f the system (13) or (14) x(t) = f(xt) we can find a k-parameter family o f solutions x(t) of the system (14) or (13) satisfying a x(t) \x*(t) a0 as t*. 6. In the case where f(t, 0) = 0 and A has at least one characteristic root with positive real part, the condition (c4) is not sufficient for the instability of the trivial solution of the system (7). EXAMPLE. We consider a equation 15 dx/dt= ax+f(t,x), (15) Where f(t,x)=(a+b+1)e-bt sin [ebtx] and a, b are positive constants. Obviously, f(t,0)=0 and f(t,x) satisfies the condition (c4) and the Lipschitz condition f(t,x) \f(t, y) (a+b+1) x-y. However, the trivial solution of the system (15) is asymptotically stable. In fact, let y = ebtx. Then, we have (16) dy/dt=(a+b)y-(a+b+1)siny=g(y). Since g(y)=-y+o(y3),the trivial solution of the system (16) is asymptotically stable, and hence so is that of the system (15). However, we shall obtain the following theorem. THEOREM 4. Under the assumptions (c4) with p=1 and (c5) f(t,0) =0 for all t 0,

13 ON THE EXISTENCE OF O-CURVES 61 if there exists a characteristic root of A with the positive real part, then the trivial solution of the system (7) is not uniform-stable. PROOF. Let a be the real part of a characteristic root of A with the positive real part. Then, by Lemma 1 there exists a non-singular matrix P(t) such that P{t} and P(t) 1 are continuous and bounded on [0, } and that the transformation,where y is a scalar and z is an (n- 1)- vector, transforms the system (7) into a system (17) Clearly, there exists a continuous function ƒé*(t, a) such that and that if y ƒ and az a ƒ, then g(t, y,z) ƒé*(t, ƒ ) and ah(t, y, z) a *(t, ƒ ) for all t 0. Now, suppose that the trivial solution of the system (7) is uniform-stable. Then, the trivial solution of the system (17) also is uniform-stable. Hence, for any ƒã>0, there exists a ƒâ(ƒã)> 0 such that for any t0 0, if y0 < ƒâ and z0 a<ƒâ, then y(t ; y0, z0, t0) < ƒã and a az(t ; y0, z0, t0) a< ƒã for all t t0, a where (y(t ; y0, z0, t0), z(t ; yo, z0, t0)) is a solution of the system (17) through (t0, y0, z0). Since ƒé we have For a given y0, y0 0, we can find a T( y0 ) 0 such that Hence, if 0 < yo <ƒâ(ƒã), az0 a <ƒâ(ƒã) and t0 T( y0 ), then we have

14 62 J. KATO y(t ; y0, z0, t0) y0 /2ea(t-to). Hence, there arises a contradiction. Thus, the trivial solution of the system (7) is not uniform-stable. REFERENCES [1] E. A. CODDINGTON AND N. LEVINSON, Theory of ordinary differential equations, McGraw-Hill, New York, [21 J. K. HALE, Linear functional-differential equations with constant coefficients, Contrib. Diff. Eqs., 2(1964), [3] J. KATO, Asymptotic behaviors in functional differential equations, Tohoku Math. Journ., 18(1966), [4] M. NAGUMO, Eine Art der Randwertaufgabe von Systemen gewohnlicher Differentialgleichungen, Proc. Physico-Math. Soc. Japan, Ser. 3, 25(1943), [5] O. PERRON, Uber Stabilit t and asymptotisches Verhalten der Tntegrale von Differentialgleichungssystemen, Math. Zeit., 29(1929), MATHEMATICAL INSTITUTE TOHOKU UNIVERSITY SENDAI, JAPAN.

RESOLVENT OF LINEAR VOLTERRA EQUATIONS

Tohoku Math. J. 47 (1995), 263-269 STABILITY PROPERTIES AND INTEGRABILITY OF THE RESOLVENT OF LINEAR VOLTERRA EQUATIONS PAUL ELOE AND MUHAMMAD ISLAM* (Received January 5, 1994, revised April 22, 1994)