0387986995Basic TheoryC

CHAPTER it

DEPENDENCE ON DATA

The initial-value problem

The right-hand side of (E) is regarded as a function of (t, to, ca, f, yj. In this chapter, we explain some basic properties of solutions with respect to (t, to, ca, f). A parameter a is used to represent the variable f. In §11-1, we explain the continuity of the solution of (P) with respect to (t, to, c"o, e) at (t, r, e, co) when the solution of

(P) is unique for the initial data (r, c', ea) (Theorem 11-1-2). Theorem 11-1-2 was first proved by I. Bendixson [Benl] for the scalar equation and later by G. Peano

(Pea2] and E. Lindelof [Lind2] for a system of equations. In §11-2, we explain the differentiability with respect to initial data (to,ca) (Theorem 11-2-1) and with respect to a parameter a (Theorem 11-2-2). The discussion of these topics can be found in [CL, pp. 22-28, 57-60] and [Har2, pp. 94-100].

II-1. Continuity with respect to initial data and parameters

Let e be a variable which takes values in a set with a separable topology. We consider RI-valued functions f (t, y, e) and fi(t) under the following assumption.

Assumption 1. For the initial-value problem (P), assume that

(i) f(t, y", e) is continuous in (t, y, e) in an open set Do in the (t, y, e)-space,

(ii)¢'(t) = f (t, 6(t), co) and (t, ¢(t), eo) E Do on an interval Zo = {t : tt < t < t2}, where eo is a fixed value of the variable e,

Do on a subinterval I of the interval Zo and if y(r) = fi(r) at a point r of the interval Z, then L,(t) = ¢(t) identically on the interval Z.

We start our discussion with the following lemma.

Lemma II-1-1. Conditions (i) and (ii) of Assumption 1 imply that there exist an open set A in the (t, y")-space and an open neighborhood llo of co in the a-space such that

(a)A x Uo C Do,

(b)(t, q',(t)) E A on the interval Zo,

28

Proof.

Let r E Zo. Since f is continuous in the open set Do and (r, ¢r(r), co) E Do, there exist an open neighborhood 0(r) of (r, m(r)) in the (t, yl-space and an open neighborhood U(r) of to in the a-space such that i(r) x U(r) C Do and

I f (t, y, t) - AT, fi(r), to) I < 1 on 0(r) x U(7-). This implies that I f (t, y e) I < 1 +Mo

on U 0(r) x U(r), where M o = m ax I f (r, fi(r), to)I. Since (t, fi(t)) E U 0(r)

on the interval Zo and the interval Zo is compact, there exists a finite set (r, : j =

1, 2, ... , N) of points on the interval Zo such that (t, m(t)) E U A(rt) for all

Then, all the requirements of Lemma II-1-1 are satisfied by 0, Uo, and M. 0

The main purpose of this section is to prove the following theorem under Assumption 1.

Theorem II-1-2. Let E be an open neighborhood of to and let Z = {t : s1 < t < s2} be a subinterval of Zo. Assume that

Assume also that there exists a real-valued function t(e) such that

(1)t(e) E I fore E E,

(2)lim t(e) = to, t-w

(3)hm 0(t(e), e) = (to)

Then:Then, -to

lim i/i(t, t)

it to

uniformly on the interval Z.

Proof.

We shall prove this theorem in two steps. In Step 1, we prove Theorem 11-1-2

assuming that there exists an open neighborhood U of Co such that (t, tb(t, e)) E 0 on Z x (E n U), where 0 is the open set in the (t, y)-space which was determined

by Lemma II-1-1. The existence of such an open neighborhood U of to is proved in Step 2.

Step 1. We can assume without any loss of generality that U is contained in the neighborhood Uo of to which was determined by Lemma 11-1-1. Furthermore, using

condition (3) of Theorem 11-1-2, we choose U in such a way that It (t((),e)I is

bounded on U.

MI t - rI if (t, c) and (r, E) E Z x (£ n U), where M is the positive constant which was determined by Lemma II-1-1. Therefore, the set F _ E) : E E £ n U} is bounded and equicontinuous on the interval Z.

Let us derive a contradiction from the assumption that ,JJ(t, e) does not converge to fi(t) uniformly on the interval I as e -+ co. This assumption means that

(II.1.3)

m Zx 1 (t,Ej) - v(t)I > p.

Here, use was made of the assumption that the topology of the E-space is separable.

Observe that the sequence {

this sequence is bounded and equicontinuous on Z. Therefore, b y Lemma 1-2-

j., -+ +oo as v - +oo and that tp(t, Ej converges uniformly on I as v -+ +oo.

Set l(t) = lim r%(t, on Z. Since (11.1.2) holds for c = Ef for all v, we obtain

sumption 1 implies that j(t) = fi(t) identically on the interval Z. This, in turn,

Step 2. We shall prove that there exists an open neighborhood U of co such that

UClloand (t,1i(t,E))EAon Ix(EnU).

Choose no > 0 so that the rectangular region

Ro(r) = {(t.: It - rl < ao, (y - (r)I <-Mao}

is contained in A for every r E I (cf. Figure 1).

FIGURE 1.

For every positive number a less than ao, set

R((T, ti7); a) _ {(t, yl: It-.I < a, Il! - i,-I «sa).

Then, ?Z((T, f ); a) C Ro(r) if IT - T 15 ao - a, I i - (r)I < M(ao - a). Fixing a positive number a less than ao, choose TI, T2,. .. , TN so that

s1 = T1 < 72 < ... < TN = s2,

Since lira t(e) = to and lim e) = (to), there exists a neighborhood U1 of

co such that It(e) - tol < ao - a and I>%,(t(e), e) - 4(to)I < M(ao - a) fore E E nU1.

Hence, R((t(e), t%i(t((), e)); a) C Ro(to) C 0ifore(t(e),E E n U1. Now, using Lemma I-1-

1, we can verify that (t, fl(t, e)) E R((t(e), e)); a) C i for It - t(e)I < a and

e E EnU1. Also, since lim t(e) = to = rjo, we have It-t(e)I < a for rjo_1 < t < rjo+i

!o

and e E E n U1, if the neighborhoodt- U1 of co is sufficiently small.

Using the same argument as in Step 1 for t (t, e) on the interval T,._ j 5 t < T3.+1

and e E E n U1, we obtain lim t(t, e) = fi(t) uniformly on rjo_ 1 < t < 7-,,.+,. In

.1 to

4(Tjo+i )), we obtain lim 1G(t, e) = Q(t) uniformly on rjo-2 - t <- rjo and r < e-co

t < rjo+2. Hence, lim ti(t, e) = q(t) uniformly on To-2 < t <- Tjo+2. Contin-

(5p

uing this process, we find an open neighborhood U of co such that U C Uo and

(t,>G(t, e)) E i on I x (E nU) in a finite number of steps. 0

Remark 11-1-3. If f (t, y, () is independent of e, Theorem 11-1-2 yields continuity

of solutions of the initial-value problem f = f (t, y-), 9(to) = 4 with respect to the initial data (to, Q.

In Theorem II-1-2, it was assumed that (t, y(t, e), e) E Do on Z x E (cf. (II.1.1)).

With regard to this assumption, we can improve Theorem 11-1-2 as follows.

Theorem II-1-4. Assume that Assumption 1 is satisfied. Let r be an open neighborhood of cc). Assume further that there exist functions t(e) and c(e) defined on E such that

(i) t(e) E Zo and (t(e), c 1c), e) E Do fore E E,

(ii) llm t(e) = to and lim dde) = 0(to).

`-'a cco

Then, we can choose a neighborhood U of eo sufficiently small so that the initial-

value problem

f (t,

has on the region lo x (E f1 Ll) a solution y = J(t, e) such that (t, 111(t, E), E) E Do.

Theorem II-1-4 can be proved by using an argument similar to Step 2 of the proof of Theorem 11-1-2 and the local existence theorem (cf. Theorem 1-2-5). Details are left to the reader as an exercise.

11-2. Differentiability

In this section, we explain the differentiability of solutions of an initial-value problem with respect to the initial data under the following assumption.

Assumption 2. For the initial-value problem (P), assume that

(i) f (t, y-) and Lf' (t, y-) (j = 1, 2, ... , n) are continuous in (t, y-) on an open set

A in the (t, yj-space,

Using Lemma 1-1-6, Theorems 1-1-4, 11-1-2, II-1-4, and Remark 11-1-3, we can show that there exists a positive number p such that the initial-value problem

has a unique solution y' = O(t, r, q') on the interval lo if r E Zo and fr) - ¢(r)j < p. The solution y = m(t, r, ri") is continuous in (t, r, q-) and satisfies the condition

where q, is the j-th entry of i), eJ is the vector in IR" with entries 0 except for I at

0Yj

speculation, we obtain the following result.

existence and continuity of L can be proved similarly.

d7),

Let +;1(t, r, ij) be the unique solution of (11.2.3) with the initial condition z(r)

- f (r, r)). From Theorems 1-3-5 and 11-1-2 and Remark 11-1-3, it follows that 1i(t, r, n) exists and is continuous in (t, T,i) on the closure D of domain (11.2.2).

Furthermore,

from (11.2.1) on D assuming that h 54 0 is sufficiently small. Using (11.2.4) and

as follows.

Theorem 11-2-2. Let a be a real variable. Assume that At, y, e), a (t, y, e) (,j =

1, 2, ... , n), and 8e (t, y', e) are continuous on an open set Do in the (t, y, e)-space. Let y = y(t, T, i), e) be the unique solution of the initial-value problem

Then, there exists an open set 11 in the (t, T, 77, e)-space such that the function (t, T, ii, e) and 00 are continuous on fl. Furthermore, i = 00 (t, r, 4J, e) is the unique solution of the initial-value problem

II-1. Given an interval I = {x : a < x < b}, show that if f ei is sufficiently small, the initial-value problem

dx

has the unique solution y = O(x, e) on Z. Show also that Ui m O(x, e) = x uniformly

on Z.

Hint. The function cos(x(y - x)) is continuously differentiable on the entire (x, y)- plane. Furthermore, I oos(x(y - x))I < 1. This shows the existence of the unique solution. Note that y = x is the unique solution in case e = 0. Complete the proof

by using Theorem 11-1-2.

11-4. Let Ax, y-) be an RI-valued function whose entries are continuously differ-

for 0 < x < 1 and it E Ao, where Do is a domain in R. Note that we must have

(0, rl E D for rj E Ao. Set A = {(1, 1 : if E Do}. Show that

(a)Al is open in lR',

(b)the mapping 4(1,r) : Ao - Al is one-to-one, onto, and differentiable with respect to i in Do,

is also differentiable with respect to S.

11-5. For the differential equation

(1 + x2) LY -_ 2xy = 2.sy2,

find

(i) the unique solution y = O(x, e, ri) satisfying the initial-condition y(£) = q;

equation:

(I+ x2) dzdz - 2xz = 4xO(x, {, q)z .

II-6. Assume that f (t, x1, x2i ... , xn) is a real-valued, continuous, and continuously differentiable function of (t, xl,... , x,) on an open set A in the (t, x1.... , xn)-

space. Assume also that o(t) is a real-valued solution of the n-th order differential equation x(") = f(t,x,x',... ,x(n-1)) and (t,Oo(t),e0(t),... ,0On-1)(t)) E A on an

interval Zo = It : t1 < t < t2}. Show that there exists a positive number p such that

(j=1,2,...,n);