variation of parameters for nonlinear differential-difference equations

Variation of Parameters for Nonlinear Differential-Difference EquationsAuthor(s): Stuart P. HastingsSource: Proceedings of the American Mathematical Society, Vol. 19, No. 5 (Oct., 1968), pp.1211-1216Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/2036062 .

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VARIATION OF PARAMETERS FOR NONLINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS

STUART P. HASTINGS

The variation of parameters formula is an important tool for the study of both ordinary and retarded differential equations. Recently, a nonlinear extension of this formula, due to Alekseev, has proved useful in studying perturbations of nonlinear ordinary differential equations. (See [1], [2], [3], and [7].) It is the purpose of this note to show that this formula may also be extended to retarded equations, thus generalizing the result given, for example, in [5, p. 366] for the linear case. This allows one to extend to equations with finite lag many of those results in [2], [3], and [7] which assume a bound on the solutions or their "derivatives with respect to initial conditions."

We first discuss the differential-difference equation

(1) *(t) = f(t, x(t), x( - 1))

where x and f are real-valued functions. We assume that f, (df/ax) (t, x, y), and (cf/cy) (t, x, y) are continuous for t ?0 and all x and y. The results to be given below can be extended to systems and to more general functional differential equations with the use of the Frechet derivative. (See, however, the note after Lemma 1 below.)

Let PC= {4/4 is a piecewise continuous real-valued function on [-1, O] }. Then it is well known that for each CPC and for each to>0, there is an E>0 and a unique function x(., to, q!) defined on [to-1, to+E] and continuous on [tot to+e] such that

x(to + , to,)=(), -1 < 0 <

and

(ax/at) (t, 4,, 4) = ?(t, t0 4) = f(t, x(t, to, 4)), x(t-1, 4, I )))

almost everywhere on [to, to+E]. (In general, a " and "a/as" will denote two-sided derivatives except at the left or right endpoints of a closed interval in which the function involved is differentiable, where these symbols will denote the appropriate right- or left-hand derivatives. We shall also have occasion to use c+/cs and d-/ds to denote right- and left-hand derivatives.) Also, x(t, to, 4) is continuous in X using the topology of uniform convergence in PC. In fact a type of weak continuity holds:

Received by the editors June 5, 1967.

1211

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1212 S. P. HASTINGS [October

LEMMA 1. Suppose that I On} is a bounded sequence of functions in PC, 0bePC, and On/-_c pointwise. Then for each t _ toO such that x(t, to, c) is defined, it is the case that for large n, x(t, to, 4n) is defined and x(s, to, 4n")-+x(s, to, 4) uniformly for to< s< t.

PROOF. It is sufficient to prove the result for to < t < to+ 1. For each 4VePC we denote by I(to, i1) the maximal interval of existence of x(-, to, i0). Also, let I+(to, ifr =I(to, ) [to, c). For tCI(to, 4) r21(to, On), let x(t) =x(t, to, 4) and xn(t) =x(t, to, On). Then on I+(to, 4) (1+(to, cn), we have

xn(t,) - X(t,) = o n(0) - 4(O)

+ J[f(s XnX(S), x(S- 1)) f(s, x(s), x(s - 1))]ds.

Since x(s- 1, to, /1) i=p(s-1) for to<s<to+1, we can use the bounded- ness of cf/cx and cf/cy on compact sets to obtain a bound of the formz

t

| xn(t) - X(t) I < I on (0) - 1(0)| + fM I Xn (S) - x(s) 1 ds to

t-I

toL on (S) -?) (S) I ds.

The bounded convergence of on to X and Gronwall's inequality can then be used to complete the proof.

NOTE. This result is not true for all functional differential equations with Lipschitz continuous right-hand sides. For instance, consider the equation i(t) =sup-,e,o I x(t+6) I. Let n(o) =O for -1 <0 _-1/n, on (O) = 1- 2NO+1 , -1/n?<?O < O. Then I 4 n(6) I < 1, n f(O)->0 for - 1 < 0< , bu t x (t, Os n) = t, O<t<l; /2 for n =1, 2,

We must now discuss the "derivative with respect to initial condi- tions" of solutions of (1). Let C = {I I | 4EPC, 4 continuous on [-1, 0 ] }.

LEMMA 2. Suppose X, f E C. Then for each s ? 0, and each t in I(s, k) except t=s, cx(t, s, O)/Os is defined. Also, limu-.+ (u, s, O)Ox/Os exists and if +,(t) = limu_.t+cx(u, s, c)/csfortCI(s,4), then4tsatisfiesthevaria- tional equation

C9f (2 )1 x(of s, ), x( -1, s,

clx (2) Cif

+ -(t, x(t, s, 0), x(! -1,s )+t

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19681 NONLINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS 1213

with initial condition

O(s + 0) = - k(0), -1 ? 0 < 0, 4b(s) =-f(S, O(M), 4(-1)).

PROOF. This is, of course, an extension of the usual result for ordi- nary differential equations, and the proof proceeds as in that special case. For small I hf, h O, define q (t, s, h) by

77(t, s, h) = (1/h)[x(t, s + h, 4)-x(t, s, )], t E I(s + h, 4) n I(s, I).

For h <O,

j(t, s, h) =-[4I(t-s-h)-4(t-s)], s-1 s + h, h

7(t, s, h) =-[1(O) + f(u, x(u, s + h, 4), x(u - 1, s + h, 4))du h L+

- s) ss+Ih?t?s,

7/(t, s, h) = U <IL[(u, x(u, s + h, 4), x(u-1, s + h, 4))

-f(u, x(u, s, 4), x(u - 1, s, 4))]du

+ J f(u, x(u, s + h, 0), x(u -1, s + h, 0))du, $+h

t E I+(s + h, 0) n I+(s,

Therefore,

lim 71(t, s, h) =-(t -s), s -1 t < s, h--O-

lim q(s, s, sh) =-f(s, 4(O), 4(-1)). h-O-

Also, for t>s and t(I+(s+h, q)C\I+(s, 4) we have that

a 1 - l(t, s, h) = - [[(t, x(t, s + h, t), x(t - 1, s + h, 0)) a1 h

(3) - At, x(t, sI 0), x( - 11 sI OM I

while at t =s the right-hand derivative O+q(s, s, h)/lt also satisfies this equation.

We now proceed just as for ordinary differential equations (see [4, pp. 25-27]). In particular, it follows from (3) that as hO-

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1214 S. P. HASTINGS [October

- rQ, S, I)

= Ir1 { [-% (1, x(t, s, 4)), x(t-1, s, t)) + o( | hrl(t, s, h) ) h7(t, s, h)

+ -(t, x(t, s, ), x(t - 1, s, ))

+ o( I h7(t -1, s, h)I -h77(t-1, s, h)4

_3f =O (t, x(t S, s), x(t s, s))71(t, s, h)

Ox

+ - (t, x(t, s, ), x(t - S1, s, -))( 1, s, i) + 0( hi ay

It now follows from known results about the theory of E-approxi- mate solutions (see [6, pp. 14-20]) that the left-hand derivative d-x(t, s, 4)/as exists for tGI(s, 4) and satisfies the conditions on 4/ in the statement of the lemma. Therefore d-x(t, s, 'k)/ds is continuous in s for each t in I(s, 45) except t=s. Thus, except at t=s, dx(t, s, 4)/ds is continuous and satisfies the conditions on i,t in the statement of the lemma, and the desired result follows.

At this point we need some more notation. Suppose y is a real- valued function which is piecewise continuous on an interval [a -1, b ] with a <b. Then for each sC [a, b] we define y8.PC by ys(O) =y(s+6).

LEMMA 3. Let y(s) be continuously differentiable on [a-1, b]. Then for sE (a, b), lx(t, to, y.)/Is exists and satisfies

{(t) = af(1, x(i, to, y8), x( - 1, to, y8))4.(t) Ox

+ (f 0 x(t) to, y,), x(t -1, to, y,))A( - 1) ay

as a function of tfor tEI+(to, y.). Also,

Os x(to + 0, to, y,) = y.(O) = y(s + ? ), -1? 0? 0.

ds

PROOF. The proof is similar to, but simpler than, that of Lemma 2 and will be omitted.

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I968] NONLINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS 1215

We now come to our main result. For given to-O, 4EPC, and

tEI+(to, 4) we define a bounded linear functional T(t, to, 4): PC-*R by setting T(t, to, 4)x equal to the value at t of the solution of (2)

with initial condition 4I'g.=x. Also, for each real-valued function g

continuous on [0, oo) and for each to _ 0 define g[t0] EPC by setting

glio](0) =0,-1 <0<0, gIto](0) =g(to).

THEOREM. With g as above, to > 0, and E4 C let y be the solution of

(4) y(1) = f(t, y(t), y(t - 1)) + g(t)

with y0 =0 . Then for all t > to such that (i) y(t) is defined,

(ii) tCI(s, y,) for to<s<t, we have

y(t) - x(t, to, )= j T(t, s, y,)g[,]ds. to

PROOF. First assume that c E C. Then y. CPC for to < s < t, and from

Lemmas 2 and 3 it follows that

-X(t, U, Y8') IUe8- = T(t, s, y.) - + g[8] Ou

while

X(t, S, Yu) |lu- = T(t, s, Ya)Y8.

We therefore obtain

(d/ds)x(t, s, y,,) = T(t, s, y,)[-y, + g[.]] + T(t, s, Y)YS = T(t, s, y,)g[.

Integrating from s=to to s=t and using the fact that x(t, t, y,) =y(t)

proves the result in this special case. Now if ckiq C, then choose {In)

with cnkECC, onr'- uniformly. Let yn(s) denote the solution of (4) with yn, =4qn. Then x(t, to, n) -yn(t) _*x(t, to,4) -y(t). Also, yn(S)-*y(S)

uniformly in s for to - 1 s< t. Therefore T(t, s, yn)g[,J-+T(t, s, y.)g[,,] uniformly in s for to< s<t and this implies the desired result.

REFERENCES

1. V. M. Alekseev, An estimate for the perturbations of the solutions of ordinary

differential equations, Vestnik Moskov. Univ. Ser. I Mat. Meh. (1961), no. 2, 28-36. (Russian)

2. F. Brauer, Perturbations of nonlinear systems of differential equations. I, J. Math. Anal. Appl. 14 (1966), 198-206.

3. , Perturbations of nonlinear systems of differential equations. II, J. Math. Anal. Appl. 17 (1967), 418-434.

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1216 S. P. HASTINGS

4. E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955.

5. A. Halanay, Differential equations. Stability, oscillations, time lags, Academic Press, New York, 1966.

6. N. Oguztorelli, Time lag control systems, Academic Press, New York, 1966. 7. A. Strauss, On the stability of a perturbed nonlinear system, Proc. Amer. Math.

Soc. 17 (1966), 803-807.

CASE WESTERN RESERVE UNIVERSITY

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