ANTOSIEWlCZ, H, A. Math. Zeitschr. 78, 44--52 (t962)

An inequality for approximate solutions of ordinary differential equations*) By

H. A. ANTOSlEWICZ

f. Let / be a mapping of a set I • H into R ~, where I ( R is an interval and H ( R " an @en set. A continuous mapping u of I into H will be called an e-approximate solution of the differential equation

(t)

x' = l(t, x)

i f u is differentiable in the complement of an at most countable subset C of I such that for every t E I - C

1I,,'(o - l(t. ,, (t))ll-<_.~ ~). It is well-known that if t is lipschitzian in I • for a constant 2 > 0 t h e n any two ez-, e2-approximate solutions u z, u 2 of (l) satisfy, at every t o and every t in I, "

(2)

l§ ~lla,~!t_tdl

11,,~(o - -~(t)ll_-< II,~(to) -,,~,.o,,,~

"1

T ~- (e~+ ,~) (w-,.~-

~).

This inequality has two important features. It implies a simple estimate for the error in I of an s-approximate solution of (t), and it shows directly the uniqueness ill I of a solution of (t) (whose existence locally can also be deduced from (2)). In the present paper we prove, b y a method due to KAMXE ([91, p. 137), a generalization of the inequality (2) which retains these two features of(2) and yet .does not require ] to be lipschitzian in / x H. Our results are in the spirit of LYAPUNOV'S Second Method. and so are sufficiently general to include the extensions of (2) in [1], [3] (p. 9), and Ell]. Moreover, they offer a unified approach to all error estimates for e-approximate solutions [8] as well as to all classical uniqueness theorems [4] (see, also, [7], [14~). The latter is a consequerme of the general comparison principle for the solutions of (I)which is implied b y our results, and which is more flexible than the one used in [/0] for the same purposeS). 9 This comparison principle is also useful in other circumstances. We show that it can be made the basis for various stabi~ty theo~) This work was done while the author was partially supported b y the Office of Naval Research under Contract N-onr 228[09J. An abstract o~ it was presented with partial support b y the National Science Foundati011 under G r a n t N SF G-13731 at the Symposium of the Provisional International Computation Centre in Rome, September 1960. 1) Throughout I]xl] denotes . t h e / / - n o r m or any-other equivalent norm of x. z) The author is indebted to the referee for pointing out to him this reference. ,

45

An i n e q u a l i t y for approxilItate solutions

rems (cf., e.g., [2], [6], [12], [13]). A special case of iris known [5] to underlie a number of criteria for the continuability of a solution of (t). 2. We assume in the sequel.that ] is continuoils in I x H although all our considerationsremain valid under weaker hypotheses (cf., e.g., [3], Lemma 1, P. 3). " Let Ho----{x -- y: x EH, y E H}. We are concerned with real valued functions V which are defined in I • and have the following property, which we shall call p r o p e r t y (L), for a continuous (real valued) function k (t) _> 0 in I : for each xE/-/o, the function V(., x) is continuous in I, and for every t e l and every x 1, x 2 in H 0

IV(t, ~1) - v(t, ~,)1 <=k(t)11~1

**ll.

Corresponding to such a function V we introduce two rea~ valued functions V;, V; in I • H x H by setting

v; (t, x, y) = (-- 1)* lim inf ~- 1,- {V It + h, x -- y + h (] (t, x) " ] (t, y))] -- V(t, x --y)} k --'. 0

.k

Observe that if V is of class C1 in I • v ; (t, x, y) -

o then in I x H x H

v,' (t, x, y) = v,(t, x -

y) + v , ( t , x -

y) . (l(t, x) - / ( t ,

y))

where V, is the gradient vector of V. LEMMA. Let V be a real valued/unction in I •

which has property (L) /or a continuous ]unction k (t)> 0 in I. I] ua, u 2 are any el-, e,-approxim, ate solutions o / ( t ) then ]or every t E I (3)

[Vi'[t,*il(t),u,(t)]--O,V[t,u~(t)--u~(t)]I<=(el+e,)k(t),

i=t,

2,

where D 1 and D~ are the upper and lower derivate, respectively. PROOF. Clearly, V(t) = V [t, u 1 (t) -- u~ (t)] and 17/'(t) = V~'it, u 1 (t), u~ (t)] are defined for all t E I. Using property (L) of V we readily verify that in I 9 [~' (t) -- D, V(t)[ < hm sup ~1 A (t, h)

n--~u

inl

where II

(t, h) -- h(t + h) Y. I[-,(t+ h) -- u,(O -- h/(t, u,(t))ll. 1

Since each u s is an ei-approximate solution of (1), we deduce from the meanvalue theorem that for every t C I and every t + h C I

ui(t + h) - u~(t) - t+h f / ( s , u,(s/) ds ! ~< *Jhl, t

~ t+h

J/

) ds--h/(t,u,(t))

=<]h} sup [I/(t + Oh, u~(t + Oh)) . /(t, ui(t))[]. o~0_<1

46

H.A.

ANTOSIEWlCZ:'

Hence we have for every t C I 9

1

h m sup ~ A (t, h) < (~s + e~) k (t). h-~0 [hi = REMARK. If U~, U2 are two solutions of (t) in I then ~'(t)=D~V(t) for t E I, and this remains true even when V is only locally lipschitzian in I • 0. T~EOREM I. Let V be a real valued/unction in I • which has property (L) /or a continuous/unction k(t)>= 0 in I, and suppose that there exist a compact interval K = [t0, ts] ( I and two continuous real valued/unctions o~1, ~o~ in K • R such that/or every t C K and every two points x, y in H

(4)

(--/)iV{(t, x, y) <= (-- t)ioJr

V(t, x -- Y)I,

i = t, 2.

Let u t, uz be any el-, s~-approximate solutions o/ (t). Let (--t)iW.: be the maximal solution o/the equation (5)

w;=(--l)ioi(t,(--t)'w,)+(el+s~)k(t),

i=1,2,

corresponding to (t o , (-- t) i Vo), where Vo ----V [to , u I (to) -- u s (to)], and suppose each Wi is defined (~t least) in K o = [to, t o + T) ( K. T h e n / o r every t E Ko

(6)

W~ (t) <-- V/t, us(t ) -- u~(t)] ~ VV~(t).

Similarly, let (--i)~l~i be the minimal solution o/ (5) corresponding to (tl, (--t)~V1), where I ~ = V[tx, us(t1)- u2(ts) ], and suppose each ~ is de/ined (at least) in K 1= (ts -- T, tl] ( K. T h e n / o r every t C K t

(7)

l ~ (t) <__V It, us (t) -- u z (t)] _--
PROOF. We only prove (6); the proof of (7) is entirely similar. Evidently, it suffices to show that (6) holds in every compact interval L = [to, to + c / ( K o. Moreover, for e~ery sufficiently small 0~> 0, the maximal solution (--~)i~2i of the equation (8)

w; = (-- t)' of (t, (-- t)' wi) + (s~ + e2) k (t) + o~,

i ---- t, 2,

corresponding to (to, (--1)iVo+~), also exists in K 0 and f2i(t, ~)--+g/~.(l) as ~--+0+ uniformly in every L ( K o. This follows from a modification of an argument by which KAMKE ([9], p. 83) proved the same assertion for the maximal solution of (8) corresponding to (to, (--t)iVo). Thus we merely need to show that in an arbitrary interval L ( K o and for all sufficiently small 0~> 0 (9)

~9~(t, ~) <= V/t, us(t) -- uo(t)] _--<~22(t, o~).

Let aga!n V-7(t)= V It, ul(t) -- u 2 (t)] for t E I and define for t C K0 and all sufficiently small ~ > 0

~At, c,.) = o,~[t, Q~(t, ~)l + ( - t) ~(~ + ~) k(t). B y definition s m)=.V(to)--~ and hence, for every sufficiently small ~r there exists a maximal interval L(oO=[to, t']CL with t ' > t o in which

An inequality for approximate solutions

47

O~(t, ~)<= V(t). If L(a). were not equal to L "for some 0c>0 there would exist a non-empty open interval (t', t " ) ( L - - L ( o O such that, by continuity, /21 (t, 00 > V(t) !or every t C (t', t'J) and Q1 (t', 0~) :

V(t') ,

" ~ff~l'(4,' ~), => O] V(t')

However, this is absurdi for by (8) we have/2~(t', e ) = ~ l ( t ' , e) - - e and, by (3) and (4), D1V(t')~'@(V, ~), Hence L(e)----L for every sufficiently small e > 0, and so/21(t, ~)=< V(t) for every t E K and these 0~. A similar argument proves the right-hand side of (9) REMARK. Our proof shows that (6) and (7) will hold when the inequalities (4) are replaced by the weaker hypothesis that for t C K (-- t) i V/' [i, u~ (t), u, (t)~ ~ (-- 1)i coi It, V(t, "i (t) --.u 2 (t))],

i

t, 2.

To prove (6) or (7) alone, it is even sufficient to assume that these inequalities be satisfied when each Vii' is defined as one-sided limit inferior only. 3. Note that the inequalities (4) are testable. They involve solely the given differential equation (t) and the functions V and osi, which are at our disposal. In general, there will be several choices of the latter for which (4) will be true. For example, if we select V(t, x) --IIx [[we find that for every t E I and e v e r y x , y in H

y,'(t,x, y)[<-_tl/(t,x)-/(t, y)ll,

i--a,2.

Thus, if /satisfies, at every t E I and every x, y in H, the classical Osgood condition

fl/(t, x) --/(t, y)ll--- ~(t)~(llx - yll) for two continuous non-negative valued functions 2, ~v, defined in I and in R, respectively, we see that by taking ,oi (t, r) = (- t)i

(4)

(r),

i = t, 2,

the inequalities (4) will hold for every t C I and every x, y in H. When 9 is non-decreasing in [0, oo) and q0(r) > 0 for r > 0, (6) and (7) include the two-sided generalization of (2) in [11] (cf., also, [1] and [3], p. 9). In the particular case ~(t) ~ > 0 , qo(r)=r the right-hand sides of ~6) and (7) together yield (2), and the left-hand sides of (6) and (7) imply that, for every to and every t in I, '(tt) Ilul (t) - u2(t)Jl >= ]]ul (to) -~ u2(to)He-a,t-t~ - -iI (el + e2) (t - e_,~lt_tu ) where the expression on the right is positive for u 1(to) ~ u 2(to) and I t-- to l small enoughS). Often a more judicious choice of V which takes into account special properties of / will result in sharper inequalities (6) and (7). This is of practical ira*) Of course, (t t) is equivalent to (2) since both inequalities hold for every to and i in I so that in either to and t may by interchanged.

48

H.A. A~zosiEwIcz:

9portance when (6) and (7) are used as estimates Ior the error in I of an g-approximate solution of (t). An illustration of t h i s point is the simple case when /(t, x)=A (t) x in I • R *, where A (t) is a (real)' continuous matrix in I. If we now take V as a positive definite quadratic form in x, say, e.g., V(t, x)= (x, M x ) where M is a constant symmetric matrix, then for e v e r y t E I and all x, y in R * Hence

Vl'(t,x,y)----Vs'(t,x,y)=2(x--y, MA(t)(x--y)).

2~1(0(x y,x y~V~(t,~;y)-v~'(t,~,y)<202(0(x-y,x-y) where 0x(t), 0z(t) are the ,niinimum and maximum eigenvalue of the matrix 89[M A (t) + A* (t) M ] . . S i n c e i~ every interval J ( I in which 01 (t), Oz(t) ~ o t h are of fixed sign, and for all x, y in R",

(-

t)~ ei(t) ( x - y, x - y)_<_ ( - t ) i ~ e ~ ( t ) v(t, x ~ - y),

where/~1,/*2 are positive constants depending on M only, it follows t h a t (4) will be satisfied with co~(t, r) -----2/z i Oi (t) r, i = 1, 2. Clearly, 0~ (t) can have negative values in I. This occurs for the example !

xl:

- - t x 1-~-x~,

t

x ~ : x 1 - t 2x 2

which was used in ~8] to exhibit the existence of what is called there a "negative Lipschitz b o u n d " for t > t. Here with V(x 1, x~)=x~+ x~ we obtain much more simply that, for all t, 20i(t ) = -- t(t + 1) + (-- t) i Et~(t - 1)'

+ 4] ~,

and hence 0 ~ ( 0 < 0 for t > 1. If l (t, x)=A x where A is a constant matrix whose eigenvalues all have negative real part, there exists a real positive definite matrix M for which 0x(t) =02(0 ~ - t. Thus, with this choice of M in V, we can always take coi(t, r) ------- 2 / ~ in (4). Similar considerations apply in more general circumstances. In fact, much of the part of LYAPTJI~OV'S Second Method which is concerned with questions of total stability can be used to derive from Theorem 1 new and generally more precise error estimates for e-approximate solutions of (4). 4. The hypotheses in Theorem 1 as regards the continuity of the functions co~ can be weakened a t the expense of restricting the class of e-approximate solutions of (t) to be admitted. As in ~10] we introduce the'following terminology. Let co be a continuous real valued function i n J • R where J = (to, tl) ( R and suppose that there exists a real valued function W of class C1. in [to, tl) which is in J a solution of the equation (l 2)

w' = co (t, w).

T h e n W will be called a maximM ~minimal~ solution of (t2) in ~t0, tl) if, for every #ro----(to, to+T)~ J, every other solution Q of (t2) in Jo that can be

49

An inequality for approximate solutions

extended to a function of class C1 in [to, to+ T) with ~9(to)=W(to), ~2'(to)= W'(to) satisfies t2 (t) < W(t) [~ (t) __>W(t)] for t E ]o. THEOREM 2. Let V be a real valued/unction in I x H o which has property (L) /or a continuous /unction k (t)>=0 in I, and suppose that there exist an open interval J = (to, tl) with [to, tl) ( I and two continuous real valued /unctions COl,cos in J • such that for every t E J and every two points x, y in H

(t3)

( - t) i v~s (t, x, y) < ( - t)~o~il't,

v(t,

x -

y)],

i = 1, 2.

Suppose, further, that/or some e >--0 each equation

(t4)

w', = ( - l)'o,,(t, ( - 0 % )

+ e k(t),

i--t,2,

has a maximal solution (--t)iWi in [to, tx)such that l/Vl(to)=Wi(to)=Wo and I/V~'(to)~ W; (to). I/ul, us are el-, es-approximate solutions of (t) with e l + e s = e for which v [to, ul (to) - us (to)] = wo and

t (t 5) D1 V [to, ua (to) - us (to)] < ~ ' (o), 3 s V [to, u 1 (to) - u s (to)] >__ w~' (to), then/or every t C J (t6) w~ct) __< v/t, ~(t) - us(t)/_-< w,(t). PROOF. Define V(t) = V It, u 1(t) -- u z (t)] for t E I . If the inequality W1 (t) =< 17(t) were false for some t* E J, let l~1 be the maximal solution of the equation (t 7) wl - o1(t, wl) - , k (t), corresponding to (t*, V(t*)). Clearly, l~x is defined in some interval L = (t* -- c, t*] ( J , and so, by (7), V(t) =< l~x (t) for t E L. Moreover, l~x (t) < W1 (t) for t E L because W1 is by hypothesis the minimal solution of (t7) in [to, tl). Hence l~1 must exist throughout (to, t*] and satisfy, at every t E (to, t*],

V(i) <__~ (t) < W~(t).

(18)

Therefore, w e can extend l~1 to a continuous function in [to, t*] by defining 1~1(to) = Wo. T h e n D 1l~x (to), D sl~x (to) are also defined and, since D 2 V-(to)> WI' (to) by (15), we deduce from (t8) that

D~ ~ (to) = D. ~(to) = ~ ' (to). Thus l~i can be extended to a function of class C 1 in [to, t*] with 1~x (to) = Wo and WI' (to)=I/V/(to).. This however contradicts the assumption that ~ is the minimal solution of (t 7) in [to, tl). Hence WI (t)_< V(t) is true for every t E J. The right-hand side of (t6) is proved similarly. REMARK. In view of (3), we may replace (t 5) by the stronger conditions t

v; fro, ~(to), us(to)] =< w; (to) - ~ k(to), v,'/to,, ul(to), us(to)] => w; (to) + ~ k(to), which involve only the values u i (to) and so require no knowledge in J of the e~-approximate solutions ui of (l). Xffnf'hemnf;crh~

?~;fc:ollr;f*

"~t~l "~'g

.4

~0

H.A. ANTOSlEWlCZ:

The significance of the hypotheses in Theorem 2 may be illustrated by assuming that/.satisfies, at every t E J aiad every x,- y in H,

tl/(t, x) -/(t, y)li__< t-to IIx- yll where 0 < 2 < I. If we take V(t, x)----~x]] and recall (t0), we find that (13) holds with ~oi(t, r) - ~~-/ ~ r, t - to so that the equations (t4) become 09)

w:-- ( - 1/~~ w i + e , t - to

i=l,2.

If o<,~< t, we easily verify that, for every e-->0, (-

t)~(t)-

. _ ( _ 1* ) i ~

(t -- to)

is the only solution of (t9)in J which can be extended to a function of class C~ in [t0, tl), Therefore it is the maximal solution of (t9I in ~to, tj). Clearly, Wdto)-~0 and (--t)~W~'(to)>0. Since V(t, x ) = 0 solely when x = 0 , it follows that only those el-, el-approximate solutions u 1, % of (t) (with el, e2 arbitrary) are admitted for which u I (to)= u S(to), and for these (t 5) is automatically satisfied. Thus, by (t6), we have then for every t E J Ilul (t) -- % (t)]I=<~1_~+e2 (t -- to). (The left-hand side of (t6) is redundant here.) If 2---1t, the situation is completely different. Now the equation (t9) with i = t still admits, for every e-->_0, a maximal solution in Eto,tl), namely - - W1(t) ~-(t--t0). However, for the equation (19)with i = 2 such a solution exists only when e = 0, and in this case it is simply. W~(t) --=0 in [to, tx). Hence the hypotheses of Theorem 2 can be satisfied only with e-----0, and this requires t h a t ul, % be two solutions of (1)with ul(to)=us(to). Then 06) asserts that u I (t)-----% (t) in J, which coincides with the assertion of NAGUMO'S uniqueness theorem (cf., TheOrem 3 below). 5. Theorems t and 2 allow us to deduce various properties of the solutions of 0) from those of theosolutions of the single equations w~=~o i(t,w3,

i=1,2.

We indicate briefly two of the many results which are direct consequences of Jthis comparison principle. The first is a general uniqueness theorem, which is an immediate corollary of Theorem 2, and the second is a stability criterion, which follows from Theorem t . THEOREM 3" Let V be a real valued/unction in I • which is locally lipschitzian and vanishes only i / x = O, and suppose that the~e exist an open interval

An inequality for approximate solutions

~1

j - - ( t o, tx) with Eto, tx) ( I and two continuous real "valued ]unctions 01, o z in J • R such that/or every t E J and every two points x, y in H

( - l ) ~ ' (t, x, y) <= ( - t)~ o, Et, v(t, x -

y)~,

i = t, 2.

Suppose, [urther, that each equation

w; = ( - i)'o,(t, ( - 1)' w,),

i -- t, 2,

has W i (t) =--0/or the maximal solution in Eto, tl). Then, ]or every x o C H, (t) has at most one solution in Eto, tl) which equals x o at t o . W i t h V(t, x ) = H , Theorem 3 contains as special case KAMKE'S general uniqueness theorem ([91, p. t39, and I10~), which in turn implies all other classical uniqueness criteria. Suppose now, for the sake of simplicity, t h a t / is defined and continuous in R + x R ~, where R § oo), and t h a t /(t, 0)=0 in R § (so t h a t x(t)----O is a solution of (t) in R§ We shall call a locally lipschitzian real valued function V in R § • R ~ positive definite if V(t, 0) ~ 0 in R § and if for every sufficiently small * > 0 there exists a # ( , ) > 0 such t h a t V(t, x ) < # implies llxll<*. THEOREM 4. Let V be a real valued ]unction in R § • ~ which is locally lipschitzian and positive definite, and suppose that there exist two continuous real valued ]unctions co1, 02 in R § • R ~ such that/or every t C R § and all x ~ R ~ (-l)'Vi'(t,x,O)<_(--t)'oiEt,

V(t,x)l,

i=1,2.

Suppose, [urther, that o , (t, O) ~ 0 in R +. and that [or every t o C R § and every su]]iciently small ]wol there exists a unique solution Wi of the equation

(20)~

w; = o,(t, w3,

i - - / , 2,

which is defined/or all t >=t o and equals w o at to. Then the stability properties of the solution x ( t ) = - 0 of (t) are the same as those o] the solution w~ (t)=--0 o/ (20)2, and the instability propel'ties of x(t)=--0 are the same as those o / t h e solution wl(t)=--0 o/ (20)1. For the proof we need only observe that, given a n y sufficiently small e > 0, there exists a # ( e ) > 0 such that, if W2 (t~) < / , at some t~_> to, then Theorem I implies I]x (to)It< e, and if W~(tl)__># at some tl>=to, t h e n ttX(tl)]t=> e. Theorem 4 includes the results in [2], ~a], [13~, and Theorem 15, 17, t8 in [12]. I n fact, Theorem 4 can be used to prove all basic theorems of LYAPU~OV'S Second Method in a simple unified way. 4) 4) Added in Proof: I have just become aware that this has already been done, based on the same comparison technique, by C. COROUNEA1NLU["Application of differential inequalities to stability theory" (in Russian), An. S,ti. Univ. 'A1. I. Cuza' Iasi, Sect. I, 6, 47--58 (1960)]. For other results using similar considerations see the paper by F. BRAUER, "(~lobal behavior of solutions of ordinary differential equations", J. Math. Anal. Appl. 2, t45-- 158 (1961). 4*

52

H . A . ANTOSIEWICZ: An inequality for approximate solutions

References [1] BINARY, I. : A generalization of a i e m m a of Bellman and its application to uniqueness problems of differential equ.ations. Acta Math. Acad. Scient. Hung. 7, 8i--94 (t956). [g] -- Researches of the boundedness and stability of the solutions of non-linear differential equations. Acta Math. Acad. Scient. Hung. 8, 26t--278-(1957). [3] BOURBAKr,BY.: Fonctions d'une variable re,lie. Act. Scient. Ind. No. 1t32, t951. [4] BRXUER, F., and S. ST~r~BERG: Local uniqueness, existence in the large, and the convergence of successive approximations, Amer. J. Math. 80, 42t--430 (t958); 81, 797 (t959). [5] CONTI, R.: Sulla prolungabilit~delle s.oiuzioni di un sistema di equazioni differenziali ordinarie. Boll. Unione Mat. Ital. 11, 5t0--5t4 (1956). [6] -- Limitazioni ,,in ampiezza" del'le,solUzioni di un sistema di equazioni differenziali e applicazioni. Boll. Unione Mat. Ital: 11, '344--350 (1956). [7] DArILgUIST, G. : Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Roy. Inst. Techn. Stockholm, No. t30, 1959. [8] ELXERMaNN, H. : Fehlerabsch~itzung bei n~herungsweiser L6sung yon Systemen yon Differentialgleichungen erster Ordnung. Math. Z. 62, 469--50t (1955). [9] KXMKE, E.: Differentialgleichungen reeller Funktionen. Leipzig t930. [10] - - ~ b e r die eindeutige Bestimmf2xeit der Integrale yon Differentialgleichungen, II. Sitzungsber. Heidelberger Akad. Wiss. Math.-naturwiss. Kt., 1930, t 7. Abh., 15 P. [11] LANGg~HOP, C . E . : Bounds for the norm of a solution of a general differential equation. Proc. Amer. Math. Soc: 11, 795--799 (t960). [~2] M~,SSERA,Jos~ L.: Contributions to stability theory. Ann. of Math. 64, t82--206 ' (1956). [18] OPIAL, Z. : Sur Failure asymptotique des solutions de certaines ~quations diff~rentielles de la mdcaniqu e non lin~aire. Ann. Pol0n. Math. 8; t05--t24 (t960). [!4] WALTW-R,W. : EindeutigkeitssRtze fiir gewOhnliche, parabolische und. hyperbolische Differentialgleicbungen. Math. Z. 74,' 191--208"(1960).

-Det~artment of Mathematics, Unive~'sity o] Southern Cali]ornia, Los Angeles, r (Eingegangen am 11. Dezember 1960)