ON AN INFINITE SET OF NON-LINEAR DIFFERENTIAL EQUATIONS By J. B. McLEOD (Oxford)

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1 ON AN INFINITE SET OF NON-LINEAR DIFFERENTIAL EQUATIONS By J. B. McLEOD (Oxford) [Received 25 October 1961] 1. IT is of some interest in the study of collision processes to consider the solution of the infinite set of non-linear differential equations where the initial conditions are!/i(0) = l, Sfi(0) = O (» = 2,3,4,...). (1.2) We are concerned with positive values of the independent variable t, which is physically the time, and the constant coefficients a if are positive and symmetric, i.e. a ti ^ 0, a^ = a it. The only work that seems to have been done on this type of problem from the pure-mathematical point of view has been restricted, in effect, to the case where o w is bounded, as in (1), (2). The extension to unbounded a y appears difficult, but this paper makes a start on the problem by solving explicitly one specific case, when a ti = ij, and by proving, in 4, an existence theorem (though not a uniqueness theorem) under fairly general conditions on a tj. I hope to return to the problem of uniqueness and other related questions in a later paper. 2. The case a {i = ij THEOEBM 1. There is one and only one solution of the equations (1.1), with the initial conditions (1.2) and with a^ = ij, for which 2*V< ** absolutely and uniformly convergent for t in some closed interval [0,&], where # < 1. Furthermore, this solution continues to be a valid solution for all positive t <C 1, but its analytic continuation, though meaningful, fails to remain a solution for t > 1. In fact, there is no solution whatever of the equations, with the given initial conditions, for which J* 2^! ** absolutely and uniformly convergent in some interval [0, e], where e ^ 1. Proof. For convenience, we divide the proof into three parts, (i) showing that the solution is unique, (ii) showing that the solution exists, and (iii) discussing the continuance of the solution. In (i), (ii), we shall, of course, be restricted to values of (in [0,#]. Quart. J. Math. Oxford (2), 13 (1962),

2 120 J. B. McLEOD (i) With the given convergence assumptions, we can multiply the ith equation by t and add to obtain 2 * = - &* provided that the second term on the right-hand side converges. In fact, it converges absolutely because it is just a rearrangement of the first term, which evidently converges absolutely. We see this by noting that, if either term is multiplied out, then the total coefficient of y t y^ is (i-\-j)ij. It follows therefore that the convergence being absolute and uniform for t in [0,#]. It follows also from (1.1) that each y ( is continuous because it is differentiable, and that dyjdt is continuous because it is the sum of a uniformly convergent series of continuous functions. Hence we may integrate (2.2) term by term, and obtain, using the initial conditions, that m t = 1. (2-3) The equations (1.1) therefore reduce to % fjii-myi-, (* = i,2,3,...). (2.4) The equations (2.4) can be solved in succession since only y 1 appears in the first, only y x and y % in the second, and so forth. This solution, and so the solution of (1.1), with initial conditions (1.2), is unique. (ii) In order to prove that there is a solution at all of (1.1), with initial conditions (1.2), we have to show that the solution of (2.4) satisfies the convergence criteria of the theorem and also (2.3). We therefore investigate the solution of (2.4). We readily obtain yx = e- 4, y* = «-*- Let us suppose in general that, forj = 1, 2,..., n, y, = A i P-^e-»IJ, (2.5) where A t is independent of t. Then, substituting in the (n+l)th equation, we have

3 ON A SET OF DIFFERENTIAL EQUATIONS 121 so that 7 n i-i 2 Hence (2.5) holds by induction for allj provided that A x = 1 and 1 "V A A A n+\ 2 ^ ( To investigate the nature of the coefficients A jt we define, formally, y(w) = fa n w»-k (2.7) n-l Then, still formally, the left-hand side of (2.6) is the coefficient of w n in o for n = 1, 2, 3,..., while the right-hand side is the coefficient of w n in to y(u)du w J o for n = 0, 1, 2,... We therefore have, formally, that, for (2.6) to be true, y(w) is a solution of w \ y i {u)du = - y{u)du A v o o Differentiating both sides, we obtain 0 and, multiplying through by w % and differentiating again, we conclude that to */ ^ whence, after a sh'ght rearrangement, we have i.e. dw, 1 dy y yw = log y+c, t

4 122 J. B. MoLEOD for some constant c. Using the fact that A x = 1, we see that y = 1 when w = 0, so that c = 0 and (2.8) This gives w as an analytic function of y in some neighbourhood of y = 1, and so, on inversion, it gives y as an analytic function of w in some neighbourhood of w = 0. Thus y can be expressed as a power series in w, and going backwards through the formal argument above, we find that the power series is given by (2.7), where A t = 1 and (2.6) is satisfied. We have therefore determined the coefficients A it given by A 1 = 1 and (2.6), as the coefficients in the expansion of a certain analytic function. We next note that, if w is given as a function of y by (2.8), then dw/dy = 0 implies logy =1, y = e, w = e- 1. Hence the radius of convergence of (2.7) is e~ x. It is now immediate that the sequence {j/y}, given by (2.5), satisfies the convergence criteria of the theorem provided that \t is so small that \te~*\ ^ e~ l 77 for some 77 > 0. It only remains to verify, in this section of the proof, that (2.3) is satisfied. If y^ is given by (2.5), and if / is sufficiently small, then 2 i-l = e-h/(te-<), (2.9) where y(w) is defined by (2.7). But, by (2.8), t/(te~0 satisfies te- t = y~ 1 \ogy, i.e. y = e*. Thus (2.9) reduces to unity, as required. (iii) The final question is whether the solution now found remains valid for all positive t. In fact, the argument in the last two paragraphs of (ii), as we have shown, holds good for all positive t such that \te~*\ < c" 1, i.e. for 0 ^ t < 1, and so the solution remains valid over this range. To investigate whether it continues to be a solution for t ^ 1, we note that it is certainly a solution of (2.4) for all positive t, and comparing (2.4) with (1.1), we see that it can continue as a solution of (1.1) if and only if J iy t = 1. In fact, this holds for t = 1, but not for t > 1.

5 ON A SET OF DIFFERENTIAL EQUATIONS 123 To consider t > 1 first, we note that then 0 < te- 4 < e~ x, and so certainly 2 *y< converges. In fact, = e-* f, Aiite- 1 )*- 1 where t* (with 0 < t* < 1) is such that t+e-'* = te-*, = e~*(f, as at the end of (ii). Hence 00 2*y< ^ * To consider t = 1, we have to investigate the coefficients -4 4 more closely. We can in fact obtain them explicitly from (2.8). For, with y = (*, (2.8) reduces to t = we?, and we can then expand e* as a power series in w by Lagrange's expansion. This gives i.e. y = 1 + > n-l and comparing this with (2.7), we obtain A n+1 = (n+l)»-vn! (n = 1, 2, 3,...). Using the asymptotic expression for n\, we conclude that J4 71 lp n and it then follows that (2.7) is uniformly convergent for 0 ^ w ^ e -1, and thus that y(w) is continuous for 0 ^ w ^ e~ l. Hence ^ iy t = 1 for < = 1. The proof of the statement in the theorem that no solution exists satisfying the convergence criteria over [0, e], where e > 1, is practically a repetition of what we have just done. For, since the convergence criteria are satisfied over [0, e], the equations (1.1) reduce to (2.4) over [0, e], as in part (i) of the proof, and (2.4) gives as the only possible solution (2.5). But (2.5) is not a solution for t > 1, and evenfor t = 1 it does not satisfy the convergence criteria since ^ i t y i diverges. 3. Theorem 1 indicates that all that we can hope to prove in any generalization is that there exists one and only one solution of the

6 124 J. B. MoLEOD equations (1.1) with initial conditions (1.2) for some interval [0,c] of t. That there is no significance in the interval [0,1] of Theorem 1 is shown by considering the case a {i = kij, for some constant k. This in fact reduces to the case a ti = ij if we replace t by T, where T = Id. From Theorem 1, we then have one and, under the convergence criteria, only one solution for the interval [0,1] of T, which is [0,1/k] for t. 4. The general case Let us first consider very briefly what happens if a ti = c i Cj, where, since a ti ^ 0, we must have c i ^ 0 for all i. In fact, we may exclude the case c i 0 for any i. For, if c t = 0 for just one value of t, then the corresponding y t appears in no equation but the tth, for it appears with the coefficient c { (= 0) in any other equation. We may therefore ignore y t and solve the remaining equations for the remaining yy The problem therefore reduces to a similar one in which all the coefficients are non-vanishing. This argument can plainly be repeated if c t = 0 for more than one value of t. Further, by making a change of scale as in 3, we may suppose that More generally, if we suppose that there are numbers c i such that c-ii ^ c<cy for all i, j, then once again we may assume without loss of generality that c t > 0 for all t. For, if c t = 0 for some particular value of t, then ay = 0 for this value of i and all j, and y t can be eliminated as before. And once again we may suppose that (^ = 1. Given a sequence of non-zero numbers {c n } (n = 1,2,3,...), with c 1 = 1, we define a related sequence {C n } by the recurrence relations We then have C, = 1, (4.1) Lf,0^ = ^1 ^ (n= 1,2,3,...). (4.2) fi C n+1 THEOREM 2. //, in the problem of (1.1) with initial conditions (1.2), we have a if <j c { Cj, where, without loss of generality, we may suppose c { > 0for alii and <^ = I, and,if the sequence {C n }defined by (4.1) and (4.2) is such that the power series J C zn n h* 13 non-zero radius of convergence R, then the problem possesses at least one solution valid in any closed interval [0, R #] of t, where # > 0. (We may remark that this is not a best-possible result since, in the case of J 2, Theorem 2 would give us a range of validity [0, e~ x #], whereas Theorem 1 tells us that it is

7 ON A SET OF DIFFERENTIAL EQUATIONS 125 actually [0,1]. It is also perhaps worth emphasizing that, although Theorem 2 places no convergence criteria as in Theorem 1, it at the same time Bays nothing about uniqueness.) Proof. Consider the N equations with the initial conditions!^(0)=l, yi N) (0) = 0 (»=2,3 N). (4.3 a) By multiplying the tth equation of (4.3) by i and adding for t = 1, 2,..., N, we find that, as at the beginning of Theorem 1, the terms on the right-hand side cancel, so that N whence 2*^ = 1. (4.4) The proof of the theorem breaks fairly naturally into three sections: (i) we show that the problem of (4.3) and (4.3 a) has one and only one solution, and that this solution is valid for all positive t and is such that yi^it) > 0 for all t; (ii) we show that, for fixed i and N -> oo, the sequence {y[ N) (t)} is uniformly bounded and equicontinuous for t in [0, R #]; (iii) we show that there is a sequence of values of N tending to infinity for which, for all i and t in [0, R &], yi N) (t) -> y^t), where t/<(<) a solution of the problem of (1.1) and (1.2). (i) We note first that the right-hand side of (4.3) satisfies a Lipschitz condition in the y[^ if the y^ are bounded. But this is certainly so initially, and so, by a familiar theorem in the theory of the existence of solutions of differential equations, there is one and only one solution of (4.3) and (4.3 a) for some interval [0,t x ], say, oft. We note secondly that, for t sufficiently small in [0, t J, y[ N) ^ 0 for ' For»r>«>> = i. 2,^(0) = 0, 5*U >0) i-o unless On = 0, when, from (4.3),

8 126 J. B. McLEOD which, with the initial condition j/^(0) = 0, implies that y^ = 0; «-* m =o, rn >», unless, arguing again as above, y^ = 0, and so forth. If we now drop from the equations all functions which are identically zero, and renumber to fill up the gaps, we can show that, except for the initial vanishing of y^,..., yffl, we have y^ > 0 for all i and all t (not necessarily sufficiently small) in [0, t{\. For, if we suppose that y]^, say, is the first to vanish as ( increases in [0, <J (if two vanish simultaneously, we choose the one with the smaller value of k) and if we suppose that it vanishes first at t = o, then we have from (4.3) that whioh is a contradiction. If y^ > 0 for all i, then it follows from (4.4) that y{ m < 1 for all t, and so we can treat t x as the initial point, apply the Lipschitz-condition argument, and extend the region of validity. As we repeat this process, we can either extend the region of validity to all positive t, which is what we want, or we find that there is a least upper bound on the values of t for which extension is possible. Let us suppose this bound is T, so that we can extend to any interval [0, T &], where & > 0, but not to any interval [0, T+&]. As t -> T, we certainly have for all i, by arguments already used, that yw > 0, and so,from (4.4), yw < 1. Hence, from (4.3), Idy^/dt] < K for all i and some constant K. Certainly, y^(<) has at least one limit as t -* T because it is bounded, but in fact the limit is unique. For, if we suppose not, then we can find two values of t arbitrarily close to T for which the values of y\ N) {t) are not arbitrarily close, and this contradicts the boundedness of the differential coefficient. If we now define, vum.,. im,.-, y\ N) (T) = km y^it), 1-+T-0 we can apply the usual Lipsohitz-condition argument to extend the solution beyond T, which contradicts the definition of T. We have thus shown that we can extend the solution to all positive t, and it is unique because we have throughout used Lipschitz-condition arguments. Furthermore the argument implies that y^ > 0 for all t except initially and except for those functions which vanish identically. This completes section (i) of the proof.

9 ON A SET OF DIFFERENTIAL EQUATIONS 127 (ii) We now investigate bounds for the solutions y^. Since certainly 1, we have, from (4.3), with t = 2, that so that ytf < #. Suppose in general that Then again from (4.3) so that yj*> < ^ t*- 1 (t = 1, 2,...,n). (4.5) v ( JV, ^ ^ <. by virtue of (4.2). Using induction, we establish the truth of (4.5) for i = 1, 2,..., N. We note that the estimate (4.5) for y^ is independent ofn. We note next that N m Ic t y^<fc t^, (4.6) i-l i-1 by virtue of (4.5), and that the right-hand side is bounded, independent of N, provided that t lies in [0, R &]. From now on we shall suppose t to lie in such an interval. With this restriction on t, we now show that y^, for fixed t, is equicontinuous for all N. We shall certainly have shown this if we prove that dyw/dt is bounded for all N, t, and this follows from (4.3) by using (4.5) and (4.6). For fixed t, therefore, but for all N, and for all f in [0, B &], the sequence {y^(0} is a uniformly bounded, equicontinuous set. Hence, by Ascoli's lemma [as, for example, in (3) 5], there is a subsequence which is uniformly convergent for t in [0, R #]. (iii) We can now employ the principle of the Helly selection theorem to show that there is a sequence of values of N for which {j/^(0} i 8 uniformly convergent, not just for one i, but for all t. For we have established a subsequence 2VJ < N\ < N\ <... such that for this subsequence {y^jia uniformly convergent. We can then find a subsequence of this subsequence, say N% < N\ < N% <..., such that {yi^} is also uniformly convergent. And, in general, for any integer i, we can find a subsequence Ng < N[ < N\ <..., which is a subsequence of that for i 1, and such that for it {y[ N) }, {yi N) },..., {y\ N) } are all uniformly

10 128 ON A SET OF DIFFERENTIAL EQUATIONS convergent. The sequence ivj, N\, N\,... is then a subsequence for which {y^1} is uniformly convergent for all t. From now on, we shall restrict ourselves to values of N in this subsequence. Then lim y^(t) = Vi(t), say. (4.7) As N -* oo through this subsequence, and for t in [0, R &], S i-l CO converges uniformly to c i!/< ^ivi ^ where N o, though large, is to be fixed. We first choose N o so that, using (4.5) and (4.6), both the second and third terms on the right-hand side do not exceed Je, for any given e > 0, and, N o once chosen, we can make N sufficiently large so that, from (4.7), the first term also does not exceed e. It thus follows that dyf^jdt converges uniformly to i-l 2 from which it follows that the functions y { (t) satisfy (1.1). This completes the proof of the theorem. REFERENCES 1. Z. A. Melzak, 'A scalar transport equation', Trans. American Math. Soc. 85 (1957) D. Morgenstern, 'Analytical studies related to the Maiwell-Boltzmann equation', J. Rational Mech. Anal. 4 (1955) E. A. Coddington and N. Levinson, Theory of ordinary differential equations (New York, 1955).