Differential Equations with Discontinuous Righthand Sides Mathematics and Its Applications (Soviet Series) Managing Editor: M. HAZEWINKEL Centre/or Mathematics and Computer Science, Amsterdam, The Netherlands Editorial Board: A. A. KIRILLOV, MGU, Moscow, U.SSR. Yu.1. MANIN. Steklov Institute o/Mathematics, Moscow, U.S.SR. N. N. MOISEEV. Computing Centre, Academy o/Sciences, Moscow, USS.R. S. P. NOVIKOV,LandauInstitute o/TheoreticaIPhysics,Moscow, US.S.R. M. C. POL YV ANOV, Steklov Institute of Mathematics, Moscow, USSR. Yu. A. ROZANOV, Steklov Institute 0/ Mathematics, Moscow, U.S.SR. A. F. Filippov Department of Mathematics. Moscow State University. U.S.S.R. Differential Equations with Discontinuous Righthand Sides edited by F. M. Arscott KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON Library of Congress Cataloging In Publication Data Fll ippov. A. F. (AlekSel Fedorovich) [Different'sia 1 'nye uravneni fa s razryvnnl pravol chast 'fu. Engllshl equations wlth discontlnuous righthand sides I A.F. Fi I lppov : edlted by F.M. Arscott. p. cm. 0 is arbitrarily small, we have for sufficiently small 0 and sufficiently large j Ix(a) - :t(a + 0)1 < e/4, l:t(a + 0) - 3:1j(a + 0) I < e/4. By Lemma 2, for a k such that Xjj(t) == :tle(t) we have (if j > j1(e)) 13:1j(a + 0) - Xjj(ale) I < e/4, by virtue of (10). Hence, Ix(a) - :tol < e. Since e > 0 is arbitrary, x(a) = :to, that is, the solution x(t) passes through the point p. Similarly, it passes through the point q. The following theorem is proved in [10J on the assumption that the func-tion f satisfies the Caratheodory equations in the whole domain G. This re-quirement is weakened in line with [12J. 1 Caratklodory Differential Equations 9 THEOREM 5. Let the function f(t, x) satisfy the OaratM6dory conditions in each closed bounded subdomain of an open domain G. Let A be a point (to, xo) (or a closed bounded set), A c G. If all the solutions of the equation :i: = f(t, x) with the initial data x(to) = Xo (or with various initial data (to, x(to)) E A) exist for a ~ t ~ 13 and if their graphs for these t lie in G, then 1) the set of points lying on these graphs (i.e., a segment a ~ t ~ 13 of an integral funnel of the set A) is bounded and closed; 2) the set M of these solutions is a compactum in the metric O[a, 13]. PROOF: Let us take a sequence of closed bounded'domains DlJ D2,. , such that A lies within D1 and D,. lies within D"+1, k = 1,2, ... , and that each bounded closed set KeG is contained in some of the domains D,.. We shall show that all the considered graphs lie in one of the domains D,.. Assume the contrary. Then for each of the domains D,. there exists a solu-tion x,.(t), 13,. E [a,pl and a,. such that (ll) From these solutions, we choose such a subsequece SoCk = k1,k2, } for which PIc -+ pEA. Since A C D1 C D2 C ... , it follows from (11) that for each m ~ 1 and for each k ~ m there exists a point qr E aDm(aDm is the boundary of the domain Dm} on the graph of the solution x,.(t) such that the arc Pleqr of this graph is contained in Dm. From the subsequence So we choose a subsequence of solutions for which q ~ -+ q1 E aD1, and from it, by Lemma 5, we choose a subsequence Sl of solutions which converges to a solution whose graph joins the points pEA and q1 E aD1. By the same method, we choose from Sl a new subsequence S2 which con-verges to a solution whose graph joins the points pEA and q2 E aD2. Pro-ceeding with this process, we obtain a solution x(t), the graph of which passes through the points pEA, ql E D1, q2 E D2, ... . By assumption this solution exists on a closed interval containing a, 13 and the abscissa tp of the point p. The graph of the solution on this interval is a bounded closed set and, accordingly, it lies in some domain Dm, that is, within Dm+1' This contradicts the fact that this graph passes through the point qm+1 E aDm+1' Thus, the assumption is incorrect, and all the considered graphs lie in one of the domains Die. Then the assertion 2} is proved as in Lemma 4, and from this assertion there follQwS the assertion 1). 4. The continuous dependence of solutions of the Caratheodory equations on initial data and on the right-hand side of the equation, or on the parameter, has been considered in a number of papers, particularly in [10] and in [14J-[27]. The differentiable dependence has been considered in [28]-[31]. Below we present two theorems on continuous dependence: the simplest (for the case where a sequence of functions f,.(t, x) on the right-hand sides of differential equations converges) and a more general one (for the case where a sequence of integrals of these functions over t converges). In 4 we deal with solutions x(t, IL) of equations, the right-hand sides of which depend on the parameter IL which changes on some set M (of a metric space) 10 Equations. .. Discontinuous only in t Chapter 1 with a limit point J.to EM and establish conditions for the convergence x(t, J.t) -+ x(t,J.to) for J.t -+ J.to, i.e., for p(J.t,J.to) -+ o. To this case one can reduce the case of the sequence Xk(t), k = 1,2, ... , if one puts x,.(t) = x(t, J.t), J.t = 11k -+ O. As is known, for differential equations with continuous right-hand sides and for Caratheodory equations, uniqueness of a solution leads to its continuous de-pendence on initial data [32]. The following lemma generalizes this assertion. It is applicable not only to solutions of differential equations, but also to solutions of differential inclusions. It specifies the sense in which one can speak of con-vergence to a set of solutions in the absence of uniqueness. This lemma makes it possible to reduce the conditions for continuous dependence of a solution to a number of simpler conditions. LEMMA 6. Let there be given a point (to, ao), numbers tl > to, eo> 0, a finite open domain D in the (t, xl-space, a set M of values of the parameter J.t, and a family S of continuous functions e(t), each of which corresponds to its initial value a = e(to) and a certain value of the parameter J.t E M. Let 1) each function e(t) be defined on some interval, its graph lying in the domain D, the endpoints of this graph being two points of the boundary of the domain Dj 2) for any a, J.t(la - aol < 0, J.t E M) there exist at least one function of the family which corresponds to these a and J.tj 3) in each sequence of functions edt) E S, i = 1,2, ... , which correspond to the values ai -+ ao, J.ti -+ J.to, all functions be equicontinuousj 4) the limit of each uniformly convergent sequence of functions of the family, for which a = a. -+ ao, J.t = J.ti -+ J.to, be a function of the family for which J.t = J.to; 5) all the functions of the family for which J.t = J.to, a = ao, be defined at least on the segment [to, tIl; the set of these functions will be defined by Xo; 6) for each of the functions eo(t) of the set Xo the eo-tube (12) Ix - eo(t) I < eo, be contained in D. Then for any e > 0 there exist a 0 > 0 and an '1 > 0 such that for all a and J.t satisfying the conditions (13) la - aol < 0, each of the functions e( t) of the family, which corresponds to these a, J.t exists on the segment [to, hJ, and differs from some function eo(t) E Xo less than bye: (14) le(t) - eo(t)1 < e (For different e(t) the functions eo(t) may be different). PROOF: Suppose for some ai -+ ao, J.ti -+ J.to there exist functions e.(t) E S defined on less than the whole segment [to, tIl. According to 1) I each of them reaches the boundary of the domain D at some point qi(ti,X.), t. E (to,td. It 1 CflrfltModory Dif/erentifll Equfltion! 11 follows from the conditions 3) and 6) that ti ~ ro > to for all i > il From the sequence {gi} we choose a subsequence which converges to some point q, and from a corresponding subsequence offunctions i(t) we choose a new subsequence which converges to the function o(t) whose graph joins the points (to, 0.0) and q as in Lemma 5. By virtue of 4), the function o(t) E Xo. Then, by virtue of 6), the eo-tube (12) is contained in D, and q is the point (tl' o(td). This contradicts the convergence of the subsequence of points qi(ti, Zi) of the boundary of the domain D to the point q because ti is l e s ~ than tl' Thus, for some 6> 0 and '1 > 0 for all a and I' which satisfy (13), the graph of the function (t) lies in D for to < t < tl' IT the lemma is not true, then for some e > 0 there exists a sequence of functions Zle (t) E 8, Ie = 2,3, ... , such that ZIe(to) = ale -. 0.0, I'le -. 1'0, and the graphs of these functions lie in D for to < t < tl, and for each Ie and each function (t) E Xo (15) Iz,.(t,.) - (t,.) I ~ e, t,. E [to, tl]' Ie = 2,3, ... j the points t,. may depend on the choice of the function (t) E Xo. By virtue of 3) and 4), one can choose from the sequence {z,.(t)} a sub-sequence which converges uniformly to some function zo(t) E Xo. This is in contradiction with (15) for (t) == zo(t). The lemma is proved. COROLLARY. Let the conditions 1)-4) of Lemma 6 be fulfilled; for a = 0.0, I' = 1'0 in the family 8 let there exist only one function o(t) and for this function let the eo-tube (12) be contained in D. Then the assertion of Lemma 6 is valid and each sequence i(t) of functions from 8, for which ai -. ao, I'i -. 1'0, converges uniformly to o(t) on the segment Ito, tl]' REMARK: For the family of solutions of the Caratbeodory equations z = f(t, z) the conditions 1) and 2) of Lemma 6 are fulfilled by virtue of Theorems 1 and 4, and the conditions 3) and 4) are fulfilled by virtue of Lemmas 2 and 3. Hence, if a solution with the initial data z(to) = flO is unique, it depends continuously on the initial data. For the Caratheodory equation z = f(t, 3:,1') with the parameter I' the conditions 1) and 2) are fulfilled, and one need only fulfillment of the conditions 3) and 4) and uniqueness of the solution for z(to) = 0.0, I' = 1'0. THEOREM 6 ([10]; [9], p. 58). Let for (t,3:) E D, I' E M, 1 f(t, 3:, 1') be measurable in t for constant 3: and 1'; 2 If(t, 3:, 1') I ~ m(t), the function m(t) being summable; 3 for almost all t the function f(t, 3:, 1') be continuous in z, and for I' = I'o-in z,l'; 4 the solution z = o(t) of the problem (16) ;; = f(t, z, 1'), 3:(to) = a for a = 0.0, I' = 1'0 be unique for t ~ to; let this solution exist for to ~ t ~ tl and let its graph have a neighbourhood ofthe type (12) which is contained in D. Then for any a and I' sulliciently near ao and 1'0 the solution of the prob-lem (16) on the closed interval Ito, ttl exists (it is not necessarily unique) and converges uniformly to o(t) as 0.-'0.0, I' -. p.o. REMARK: The condition 3 leads to the fact that for almost all t the function f(t, z, 1') tends to f(t, z, 1'0) uniformly in 3: (on any compactum) as I' -. 1'0. 12 Equations. .. Discontinuous only in t Chapter 1 PROOF: By virtue of 1-3, equation (16) with the J.t-independent majorant met) is a Caratheodory equation, and the conditions 1)-3) of Lemma 6 are therefore satisfied for its solutions; the conditions 5) and 6) are satisfied by virtue of 4. The solution x = e(t, a, J.t) of the problem (16) satisfies the integral equation (17) 't e(t; a, J.t) = a + r I(s, e(s; a, J.t); J.t)ds. , ito If the sequence of solutions e(t, ai, J.ti) converges uniformly as ai --+ ao, J.ti --+ J.to, then, from 3 , for almost all s I (s, e(s; ai, J.t.); J.ti) --+ I (s, e(s; a, J.to); J.to) . In this case, by virtue of 2 a limit transition is legitimate under the integral sign in an integral equation for e(t; ai, J.ti). Thus, the limiting function e(t; ao, J.to) satisfies the equation (17) for a = ao, J.t = J.to arid is a solution of the problem (16) with a = ao, I-' = 1-'0. That is, the condition 4) of Lemma 6 is fulfilled. The assertion of the theorem follows from this lemma. A further generalization of the continuous dependence theorem consists in the replacement of the requirement of the convergence I(t, X; 1-') --+ j(t, x; 1-'0) (i.e., continuity of the function I in I-' for I-' = 1-'0) by the requirement of con-vergence of the integral of j(t, x; 1-') over t to the integral of I(t, Xi 1-'0) ([14], [17] and others). ' THEOREM 7 [17]. Let for J.t E M, to ~ t ~ tll X E B (B being a finite open region in R"') 1 the function l(t,x;l-') be measurable in t for constant X,I-'j 2 I/(t, x; J.t) I ~ met, 1-'), the function met; 1-') being summable in tj 3 there exist a summable function l(t) and a monotone function ,per) --+ 0 for r --+ 0 such that for each r > 0, if Ix - YI ~ r, and for almost all t (18) I/(t, Xj 1-') - j(t, Yi 1-') I ~ l(t),p{r); 4 for each x E B for I-' --+ 1-'0 (19) rt I(s, x; I-')ds --+ it I(s, x; I-'o)ds ito to be uniform in t on the segment [to, tl]j 5 the solution x = eo(t) of the problem (16) for a = ao E B, I-' = 1-'0 be unique for t ~ to and lie in the domain B for to ~ t ~ tl. Then for any a and I-' sufliciently near ao and 1-'0 the solution of the prob-lem (16) on the interval [to, tl] exists (it is not necessarily unique) and converges uniformly to eo(t) as a --+ 0,0, I-' --+ 1-'0. PROOF: First we will show that for each sequence I-'i --+ 1-'0 and each sequence of continuous functions Xi(t) E B, i = 1,2, ... , uniformly convergent to xo(t) we have, for all t E [to, tl], (20) r [/(s, Xi( s); I-'i) - f(s, xo( s); 1-'0)] ds --+ 0 ito (i --+ 00). 1 Carathlodory Differential Equations 13 Since the function xp(t) (p = 1,2, ... ) is continuous, the sequence of piece-wise constant functions to + (i - l)hq ~ t < to + ihq, i = 1,2, ... ,2'1, where hq = 2-'I(tl -to), q = 1,2, ... , converges uniformly to xp(t). There exists such a q(p) that for the function zp(t) == !lP,'I(p)(t) (21) Since xp(t) --. xo(t) as p --. 00, so does zp(t). By virtue of 1 and 2, r [/{8, Zp(8)j JLo) - 1(8, Xo(8), JLo) I d8 --.0 ito (p --. 00). It follows from 4 that a relation similar to (19) holds also for integrals over any interval contained in [to, tll. In such a relation one can replace x by zp(t) on any interval where the function zp(t) is constant. Summing up over such intervals, we obtain for p = const Ji,p(t) == r [/(8,zp{8)jJLd - l(s,zp(s);JLo)] ds --. 0 ito Thus, for each t E [to,tlJ and each p ~ 1 there exists anip{t} such that IJi,p(t)1 < 2-P for all i > ip(t). The number ip(t) can be increased and we assume therefore that ip+1(t) > ip(t). Suppose v(i, t) = p for ip(t) < i ~ ip+1(t), p = 1,2, .... Then if i --. 00, we have for each t = const (22) v(i, t} --. 00, IJi,v(i,t) (t}1 < 2-v(i,t) --. O. By virtue of the condition 3, for almost all 8 E (0, t) The right-hand side does not exceed the summable function 1(8}tjJ{d}, where d is the diameter of the domam B, and for almost all s it tends to zero as i --. 00 since X'(8} --. XO(8}, ZV(i,t) (8) --. XO(8). Hence, (23) as i --. 00. From (21)-(23) there follows (20). Now let x.(t) be the solution of the problem (16) with a = a. --. ao, JL = JLi --. JLo (i --. 00) and with a function f which meets the requirements of Theorem 7. Then (24) Xi(t) = at + r 1 (8, X.(8)j JL.) d8. ito 14 Equations. .. Discontinuous only in t Chapter 1 IT x.(t) tends to xo(t) uniformly on some interval [to, t*], then, by virtue of (20), one can pass to the limit in the equality (24), and the function xo(t) is a solution of the problem (16) with a = ao, I-' = 1-'0. The condition 4) of Lemma 6 is thus fulfilled for the family of solutions of the problem (16) with different a and 1-'. Let us verify fulfillment of condition 3) of Lemma 6. Since the functions f(s, ao, 1-'0) and l(s) are summable, for each e > 0 there exists a 0 > 0 such that for any a, P E [to, til it follows from IP - al < 0 that (25) where d is the diameter of the domain B. For a certain pde) the difference between the left- and the right-hand sides of the relation (19) is less than e for p(l-', /Lo) < p1(e) and for all t E [to, t1l. From this and from (25) there follows (26) liP f(s,aO;I-')dsl < 3e Let Xi(t) (i = 1,2,.,.) be a sequence of solutions of the problem (16) with a = ai ~ ao, I-' = I-'i ~ 1-'0, From (24) we have (27) By virtue of (18) the integrands in (26) for /L = /Lo and in (27) differ by not more than l(s),p(d) and therefore, on account of (25), the integrals differ by not more than e. Now for p(l-', 1-'0) < pde) we have from (26) and (27) (28) Since I-'i ~ 1-'0, the inequality P(I-'i, 1-'0) < pde) may fail to hold only for a finite number of i values. By virtue of continuity of the functions Xi(t), there exists a 01 for these i such that (28) holds for all a,p E [to,t1], IP-al < 01. For IP - al < min {o; 01} the inequality (28) is thus satisfied for all i. Since e is arbitrary, the solutions under consideration are equicontinuous, and the condition 3) of Lemma 6 is fulfilled. The conditions 1) and 2) of this lemma are satisfied by virtue of 10_30 and Theorems 1 and 4. The assertion of the theorem is valid by virtue of the corollary of Lemma 6. REMARK: The assertion of Theorem 7 remains true if the condition 30 is re-placed by the following: 3*, There exist functions l(t,r,l-') and oo(e) > 0, 0 < e < eo, such that for each r > 0 for almost all t and Ix - YI ~ r (29) If(t,x,l-') - f(t,y,I-')1 ~ l(t,r,I-'), let, r, 1-') ~ 0 and for any a,p E [to,td, IP - al < oo(e), for all r,l-' (30) (31) I[a,p] '= f: l(s,r,l-')ds < e, I[tO,t1] ~ 0 (r ~ 0, I-' ~ 1-'0). (r ~ 0), 1 CaratModorll Differential Equations 15 In this case only two changes must be introduced into the proof of Theo-rem 7. By virtue of (29), the integrand in (23) is not greater than I (s, l.zi(8) - ZV(i,t)(s)l,iJi) and by virtue of (Sl) the integral (2S) therefore tends to zero as i -+ 00. IT IP - al < min{ojoo(e)} the integrands in (26) for iJ = iJi and in (27) differ by not more than l(8, d, iJi), and therefore, by virtue of (30), the integrals differ by not more than e. The rest of the argument in the proof of Theorem 7 remains the same. The following eXaJRple shows that neither the conditions 3 in TheoreJRS 6 and 7, nor the condition (Sl) of the remark can be omitted, even in the case where for each iJ the function f(t,.z, iJ) is continuous in t,.z and is bounded by a iJ-independent constant. Let iJ = 11k -+ 0, k = 1,2, ... j.z e Rl, 1 f(t,.z,iJ) = II.(t,.z) = [k2(.z _ t) _ kJ2 + 1 -+ O. The function .z1o(t) = t + 11k is a solution of the equation :i: = II.(t, .z), but the limit lim,. ... oo .z,. (t) = t does not satisfy the equation :i: = O. THEOREM 8. Let elements bij(t, iJ) of a matrix B(t, iJ) and a vector-valued function get, iJ) for iJ E M be absolutely continuous on the segment [to, tIl and uniformly in t . (S2) get, iJ) -+ get, iJo) Let there exist a o(e) > 0 (0 < e < eo) such that for an i, i = 1,2, ... , n, all iJ E M and all a, P E [to, tlJ, IP - al < See) we have (SS) ifl!bii(t,iJ)! dt < e. Then on the segment [to, tIl the solution of the problem (S4) for a -+ ao, P -+ iJo converges uniformly to the solution of the same problem with a= ao, PROOF: Since equation (S4) is of CaratModory type, then, by Theorem 3, its solution .z(tj ao, Po) for a = ao, P = Po exists on the segment [to, tlJ and is unique. Then for the problem (34) in the region l.zl < 1 + max l.z(tj ao, iJo) 1 Ito,hl the requirements of Theorem 7 and of the remark are met. Thus, the assertion is valid. 16 Equations. .. Discontinuous only in t Chapter 1 COROLLARY. If, for a sequence of linear Caratheodory systems, coefficients and free terms converge in the metric L1, and a sequence of initial data converges, then the sequence of solutions converges uniformly on a given segment. Using the estimate obtained in [18] for the difference of solutions of two linear systems, one can evaluate the rate of this convergence in terms of the norms (in Ld of differences of their coefficients and of the difference of free terms. 5. The properties of integral funnels investigated in [12] for differential equa-tions with continuous right-hand sides remain the same also for Caratheodory differential equations [10]. For the differential equation x = !(t, x) (x ERn) an integral funnel of a point (to, xo) (or of a set A) is a set of the (t, x )-space points lying on all solutions which pass through the point (to, xo) (or respectively through the points of the set A). A funnel segment is a part of the funnel lying in the interval a ~ t ~ f3. In the following theorems we assume that the equation x = !(t, x) satisfies the Caratheodory conditions in each finite part of the domain under considera-tion and that all the solutions with the initial data x(to} = Xo (or all the solutions which pass through the points of a given closed set A) exist for a ~ t ~ f3, and the point (to, xo) (correspondingly, the set A) is contained in the layer a ~ t ~ f3. Compactness of a funnel segment was proved in Theorem 5. THEOREM 9. If A is a point or a connected compactum, the cross-section of a funnel by any plane t = t1 E [a, f3] is a connected compactumj the set of solutions passing through points of the set A is a connected compactum in the metric C[a, f3]. The first assertion for a funnel of the point (to, xo) in the case of a sufficiently small segment [a, f3] is proved in [10]. The second assertion is proved similarly with the help of the metric C[a, f3]. In the case of a segment of any length and any connected compactum A the assertions are extended by the methods presented in [12] and [33]. THEOREM 10. An arbitrary point (t1' xd of a funnel boundary can be joined to a point (to, xo) by such an arc of the graph of a solution which passes along the funnel boundary. This assertion is proved in [10]. THEOREM 11. Let !(t, Xi 1-'0) and A satisfy the conditions formulated before Theorem 9, and the functions !(t, Xi 1-'), I-' = I-'k - 1-'0, k = 1,2, ... , meet the requirements of Theorem 6 or Theorem 7, except the requirement of solution uniqueness. Let Ak, k = 1,2, .. , be a sequence of sets such that for each e > 0 all Ak, beginning with a certain one, are contained in the e neighbourhood of the set A. Then the same property is inherent in the segments a ~ t ~ f3 of funnels of the set Ak for the equations x = !(t, Xi I-'k) with respect to the segment of the funnel of the set A for the equation x = !(t, Xi 1-'0)' 2 Equations with Distributions Involved as Summands 17 The assertion follows from Lemma 6 because if the requirements of The-orem 6 or of Theorem 7 (without the uniqueness requirement) are met, the requirements of Lemma 6 for the family of solutions are met also. 2 Equations with Distributions Involved as Summands We deal here with different classes of differential equations with additively involved distributions, including differential equations with impulses, linear (and simple nonlinear) equations with distributions on the right-hand sides, and lin-ear systems not solved with respect to derivatives and possessing discontinuous solutions. We present the methods of reducing such equations and systems to Caratheodory systems, which enables us to prove the existence and to investigate the properties of solutions. 1. In [34] (pp. 169-179) the equations (1) a: = f(t, z) + p(t), are analyzed, where z eRn, the function f(t, z) satisfies the Caratheodory equations, and p(t) is a distribution or an ordinary, but not Lebesgue integrable, function. In 1 the function p(t) is assumed to be a distributional derivative of some measurable function q(t) bounded on each finite interval, that is (2) p(t) = q(t), Iq(t) I ~ 'Y (a < t < ,8). In particular, p(t) can be a usual function integrable in one or another sense, and q(t) can be an integral of p(t) (Perron, Denjoy, Denjoy-Khintchine integrals; for a more general formulation of the problem see [35]); p(t) can be a delta-function (in this case equation (1) belongs to the class of equations with impulses, encountered in applications) or a distributional derivative of a continuous or a discontinuous function of bounded variation. In all these cases, one can reduce equation (1) to the Caratheodory equation (3) iJ = f(t, y + q(t)), on making the substitution z = y + q(t). Measurability of the right-hand side of (3) in t for any constant y follows from Lemma 1, 1. A solution of equation (1) is any function of the form z(t) = y(t) + q(t), where y{t) is a solution of equation (3). Such a function z{t) satisfies equation (1) if the derivative a: is understood in the sense of the theory of distributions (note that the derivative of the Perron and Denjoy integrals and the approximative derivative of the Denjoy-Khintchine integral exists almost everywhere and is a derivative in the sense of the theory of distributions; this follows from [36], Ch. 8, 2). Since equation (3) has a solution with any initial data of the form y(to) = a, equation (1) has a solution for initial data of the form (4) z(t) - q(t)lt=to = a. 18 Equations. .. Discontinuous only in t Chapter 1 If the function q is continuous at the point to, the condition (4) is equivalent to the initial condition (5) x(to) = b (b = a + q(to)) . If the function q is discontinuous at the point to, all solutions of equation (1) are also discontinuous at this point and the condition (5) has no uniquely defined meaning. If there exists limt_to-O q(t) = q(to - 0) or limt_to+o q(t) = q(to + 0), the condition (5) can be replaced by the condition x(to - 0) = a + q(to - 0) or x(to + 0) = a + q(to + 0). In other cases one has to restrict oneself to setting the initial data in the form (4). Knowing the properties of the solutions ofthe Caratheodory equation (3), 1, and using the change x = 11 + q(t), we obtain corresponding properties of solu-tions of equation (1): existence of the solution, compactness of the set of solutions contained in a closed bounded domain, and uniqueness under the conditions (6) or (7), 1. The behaviour of the solution near the ends of its interval of existence is examined in [34] (pp. 176-179). Let us make a more detailed analysis of equations with impulses. Consider the equation (6) z = I(t, x) + P6(t), where I(t, x) is a known function; with regard to the function Pe(t), it is known only to be equal to zero outside a small interval (tl - e, tl + e), and its integral over this interval is known to be equal to tI. Such equations arise from problems of body motion in the presence of pushes and knocks if such a push or knock is known to occur at the moment t = h, to be of short duration, and if the total impulse, i.e., the impulsive force integral over the time interval during which the knock lasts, is known. To exclude from consideration the unknown values of the function P6(t) in the interval (tl - e, tl + e), one has to make a limit transition e -+ 0 with a retained constant value tI of the integral of P6(t). In the limit one obtains the equation (7) z = I(t,x) + tlo(t - tl), where 0 is a delta-function. In the theory of distributions oCt) = F1'(t) , where F1(t) = 0 (t < 0), F1(t) = 1 (t> 0). Hence, by the change x = 11 + tlF1(t - td equation (7) is reduced to equation (3) with q(t) = tlF1(t - tl)' The solutions of equation (3) are absolutely continuous. Thus, the solutions of equation (7) are functions which for t < tl and t > tl are absolutely continuous and almost everywhere satisfy the equation z = I(t, x), and for t = tl have a jump x(t + 0) - x(t - 0) = tI. Similarly, at points ti all solutions of the equation 00 (8) z = I(t, x) + L tliO(t - til ;=1 2 Equations with. Distributions Involved as Summands 19 have jumps equal to tli (i = 1,2, ... ), and in the intervals between these jumps they are absolutely continuous and satisfy the equation a: = J(t, x). The following theorem motivates the transition from equation (6) to equa-tion (7) and the corresponding step in more general cases. The case where vectors tli in (8) depend on z, i.e., the values of the jumps depend on z, will be discussed in 3, 3. THEOREM 1. In a bounded closed domain D consider the equations (9) k = 1,2, ... , where the function J satisfies the Caratheodory conditions, p",(t) = q",(t), (10) (k=l,2, ... ), Then each function x( t), being the limit of some sequence of solutions x", (t) of equations (9) is a solution of equation (1) with pet) = q(t). PROOF: We pass over from equation (1) to (3), and using a similar change of variables z", = y", + q",(t), from equations (9) to the equations (11) k = 1,2, .... For these equations the Caratheodory condition 2) (I, 1) holds by virtue of Lemma I, 1, the condition 3) holds with one and the same function met) for all k. For almost all t the function J(t, x) is continuous in z. Taking into account (10) and making a limit transition k -+ 00 in the integral equation equivalent to (11), we find that the function yet) = limy",(t) satisfies a similar integral equation, but with the function q(t) instead of q",(t). Therefore, yet) is a solution of equation (3). Then z(t) = yet) + q(t) is a solution if equation (1). Differential equations with impulses have been examined in many papers, which cannot all be referenced here (below we refer only to some of them). Equations with impulses at given time instants, as in (8), or when solutions reach given surfaces in the (t, z)-space have been investigated. The magnitudes of the jumps of solutions are either given in advance or depend on the point at which the jump occurs. Consideration has been given to existence and uniqueness of solutions, con-tinuation of solutions, continuous dependence of solutions [37]-[40], the prop-erties of integral funnels and absorbing sets from which the solutions do not go out [37], stability [41]-[47], existence and stability of periodic solutions [37], [48]-[52], application of the averaging method to equations with small parameter [39], [53]-[56], and the use of jumps for retaining solutions in the domain in case these solutions reach the boundary of the domain [57]. See also the book [206]. 2. The cases with distributions more complicated than in 1 have been studied in linear equations and systems and only in a few nonlinear cases. In applications the equation 20 Equations . .. Discontinuous only in t Chapter 1 often appears, where m ~ n, the coefficients ai, bi are constants or sufficiently smooth functions of t,z = z(t) is a known function, y = y(t) is the unknown function. li z E em, then (12) is a familiar and thoroughly investigated linear equation. lithe function z is less smooth, the right-hand side of (12) may involve distributions. It will be shown below that if z is an ordinary (locally summable) function, the solutions of equation (12) are ordinary functions and can be found without the use of the theory of distributions. To pick a unique solution, one may impose the usual initial data y(to) = Yo, y'(to) = y ~ , ... , (n-l) (t ) _ (n-l) y 0 - Yo , only for such a to for which the functions z, z', . .. ,z(m) are continuous. One often considers the following problem. We"must find the solution y(t) for t ~ 0 if it is known that y(t) == z(t) == 0 for t < 0 (that is, we wish to find the reaction of the system to the external action which starts only at the moment t = 0). Different methods can be applied to the solution of this problem. li all ai and bi are constant, then, for example, one can integrate the function z as many times as is needed for the derivative u(m) of the obtained function u to be continuous or at least Lebesgue-integrable on the interval 0 ~ t ~ tl, that is i = 1,2, ... , k. The lower limit of integration a: < 0 is arbitrary since z == 0 for t < OJ the number k is such that the function u = Zk(t) E em. Let us seek the solution x(t) of the equation (13) x(n) + an_lX(n-l) + ... + aox = bmu(m) + bm_lu(m-l) + ... + bau, vanishing for t < O. Having differentiated both sides of the equality (13) k times, we see that the function y = x(k) is a solution of equation (12) which vanishes for t < O. Another way [58] of solving equation (12) consists in reducing this equation to a system of equations containing no distributions. Let all the ai and bi be constantj if m < n, then bi = 0 for i > m, the function z(t)is continuous or Lebesgue-integrable on every finite interval. Let us introduce new unknown functions Xl, ,Xn by the formulae (14) i = n - 1, n - 2, ... , 1. Successively substituting Xn into the formula for Xn-l, and then Xn-l into the formula for Xn-2, etc., and using equation (12), we obtain the first equation of the following system (the rest of the equations are presented in (14)): X ~ = boz - aoy, (15) x ~ = blZ - alY + Xli 2 Equations with. Distributions Involved as Summands 21 Substituting 11 = 3IA + bAz into the system (15), we derive a system of the normal form. H the function z is continuous, this is an ordinary linear system with constant coefficients, and if z is Lebesgue-integrable, this is a linear Caratheodory system. Having found the solution of this system, we obtain the solution of equation (12) by the formula 11 = 3IA + bAz. We shall prove this. Differentiating the equality y = xA + b,.z n times and replacing, after each differentiation, 3 I ~ , 3l'A-1,"" x ~ by the right-hand sides of equation (15), we obtain y' = b,.z' + b,._lZ - 4,.-lY + 31,.-1, (16) 11" = b,.z" + b,._lZ' + bn - 2 z - 4 n-l11' - a,.-2Y + X,.-2. Therefore, y satisfies equation (12). H the function z is bounded on the interval a < t < P and if for t = ., E (a, P) the functions z, z', . .. have jumps z( l' + 0) - z( l' - 0) = [z]. z'(1' + 0) - z'(1' - 0) = [z'] .. , then for t = l' the jumps fyI, [11'],'" of the functions y. y' ... can be expressed [59] in terms of Iz]. [z'] ... and of the coefficients lli. b,. We should note in this connection that in the system (15) the functions Xlo" x,. are continuous. Hence, from the relations 11 = 3In + bAz and (16) we have [y] = b,.[z]. (17) [y'] = bA[z'] + bn- 1[z]- a,.-l[y], [1I"J = bn[z"J + bn- 1[z'J + b"_2[Z]- 4n -1[y'J - an-2[1I), The jumps [11(10)1 a.re therefore expressed, through the jumps [z), [z'I, . . , [z("')J. by the formula (18) k = 0, 1.2, .... The coefficients Co, Cl, .. are determined successively: Co = bn C1 = b,.-1 - C04,.-I, C2 = bn - 2 - COa,.-2 - Clan-I, .. , Ci = b"-i - COa,.-i - Clan-i+1 - .. - Ci-1a,.-lJ i = 1,2, .... H the function z = 0 for t < 0, and z E em for t > 0, m is the same as in (12), then y(t) for t > 0 can be found as a. solution of equation (12) with the initial data y(+O) = [1IJ, 11'(+0) = [11'],'" , 22 Equations . .. Discontinuous only in t Chapter 1 where [y], [y'], ... are given by formulae (17) or (18) in which [z] = z(+O), [z'] = z'(+O}, .... Now let the coefficients ai,bi in (12) depend on t,z(t) E L1(loc), that is, the function z(t) is Lebesgue-integrable on each finite interval contained in its domain of definition. Then z', z", ... are distributions. For the products b.(t)z(i) and ai(t)y(i) to have sense, we require that (19) bi E 0', i = 0,1, ... ,no Then the product b(t)z(k) (t) is given by the formula 160] (20) bz(k) == z(k)b == t(-l)k-i0. (b(k-i lz) (iI, i=O where Ck are binomial coefficients, and the derivatives are understood in the sense of theory of distributions. We shall prove formula (20) first for the case b E Coo. Using the definitions of the product of the distribution by the function from COO and the derivative of the distribution ([61], Chapter 1), we have for any test function IP E K Expressing (bIP)(k) by means of the Leibniz formula, we obtain (bZ(k),IP) = (Z,(-l)ktCkb(k-i)IP(i)) = (-l)kt Ck (z,b(k-'IIP(il). i=O i=O Using again the definitions of the product and of the derivative for the distribu-tions, we get (21) k k (bz(kl,IP) = (-l)kE0. (b(k-i)z,IP(il) = E ((-l)k-iCk(b(k-i)z)(il,IP). i=O i=O From this there follows (20) for the functions bE Coo. Let now bECk. Approximating the function b by the sequence of functions bi -to b (convergence in Ck), we find that the expressions derived from the right-hand side of (21) by the replacement of b by bi converge to the right-hand side of (21) for any function IP E K. For bECk the right-hand side of (21) is a linear continuous functional on the functions IP E K, that is, a distribution. The product bz(kl can therefore be defined as a distribution which satisfies the equality (21) for any function IP E K. This is equivalent to (20). To find the solution of equation (12) under the condition (19), one may either reduce it to a linear Caratheodory-type system or use the representation of the solution in the integral form [62] (for the case y(t) = z(t) = 0 for t < 0) r n y(t) = bn(t)z(t) + 10 ECI(S)tll(t,S)z(s)ds, o 1=1 t ~ 0, 2 Equations with Distributions Involved as Summands 23 where VI(t, 8) are solutions of a linear homogeneous equation with particular ini-tial data, and CI (8) are expressed through the coefficients /Ii, bi and their deriva-tives (see [62]). To reduce equation (12) under the conditions (19) to a linear Caratheodory system, transform each product bkz(k) and aky(k) by formula (20). Combining terms with the same i value, reduce equation (12) to the form (22) Un = Y - bnz, Un-l = an-lY - (bn - l - ... , ( Ck, Ck " (1)n-l-kck (,.-l-k) Uk = ale: - Ie:+l ale:+l + k+2ale:+2 - ... + - n-l an-l Y - (bk - C:+1bk+1 + C:+2bZ+2 - ... + z, Mter the introduction of new unknowns :Z:n-l = + Un-I, , equation (22) takes the form + Uo = O. Thus, equation (12) is reduced to the system (23) i = 1, ... ,n -1, where Uo, Ul, ... , Un -l are expressed through Y and Z by means of the above formulae, and 1/ == Un + bnz should be replaced by :Z:n + bnz. Under the above assumptions concerning the functions z, Gi, bi the system so obtained is a linear Caratheodory system. REMARK: IT one adds an ordinary function (not a distribution) I(t, 1/, z) to the right-hand side of equation (12), the first equation of the system (23) will have the form = -uo + I, and the other equations will remain unchanged. A linear equation with the coefficients ai E cm-n+i with a right-hand side which is a derivative of any order m > n of the integrable function (24) can be reduced to similar equations with smaller values of m. The change of variables 1/ = z + g(m-n) gives Each term of the right-hand side can be transformed by formula (20). After this the right-hand side will have the form + ... + hm(t), where hi(t) are integrable functions. By virtue of the linearity of the equation, its solution is a sum of solutions of equations of the form (24), but with smaller m values. By means of a finite number of such transformations, equation (24) is reduced to similar equations, but with m n, that is, to equations of the form (12), but with variable coefficients. 24 Equations . . , Discontinuous only in t Chapter 1 3. The linear equations (25) X' = A(t)x + I(t) with distributions I(t); x E Rn, A is a matrix, are considered in [34], [66], and [63]. If A(t) belongs to eoo, then I(t) can be any distribution vanishing for t < (see [63]). Let now I(t) = g(m+l) (t) be an (m + l)-th derivative of the function g(t). Let us consider two cases: a) the function g(t) is measurable and locally bounded, A(t) belongs to Wi" (loc), that is, A (m-l) (t) is locally absolutely continuous and A (m) (t) belongs to Ldloc), m ~ 0; b) the function g(t) is of locally bounded variation, and A(t) belongs to em-I. After the change of variables x = Y + g(m) (t) we obtain from (25) (26) y' = A(t)y + A(t)g(m) (t). Now consider the case a). If m = 0, A(t) E Lt{loc), then (26) is a linear Caratheodoryequation. If m ~ 1, then, by virtue of (20), we have A solution of equation (26) can therefore be expressed in the form y = Yo + YI + ... + Ym, where Yk (k = 0,1, ... ,m) is a solution of the equation (28) For k = m, equation (28) is a Caratheodory equation. For k = 0, 1, ... , m-l the function hk(t) is measurable and locally bounded. Equation (28) is therefore of the same type as (25), but the number m + 1 is replaced by a smaller number m-k. Here one can again make the change Y = z+ (hk(t))(m-k-l), etc. After a finite number of changes of variables, equation (25) is reduced to Caratheodory equati6ns. Consider the case b). Since A E em-I, m ~ 1, the product A(m)g in (27) is a distribution equal to (29) The integral is understood in the sense of Stieltjes. Formula (27) remains valid. To prove this formula, we consider a subse-quence Ai(t) --+ A(t) such that on every finite interval the derivatives A ~ k ) (t), k = 0, 1, ... , m - 1 are absolutely continuous and converge uniformly to A (k) (t) as i --+ 00. The possibility of a limit transition under the Stieltjes integral sign in (29) follows from [64] (pp. 250 and 254). In formula (27), with Ai instead of A and with the last term transformed according to (29), one can pass to the limit 2 Equations with Distributions Involved as Summands 25 in the sense of the theory of distributions. We obtain that formula (27) holds also for A E cm-l, and for a function g of locally bounded variation. By virtue of (29), equation (28) for k = m has the form where p(t) is a continuous function equal to the integral in (29). By the change Ym = Z + (-l)mp(t) this equation is reduced to a Caratheodory equation. For k < m, equation (28) is reduced to a Carathedory equation in the same way as in the case a). Thus, in both cases equation (25) can be reduced to Caratheodory equations. By virtue of Theorem 3, 1, its general solution has the form where Ul! , u,. is a fundamental system of solutions of the homogeneous equa-tion u' = A(t)u, and :z:t(t) is a particular solution of equation (25); the function Xl (t) may be a distribution. Consider the system (30) X' = A(t):z: + B(t)y + C(t)y', where y(t) is a known vector-valued function, possibly a distribution, :z:(t) is an unknown, and the coefficients belong to the class Coo. SiIch a system can be reduced to a Caratheodory system in the same way as the system (25). In particular, the change :z: = z + C(t)y reduces the system (30) to the form z' = Az + (AC + B - C/)y. In [65], the well-known Cauchy formula, which expresses the solution of a lin-ear inhomogeneous equation through its right-hand side and the reaction of a homogeneous equation to an impulse, is extended to linear equations of the gen-eral form with distributions. The general integrodifferential representation of solutions of such equations is given. Stability of solution of equation (25) in a special topology of the input-output operator, i.e., the operator transforming a given function !(t) into a solution x(t), is investigated in [66], [63], and [34] (pp. 18D-187). Some nonlinear equations, for example those derived from (25) by adding to the right-hand side the function Ip(t, x), and derived from (12) for z = Ip(t, y) + !(t), are analyzed in [63] and in [66]-[68]. In particular, sufficient existence and stability conditions for periodic solutions are obtained. 4. Solutions of a linear system with constant coefficients, which is not solvable for higher derivatives of each of the unknown functions, may appear to be discontinuous functions or distributions even if the right-hand sides are ordinary continuous functions. The general method of solving such systems, using reduction of a bundle of matrices to a canonical form, is presented in [69] (Chapter 12, 7). Some other methods are proposed in [10]. [71] (elementary methods and application of the Laplace transform) and in [12]-[15] (application of generalization of the inverse matrix concept). Below we present an elementary method of solving such systems, which is applied in the general case and is a development of the method proposed in [10). 26 Equations . .. Discontinuous only in t Chapter 1 Consider a system of m equations with n unknown functions and with constant coefficients which is written in the vector form (31) A:i: + Bz = Itt). Here z and I are n-dimensional vectors, A and Bare m X n matrices. A system with derivatives of any order AIoII 1) and is defined by means of the Stieltjes integral (4) d (t f(t)g'(t) = dt}c f(s)dg(s), where c is any point of continuity of the function g(t). We introduce the norm" IIa for the functions defined on the interval (c, d) and belonging to the class M(a), a ~ 1. If a = -m + 0, m ~ 0 is an integer, 0< 0: ~ I, P = 1/0:, then (m ~ I). Using the norm introduced, we define convergence. For a ~ 1 the conver-gence Ii --+ f in M(a} implies that IIIi - flia --+ O. For a = k + 0: (k is an integer), 0 < 0: ~ 1, the convergence Ii .-+ f in M(a) implies that for some gi E M(o:), 9 E M(o:) we have f. - g(k) \ - i , g. --+ 9 in M(o:). Then for any a and integer m > 0, from the convergence Ii --+ f in M(a) there follows convergence f.(m) --+ f(m) in M(a + m). 3 Differential Equations with Distributions in Coefficients 31 LEMMA 2. If 0 < max{a;b} 1 and a + b 1, then for f e M(a), g e M(b), a 0 b = "I. we have (5) PROOF: IT b 0 < a 1, then 7 = a, Ilglib Ilgllo = max Igl and the inequality (5) holds. The case a 0 < b 1 is similar. IT a > 0, b > 0, a + b I, then 7 = a + b. The inequality (5) takes the form Raising both the sides of the inequality to the power v = 1/ (a + b) and writing IglV = g1, a+b -a-=P' a+b -b-=q, we obtain the known Holder's inequality. REMARK: IT max{a; b} 0, then there holds an inequality similar to (5), but with an a-, b-, and Id - cl-dependent numerical factor in the right-hand side. LEMMA 3. 1 and It - f in M(a}, g, - gin M(b) as i - 00, then fig, - fg in M(a 0 b). PROOF: IT max{a;b} I, then, but virtue of Lemma 2 and the remark, the norm in M(a 0 b) of each summand in the right-hand side of the equality "g, - fg = (t, - f)g, + f(gi - g) tends to zero as i tends to infinity. IT a = k + a, k 1 is an integer, 0 < a 1, then, by virtue of the above it follows from the relations f = h(le) , ,,= hi - h in M(a), g, - g in M(b) that for i k _ g 0), (7) f(t) ... w.(t) = f.(t) e 000, f.(t) - f(t) r1 w(t)dt = I, 1-1 32 Equations. .. Discontinuous only in t Chapter 1 Convergence is understood in the same sense as in the theory of distributions. If f E Lp, 1 ~ p ~ 00, or fEW;", then f. -+ f in Lp , correspondingly, in W;". Therefore, if f E M(a), then f. -+ f in M(a). 2. Some equations and systems with distributions in coefficients are reduced to Caratheodory systems by a change of the unknown functions. Consider the equation [77] (8) y" = a(t)y' + b'(t)y = g'(t). Let the functions a, b, g, and (b-a)b, (b-a)g be summable and let the derivatives b'(t) and g'(t) be understood in the sense of distributions. One can write this equation in the form (y' + by - g)' = (b - a)y' and using the change y' + by - 9 = z reduce it to the Caratheodory system (9) y' = z - by+ g, z' = (b-a)(z-by+g). To the system (9) one can set the usual initial data y(to) = Yo, z(to) = zoo To the equation (8) one can therefore add the following initial data (10) y(to) = Yo, (y' + by - g) It=to = ZOo Here to, Yo, Zo are arbitrary numbers. Initial data of the form y(to) = Yo, y'(to) = yb cannot be given for arbitrary to, but only for those for which the functions b(t) and g(t) are continuous. As is seen from the above, a reduction of differential equations with distri-butions to Caratheodory equations makes it possible to prove the existence of a solution and to establish the form of admissible initial data. The same method enables us to show that the solution of an equation with distributions in its co-efficients depends continuously on the coefficients and is the limit of a sequence of solutions of equations with smooth coefficients ai (t), b.(t), ... , which tend to given distributions as i tends to infinity. This is proved below for equation (8), but the same method can also be applied to the more complicated equations considered in the following theorems. LEMMA 4. Let a(t), b(t), g(t) E L2 on a closed interval [c,d] (11) b.(t) -+ b(t), gi(t) -+ g(t) in L2 and let YOi -+ Yo, ZOi -+ zoo Then the sequence of the solutions of the problems (12) y ~ ' + ai(t)y; + bHt)Yi = gHt) , i = 1,2, ... , Yi(tO) = YOi, (y; + biYi - gi)lt=to = ZOi 3 Differential Equations with Distributions in Coefficients converges in WJ to the solution of the problem (8), (10). PROOF: Using the change (13) 33 we reduce each equation under consideration to a Caratheodory system similar to (9). As i -+ 00, the coefficients of this system converge in L1 to coefficients of the system (9), the initial data 1/0. and zo., converging also. By virtue of the corollary to Theorem 8, 1, the sequence of solutions 1/., z. (i = 1,2, ... ) of these systems converges uniformly to the solution of the system (9). Then it follows from (11) and (13) that 1/; -+ 1/' in L2 Hence, 1/. -+ 1/ in WJ, so the lemma is proved. For any functions a, b, g E L2 we construct the sequence of smooth functions a.(t), b.(t), g.(t), like (7), taking s = 1/i, i = 1,2, .... Then by Lemma 4, the sequence of ordinary solutions 1/. of the problems (12) converges in WJ to the solution of the problem (8), (10) which contains distributions b'(t), g'(t) (the case aft) == get) == 0 is considered in [77)). This makes it possible to extend some known results from the qualitative theory of linear equations [78] to equations with distributions. Eigenvalue problems for the equation x" + Ap(t)X = 0, where pet) = q'(t), the function q(t) being nondecreasing, have also been discussed in [79], [80j. In the following theorem some of the components of the solution Xl, , Xn are distributions of different classes. Although this property is not retained under linear transformation of coordinates, such systems should be considered, for instance, for the reason that the equation of order n is reducible to a system of this kind. THEOREM 1 [60]. Consider a linear system (14) dx' dt' = L. a'i(t)xi + f.(t), i=1 i = 11 , n, where (15) ati E M(a'i) , It E M(Ip.). If there exist numbers A1,"" An, such that (16) (17) 0 Ai)} A. + 1, i = 1, ... , n , , ali + Ai 1, i, i = 1" .. , n (for the notation of a 0 A see (3), then using a linear change of the unknown functions the system (14) is reduced to a linear Caratheodory system. The system (14) has an n-dimensional linear manifold of solutions for which x. E M(A.), i= l, ... ,n. 34 Equations. .. Discontinuous only in t Chapter 1 PROOF: Let the functions Zi be enumerated in such an order that >'1 >'2 >'n. If all >'i 0, then it follows from (16) that all 'Pi I, OI.ii I, and the system (14) is a Caratheodory system. Hence, let (18) (if >'1 > 0, then m = 0 and the sums over indices less than or equal to m will be omitted from the following formulae). (19) (20) (21) It follows from (16) and (17) that for all i and i OI.ii 1 + >'i, OI.ii 1 + >'i - >'i' OI.ii 0 >'i 1 + >'i, 'P; 1 + >.;; if O;j ;'i >.; + 1. We will show that by a change of variables the system (14) can be reduced to a system in which the number >'n is replaced by a smaller number (22) /Ion = max{>.n - 1; O}, and the rest of the >.; remain unchanged. Let bni and gn be functions such that = an;' = In. We rewrite the last equation of the system (14) in the form 1t (zn - f)niZi - gn) = - t bnl;Zk + t njZj' i=1 1;=1 i=m+l In the right-hand side we replace zk(/c = 1, ... , m) by the corresponding right-hand sides from (14). The products so obtained, bnl:ol:j and b"TelTe are meaningful since bnl; E M(OI."Te - I), and by virtue of (19) and (20) 01.,,1; - 1 ->"a OI.l:i 1 + >'1;, 'PI; 1 + >'1;. Sincek m, >.1; 0, and in the case OTei;..1: 1->'i, OI..l:i 1 + >'1; - >'i' then by Lemma 1 (23) b"kaTej E M(vi), Vi = (OI."Te - 1) 0 OI..l:j 1 - >'i' If we take into account the fact that anI: - 1 >'n, Ct.l:i 1 + >'1: I, then we have (24) Vj max{>'"i I} = /Ion + 1. Similarly, bnTeh E M(lln + 1). (25) In the system (14) with the last equation already transformed, we make the change ZI=II1,,,, Zn-l = lin-I, m Z" = lin + L bnjlli + gn' i=1 We derive the system (26) (27) n + L aiilli + aingn + ,;, i=m+l m d;; = L [(ann - dnn)bni - dni] IIi j=1 n m i = 1,2, ... , n - Ii' + L (anj - dni)lIi - L bnl;/I: + (an,. - dnn)g,., i=m+l 10=1 m dni = L bnl:aTei E M(Vj), 1:=1 Vi 1- >'i, Vi /Ion + I, 3 Differential Equations with Distributions in Coefficients 35 d,.i == 0 if Ai > 1 since in this case a1r.i == O. Ie m (see (21. Using the inequalities (18)-(21) and (27). we can prove. as we have done above. that the products occurring in the coefficients of the system (26) are meaningful and that (28) (29) ai,.bni. ainU,. E M(ai .. 0 A .. ) C M(l + A.). (ann - dnn) Un EM 1- An) 0 An) C M(I' .. + 1). Let us estimate ai .. bni in a different way. Since An > O. i m. Ai O. then. by virtue of (19). (30) lI'i = ain 0 (ani -1) (1- An) 0 (-Ai) < 1- Ai' Similarly we derive (31) (ann - d .. n)bni E M(l - Ai)' If we take into account that ani - 1 An. then instead of (30) and (31) we have (32) dnnbni E M(max{>.ni I}) = M(l'n + 1). We write the system (26) in the form (33) dll' En -' = C'i(t)lIi + h.(t). dt i = 1, ... n. i=1 In this case Cii E M("tii). h. E M(i). Let (34) 1'. = Ai. i < ni I'n = maX{An - 1; O} < An. We will show that the numbers 'l'i .pi, I'i satisfy the inequalities analogous to (16) and (17). For i < n the inequality 'l'i + I'i 1 follows from (17) and (30). and for i = n from (17). (27) and (31) using the fact that 0 1' .. < A ... For i < n the inequality.pi 1'.+ 1 follows from (20) and (28). and for i = n from (29) and from the obtained estimate of bnkl1r.. For i < n the inequality 'lii 0 I'i I'i + 1 follows from (20) and (28). and fori = n from (17). (27). and (31). It follows from the inequalities obtained that (35) max{ ii ml;\X('l'i 0 I'il} 1'. + 1. 1 i = 1 ..... n. Thus. in the system (33), the coefficients possess the same properties as in the system (14). Consequently, the system (33) can be treated like the system (14). By virtue of (34), under each such transformation of the system the largest of the numbers Ai decreases either by 1, if it was not smaller than 1, or to 0, if it had values between 0 and 1. (36) Hence, after a finite number of transformations we obtain the system n dZi 0 ( ) O( ) de = a'i t zi + Ii t, i=1 i = 1, ... ,n, for which all the numbers A? obtained from Ai by successive decrease are non-positive. Like the numbers Ai and I'i, they satisfy the inequalities similar to (16) and (35), (37) i = l, ... ,n, where fP? and a?i are such that a?i E I? E Since all A? 0, it follows from (37) that a?i 1, fP? 1. In other words. a?i E M(l), If E M(l), and (36) is a 36 Equations . .. Discontinuous only in t Chapter 1 Caratheodory system. It follows from Theorem 3, 1 that this system has an n-dimensional linear manifold of solutions. We will show that for any solution of this system z, E M(A?), i = 1, ... , n. The functions z, being absolutely continuous, Zi E M(O). Hence a?izj E M(a?i 00), a?i 00= max.{a?i; OJ. Thus the right-hand side of (36) belongs to Mhf) (38) Then Zi E Mh;- 1). i = 1, ... , n. Using this more precise estimate of Zi, we deduce that the right-hand side of (36) belongs to Mhll, (39) Consequently, Zi E Mb; - 1). This procedure of specifying estimates leads successively to Zi EMbn, k= 1,2, ... , ( 40) We now show that for some k, " " I ~ - 1 :::;;; A? For A :::;;; 0 we have a 0 A = max{a; A}. For those i, for which the right-hand side of (40) is nonnegative, it is equal to ""If in (38) and to the left-hand side of (37). Therefore, for these i, ""Ii - 1 :::;;; A? For the remaining i the right-hand side of (40) is negative and, accordingly, in (38) ""If = O. Thus, ""1;- 1 :::;;; max.{>.?i -I}, Then, comparing (39) and (37), we have i = 1, .. . ,n. i = 1, ... n. Continuing, we have " " I ~ - 1 :::;;; max{>.?; -k}, i = I, ... ,n; k = 1,2, .... Hence, for a sufficiently large k (41) Zi E M b ~ - 1) C M(A?), i = 1, .. . ,n. Returning from the system (36) by successive changes of variables to the initial sys-tem (14), we conclude that the system (14) has an n-dimensional linear manifold of solutions. We will show that it follows from (41) that ZI E M(A,). It suffices to consider one of the single-type transitions that arise in going back from the system (36) to (14). Let it be proved for the system (33) that ( 42) III E M(Jl;) , i = l, ... ,n.. The transition from the system (33) to (14) proceeds by formulae (25). From (42) and (34) for i :::;;; m we have IIi C M(Ai), and from (18) and (19) we have Ai :::;;; 0, bni E M(an -1) C M(-Ai)' 3 Differential Equations with Distributions in Coefficients 37 The product bnjllj is therefore meaningful. Next, an; - 1 ~ An, bn;lI; E M(>'n 0 >.;) = M(>'n), 9n E M('Pn - 1) C M(>'n). Taking into accoun.t (26) and (34), we obtain z. = tI. e M(A.) (i < n), z,. E M(>.,.). The theorem is proved. REMARK 1: The best estimate of the form x. E M(>..) is obtained if there is equality in all the relations (16). REMARK 2: The transition from the system (14) to a Caratheodory system makes it possible to indicate the initial data under which this system has a unique solution. Some other conditions ensuring uniqueness of a solution are mentioned in [601. REMARK 3: Some nonlinear systems can be reduced to a Caratheodory system in the same way as the system (14). For instance, those a.j(t) in the system (14), for which CIt,j = -"'ij + 'Yij, "'ij ~ 0 is an integer, 0 < 'Yi; ~ 1, can be replaced by the bounded functions P'j(t, ... , Xlo,"') of class om;; (with respect to its arguments) dependent on t and on those x,. for which >.,. ~ CIt,j' In the other terms of the equations ofthe system those x;. for which >'j = -lj+6;. lj ~ 0 is an integer,O < 6; ~ 1, can be replaced by the bounded functions W'j(t, ... , x,., ... ) of class O'i (in its arguments) dependent on t and on those x,. for which >.,. ~ >'j. In the next theorem the assumption (44) is equivalent to the assumption of a similar theorem from [60], but is expressed in a much simpler form. THEOREM 2 [601. Oonsider the linear equation (43) where a. E M(CIt.), i = 1, ... , nj f E Mb). Let us denote max{Cltl,"" Clt n , 'Y} = 1'. If (44) CIt, - i ~ 1 - 1', i = l, ... ,n, then equation (43) is reduced to a Oaratheodory system and has an n-dinlensional linear manifold of solutions ( 45) where Ydt) is a a partial solution of equation (43), Clt , Cn are arbitrary con-stants, and Ul (t), .. , un(t) are linearly independent solutions of a corresponding homogeneous equation. The solutions (45) belong to M(p. - n), and (46) U,(t) E M(p.o - n), i = 1, ... ,n. 38 Equations... Discontinuous only in t Chapter 1 PROOF: By the usual change of variables ... , y(n-1) = Xn we reduce equation (43) to the system (47) i = 1, .. . ,n -1; We will show that in the case when the conditions (44) are fulfilled, there exist numbers A1, ,.An satisfying the conditions (16) (with equality signs) and (17) of Theorem 1 as applied to the system (47), that is, the conditions (48) Ai+1 = Ai + 1, i = 1, ... , n; max {-y; an 0 .Al; an-1 0 A2; ; al 0 An} = An + 1. Expressing all Ai in terms of the number A = An + 1 we obtain (49) max {-y; an 0 (A - n); an-1 0 (A - n + 1); ... ; a1 0 (A - I)} = A. We will show that the number .A = J.t satisfies this equation. For A = J.t each of the expressions ai 0 (J.t - i) is meaningful by virtue of (44) and is equal to max{ ai; J.t - i; ai + J.t - i}. It is clear that the sum ai + J.t - i should be included only in the case J.t - i > 0, that is, J.t > i ~ 1. But in this case, by virtue of (44), ai + J.t - i ::;;; 1 < J.t. Therefore, for A = J.t = max{ a1, , an, 'Y} the left-hand side of (49) is equal to J.t, thus A = J.t satisfies equation (49). Thus, for .Ai = J.t - n + i-I the conditions (48), i.e., the conditions (16) for the system (47), are satisfied. The conditions (17) for i = n take the form an -H1 + J.t - n + i-I::;;; 1, i= 1, ... ,n, and hold by virtue of (44). For i < n the coefficients in equations (47) are constant and, therefore, ai; in (17) can be taken to be less than any negative number, and the conditions (17) are fulfilled. Thus, the system (47) meets the requirements of Theorem 1. It has, there-fore, an n-dimensional linear manifold of solutions, and in this case (50) Y = Xl E M(Ad = M(J.t - n). From this there follows the assertions of Theorem 2, except (46). To prove (46), one must apply the result (50) to equations of the form (43), but with f == O. REMARK 1: In practice it is, as a rule, more convenient to reduce equation (43) to a Caratheodory system without first using the system (47). With this objec-tive, those products ai(t)y(n-i) in which one of the factors is a distribution (for yen-i) this question is settled by use of (50)) are transformed by means of for-mula (2). Next, as in 2, 2 one combines the terms represented in the form of one and the same order derivatives of some functions. From the derived equation of the form (22), 2 (but with other tli), one can pass over to the system (23), 2. 3 Differential Equations with Distributions in Coefficients 39 In this way one can not only reduce formally equation (43) to a Caratheodory system, but also prove Theorem 2 by a method independent of Theorem 1. REMARK 2: Using the method proposed in Remark 1, one can also reduce to a system 80me nonlinear equations of order ", for instance, the equation n-1 (51) lien) = L "WPi ("",,,', .. . ,,,(2I:-i) + P1 ("",,,', ... ,,,(A:) i=1+1 A:-1 + L 6i(')lOd',,,,,,' ... ,,,(i) + 1('). i=o Here 0 k n - 1, Pj E ci-A:, lOj E C"-i (in its arguments), the functions IIj and I may be distributions, I E M(n - k + a), 0 < a 1, (Ji min{k - i + 1 - aj n - k + a}, for i > 2k the functions Pi depend on , only, p. EC. In the case at > 1/2 the functions P' (i > k) are independent of II (2I:-i) Then equation (51) is reduc:;{ to a Carath40dory system and has an n-dimensional set ofsolutions which belong to M(a - k). To reduce this equation to a ayltem, we assume that II E M(a - k). Then ,,(A:) E M(OI) and the composite functions lOi (t,,,(,),. ,,,W(t) E M(a - k + i), I'A: (',II(t), ... , II (A:) (t E M(l), Pi E M(OI + k - i), i> k, where 01 = a (01 1/2), 01 = a - 1 (01 > 1/2). Therefore, in (51) the products ,,(i)Pi and lIilOj are meaningful and can be transformed by formula (2). Next, as in Remark 1, we derive the eqUation (n) + (n-i) , 0 " tin_A: + ... , +tl1 + tlo = , and then a Carathfodory system. Estimating smoothness of its solution and going back to equation (51), we get" E M(OI - k). Note that in equation (51) each product 6ilOi can be replaced by the sum of a finite number of summands 6imlOim, where the functions 6im and lOi". satisfy the same conditions as 6i and lOi' Nonlinear equations with distributions, but simpler than those in (51), were considered in (81]. EXAMPLE: Let us reduce the equation (52) ,,(.) = ""'Pa(t,,,,,,') + 1'2(""''''''''') + h(')101(t,,,,,,') + 60(t)lOO(t,,,) + I(t). to a system. Let 1'2 E Cj 1'3 E C1, 101 E C1, 100 E C2, 111 E M(S/2), "0 E M(5/2), IE M(5/2), 11'21 ,,"21'.(""'''')' p. E C. Then the conditions of Remark 2 are fulfilled, n = 4, k = 2, a = 1/2. Hence,,,' E M(1/2) = L2(loc). Using the identity (2), we write equation (52) In the form 40 Equatwns ... Discontinuous only in t Chapter 1 where = h. = boo g" = f. and the derivatives w&. are total derivatives of the composite functions with respect to t. Assuming tJ2 = -110WO - g. we derive the equation 1/(4) + + + tJo = O. Introducing new unknowns we obtain the system The functions tJi depend on the variables t. II. II',I/"i here 1/.1/'.11" should be replaced by the formulae II' = 2:3. II" = 2:2 - tJ2 = 2:2 + 110(t)wo(t. 2:4) + g(t). This system satisfies the Caratheodory conditions. 3. In [821 the following linear system in vector notation (53) y' = B'(t)y + g'(t), is considered. Here the matrix B(t) and the vector-valued function g(t) are of bounded variation, B(t) is continuous, and the derivatives are understood in the sense of the theory of distributions. The integral equations (54) y(t) = + fat (dB(s)) y(s) + g(t) - g(a) (where the integral is understood in the Stieltjes sense) equivalent to (53) and several more general equations were considered earlier [83], [841. In [82] exis-tence and uniqueness of a solution with initial data y(a) = Yo are proved, the fundamental matrix is shown to be continuous and of bounded variation. The solution is expressed in terms of a fundamental matrix and of the function g(t). The existence of a solution can be proved, for instance, by applying to (54) the successive approximation method and by using the known estimates of Stieltjes integrals ([641, p. 254). Another method of investigating the system (53) is to reduce it to a , Caratheodory system. Using the change y(t) = z(t) + g(t) from (53) and (54), we obtain (55) z' = B'(t)z + h'(t), h(t) = it (dB(s)) g(s), z(t) = z(a) + it (dB(s)) z(s) + h(t). 3 Differential Equations with Distributions in Coefficients 41 Using the estimates taken from [64], (p. 254) we find that the function h(t) is continuous. Denote the elements of the matrix B(t) by b'i(t) and the continuous function n t + L var bii(S) .. ',J=l by ret). For any t1, t2 > t1, we have (56) Let t(r) be inverse to ret). It follows from (56) that the functions t(r) and b'i(t(r)) are absolutely continuous. Therefore, the last integral in (55) is equal to ([64], p. 290) z(t(r))dr and equation (55) is equivalent to the Caratheodory system (57) dz _ dB(t(r)) dB(t(r)) (()) dr - dr z + dr g t r . From this there follow the above assertions on solutions of the system (53). All this is not extended directly to the case where B(t) is a discontinuous function of bounded variation, even for get) == O. In this case the solutions, generally speaking, have discontinuities at the same points as B(t), and the Stieltjes integral in (54) and (55) may not exist (example in [64], p. 249). The theory of distributions does not work here either since, for instance, the product of the delta-function and its indefinite integral is not defined. H for functions of bounded variation one distinguishes the values y( t -0), yet), y(t+O) and considers separately the left and the right jumps yet) -y(t-0) and y(t+O) -yet), then one can define the integral in (54) where B(t) and y(t) are discontinuous functions of bounded variation. Under different assumptions (for instance, if yet) = yet - 0) or yet) = [yet - 0) + y(t+ 0)]/2, etc.) one obtains different conditions for the existence of the solution of equation (53) or (54). the solutions themselves being also different. Equations of this kind written in a differential or in an integral form were considered in [85]-[90]. Several nonlinear equations and systems of this type were analyzed in [90]-[92]. Equations with impulses belong to this type in the cases where the magnitude of the jump of a solution depends not only on t, but also on the value of the solution before the jump. We will show by a very simple example that in such cases different ap-proaches to the definition of a solution yield different results. Consider the linear equation (58) y' = k6(t)y (k = const). Since y' = 0 for t < 0 and for t > O. then yet) = c yet) = e(l + a) (t < 0). (t > 0), 42 Equations. .. Discontinuous only in t Chapter 1 that is, y(t) = c(l + a71 (t)) , where (59) 71(t) = 0 (t < 0), '1(t) = 1 (t> 0). To find a, we substitute yet) into (58); '1'(t) = o(t) (60) ao(t) = ko(t) + kao(t)'1(t). In the theory of distributions, the product o(t)'1(t) is not defined. If one takes the sequences of smooth functions o,(t) -+ o(t), '1,(t) -+ '1(t), the limit of the product Oi(t)'7,(t) does not, generally speaking, exist. In the cases where it does exist, it depends on the choice of the sequences o,(t), '7i(t). Under natural assumptions this limit, if it exists, has the form -yo(t), where -y may be any number from the closed interval [0,1]. If we assume that o(t)'1(t) = -yo(t), we obtain from (60) (61) k a=--. 1- k"l Consider different approaches to the choice of the value of "I. 1 0 If "I is assumed to be zero, i.e., 0 (t)'7 (t) = 0 (this is equivalent to the statement that by definition the solutions of equation (58) must be continuous on the left), then a = k, (62) y(t) = c(l + k'1(t)) , c being an arbitrary constant. The same result is suggested by the limit transition Yi(t) -+ y(t), where y,(t) is a solution of the delay equation (63) T. -+ +0 (i-+oo); the function o.(t) being different from zero only on the interval (a" (ii), not longer than T., which contracts to the point t = 0 as i -+ 00, and for some q = const (64) rfi' J ai' o.(t)dt -+ 1, In the case -y = 0, k = -1, from (62) we have [92] y(t) = c (t < 0), y(t) = 0 (t> 0). This implies that all the solutions which exist for t < 0 jump to the point y = 0 at the moment t = 0 and remain at that point. Under the initial data y(to) = Yo, to > 0, Yo =F 0, we have the solution y(t) = yo(t > 0) which is not continued to the region t < O. 20 If "I is assumed to be equal to 1/2, i.e., o(t)'1(tj = 1/2o(t) (for instance, from symmetry considerations or taking by the definition the solutions of equa-tion (58) to be such that y(t) = [y(t - 0) + y(t + 0)l/2), then from (61) we obtain a = 2k/(2 - k). For k = 2, this result becomes meaningless. More precisely, for 3 Differential Equations with Distributions in Coefficients 43 Ie = 2 a solution with any initial data of the form veto) = Yo =I- 0, to < 0 cannot be continued to the region t > o. For Ie > 2 we obtain a solution yet) which changes sign. This is unnatural for equations of the form y' = !p(t)y (in the case of any continuous or summable functions !pet) solutions do not change sign). 3 We shall consider equation (58) as a limit of the equations y ~ = kO.(t)y., where for each i the function S. is summable, vanishing outside the interval (a.,.8.), and meets the requirements (64); a.,p. -+ 0 as i -+ 00. Then y.(t) = cexp (Ie fa: 6. (s)dS) . For i -+ 00 we obtain Yi(t) -+ yet) (t =I- 0) (65) yet) = c (t < 0), yet) = celc (t> 0). IT the functions (65) are assumed to be solutions of equation (58), then for all values of Ie all the solutions exist both for t < 0 and for t > o. The choice of such a definition of a solution corresponds to the case where in (61) elc -1-1e 'Y = Ie(elc - 1) . Note that 'Y -+ 1/2 as Ie -+ o. Using these arguments, one may come to the following conclusion. Let equation (58) be interpreted as an idealization of equation (63), where " ~ 0, (66) 6.(t) = 0 (t ~ ai, t ~ .8.), The integral of 6.(t) over the interval (a .8.) is equal to 1, and the numbers '" a., P. are small. Then in the case " ~ P. - a. the solution is close to the function (62), and in the case " = Q-to the function (65). Hence, in the case 1". ~ P. - a. it is more convenient to write the limit equation not in the form (58), but as yl(t) = k6(t)y(t - O)i Solutions of such an equation are the functions (62). In [921 such limit transitions are considered for more complicated equations (67) i = 1,2, .... Let the functions J(t, z), g(z) and S.(t) be continuous (to ~ t ~ tl, Z ERR), J,g ERR, and 6.(t) satisfy the conditions (64) and (66), a. -+ 0, .8. -+ 0, z.o -+ Zo (i -+ 00), The solution of the problem (67) is not necessarily unique. 44 Equations. .. Discontinuous only in t THEOREM 3 [92]. Let the problems (68) (69) (70) u' = f(t, u) v' = g(v) w' = f(t, w) (0 t 1), have unique solutions u(t), v(t), w(t). u(to) = Xo v(O) = u(O), w(O) = v(l) Chapter 1 Then for an arbitrary sequence of solutions of the problems (67) (i = 1, 2, ... ) we have x.(t) -+ u(t) x.(t) -+ w(t) (to t < 0), (0 < t tIl. THEOREM 4. Let 7. -+ 0, 0 < P. - Q. 7., i = 1,2, ... , and let the prob. lems (68) and (71) z' = 1ft, z) z(O) = u(O) + g(u(O}) have unique solutions u(t), z(t). Then for each sequence of solutions of the problems (72) we have for i -+ 00 x.(t) -+ u(t) (to t 0), Xi(t) -+ z(t) (0 < t tIl. PROOF: For i > i l we have to < Q. < P. < tl, and the solution of the prob lem (72) satisfies the equation (73) xHt) = ! (t, x.(t)) Since for some a > 0, b > 0 we have 1!(t,x)1 m for It I a, Ix - u(O)1 2b, then for d = min{a; bm-l} all the solutions of the equation x' = 1ft, x) for which Ix(o) - ufO) I b, exist on the interval It I d and satisfy the inequality (74) Ix(t) - x(O)1 m Itl. By virtue of the remark to Lemma 6, 1, the solutions of this equation with the initial data x(to) = XiO for XiO -+ Xo converge uniformly to u(t), to t 0, that is, for any '1 > 0 these solutions for i > i2 ('1) satisfy the inequality Ix(t) - u(t) I '1 From this and from (74) it follows that for i > i2('1), It I < '1m-l (75) Ix(t) - u(O)1 < 2'1. 3 Differential Equations with Distributions in Coefficients 45 By virtue of (73), on the interval to ~ t ~ ai, these solutions are solutions of the problem (72). Since f3i - a. ~ Ti -+ 0, then f3i - Ti ~ a., and for i > i3(r7) and a. ~ t ~ f3i we have Ix(t - Ti) - u(O) 1 ~ 2'7. Since the function g(x) is continuous, then for any e > 0, for a sufficiently small '7 and i > i 3 ('7) (76) Introduce the notation Ig(u(O))/ = go, Then, by virtue of (64) and (76), for ai ~ t ~ f3i, (77) IJi(t)/ ~ q(gO + e), Ji (f3i) -+ g (u(O)) (i -+ 00). We will show that for a sufficiently large i, and for all i > i4 the solution Xi(t) of the problem (72) for ai ~ t ~ f3i remains in the ball K{/x - u(O)/ ~ q(gO + e:) + 2}. For/tl ~ a, x E K, we have If(t,x)1 ~ mI. By virtue of (75), IXi(aJ) - u(O) I < 1 for large i. Next, (78) If for 0 < t - ai ~ f3i - ai < mIl the solution had at some point reached the boundary of the ball K, the left-hand side of (78) would have been greater than q(gO + e) + 1 at that point, and the right-hand side (by (77)) would have been less than this number. This is impossible. According to (75), xi(ai) -+ u(O) as i -+ 00. Hence, for t = f3i and (77), we obtain from (78), as i -+ 00, (79) Xi(f3i) -+ u(O) + g(u(O). For t ;;?; f3. the solution of the problem (72) coincides with the solution of equation (73) with the initial data Xi = Xi(f3.) for t = f3i. If f3i > 0, this solution, as the solution of equation (73), can be continued up to t = O. For such a solution of equation (73), we obtain from (75) and (79), both for f3i > 0 and for f3i ~ 0 X.(O) -+ u(O) + g(u(O)). By virtue of the remark to Lemma 6, 1, the solution x.(t) converges uniformly to the solution of the problem (71) on the interval 0 ~ t ~ tl. This solution Xi(t) coincides with the solution of the problem (72) for f3i ~ t ~ tl, and the result follows. 46 Equatt"ons ... Discontinuous only in t Chapter 1 It is obvious that the continuity condition for f(t, x) can be replaced by the CaratModory conditions; the function f(t, x) in (72) may also depend on the parameter /-li, /-li --+ /-lo, as in Theorem 6, 1; the function g(x) can be replaced by the continuous function g(t, x). Very general theorems (but with complicated formulations) on continuous dependence on the parameter for differential equations with discontinuous solu-tions are presented in [92J. Summarizing, one can say that the concept of solution for equations of the form x' = f(t, x) + ~ ( t ) g ( t , x), where ~ ( t ) is a delta-function or a derivative of a discontinuous function of bounded variation, the same as for equations (53) with a discontinuous matrix B(t), is not uniquely defined. In the choice of the definition of a solution one must pay close attention to the character of the limit transition which has led to a given equation. 4. The generalized differential equations dz dt = DF(z, t). are considered in the papers by J. Kurzweil ([15]. [16J, [92J, and others). It is pointed out that under certain assumptions such an equation can be written in the form dz a - = -F(z t) dt at " the derivatives being understood in the sense of distributions. Under certain conditions (dif-ferent in [15J and in [92]) the author proves theorems on existence and uniqueness of solutions, on continuous dependence of a solution on initial data and on the parameter. The solution of equation (80) is defined by means of a generalization of the Perron integral [15J. In [161, [92] it is assumed that (81) IF(z, t2) - F(z, tl) I ~ Ih(t2) - h(tl)I, IF(z, t2) - F(z, tl) - F(tI, t2) + F(tI, h)1 ~ w (Iz - tiD Ih(t2) - h(ttll, where the function h(t) is nondecreasing, continuous on the left, and w('7) is nondecreasing, continuous, and w(O) == O. Under these assumptions, the solution of equation (80) with the initial data z(to) = Zo is proved to exist on some dosed interval [to, to + ul, u > O. The solution satisfies the condition and is therefore continuous on the left. In the case w('7) = k'7 the solution is unique for t ~ to. EXAMPLE [921: In the region Izl < I, It I < I, the function (82) F(z, t) = 2: (t ~ 0), F(z, t) = 0 (t > 0) satisfies both the conditions (81) with w('7) = '7, h(t) = 0, (t ~ 0) h(t) == 1 (t> 0). With the initial condition z(to) = zo. to < O. we have the solution z(t) == Zo (t ~ 0). z(t) = 0 (t > 0). 3 Differential Equations with Distributions in Coefficients 47 With the initial condition z(to) = zo, to > 0, the solution z(t) = Zo (t > 0) cannot be continued to the region t ~ 0 if Zo ;i: O. In this example, equation (80) with the function (82) can also be written in the form Zl(t) = -6(t)z(t - 0). We will show that under the con(}jtions (81) equation (80) can be reduced to an equation with impulses, similar to equation (72). As is known ([64), p. 290), the function h(t) of bounded variation on the closed interval [to, tIl can be represented as (83) h(t) = tp(t) + ret) + ,(t), where tp(t) is absolutely continuous, ret) is a continuous function of bounded variation which almost everywhere has r'(t) = 0, ,(t) is the algebraic sum of the jumps of the function h(t) on the interval [to, f). If the function h(t) is nondecreasing, so are the functions tp, r,'. Let the function h(t) from (81) be represented in the form (83). Then the change of the independent variable t + ret) = r, z(t) = z(r) reduces equation (80) to the equation (84) d ~ ~ ) = I (r,z(r + E 6(t - tj)gj (z(r - 0, j a I(r,z) = a; F (z,t(r , gj(z) = F(z, rj + 0) - F(z, rj), the sum being taken over all the points rj of discontinuity of the function h(t(r. It can be shown that for almost all r I/(r, z)1 ~ tp' (t(r + 1, I/(r,z) - l(r,tI)1 ~ w (Iz - til) (tpl(t(r + 1). Hence, the function I satisfies the Caratheodory conditions, and if we,,) = Ie", it also satisfies the Lipschitz condition (6), 1. CHAPTER 2 EXISTENCE AND GENERAL PROPERTIES OF SOLUTIONS OF DISCONTINUOUS SYSTEMS Various definitions of solutions of differential equations and systems with discontinuous right-hand sides are considered for the cases where, by contrast with Chapter 1, the right-hand sides are not continuous in 2:. The range of applicability of different definitions is indicated. For differential equations with discontinuous right-hand sides and for differential inclusions, existence ofsolutions is proved and the properties of these solutions are analyzedj in particular, the dependence of solutions on initial data and on right-hand sides of equations, and the properties of integral funnels are examined. 4 Definitions of Solution Several definitions of solutions of differential equations with discontinuous right-hand sides are proposed. The connection between such equations and dif-ferential inclusions is established. The velocity of motion along a surface of dis-continuity (the derivative dx/dt for a solution lying on a surface of discontinuity) is determined for different cases. The velocity of motion along an intersection of surfaces of discontinuity is found in two main cases. 1. The definition of a solution (given in 1) as an absolutely continuous function satisfying the equation almost everywhere is not always applicable for equations, the right-hand sides of which are discontinuous on an arbitrary smooth line or on a surface S. This definition can be applied in the case where the so-lutions approach S on one side and leave S on the other side. Here the solution passes through S and satisfies the equation everywhere except at the intersec-tion point at which the solution does not have a derivative (Example 1 of the Introduction) . In the other case where on both sides of a line or a surface of discontinu-ity S the solutions approach S, this definition is unsuitable because there is no indication of how a solution which has reached S may be continued (Example 2 of the Introduction). To provide the existence and the possibility for solutions to be continued in this case, it is necessary either to change the value of the right-hand side of the equation in some way at its points of discontinuity or to define it at these points in case it has not already been defined there. 48 4 Definitions of Solution 49 It is necessary to have a definition of the solution which will cover both these main cases and be formulated irrespective of the position of lines and sur-faces of discontinuity. The solutions thus obtained must meet the requirements mentioned in the I