By means of Schauder's fixed point theorem sufficient conditions for asymptotic equivalence of impulsive equations in a Banach space are found

Bainov, D.

Kostadinov, S. I.

Myshkis, A. D.

Diposit Digital de Documents de la UAB

Publicacions Matemátiques, Vol 34 (1990, 249-257 .A bstractASYMPTOTIC EQUIVALENCEOF IMPULSIVE DIFFERENTIAL EQUATIONSIN A BANACH SPACED .D . BAINOV, S .I . KOSTADINOV AND A .D . MYSHKISBy means of Schauder's fixed point theorem sufficient conditions forasymptotic equivalence of impulsive equations in a Banach space arefound .1 . IntroductionImpulsive differential equations are a powerful apparatus for mathematicalsimulation in physics, chemical technology, ecology, biotechnologies, impulsetechnology and industrial robotics . That is why the theory of impulsive dif-ferential equations develops actively in the last 20 years [4], [8-13], [15] . Thepresente of phenomena specific for the impulsive differential equations such asbeating of the solutions, merging of the solutions, bifurcation, dying of the so-lutions, loss of the property of autonomy, etc . makes this theory richer thanthe theory of ordinary differential equations and, on the other hand, in theinvestigation of impulsive differential equations a preliminary modification ofthe methods used is necessary.In relation to the mathematical simulation in theoretical physics recently theinvestigation of impulsive differential equations in abstract spaces began [1],[2] . In the present paper the asymptotic equivalence of impulsive differentialequations in a Banach space is investigated . We shall note that the asymptoticequivalence of systems of ordinary differential equations has been investigatedby many authors [3], [5-7], [14], [16] .The present investigation is supported by the Ministry of Culture, Science and Eclucation ofPeople's Republic of Bulgaria under Grant 61 .250	D.D. BAINOV, S .I . KOSTADINOV, A.D . MYSHKISandConsider the impulsive equations2 . Preliminary notesdx _dt A( t )x (t 1 {tj}J° 1)(2) x(t~- ) = Ljx(t~) (j = 1, 2,3 . . . . )dt A(t)y + f(t, y)(t ~ {ti }39'=1)(4) y(t;) = (Lj + Hj)y(t_) (j = 1, 2, 3 . . . . )in a real or complex Banach space X. Here t varies in the interval Ft, oo) ; A(t)X -> X (for each t E r, oo)) is a linear bounded operator and the operator-valued function A( .) is continuously continuable from (t .i_1, tj) to [t.i_1, tj] (j =1, 2, 3, . . ., to = t) ; Lj : X -> X (j = 1, 2, 3 . . . . ) are linear bounded operatorshaving bounded inverse ones and LUX = X ; the function f : r, oo) x X -> X iscontinuosly continuable from (tj_ 1 , tj) x X to [tj_ 1 , tj] x X (j = 1, 2,3 . . . . ) ; HjX --> X (j = 1, 2,3 . . . . ) ; the sequence T = {tj}~°_ 1 satisfies the conditiont < ti < t2 < --- -+ 00We assume that the solutions of equations (1), (2), (3) and (4) are continuousfrom the left, thus in the conditions of a jump (2) and (4) instead of tj we shallwrite tj . If x is a solution of equation (1), (2) on the interval [ce, 00), we shallwrite x E ((1), (2))« and analogously of equation (3), (4) . An arbitrary solutionof impulsive equation (1), (2) defined on some subinterval of [t, a) is continuableto the whole interval which for impulsive equation (3), (4) in general is not true .Denote by W(t, T)(t <_ t, T < oo) the evolutionary operator of impulsiveequation (1), (2) associating with each element xo E X the solution x(t) _W(t,T)xo of (1), (2) with condition x(T) = xo .The following equalities are validW(t,T) = A(t)W(t,T)(t 1 {tj};°-_1),W(tt ,T) = LjW(tj,-r)Assume that the space X is split up in a direct sum of the subspaces Xland X2, i .e . X = X1-1-X2 and let PI, P2 be the corresponding complementaryprojectors (Pl + P2 = I, where I is the identity operator) . Set Wi(t1,t2) _W(t 1 )P;W-1 (t2) (i = 1, 2), where W(t) = W(t,t) .Let K C r, oo) . Denote by S(K, X) the Banach space of all bounded func-tions x : K -+ X which are continuous for t qÉ tj E K continuous from the left,have a limit from the right, provided with the norm 111xIII = sup tc K 11x( 1 )11 .IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE	251Lemma 1. Le¡ F C S([a,oo),X)(a >_ t) and le¡ ¡he following conditionshold :1 . There exists a functioncpo E S([a,oo),X) such that llcp(t)-cpo(t)11 t+0uniformly with respect to cp E F.2 . The set F is equicontinuous on each interval [c¿, w) fl (t i,ti] (j =1,2,3, . . .) .3. For each t E [a, oo) ¡he set {cp(t) : cp E F} is relatively compact in X .Then F is relatively compact in S([a, oo), X) .Lemma 1 is proved in a standard way by the construction for any e > 0 ofan s-lattice for F in the space S([a, oo), X) .3. Main resultsSet hj = L) 'Hj and xj = x(tj) where x(t) is an arbitrary function of t.Definition 1 . The impulsive equations (1), (2) and (3), (4) are called asymp-totically equivalent if for any bounded solution x E ((1), (2))7 there existsa E r, oc) and a solution y E ((3), (4»,,, such that y(t) - x(t) t> 0 and,conversely, for any bounded solution y E ((3), (4)r there exists a solutionx E ((1), (2))t for which y(t) - x(t) t 0 .-00Theorem 1. Let the following conditions hold :1 . There exists a bounded solution x E ((1), ( 2))i .2 . There exists a constant R > 0 for which(5 )J-tsup ilWi(t,s)f,z»ds+ sup"11Wl(t,tj)hj(z)11+t zz¡<ti <tX .00sup'11W2(T,s)f(s,z)JJ ds+ sup',11W2(T,ti)hj(z)~~Jt>0t<ti <oo zwhere sup' is taken over ¡he set {z E X : liz - x(s)JJ _< R} and sup"oven ¡he set {z E X : liz - xjll < R} .3. The set f{(t, z) : t <_ t < oo, liz - x(t)JJ <_ R} is relatively compact in X.4 . For j = 1, 2,3 . . . . the mappings Hj are continuous on the balls Bj =00{z E X : liz - xj1l <_ R} and the set U hj(Bj) is relatively compact inj=lThen for some a E [t, oo) there exists a solution y E ((3, (4))« for whichy(t) - x(t) -> 0. If, moreover, the left-hand side of the relation in (5) (as at-00252	D.D . BAINOV, S .I . KOSTADINOV, A.D. MYSHKISfunction of t) belongs to the space Lp(t, oo) for some p E [1, oo], then IIy(t) -x(t)II E Lp(a>oo) .Proof. Denote the left-hand side of relation (5) by F(t) and choose a > tso large that F(t) _< R for t >_ a . The set D = {z E S([a, 00), X) :IIIz -xI[a,,,) I I I < R} is nonempty, convex, closed and bounded .On D define the operator Q by means of the formula(6) (Qz)(t) = x(t) -}- J t W1(t, s)f(s, z(s» ds + 5-~ W1(t, tj)hj(zj)-a<t ; <t00W2(t,s)f(s,z(s))ds- E W2(t,tj)hj(zj)t<t; <oowhere z E D, t E [a, oo) . The convergence of the improper integral and theseries in (6) follws from the fact that F(t) < oo .We shall show that the operator Q satisfies the conditions of Schauder's fixedpoint theorem . .1 . Q : D -> D. In fact, from the uniform convergence of the improperintegral and the series in the right-hand side of equality (6) and from the choiceof a it follows that Qz E S([a, oo), X) and that II(Qz)(t) - x(t)II < R, a < t <oo .2 . The operator Q is continuous . In fact, let e > 0 be an arbitrarily chosennumber and let > a be choosen so that F(s) < á for s >_ ¿ . Let z, z1 E Dand a < t < f . ThenII(Qz)(t) - (Qz1)(t)II <-JzIIW1(t, s)[f(s, z(s)) - f(s, z1(s))JII ds+1+ f IIW2(t,s)[ .f (s,z(s)) - f(s, z1(S))JII ds+t+ E IIWI(t,ti)[hj(zj) - h.i(zi;)III+a<t; <t+ E IIW2(t,tj)[hi(zj)-hj(z1 ;)JII+j :5 ti <f+J' IIW2(t,s)[f(s,z(s))-f(s,z1(s))JIIds+f+ IIW2(t,tj)[hj(zj)-h;(z1 ;)JII~<tj <ooDue to the continuity of the functions f and hj for a fixed function z1 anumber 71 > 0 can be found such that for 111z - zi 111 < T7 each of the first fouraddends in the right-hand side of the last inequality should be smaller thanIMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE	253é . But then the right-hand side of the last inequality becomes smaller than48 + 2$ + 28 = e . For t > 1 the following inequalities are validII(Qz)(t) - (Qzi)(t)II -< II(Qz)(t) - x(t)II + II(Qzi)(t) - x(t)II < 28< eHence the operator Q is continuous .3 . We shall show that the set QD is relatively compact in D. For thispurpose we shall show that the conditions of Lemma 1 are fulfilled for F =QD, cp o = x . Condition 1 of Lemma 1 follows immediately from relation (5) .We shall show that condition 2 of Lemma 1 holds . Let z E D and t', t" E(tk-1,tk] for some K where t' < t" . Then, after elementary transformationswe obtain the equality(Qz)(t') - (Qz)(t',) =x(t,)- x(t") -0000from which we deduce the estimatet"W, (t", s)f(s, z(s» ds--J L[WI (t", s) - Wl (t', )If(s, z(s)) ds+a+ [WI(t',tj) -W,(t" tj)]hj(zj)-a<tj <tkW2(t', s)f(s, z(s)) ds-[W2(t', s) - W2 (t", s)]f(s, z(s)) ds-- E[W2(t',tj)-W2(t" tj)]hj(zj),j-k_ t 11II(Qz)(t') - (Qz)(t'%I :5 II x(t') - x(t")II + IIWI(t" s)f(s, z(s»II ds+J¡t ,ft1+ IIW(t') - W(t")II ' IIPI W-'(s)f(s, z(S))II ds+a+ IIW(t')-W(t")IIJIPI W -' (tj)hj(zj)II+«<tj <tkft11+ IIW2(t',s)f(s,z(s))II ds+t+f 00 II[W(t') -W(t")]P2W-'(s)f(S, z(s))II ds+t"+ 1: IIW(t ' ) - W(t")]P2W-'(tj)hj(zj)IIj-k254	D .D . BAINOV, S .I . KOSTADINOV, A .D . MYSHKISDenote by I1, . . . , 17 the successive addends in the right-hand side of thelast inequality and assume that t"-t' --> 0. Then I1 -> 0 since x E S([t, oo), X) .12, IS --> 0 because of the uniform (with respect to z, t', t" ) boundedness ofthe integrand . 13 -> 0 and 14 -> 0 since under the signs of the integral and ofthe sum the first factor tends uniformly to zero and the second one is uniformlybounded . Represent 16 for t > tk in the formt tIs = f + f :51 IIW(t~) - W(t)II ' IIP2W-1 (S)f(S, z(S))II d.s + 2F(t)The second addend in the last inequality can be made arbitrarily smallfor a sufficiently large value of t. For t fixed and t" - t' small enough the firstaddend can also be made uniformly arbitrarily small (analogously to I3) . Hence16 -> 0 . Analogously it is proved that 17 -> 0 .Thus the equicontinuity of the set QD on the integral (tk_1, tk] is proved .In order to verify condition 3 of Lemma 1, first we shall show that for t Ela, oo) the set {f,00 W2(t, -s)f(s, z(s» ds : z E D} is relatively compact in X. Lete > 0 be arbitrarily chosen. Choose > t so that II fE W2(t, s)f(s, z(s)) dsll <3 for all z E D. The set {W-1 (s)f(s, z(s)) : t < s < j, z E D} is rela-tively compact in X . Hence the sets {ft' W -1(s)f(s, z(s» ds : z E D} and{f~ W2(t, s)f(s, z(s)) ds : z E D} are also relatively compact subsets of X.For the last set we choose a finite á -lattice- {ft{ W2(t, s)f(s, zi(s) ds : i =1, . . . , N; zi E D} and obtain that the set { ft°° W2(t, s)f(s, zi(s)) ds1, . . .,N} forms an E-lattice for the set { ft°° W2(t, s)f(s, z(s)) ds : z E D}which is therefore relatively compact in X .Analogously the relative compactness in X of the remaining addends isverified, hence of the sum in the right-hand side of equality (6) for arbitraryfixed t E Ice, oo) as well when z runs over the set D. This enables us to applyLemma 1 .From Schauder's theorem it follows that the operator Q has a fixed pointy E D. We shall show that y E ((3), (4))« .Let t E la, oo)\{tj}j?=1 . Then, in view of the fact that x is a solution ofimpulsive equation (1), (2) and of the equality W1 (t, t)+W2(t, t) = I, we obtain(7) y' (t) = (Qy) '(t) == x'(t) + W1 (t, t)f(t, y(t» + A(t)ftW1 (t, s)f(s, y(s)) ds++A(t) 1: W1(t,tj)hj(yj)+W2(t,t)f(t,y(t))-«<ti <t-A(t)1,00W2(t,s)f(s,y(s))ds-A(t) ~ W2(t,ti)hi(yi)=t<t;<t <OO= A(t)y(t) + f(t, y(t))IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACII SPACE	255Let t = tk E [ce, oo) . Then(8) y(tk) = (Qy)(tk) _tk= Lkxk + 1.aLkW1(tk, s)f(s, y(s)ds + 1: LkW1(tk, ti) - hi(yi)-a<t; <tk00 00LkW2(tk, s)f(S, y(s)) ds - 1: LkW2(tk, t7)hj(yj) =tk j-k+1= Lk[yk + W1(tk,tk)hk(yk) + W2(tk,tk)hk(yk)] =Finally, as a consequence of the definition of the operator Q we obtain theestimatelly(t) - x(t)jj = 11(Qy)(t) - x(t)jj < F(t) (a < t < oo)which implies all assertions of Theorem 1 .Remark 1 . For dim X < oo Theorem 1 still holds without condition 3and the second part of condition 4 .Theorem 2 . Let a solution y E ((3), (4))i exist satisfying the conditiontM- 17 llWl(t,s)f,y(s)i1ds+1: 11Wi(t,ti)hi(yj)11+7<t; <t= Lkyk + Hkykw+ f 11W2(t,s)f(s,y(s))jj ds+ E 11W2(t,ti)hi(y.i)jj t~0t t<t; <OOThen there exisis a soluiion x E ((1), (2))t for which x(t) - y(t)o00 .tIf for some p E [1, oo] the left-hand side of relation (9) belongs to the, spaceLp (7, oo), then lix(t) - y(t)jj E L,,(t, oo) too.Proof.- For t > t define the function x by means of the formulax(t) =y(t)- f t W1(t,s)f(s,y(s))ds- E W1(t,tj)hj(yj)+t7<t; <t+ J - W2(t,s)f(s,y(s))ds+ E W2(t,ti)hi(yj)t t<t ; <OOBy a straightforward verification, analogously to (7) and (8), it is estab-lished that x E ((1), (2))t and has the necessary properties .Theorems 1 and 2 immediately imply the following theorem of asymptoticequivalence of impulsive equations (1), (2) and (3), (4) :256	D.D . BAINOV, S.I . KOSTADINOV, A.D . MYSHKISTheorem 3. Let for any R > 0 the following conditions hold :1 .t(10) sup llWi(t,sfs,z)i1ds+ 1: sup 11Wl(t,tj)hl(z)11+J- Ilzll<<R 7<t; <t Ilzll<Rsup [1 00 sup 11 Wz(T, s)f(s, z)i1ds -t- E suP 11W2 Ttjh z7<T<t t Ilzll<R t<t; <,,. Ilzll<<RX .References2. The set f{(t, z) : t < t < oo, liz11 < R} is relatively compact in3. The mappings Hj (j = 1, 2,3. . . . ) are continuous and the set00U hj{z E X : liz1l < R} is relatively compact in X.j=iThen ¡he impulsive equations (1), (2) and (3), (.4) are asympiotically equiv-alent. Moreover, if the left-hand side of relation (10) belongs for come p E [1, oo]to the space Lp (t, oo), then lly(t) - x(t)jj E Lp (a, oo) as well . .Remark 2. For dim X < oc Theorem 3 still holds without condition 2and the second part of condition 3.1. BAINOV D.D ., KOSTADINOV S .I . AND MYSHKIS A.D., Bounded and pe-riodic solutions of differential equations with impulse effect in a Banachspace, Differential and integral equations 1, no . 2 (1988), 223-230.2. BAINOV D.D., ZABREIKO P.P . AND KOSTADINOV S .I ., Stability of thegeneral exponent of nonlinear impulsive differential equations in a Banachspace, International J. of theoretical physics 27, no. 3 (1988), 373-380.3. BROUWER, F ., On the asymptotic relationships between solutions of twosystems of ordinary differential equations, J. of differential equations 6(1969),527-543 .4. DISHLIEV A.B . AND BAINOV D.D ., Sufficient conditions for absence ofbeating in systems of differential equations with impulses, Applicable Anal .18 (1985), 67-73.5. FUTAK J ., Asymptotic and integral equivalence of functional and ordinarydifferential equations, Fasciculi Mathematici 15 (1984), 97-108 .6. HASCAK AND SVEC M., Integral equivalence of two systems of differentialequations, Czechoslovak Mathematical Journa132 (107) (1982), 423-436.7. HASCAK A., Integral equivalence between a nonlinear system and its non-linear perturbation, Math . Slovaca 34, no. 4 (1984), 393-404.8. HU, SCHOUCHUAN AND LAKSHMIKANTHAM V., Periodic boundary valueproblems for second order impulsive differential systems, To appear in J.of Nonlinear Analysis .IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE	2579. HU, SCHOUCHUAN, LAKSHMIKANTHAM V., Existence and continuationresults for impulsive differential equations, To appear.10 . KULEv G.K . AND BAINov D .D ., Asymptotic stability of systems withimpulses by the direct method of Lyapunov, Bull. Austral. Math . Soc. 38(1988),113-123 .11 . KULEv G.K . AND BAINov D .D ., Second method of Lyapunov and com-parison principle for systems with impulse effect, J . of Computational andApplied Mathematics 23 (1988), 305-321 .12 . LAKSHMIKANTHAM V.AND LIU XINZHI, Stability for impulsive differentialsystems in terms of two measures, Appl . Math . Comp . (to appear) .13 . LAKSHMIKANTHAM V . AND LIU XINZHI, On quasi-stability for impulsivedifferential systems, To appear in J. Nonlinear Analysis .14 . MARLIN J.A. AND STRUBLE R.A ., Asymptotic equivalence of nonlinearsystems, J. of differential equations 6 (1969), 578-596 .15 . MIL'MAN V.D . AND MYSHKIS A .D ., On the stability of motion in thepresence of impulses, Sibirsk. Mat. J. 1 (1960), 223-237 (in Russian) .16 . SIMEONOV P .S . AND BAINov D .D., Asymptotic equivalence of two sys-tems of differential equations with impulse effect, Systems control letters3 (1983), 297-301 .P.O . Box 451504 SofiaBULGARIARebut el 16 de Juny de 1989

Asymptotic equivalence of impulsive differential equations in a Banach space

D. D. Bainov

S. I. Kostadinov

A. D. Myshkis

Crossref

Publicacions Matemàtiques

Kostadinov, Stepan I.

Revistes Catalanes amb Accés Obert

Publicacions Matemátiques, Vol 34 (1990, 249-257 .
A bstract
ASYMPTOTIC EQUIVALENCE
OF IMPULSIVE DIFFERENTIAL EQUATIONS
IN A BANACH SPACE
D .D . BAINOV, S .I . KOSTADINOV AND A .D . MYSHKIS
By means of Schauder's fixed point theorem sufficient conditions for
asymptotic equivalence of impulsive equations in a Banach space are
found .
1 . Introduction
Impulsive differential equations are a powerful apparatus for mathematical
simulation in physics, chemical technology, ecology, biotechnologies, impulse
technology and industrial robotics . That is why the theory of impulsive dif-
ferential equations develops actively in the last 20 years [4], [8-13], [15] . The
presente of phenomena specific for the impulsive differential equations such as
beating of the solutions, merging of the solutions, bifurcation, dying of the so-
lutions, loss of the property of autonomy, etc . makes this theory richer than
the theory of ordinary differential equations and, on the other hand, in the
investigation of impulsive differential equations a preliminary modification of
the methods used is necessary.
In relation to the mathematical simulation in theoretical physics recently the
investigation of impulsive differential equations in abstract spaces began [1],
[2] . In the present paper the asymptotic equivalence of impulsive differential
equations in a Banach space is investigated . We shall note that the asymptotic
equivalence of systems of ordinary differential equations has been investigated
by many authors [3], [5-7], [14], [16] .
The present investigation is supported by the Ministry of Culture, Science and Eclucation of
People's Republic of Bulgaria under Grant 61 .
250
	
D.D. BAINOV, S .I . KOSTADINOV, A.D . MYSHKIS
and
Consider the impulsive equations
2 . Preliminary notes
dx _
dt A( t )x (t 1 {tj}J° 1)
(2) x(t~- ) = Ljx(t~) (j = 1, 2,3 . . . . )
dt A(t)y + f(t, y)(t ~ {ti }39'=1)
(4) y(t;) = (Lj + Hj)y(t_) (j = 1, 2, 3 . . . . )
in a real or complex Banach space X. Here t varies in the interval Ft, oo) ; A(t)
X -> X (for each t E r, oo)) is a linear bounded operator and the operator-
valued function A( .) is continuously continuable from (t .i_1, tj) to [t.i_1, tj] (j =
1, 2, 3, . . ., to = t) ; Lj : X -> X (j = 1, 2, 3 . . . . ) are linear bounded operators
having bounded inverse ones and LUX = X ; the function f : r, oo) x X -> X is
continuosly continuable from (tj_ 1 , tj) x X to [tj_ 1 , tj] x X (j = 1, 2,3 . . . . ) ; Hj
X --> X (j = 1, 2,3 . . . . ) ; the sequence T = {tj}~°_ 1 satisfies the condition
t < ti < t2 < --- -+ 00
We assume that the solutions of equations (1), (2), (3) and (4) are continuous
from the left, thus in the conditions of a jump (2) and (4) instead of tj we shall
write tj . If x is a solution of equation (1), (2) on the interval [ce, 00), we shall
write x E ((1), (2))« and analogously of equation (3), (4) . An arbitrary solution
of impulsive equation (1), (2) defined on some subinterval of [t, a) is continuable
to the whole interval which for impulsive equation (3), (4) in general is not true .
Denote by W(t, T)(t <_ t, T < oo) the evolutionary operator of impulsive
equation (1), (2) associating with each element xo E X the solution x(t) _
W(t,T)xo of (1), (2) with condition x(T) = xo .
The following equalities are valid
W(t,T) = A(t)W(t,T)(t 1 {tj};°-_1),W(tt ,T) = LjW(tj,-r)
Assume that the space X is split up in a direct sum of the subspaces Xl
and X2, i .e . X = X1-1-X2 and let PI, P2 be the corresponding complementary
projectors (Pl + P2 = I, where I is the identity operator) . Set Wi(t1,t2) _
W(t 1 )P;W-1 (t2) (i = 1, 2), where W(t) = W(t,t) .
Let K C r, oo) . Denote by S(K, X) the Banach space of all bounded func-
tions x : K -+ X which are continuous for t qÉ tj E K continuous from the left,
have a limit from the right, provided with the norm 111xIII = sup tc K 11x( 1 )11 .
IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE
	
251
Lemma 1. Le¡ F C S([a,oo),X)(a >_ t) and le¡ ¡he following conditions
hold :
1 . There exists a functioncpo E S([a,oo),X) such that llcp(t)-cpo(t)11 t+
0
uniformly with respect to cp E F.
2 . The set F is equicontinuous on each interval [c¿, w) fl (t i,ti] (j =
1,2,3, . . .) .
3. For each t E [a, oo) ¡he set {cp(t) : cp E F} is relatively compact in X .
Then F is relatively compact in S([a, oo), X) .
Lemma 1 is proved in a standard way by the construction for any e > 0 of
an s-lattice for F in the space S([a, oo), X) .
3. Main results
Set hj = L) 'Hj and xj = x(tj) where x(t) is an arbitrary function of t.
Definition 1 . The impulsive equations (1), (2) and (3), (4) are called asymp-
totically equivalent if for any bounded solution x E ((1), (2))7 there exists
a E r, oc) and a solution y E ((3), (4»,,, such that y(t) - x(t) t> 0 and,
conversely, for any bounded solution y E ((3), (4)r there exists a solution
x E ((1), (2))t for which y(t) - x(t) t 0 .-00
Theorem 1. Let the following conditions hold :
1 . There exists a bounded solution x E ((1), ( 2))i .
2 . There exists a constant R > 0 for which
(5 )
J-
t
sup ilWi(t,s)f,z»ds+ sup"11Wl(t,tj)hj(z)11+
t z
z¡<ti <t
X .
00
sup'11W2(T,s)f(s,z)JJ ds+ sup',11W2(T,ti)hj(z)~~J
t
>0
t<ti <oo z
where sup' is taken over ¡he set {z E X : liz - x(s)JJ _< R} and sup"
oven ¡he set {z E X : liz - xjll < R} .
3. The set f{(t, z) : t <_ t < oo, liz - x(t)JJ <_ R} is relatively compact in X.
4 . For j = 1, 2,3 . . . . the mappings Hj are continuous on the balls Bj =
00
{z E X : liz - xj1l <_ R} and the set U hj(Bj) is relatively compact in
j=l
Then for some a E [t, oo) there exists a solution y E ((3, (4))« for which
y(t) - x(t) -> 0. If, moreover, the left-hand side of the relation in (5) (as a
t-00
252
	
D.D . BAINOV, S .I . KOSTADINOV, A.D. MYSHKIS
function of t) belongs to the space Lp(t, oo) for some p E [1, oo], then IIy(t) -
x(t)II E Lp(a>oo) .
Proof. Denote the left-hand side of relation (5) by F(t) and choose a > t
so large that F(t) _< R for t >_ a . The set D = {z E S([a, 00), X) :IIIz -
xI[a,,,) I I I < R} is nonempty, convex, closed and bounded .
On D define the operator Q by means of the formula
(6) (Qz)(t) = x(t) -}- J t W1
(t, s)f(s, z(s» ds + 5-~ W1(t, tj)hj(zj)-
a<t ; <t
00
W2(t,s)f(s,z(s))ds- E W2(t,tj)hj(zj)
t<t; <oo
where z E D, t E [a, oo) . The convergence of the improper integral and the
series in (6) follws from the fact that F(t) < oo .
We shall show that the operator Q satisfies the conditions of Schauder's fixed
point theorem . .
1 . Q : D -> D. In fact, from the uniform convergence of the improper
integral and the series in the right-hand side of equality (6) and from the choice
of a it follows that Qz E S([a, oo), X) and that II(Qz)(t) - x(t)II < R, a < t <
oo .
2 . The operator Q is continuous . In fact, let e > 0 be an arbitrarily chosen
number and let > a be choosen so that F(s) < á for s >_ ¿ . Let z, z1 E D
and a < t < f . Then
II(Qz)(t) - (Qz1)(t)II <-
J
z
IIW1(t, s)[f(s, z(s)) - f(s, z1(s))JII ds+
1
+ f IIW2(t,s)[ .f (s,z(s)) - f(s, z1(S))JII ds+
t
+ E IIWI(t,ti)[hj(zj) - h.i(zi;)III+
a<t; <t
+ E IIW2(t,tj)[hi(zj)-hj(z1 ;)JII+
j :5 ti <f
+
J
' IIW2(t,s)[f(s,z(s))-f(s,z1(s))JIIds+
f
+ IIW2(t,tj)[hj(zj)-h;(z1 ;)JII
~<tj <oo
Due to the continuity of the functions f and hj for a fixed function z1 a
number 71 > 0 can be found such that for 111z - zi 111 < T7 each of the first four
addends in the right-hand side of the last inequality should be smaller than
IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE
	
253
é . But then the right-hand side of the last inequality becomes smaller than
48 + 2$ + 28 = e . For t > 1 the following inequalities are valid
II(Qz)(t) - (Qzi)(t)II -< II(Qz)(t) - x(t)II + II(Qzi)(t) - x(t)II < 28
< e
Hence the operator Q is continuous .
3 . We shall show that the set QD is relatively compact in D. For this
purpose we shall show that the conditions of Lemma 1 are fulfilled for F =
QD, cp o = x . Condition 1 of Lemma 1 follows immediately from relation (5) .
We shall show that condition 2 of Lemma 1 holds . Let z E D and t', t" E
(tk-1,tk] for some K where t' < t" . Then, after elementary transformations
we obtain the equality
(Qz)(t') - (Qz)(t',) =x(t,)
- x(t") -
00
00
from which we deduce the estimate
t"
W, (t", s)f(s, z(s» ds-
-
J L
[WI (t", s) - Wl (t', )If(s, z(s)) ds+
a
+ [WI(t',tj) -
W,(t" tj)]hj(zj)-
a<tj <tk
W2(t', s)f(s, z(s)) ds-
[W2(t', s) - W2 (t", s)]f(s, z(s)) ds-
- E[W2(t',tj)-W2(t" tj)]hj(zj),
j-k
_ t 11
II
(Qz)(t') - (Qz)(t'%I :5 II x(t') - x(t")II + IIWI(t" s)f(s, z(s»II ds+J
¡
t ,
f
t
1
+ IIW(t') - W(t")II ' IIPI W-'(s)f(s, z(S))II ds+a
+ IIW(t')-W(t")IIJIPI W -' (tj)hj(zj)II+
«<tj <tk
f
t11
+ IIW2(t',s)f(s,z(s))II ds+
t+f 00 II[W(t') -W(t")]P2W-'(s)f(S, z(s))II ds+
t"
+ 1: IIW(t ' ) - W(t")]P2W-'(tj)hj(zj)II
j-k
254
	
D .D . BAINOV, S .I . KOSTADINOV, A .D . MYSHKIS
Denote by I1, . . . , 17 the successive addends in the right-hand side of the
last inequality and assume that t"-t' --> 0. Then I1 -> 0 since x E S([t, oo), X) .
12, IS --> 0 because of the uniform (with respect to z, t', t" ) boundedness of
the integrand . 13 -> 0 and 14 -> 0 since under the signs of the integral and of
the sum the first factor tends uniformly to zero and the second one is uniformly
bounded . Represent 16 for t > tk in the form
t t
Is = f + f :51 IIW(t~) - W(t

)II ' IIP2W-1 (S)f(S, z(S))II d.s + 2F(t)
The second addend in the last inequality can be made arbitrarily small
for a sufficiently large value of t. For t fixed and t" - t' small enough the first
addend can also be made uniformly arbitrarily small (analogously to I3) . Hence
16 -> 0 . Analogously it is proved that 17 -> 0 .
Thus the equicontinuity of the set QD on the integral (tk_1, tk] is proved .
In order to verify condition 3 of Lemma 1, first we shall show that for t E
la, oo) the set {f,00 W2(t, -s)f(s, z(s» ds : z E D} is relatively compact in X. Let
e > 0 be arbitrarily chosen. Choose > t so that II fE W2(t, s)f(s, z(s)) dsll <
3 for all z E D. The set {W-1 (s)f(s, z(s)) : t < s < j, z E D} is rela-
tively compact in X . Hence the sets {ft' W -1(s)f(s, z(s» ds : z E D} and
{f~ W2(t, s)f(s, z(s)) ds : z E D} are also relatively compact subsets of X.
For the last set we choose a finite á -lattice- {ft{ W2(t, s)f(s, zi(s) ds : i =
1, . . . , N; zi E D} and obtain that the set { ft°° W2(t, s)f(s, zi(s)) ds
1, . . .,N} forms an E-lattice for the set { ft°° W2(t, s)f(s, z(s)) ds : z E D}
which is therefore relatively compact in X .
Analogously the relative compactness in X of the remaining addends is
verified, hence of the sum in the right-hand side of equality (6) for arbitrary
fixed t E Ice, oo) as well when z runs over the set D. This enables us to apply
Lemma 1 .
From Schauder's theorem it follows that the operator Q has a fixed point
y E D. We shall show that y E ((3), (4))« .
Let t E la, oo)\{tj}j?=1 . Then, in view of the fact that x is a solution of
impulsive equation (1), (2) and of the equality W1 (t, t)+W2(t, t) = I, we obtain
(7) y' (t) = (Qy) '(t) =
= x'(t) + W1 (t, t)f(t, y(t» + A(t)f
t
W1 (t, s)f(s, y(s)) ds+
+A(t) 1: W1(t,tj)hj(yj)+W2(t,t)f(t,y(t))-
«<ti <t
-A(t)
1,00
W2(t,s)f(s,y(s))ds-A(t) ~ W2(t,ti)hi(yi)=
t<t;<t <OO
= A(t)y(t) + f(t, y(t))
IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACII SPACE
	
255
Let t = tk E [ce, oo) . Then
(8) y(tk) = (Qy)(tk) _
tk
= Lkxk + 1.
a
LkW1(tk, s)f(s, y(s)ds + 1: LkW1(tk, ti) - hi(yi)-
a<t; <tk
00 00
LkW2(tk, s)f(S, y(s)) ds - 1: LkW2(tk, t7)hj(yj) =
tk j-k+1
= Lk[yk + W1(tk,tk)hk(yk) + W2(tk,tk)hk(yk)] =
Finally, as a consequence of the definition of the operator Q we obtain the
estimate
lly(t) - x(t)jj = 11(Qy)(t) - x(t)jj < F(t) (a < t < oo)
which implies all assertions of Theorem 1 .
Remark 1 . For dim X < oo Theorem 1 still holds without condition 3
and the second part of condition 4 .
Theorem 2 . Let a solution y E ((3), (4))i exist satisfying the condition
tM- 17 llWl(t,s)f,y(s)i1ds+
1: 11Wi(t,ti)hi(yj)11+
7<t; <t
= Lkyk + Hkyk
w
+ f 11W2(t,s)f(s,y(s))jj ds+ E 11W2(t,ti)hi(y.i)jj t~0
t t<t; <OO
Then there exisis a soluiion x E ((1), (2))t for which x(t) - y(t)
o0
0 .
t
If for some p E [1, oo] the left-hand side of relation (9) belongs to the, space
Lp (7, oo), then lix(t) - y(t)jj E L,,(t, oo) too.
Proof.- For t > t define the function x by means of the formula
x(t) =y(t)- f t W1(t,s)f(s,y(s))ds- E W1(t,tj)hj(yj)+
t
7<t; <t
+ J - W2(t,s)f(s,y(s))ds+ E W2(t,ti)hi(yj)
t t<t ; <OO
By a straightforward verification, analogously to (7) and (8), it is estab-
lished that x E ((1), (2))t and has the necessary properties .
Theorems 1 and 2 immediately imply the following theorem of asymptotic
equivalence of impulsive equations (1), (2) and (3), (4) :
256
	
D.D . BAINOV, S.I . KOSTADINOV, A.D . MYSHKIS
Theorem 3. Let for any R > 0 the following conditions hold :
1 .
t
(10) sup llWi(t,sfs,z)i1ds+ 1: sup 11Wl(t,tj)hl(z)11+J- Ilzll<<R 7<t; <t Ilzll<R
sup [1 00 sup 11 Wz(T, s)f(s, z)i1ds -t- E suP 11W2 Ttjh z
7<T<t t Ilzll<R t<t; <,,. Ilzll<<R
X .
References
2. The set f{(t, z) : t < t < oo, liz11 < R} is relatively compact in
3. The mappings Hj (j = 1, 2,3. . . . ) are continuous and the set
00
U hj{z E X : liz1l < R} is relatively compact in X.
j=i
Then ¡he impulsive equations (1), (2) and (3), (.4) are asympiotically equiv-
alent. Moreover, if the left-hand side of relation (10) belongs for come p E [1, oo]
to the space Lp (t, oo), then lly(t) - x(t)jj E Lp (a, oo) as well . .
Remark 2. For dim X < oc Theorem 3 still holds without condition 2
and the second part of condition 3.
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P.O . Box 45
1504 Sofia
BULGARIA
Rebut el 16 de Juny de 1989


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Asymptotic equivalence of impulsive differential equations in a Banach space

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