On the asymptotic behavior of the pantograph equations. G. Makay and J. Terjéki

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1 On he asympoic behavior of he panograph equaions G. Makay and J. Terjéki Bolyai Insiue, Aradi véranúk ere 1, H-6720 Szeged, Hungary Dedicaed o Professor J. Kao on his 60h birhday 1. Inroducion Our aim is suding he asympoic behaviour of he soluions of he equaion ẋ( = a(x( + a(x(p, (1.1 where a( is a nonnegaive coninuous scalar funcion on R + := [0, and 0 < p < 1 is a consan. This equaion is a special case of he so called panograph equaions arising in indusrial applicaions [5,11]. The only soluion of equaion (1.1 wih iniial daa x(0 = x 0 is x( x 0. However, if 0 > 0 and ϕ( is a given coninuous funcion on [p 0, 0 ] hen he soluion x( wih x(s = ϕ(s for s [p 0, 0 ] is defined for and i differs from any consan soluion if ϕ is no consan. Equaion (1.1 can be ransformed o he equaion ẏ( = a 1 (y( + a 1 (y( h, (1.2 by y( = x(e, where p = e h and a 1 ( = a(e e or o he equaion ż( = a 2 (z( + a 2 (z(p(, (1.3 wih a given reardaion p( choosing he ransformaion z( = x(g(, where g( saisfies he equaion pg( = g(p( and a 2 ( = a(g(ġ(. Therefore some resuls Suppored by he Hungarian Naional Foundaion for Scienific Research wih gran numbers T/ and F/016226, and by he Foundaion of he Hungarian Higher Educaion and Research. EJQTDE, 1998 No. 2, p. 1

2 can be concluded for he equaion (1.1 from resuls for (1.2, (1.3 or heir generalizaions. T. Kriszin [9] invesigaed he equaion ẋ( = f(, x wih infinie delay. The applicaion of his resul for (1.1 gives ha if a(s ds is p bounded on R + hen all soluion of (1.1 ends o a consan as. N. G. De Bruijn [3, 4] sudied linear scalar equaion w(ẋ( = c(x( + d(x( 1 + r(. From his resuls i can be proved ha if n=1 p n exp{ a(u du} = p (n 1 hen every soluion of (1.1 has a finie limi if. On he oher hand if a( is wice coninuously differeniable and here exiss a coninuous nonincreasing posiive funcion Φ such ha 1 Φ(s ds < and for w( := 1/a( he condiions w(, w (, w ( < e Φ( hold, hen here exiss a coninuous periodic funcion ψ of period 1 and a posiive consan c such ha x( ψ { } log 1/ log(1/p log 1/p 1/a(s ds < c Φ(s ds. 1 log / log(1/p 1 The scalar equaion ẋ( = ax( + bx(p, (where a and b are consans, a > 0 is also sudied. The exac asympoic behaviour of he soluions as is known [1, 2, 8]. In he special case a = b he following asserion is proved. For any soluion x( here exiss an infiniely many imes differeniable, periodic funcion ψ of period 1 such ha x( ψ(log 0 as. (1.5 We give an exension of he las resuls for (1.1. We need some ligh monooniciy like condiions for a( ha resric oo fas changes of a(. Our condiion works for EJQTDE, 1998 No. 2, p. 2

3 he funcion a( = α if α > 1, or for he funcion a( = a + sin b, if he consans a, b saisfies b < (a b 2. We show by an example ha ψ may be non-consan funcion. In he proof of he resuls we need o know he decay rae of soluions. This argumen works for more general equaion. Therefore in he second par of he aricle we sudy he equaion ẋ( = a(x( + b(x(p( and give condiions such ha x( C k [ 0,, (or a similar esimae is rue. Esimaion of such ype was given by J. Kao [6,7] and his resuls were sharpened by T. Kriszin [10]. However hese resuls and our ones in his paper canno be compared and mehods are differen, oo. 2. An asympoic esimae of he soluions Le us consider he equaion ẋ( = a(x( + b(x(p(, (2.1 where a(, b(, p( are coninuous funcions on R +, p( and lim p( =. Le us define he funcion m( := inf{s : p(s > } on R +. Then m(, p(m( = and m( is increasing. Le be given 0 0 such ha p( 0 < 0 and inroduce he qualiies q 1 = inf{p(s : s 0 }, q 0 = 0, q n = m(q n 1, n = 1, 2,..., q = lim n q n and he inervals I n := [q n 1, q n ], n = 0, 1, 2,.... EJQTDE, 1998 No. 2, p. 3

4 Then n=1i n = [ 0, q, and p(q = q, if q <. Moreover we also have p(i n+1 n k=0 I k. For a given funcion ρ : R + (0, having bounded differenial on finie inervals le us inroduce he numbers ρ n := max I n ρ( q n 1 exp{ a(z dz}ρ 2 (s ρ(s ds n = 1, 2,.... Since a(s exp{ a(z dz} ρ(s ds = exp{ a(z dz} ρ(s and a( is nonnegaive, we ge for I n ha + ρ(s exp{ a(z dz} ρ 2 (s ds (2.2 0 ρ( q n 1 = 1 + ρ( a(s exp{ a(z dz} ρ(s q n 1 Hence ρ n for all n = 1, 2,.... ds + ρ( exp{ q n 1 ρ(s exp{ a(z dz} ds. ρ 2 (s a(z dz} ρ(q n 1 = (2.3 Theorem 1. Suppose ha here exiss a differeniable funcion ρ : [q 1, q (0, such ha ρ( is locally bounded, and If M 0 := max I0 ρ( x(, hen b( ρ( a(ρ(p( [ 0, q (2.4 P := (1 + ρ n <. n=1 x( M 0P ρ(, [ 0, q. Proof. Inroduce he funcion Then y( saisfies he equaion ( ẏ( = y( := x(ρ(. a( ρ( ρ( y( + b(ρ( ρ(p( y(p( EJQTDE, 1998 No. 2, p. 4

5 which is equivalen o ( d y( exp{ d 0 a(sds} ρ( = y(p(b( exp{ 0 a(sds} ρ(p( Inegraing his equaliy on [q n, ] and using (2.4 we ge y( exp{ 0 a(sds} ρ( y(q n exp{ q n 0 ρ(q n q n a(sds} + a(s exp{ 0 a(zdz} y(p(s ds. ρ(s Le m n := max In y( n = 0, 1, 2, 3,... and M n := max{m 0, m 1,..., m n }. Since y(p( M n for I n+1, we have ρ( exp{ q n a(s ds} y( M n ρ(q n + M n ρ( q n for all I n+1, n = 0, 1, 2,.... Using he formula (2.3 we ge y( M n (1 + ρ( q n a(s exp{ a(z dz} ds ρ(s ρ(s exp{ a(z dz} ds M ρ 2 n (1 + ρ n+1 (s for all I n+1. Hence M n+1 M n (1 + ρ n+1 for all n = 0, 1, 2,... ha implies n+1 M n M 0 (1 + ρ k M 0 (1 + ρ k. k=1 k=1 This inequaliy is equivalen o he asserion of he heorem. Corollary 1. Suppose here exis 0 < Q 1, 0 < p 1 p 2 < 1, m > 0, 0 α < 1 and 0 > 0 such ha (1 α log 1 p 2 log 1 p 1 > log 1 Q (log 1p1 log 1p2. p 1 p( p 2, b( a(q, m α a( [p 1 0,. If k = log Q/ log p 1, x : [p 1 0, R is a soluion of (2.1 on [ 0, hen x( MC k [ 0,, (2.5 EJQTDE, 1998 No. 2, p. 5

6 where C = n=0 (1 + k m 1 α 0 p k 1 ( p k+1 α n 2 p and M = k max [p1 0, 0 k x(. 1 Proof. Firs of all we remark ha he produc in he definiion of C exiss since p k+1 α 2 /p k 1 < 1 by he definiion of k. The relaions p(m( = and p 1 p( p 2 imply ha p 2 m( p 1. Hence 0 p n 2 q n 0 p. Also n 1 ( p( b( a(q = a(p k 1 a( and hence (2.4 is valid wih ρ( = k. Moreover ρ n+1 max k k I n+1 q n e mz α dz s k+1 max I n+1 k k e m(1 α 1 1 α = max I n+1 k k e m(1 α 1 1 α k 1 α ( 0 k 1 α 0 mp k 1 p n+1 1 ( k ( 0 p k+1 α 2 p k 1 p n 2 n So, Theorem 1 implies he asserion. ds 0 p n 2 1 α ( 0 p n 2 (k+1 α max I n+1 e m(1 α 1 s 1 α k s k+1 ds = em(1 α 1 u 1 α (1 αu (k+1 α(1 α 1 du 1 α ( 0 p n 2 1 α e m(1 α 1 (u 1 α du Corollary 2. Suppose ha p( = k, a a( and b( θa( hold on he inerval [ 0,, where 0 > 1, k > 1, 0 < a, 0 < θ < 1 are consans. Then here exiss a posiive consan C such ha for any soluion of (2.1 on ( k 0, we have log θ x( C M(log log k ( [ 0,, where M = max{ x( : [ k 0, 0 ]}. Proof. Apply again Theorem 1 by ρ( = log α, where α = log θ/ log k. Then q 1 = k 0, q n = kn 0 for n = 0, 1, 2,... and ρ( exp q n 1 a(zdz ρ(sρ 2 (sds k αn log α 0 kn 1 0 e a( s α s log α+1 s ds EJQTDE, 1998 No. 2, p. 6

7 Hence αk α 1 e a( 0 k n 1 log 0 a kn 1 ρ n αk α kn 1 0 k n 1 a log 0 k n 1 0 ha is n=1 (1 + ρ n < and he asserion follows wih C = (1 + n=1 αk α. 0 kn 1 k n 1 a log 0 Remark. If a a( and b( θa(, where 0 < a and 0 < θ < 1 hen Corollary 1 and Corollary 2 imply ha if p( = p, 0 < p < 1 hen x( MC 1 log θ log p, (2.6 if p( = k log θ, 1 < k hen x( MC 2 (log log k (2.7 for 0. T. Kriszin [10] applied his resuls for he cases p p( and k p( and he gave he condiions if p( p, 0 < p < 1 hen x( MC 3 p(p 1 log µ, (2.8 if p( k, 1 < k hen x( MC 4 (log log µ log p (2.9 for 0, where µ ( 1, 1 θ. I is easy o see, ha (2.6 and (2.7 are sharper han (2.8 and (2.9. On he oher hand we required ha p( is far from, and he assumpions ha p( = p and p( = k canno be changed o p( p and p( k. Therefore, our resuls and he ones in Kriszin s paper are independen. Corollary 3. Suppose here exis 0 < p < 1, Q > 1 such ha p( = p, b( a(q on [p 0,. If k = log Q/ log p and M = max [p0, 0 k x(, hen x( k [ 0,. M on Proof. Now, we have q n = 0 /p n and (2.4 is valid wih ρ( = k. Then ρ( 0, so 1 + ρ n 1 and Theorem 1 can be applied. EJQTDE, 1998 No. 2, p. 7

8 3. The asympoic behavior Consider he equaion ẋ( = c(x( + c(x(p, (3.1 where 0 < p < 1. Le c( be nonnegaive, coninuously differeniable on R +. Then he soluions are wice differeniable and y( = ẋ( saisfies he equaion ẏ( = ( c( ċ( c( Now, we apply he above resuls o his equaion. y( + pc(y(p. Theorem 3. Suppose ha c( is coninuously differeniable on R + and here exis 0 > 0, 0 < k 1, m > 0 and 0 α < 1 such ha m α c( ċ( c 2 ((1 p 1 k ( [ 0,. Le x( be a soluion of (3.1 on [p 0, hen where C = ( n=0 1 + kp(n+1(1 α k m 1 α 0 ẋ( CM k ( [ 0 p,, and M = sup [0, 0 /p k ẋ(. Proof. Since a( = c( ċ( c(, b( = c(p, Q = pk he condiion b( a(q is equivalen o ċ( c 2 ((1 p 1 k. Corollary 1 is applicable wih p 1 = p 2 = p and replacing 0 by 0 p. Now le us ransform equaion (3.1 in a differen way. Le y( = ẋ(/c(, hen ẏ( = c(y( + pc(py(p. Theorem 4. Suppose ha here exis 0 > 0, 0 < k 1, m > 0 and 0 α < 1 such ha m α c( p 1 k c(p c( ( [ 0, EJQTDE, 1998 No. 2, p. 8

9 Le x( be a soluion of (3.1 on [p 0, hen ẋ( CMc( k ( [ 0 p,, where C is he same as in Theorem 3 and M = sup [0, 0 /p k ẋ( c(. Proof. Use Corollaries 1 and 3 as in Theorem 3. Noe ha i is only a echnical deail ha we esimae he derivaive on he inerval [ 0 /p, in Theorems 3 and 4. If we choose an iniial funcion so ha he soluion is coninuously differeniable a he poin 0, hen we can prove a similar esimae using M as he supremum of he appropriae funcion on he inerval [p 0, 0 and a lile bi differen C s. We will use his commen laer. Noe also, ha i is easy o see ha if x is a soluion of (3.1, M 0 = sup [p0, 0 x( and m 0 = inf [p0, 0 x(, hen m 0 x( M 0 ( [ 0,. Definiion. We say ha he funcion x( is asympoically logarihmically periodic, if x(e is asympoically periodic, i.e. here is a periodic funcion φ( such ha x(e φ( 0 as. Theorem 5. Suppose ha all he condiions of Theorem 3 are saisfied and k > α. Then all he soluions of equaion (3.1 are asympoically logarihmically periodic. Proof. Le φ : [p 0, 0 ] R be given and consider x( = x(, 0, φ, he soluion of (3.1 saring a 0 wih he iniial funcion φ. To simplify our noaion le us assume ha 0 = 1, for oher 0 s he proof is similar. Le M and C be he consans appearing in Theorem 3 and hence we have ẋ( CM/ k on he inerval [1/p,. Le us ransform he equaion by replacing = e s and x( = y(ln( = y(s. From (3.1 we obain ẏ = c(e s e s ( y(s + y(s + ln(p = c(e s e s ( y(s + y(s h, (3.2 where h = ln(p (here we use ha 0 = 1 and hence he soluion y corresponding o x sars from ln( 0 = 0. We also have ẋ( = ẏ(ln(/ for 1/p and hence ẏ(s CMe s(1 k for s h. Then we use he equaion o have y(s y(s h CMe sk c(e s s h EJQTDE, 1998 No. 2, p. 9

10 Using ha k > α i is easy o prove ha he sequence e (s+ihk c(e s+ih e (s+ihk e (s+ihα /m e ih(k α /m (s [0, h] is summable. Therefore y(s + lh y(s + nh CM l i=n+1 e (s+ihk c(e s+ih s [0, h] and l n 0 (3.3 and hence he funcion sequence z n (s := y nh (s = y(s + nh (for s [ h, 0] is a Cauchy-sequence in he supremum norm. Thus i converges o a funcion χ. Consider χ(s o be an h-periodic funcion, and hen we have y(s χ(s 0 as s. Therefore all soluions of (3.2 are asympoically h-periodic, which means ha all soluions of (3.1 are asympoically logarihmically periodic. Theorem 6. Suppose ha all he condiions of Theorem 4 are saisfied. Then all he soluions of equaion (3.1 are asympoically logarihmically periodic. Proof. The proof is very similar o ha of Theorem 5. The only difference is ha afer he ransformaion we have y(s y(s h CMe sk s 0 since he c( in he esimae on ẋ( and he c(e s coming from (3.2 cancel each oher. The res of he proof is he same. In his secion we esablished condiions under which we can prove asympoical logarihmical periodic behavior of he soluions of equaion (3.1. Boh he condiions of Theorem 3 and 4 are reasonable, hey require c( no o be oo small or decrease oo fas. Clearly, all consan funcions are soluions of equaion (3.1, and asympoic logarihmic periodiciy includes he special case of he soluions being asympoically consan. We now show by an example ha here is an equaion of he form (3.1 which has an asympoically non-consan soluion. Le c( = 1, 0 = 1, k = 1, m = 1, α = 0 in Theorem 6. Le φ : [p, 1] R be given (i will be specified laer, bu i saisfies he condiion ha he soluion is coninuously differeniable a 0 and hence we have an esimae on he derivaive on he inerval EJQTDE, 1998 No. 2, p. 10

11 [1,. We do he same ransformaion as we did in he proof of Theorem 6. Then we have By inducion we ge y(s y(s h CMe s s 0. y(s + lh y(s h CMe s 1 e h. Le ψ(s := φ(e s = φ( for [p, 1]. Define s max and s min so ha ψ(s max is a maximum and ψ(s min is a minimum of ψ in he inerval [ h, 0]. The above inequaliy gives (as a special case ha and Puing hese ogeher we obain y(s max + lh y(s max CM e (s max+h 1 e h y(s min + lh y(s min + CM e (s min+h 1 e h. y(s max + lh y(s min + lh (y(s max y(s min CM e (s max+h + e (s min+h 1 e h Now we define φ a lile more precisely. Le φ be sricly increasing on he inerval [p, (1 + p/2] and decreasing on [(1 + p/2, 1], hence s min = 0, s max = ln((1 + p/2 and ψ(s max ψ(s min > 0. Then we have and hence e s max + e s min 1 e h = 3 + p 1 p 2 y(s max + lh y(s min + lh ψ(s max ψ(s min CMp(3 + p 1 p 2 γ > 0 if we choose p small enough. This shows ha y a he shifs of s max and s min differs by a fixed posiive consan and hence y canno end o a consan. Since x( = y(ln(, we also proved ha x does no end o a consan. EJQTDE, 1998 No. 2, p. 11

12 References [1.] J. Carr J. Dyson, The funcional differenial equaion y (x = ay(λx + by(x. Proc. Roy. Soc. Edinburgh, 74A, 13, (1974/75, [2.] J. Carr J. Dyson, The marix funcional differenial equaion y (x = Ay(λx + By(x. Proc. Roy. Soc. Edinburgh, 75A, 2, (1975/76, [3.] N. G. De Bruijn, On some linear funcional equaions, Publ. Mah. Debrecen, 1950, [4.] N. G. De Bruijn, The asympoically periodic behavior of he soluions of some linear funcional equaions, Amer. J. Mah. 71(1949, [5.] L. Fox, D. F. Mayers, J. R. Ockendon and A. B. Tayler, On a funcional differenial equaion, J. Ins. Mahs Applics, 8(1971, [6.] J. Kao, Asympoic esimaion in funcional differenial equaions via Liapunov funcion, Colloq. Mah. Soc. J. Bolyai, 47. Differenial equaions: Quliaive heory, Norh-Holland, Amserdam- New york, 1987, [7.] J. Kao, An asympoic esimaion of a funcion saisfying a differenial inequaliy, J. Mah. Anal. Appl. 118(1986, [8.] T. Kao and J. B. McLeod, The funcional-differenial equaion y (x = ay(λx + by(x, Bull. Amer. Mah. Soc., 77(1971, [9.] T. Kriszin, On he convergence of soluions of funcional differenial equaions wih infinie delay, Aca Sci. Mah. 109(1985, [10.] T. Kriszin, Asmpoic esimaion for funcional differenial equaions via Liapunov funcions, Colloquia Mah. Soc. J. Bolyai, 53. Qualiive Theory of Differenial Equaions, Szeged, 1988, [11.] J. R. Ockendon and A. B. Taylor, The dynamics of a curren collecion sysem for an elecric locomoive, Proc. Roy. Soc. London, Ser. A, 322(1971, EJQTDE, 1998 No. 2, p. 12