Asymptotic equivalence of Volterra difference systems

Publicacions Ma em`a iques, Vol 39 (1995), 301–312. ASYMPTOTIC EQUIVALENCE OF VOLTERRA DIFFERENCE SYSTEMS J. Mo chalo Abs ac The pu pose o his pape is o gi e some esul s on he asymp o ic ela ionship be ween he solu ions o a linea diffe ence equa ion and i s pe u bed nonlinea equa ion. 1. In oduc ion The p oblem o he asymp o ic equi alence o sys ems o o dina y diffe en ial equa ions has been s udied by many au ho s, as e.g. B aue [3], B aue and Wong [4], Boundo ides and Geo giou [2], Lowell Lo elady [11], Mo cha#lo [13], ˘ S ec [17], Szufla [18], and o he s. The p oblem o he asymp o ic equi alence o in eg odiffe en ial equa- ions has been s udied by Mo cha#lo [14], Razapo [16], Talpala u [19]. The p oblem o he asymp o ic beha io o solu ions o o dina y di - e ence equa ions has been s udied by Benzaid [1], Conffman [5], D oz- dowicz, Popenda [6], Elaydi, Gyo i [8], Li [10] and Pin o [15]. In his pape , we shall conside some esul s on he asymp o ic ela- ionship be ween he solu ions o a linea Vol e a diffe ence equa ion and i s pe u bed nonlinea equa ion. The au ho knows only he wo ks o Talpala u [19], Ved and Go#lo ina [21], Ved and Kap agae [20], deal- ing wi h he abo e p oblem o special case. 2. No a ions and Defini ions He e N(n0)={n0,n 0+1,...}, whe e n0is a na u al numbe o ze o; Rk- he k- dimensional eal euclidean space wi h he no m |x|= k  i=1 |xi|,x=(x1,... ,x k); 302 J. Mo chalo Mk- he space o all k×kma ices D=(dij) wi h he no m |D|= max j k  i=1 |dij|,I- iden i y ma ix. We deno e by Φ(N,Rk) he space o all unc ions om N(n0)in oRk. Le Φ1=Φ 1(N,Rk) be he Banach space in Φ o all bounded unc ions u:N(n0)→Rkwi h no m x=|x(n)|Φ1= sup{|x(n)|:n∈N(n0)}. In his pape we conside he ollowing sys ems o diffe ence equa ions (2.1) x(n+1)=[A+B(n)]x(n)+ n  =0 [K(n− )+Q(n, )]x( ), (2.2) x(n+1)=Ax(n)+ n  =0 K(n− )x( )+ (n)+F(n, x(n)) hough e as pe u ba ions o y(n+1)=Ay(n)+ n  =0 K(n− )y( ),(2.1’) y(n+1)=Ay(n)+ n  =0 K(n− )y( )+ (n)(2.2’) whe e x,y, a e k-dimensional ec o s, A- is a cons an ma ix k×k, B, K :N(n0)→Mk Q:N(n0)×N(n0)→Mk, F:N(n0)×U→Rkis o any n∈N(n0) con inuous as a unc ion o x∈U (U- a egion in Rk). We define he esol en ma ix R(n, m) o he equa ion (2.3) x(n+1)=[A+B(n)]x(n)+ n  =0 [K(n− )+Q(n, )]x( )+ (n) as he unique solu ion o he ma ix diffe ence equa ion [7] (2.4) R(n+1,m)=[A+B(n)]R(n, m) + n  =m [K(n− )+Q(n, )]R( , m),n≥m, Asymp o ic equi alence o Vol e a di e ence sys ems 303 wi h R(m, m)=I. Using he esol en ma ix R(n, m) we can es ablish he ollowing e- la ion [7] (Va ia ion o Cons an s Fo mula) (2.5) x(n, 0,x 0)=R(n, 0)x0+ n−1  =0 R(n, +1) ( ), whe e x(n, 0,x 0) is he unique solu ion o he equa ion (2.3) sa is ying x(0,0,x 0)=x0. Le Y(n) deno e he undamen al ma ix o he sys em (2.1’) [7]. No- ice ha Y(0) = Iand y(n, 0,y 0)=Y(n)y0is he unique solu ion o (2.1’) wi h y(0,0,y 0)=y0. Mo eo e , (2.6) Y(n+1)=AY (n)+ n  =0 K(n− )Y( ). Rema k [7]. We ema k he e ha he esol en ma ix R(n, m) o equa ions o noncon olu ion ype is closely ela ed o he undamen- al ma ix Y(n). By uniqueness o solu ions, i is easy o see ha o equa ions o con olu ions ype such as (2.1’), R(n, 0) = Y(n) and R(n, m)=Y(n−m). In his pape we conside he no ion o asymp o ic equi alence gi en by, Defini ion. We say ha he equa ions (2.1) and (2.1’) o (2.2), (2.2’) a e asymp o ically equi alen i , co esponding o each solu ion x=x(n) o (2.1), ((2.2)), he e exis s a solu ion y=y(n) o (2.1’), ((2.2’)) wi h he p ope y (2.7) lim[x(n)−y(n)] = 0 as n→∞and con e sely. 3. Asymp o ic equi alence We s a e he ollowing lemma. Lemma 3.1. I 1. ϕ(n)is bounded on N(n0)and lim n→∞ ϕ(n)=ϕ(∞)exis s, 2. ∞  k=n0 |g(k)|<∞, 304 J. Mo chalo hen lim n→∞ n  k=n0 ϕ(n−k)g(k)=ϕ(∞) ∞  k=n0 g(k). Theo em 3.2. Assume ha 1. all solu ions o he sys em (2.1’) end o fini e limi s as n→∞, 2. ∞  =0 |B( )|+  s=0 |Q( , s)|<∞, 3. de P=0, whe e P= lim Y(n)as n→∞,Pis a cons an ma ix, 4. q= ∞  =0 |B( )||R( , 0)|+  s=0 |Q( , s)||R(s, 0)|<1. Then, a) co esponding o each solu ion x=x(n)∈Φ1o (2.1), he e exis s a solu ion y=y(n)∈Φ1o (2.1’) such ha (2.7) is sa isfied p o ided ha Condi ions 1, 2 hold, b) in Rela ion (2.7) he solu ion y=y(n)o (2.1’) is unique i Con- di ions 1, 2 and 3 a e sa isfied, c) o each non-ze o solu ion x=x(n)∈Φ1o (2.1) he e co esponds in Rela ion (2.7) a non-ze o solu ion y=y(n)∈Φ1o (2.1’), i Condi ions 1, 2 and 4 hold and con e sely, d) in Rela ion (2.7) he solu ion x=x(n)o (2.1) is unique i Con- di ions 1, 2, 3 and 4 a e sa isfied. P oo : By Fo mula (2.5) he solu ions x(n) o (2.1) and y(n) o (2.1’) can be w i en as (3.1) x(n)=Y(n)x0+ n−1  =0 Y(n− −1) B( )x( )+  s=0 Q( , s)x(s) and (3.2) y(n)=Y(n)y0,n∈N. Fu he mo e, om he Rela ions (3.1) and (3.2) we ob ain (3.3) x(n)−y(n)=Y(n)[x0−y0] + n−1  =0 Y(n− −1) B( )x( )+  s=0 Q( , s)x(s). Asymp o ic equi alence o Vol e a di e ence sys ems 305 F om Assump ions 1, 2 and (3.1) we ob ain |x(n)|≤|Y(n)||x0| + n−1  =0 |Y(n− −1)||B( )||x( )|+  s=0 |Q( , s)||x(s)|. Hence and diffe ence inequali y [9] we can easily ob ain ha all solu ions o (2.1) a e bounded. Thus (3.4) ∞  n=0  B( )x( )+  s=0 Q( , s)x(s) <∞. By Assump ion 1, Lemma 3.1 and Rela ions (3.3) (3.4) we ge (3.5) lim n→∞[x(n)−y(n)] =Px0−y0+ ∞  n=0 B(n)x(n)+ n  s=0 Q(n, s)x(s). This shows ha o a bi a y solu ions x(n) and y(n) o (2.1), (2.1’) espec i ely Rela ion (2.7) hold iff (3.6) Px0−y0+ ∞  n=0 B(n)x(n)+ n  s=0 Q(n, s)x(s)=0. Equali y (3.6) defines a ela ion be ween all solu ions x(n), y(n)o (2.1), (2.1’), espec i ely, o which (2.7) holds. I P= 0, hen (3.6) means ha (2.7) holds o a bi a y solu ions x(n), y(n) o (2.1), (2.1’) espec i ely. On he o he hand i P= 0, hen o a bi a y solu ion x(n) o (2.1) we ha e (3.7) y0=x0+ ∞  n=0 B(n)x(n)+ n  s=0 Q(n, s)x(s). Hence o sui able solu ion y(n) o (2.1’) we conclude ha (2.7) holds. Since Condi ion 3 holds, we claim ha he solu ion y(n) wi h he ini ial condi ion y0defined by (3.7) is unique in (2.7). 306 J. Mo chalo F om (2.5) o (n) = 0 and (3.7) we ha e y0=x0I+ ∞  n=0 B(n)R(n, 0) + n  s=0 Q(n, s)R(s, 0) o (3.8) (I+P0)x0=y0 whe e P0= ∞  n=0 B(n)R(n, 0) + n  s=0 Q(n, s)R(s, 0). Assume ha (I+P0)−1exis s and x(n)= 0 o n∈N(x0= 0). Then, by (3.8), we ha e y0=0(y(n)= 0 o n∈N). Such a ma ix exis s i , o example, |P0|<1[12] (Banach Theo em’s). F om Assump ion 4 i ollows ha |P0|<1. Le he ini ial condi ion y0o solu ion y(n)=y(n, 0,y 0) be a bi a y. F om (3.8) we ob ain x0=(I+P0)−1y0. Hence o e e y solu ion y(n)=0onN he e exis s a unique solu ion x(n)=0onNsuch ha (2.7) holds and con e sely. Rema k. I all solu ions o (2.1’) ends o ze o as n→∞and Con- di ion 2 hold, hen all solu ions o (2.1) end o ze o as n→∞. Now, we conside asymp o ic equi alence be ween Equa ions (2.2) and (2.2’). Lemma 3.3. Suppose ha he ollowing condi ions hold: 1. e e y solu ion o (2.2’) is bounded on N, 2. |F(n, x1)−F(n, x2)|≤g(n)||x1−x2| o n∈N,x1,x2∈U, 3. ∞  n=0 g(n)<∞and ∞  n=0 |F0(n)|<∞whe e F0(n)≡F(n, 0). Then e e y solu ion o (2.2) is bounded on Nand (3.9) |x(n)|≤LM(n),n∈N Asymp o ic equi alence o Vol e a di e ence sys ems 307 whe e Y0= sup N |Y(n)|, L= sup N y0(n)+ n−1  =0 Y(n− −1)F0( ) <∞, M(n) = exp Y0 n−1  =0 g( ), y0(n)is a solu ion o (2.2’). P oo : By he o mula (2.5) he solu ion o (2.2) can be w i en as (3.10) x(n, 0,x 0)=y0(n)+ n−1  =0 Y(n− −1)F( , x( )) whe e y0(n)=Y(n)x0+ n−1  =0 Y(n− −1) ( ),n∈N. Fu he mo e, i ollows om (2.5) and in iew 1 ha all solu ions o (2.1’) a e bounded on N. Now, using he Rela ion (3.10) and he Condi ion 2, we ge (3.11) |x(n)|≤L+Y0 n−1  =0 g( )|x( )|, which implies, by G onwall inequali y |x(n)|≤Lexp Y0 n−1  =0 g( )=LM(n). Because he unc ion M1(n) is bounded on N, we can conclude ha he solu ion x(n) o (2.2) is also bounded on N. Rema k. F om (3.9), we ha e (3.12) |x(n)|≤[Y0|x0|+ 0+Y0F1]M(n) whe e 0= sup N n−1  =0 Y(n− −1) ( ) <∞, F1= ∞  n=0 |F0(n)|<∞. 308 J. Mo chalo Theo em 3.4. Le 1. all solu ions o he sys em (2.1’) end o fini e limi s as n→∞, 2. Condi ions 2 and 3 o lemma 3.3 hold, 3. sup n∈Nn−1  =0 |Z1(n, )||g( )|+ ∞  =n |Z2(n, )||g( )|<1whe e Z1(n, )=Y(n− −1) −Y(n),Z 2(n, )=−Y(n). Then o each solu ion x(n)o (2.2) he e co esponds a solu ion y(n)o (2.2’) such ha (2.7) holds. Mo eo e , suppose ha Condi ion 3 o Theo em 3.2 holds. Then he solu ion y(n)o (2.2’) in Rela ion (2.7) is unique. Le Condi ions 1-3 hold and 4. q1=Y0 ∞  n=0 g(n)M1(n)<1. Then o each solu ion o (2.2) wi h x0=0and |x0|>(1 −q1)−1[F2+q1(F1+Y−1 0 0)] whe e F2= ∞  n=0 F0(n) <∞ he e co esponds a solu ion y(n)o (2.2’) wi h y0=0such ha (2.7) holds and con e sely. I , in addi ion he Condi ion 3 o Theo em 3.2 holds, hen he solu ion x(n)o (2.2) in Rela ion (2.7) is unique. P oo : By 1 i ollows ha solu ions o (2.2’) a e bounded on N. Then, he bounded p ope ies o solu ions o (2.2’) imply ha solu ions o (2.1’) a e bounded oo. The fi s wo pa s o Theo em a e easily e ified (see Theo em 3.2 and Lemma 3.3). Since P= 0, hen we can find a ini ial condi ion y0o he solu ion y(n) o (2.2’) such ha (3.13) y0=x0+ ∞  n=0 F(n, x(n)), whe e x(n) is a gi en solu ion o (2.2). Asymp o ic equi alence o Vol e a di e ence sys ems 309 Le x0= 0, hen om (3.13), (3.12) we ha e (3.14) |y0|≥|x0|− ∞  n=0 F(n, x(n)) =|x0|− ∞  n=0 F(n, x(n)) −F(n, 0)] + ∞  n=0 F(n, 0) ≥|x0|− ∞  n=0 |F(n, x(n)) −F0(n)|−F2 ≥|x0|− ∞  n=0 g(n)M1(n)[Y0|x0|+ 0+Y0F1]−F2 ≥|x0|(1 −q1)−[(F1+Y−1 0 0)q1+F2]>0. Hence y0=0. Le he ini ial condi ion y0o he solu ion y(n) o (2.2’) be a bi a y selec ion. Then by Condi ions 1, 2 and (3.13) he solu ion x(n) o (2.2) be defined o all n∈Nand (2.7) be hold. By his means we gi e some condi ions o exis ence and uniqueness o he solu ion x(n) o (2.2) in Φ1which sa isfied (3.13). Since he equa ion (2.2) wi h ini ial condi ion x0is equi alen o he equa ion (3.10), hen subs i u ing o x0 om (3.13) in o (3.10) (3.14) x(n)=y(n)+ n−1  =0 Z1(n, )F( , x( )) + ∞  =n Z2(n, )F( , x( )) whe e y(n)=Y(n)y0+ n−1  =0 Y(n− −1) ( ) is a bi a y solu ion o (2.2’), Z1(n, )=Y(n− −1) −Y(n) Z2(n, )=−Y(n). Le Tbe he ope a o defined o each x∈Φ1by he equa ion Tx(n)= n−1  =0 Z1(n, )F( , x( )) + ∞  =0 Z2(n, )F( , x( )).