Asymptotic equivalence of Volterra difference systems

Publicacions Matem`atiques, Vol 39 (1995), 301–312. ASYMPTOTIC EQUIVALENCE OF VOLTERRA DIFFERENCE SYSTEMS J. Morchalo Abstract The purpose of this paper is to give some results on the asymptotic relationship between the solutions of a linear difference equation and its perturbed nonlinear equation. 1. Introduction The problem of the asymptotic equivalence for systems of ordinary differential equations has been studied by many authors, as e.g. Brauer [3], Brauer and Wong [4], Boundorides and Georgiou [2], Lowell Lovelady [11], Morcha#lo [13], ˘ Svec [17], Szufla [18], and others. The problem of the asymptotic equivalence for integrodifferential equations has been studied by Morcha#lo [14], Razapov [16], Talpalaru [19]. The problem of the asymptotic behavior of solutions of ordinary difference equations has been studied by Benzaid [1], Conffman [5], Drozdowicz, Popenda [6], Elaydi, Gyori [8], Li [10] and Pinto [15]. In this paper, we shall consider some results on the asymptotic relationship between the solutions of a linear Volterra difference equation and its perturbed nonlinear equation. The author knows only the works of Talpalaru [19], Ved and Go#lovina [21], Ved and Kaptagaev [20], dealing with the above problem for special case. 2. Notations and Definitions Here N(n0)={n0,n 0+1,...}, where n0is a natural number or zero; Rkthe kdimensional real euclidean space with the norm |x|= k  i=1 |xi|,x=(x1,... ,x k); 302 J. Morchalo Mkthe space of all k×kmatrices D=(dij) with the norm |D|= max j k  i=1 |dij|,Iidentity matrix. We denote by Φ(N,Rk) the space of all functions from N(n0)intoRk. Let Φ1=Φ 1(N,Rk) be the Banach space in Φ of all bounded functions u:N(n0)→Rkwith norm x=|x(n)|Φ1= sup{|x(n)|:n∈N(n0)}. In this paper we consider the following systems of difference equations (2.1) x(n+1)=[A+B(n)]x(n)+ n  r=0 [K(n−r)+Q(n, r)]x(r), (2.2) x(n+1)=Ax(n)+ n  r=0 K(n−r)x(r)+f(n)+F(n, x(n)) thought et as perturbations of y(n+1)=Ay(n)+ n  r=0 K(n−r)y(r),(2.1’) y(n+1)=Ay(n)+ n  r=0 K(n−r)y(r)+f(n)(2.2’) where x,y,fare k-dimensional vectors, Ais a constant matrix k×k, B, K :N(n0)→Mk Q:N(n0)×N(n0)→Mk, F:N(n0)×U→Rkis for any n∈N(n0) continuous as a function of x∈U (Ua region in Rk). We define the resolvent matrix R(n, m) of the equation (2.3) x(n+1)=[A+B(n)]x(n)+ n  r=0 [K(n−r)+Q(n, r)]x(r)+f(n) as the unique solution of the matrix difference equation [7] (2.4) R(n+1,m)=[A+B(n)]R(n, m) + n  r=m [K(n−r)+Q(n, r)]R(r, m),n≥m, Asymptotic equivalence of Volterra difference systems 303 with R(m, m)=I. Using the resolvent matrix R(n, m) we can establish the following relation [7] (Variation of Constants Formula) (2.5) x(n, 0,x 0)=R(n, 0)x0+ n−1  r=0 R(n, r +1)f(r), where x(n, 0,x 0) is the unique solution of the equation (2.3) satisfying x(0,0,x 0)=x0. Let Y(n) denote the fundamental matrix of the system (2.1’) [7]. Notice that Y(0) = Iand y(n, 0,y 0)=Y(n)y0is the unique solution of (2.1’) with y(0,0,y 0)=y0. Moreover, (2.6) Y(n+1)=AY (n)+ n  r=0 K(n−r)Y(r). Remark [7]. We remark here that the resolvent matrix R(n, m) for equations of nonconvolution type is closely related to the fundamental matrix Y(n). By uniqueness of solutions, it is easy to see that for equations of convolutions type such as (2.1’), R(n, 0) = Y(n) and R(n, m)=Y(n−m). In this paper we consider the notion of asymptotic equivalence given by, Definition. We say that the equations (2.1) and (2.1’) or (2.2), (2.2’) are asymptotically equivalent if, corresponding to each solution x=x(n) of (2.1), ((2.2)), there exists a solution y=y(n) of (2.1’), ((2.2’)) with the property (2.7) lim[x(n)−y(n)] = 0 as n→∞and conversely. 3. Asymptotic equivalence We state the following lemma. Lemma 3.1. If 1. ϕ(n)is bounded on N(n0)and lim n→∞ ϕ(n)=ϕ(∞)exists, 2. ∞  k=n0 |g(k)|<∞, 304 J. Morchalo then lim n→∞ n  k=n0 ϕ(n−k)g(k)=ϕ(∞) ∞  k=n0 g(k). Theorem 3.2. Assume that 1. all solutions of the system (2.1’) tend to finite limits as n→∞, 2. ∞  r=0 |B(r)|+ r  s=0 |Q(r, s)|<∞, 3. det P=0, where P= lim Y(n)as n→∞,Pis a constant matrix, 4. q= ∞  r=0 |B(r)||R(r, 0)|+ r  s=0 |Q(r, s)||R(s, 0)|<1. Then, a) corresponding to each solution x=x(n)∈Φ1of (2.1), there exists a solution y=y(n)∈Φ1of (2.1’) such that (2.7) is satisfied provided that Conditions 1, 2 hold, b) in Relation (2.7) the solution y=y(n)of (2.1’) is unique if Conditions 1, 2 and 3 are satisfied, c) to each non-zero solution x=x(n)∈Φ1of (2.1) there corresponds in Relation (2.7) a non-zero solution y=y(n)∈Φ1of (2.1’), if Conditions 1, 2 and 4 hold and conversely, d) in Relation (2.7) the solution x=x(n)of (2.1) is unique if Conditions 1, 2, 3 and 4 are satisfied. Proof: By Formula (2.5) the solutions x(n) of (2.1) and y(n) of (2.1’) can be written as (3.1) x(n)=Y(n)x0+ n−1  r=0 Y(n−r−1) B(r)x(r)+ r  s=0 Q(r, s)x(s) and (3.2) y(n)=Y(n)y0,n∈N. Furthermore, from the Relations (3.1) and (3.2) we obtain (3.3) x(n)−y(n)=Y(n)[x0−y0] + n−1  r=0 Y(n−r−1) B(r)x(r)+ r  s=0 Q(r, s)x(s). Asymptotic equivalence of Volterra difference systems 305 From Assumptions 1, 2 and (3.1) we obtain |x(n)|≤|Y(n)||x0| + n−1  r=0 |Y(n−r−1)||B(r)||x(r)|+ r  s=0 |Q(r, s)||x(s)|. Hence and difference inequality [9] we can easily obtain that all solutions of (2.1) are bounded. Thus (3.4) ∞  n=0  B(r)x(r)+ r  s=0 Q(r, s)x(s) <∞. By Assumption 1, Lemma 3.1 and Relations (3.3) (3.4) we get (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 that for arbitrary solutions x(n) and y(n) of (2.1), (2.1’) respectively Relation (2.7) hold iff (3.6) Px0−y0+ ∞  n=0 B(n)x(n)+ n  s=0 Q(n, s)x(s)=0. Equality (3.6) defines a relation between all solutions x(n), y(n)of (2.1), (2.1’), respectively, for which (2.7) holds. If P= 0, then (3.6) means that (2.7) holds for arbitrary solutions x(n), y(n) of (2.1), (2.1’) respectively. On the other hand if P= 0, then for arbitrary solution x(n) of (2.1) we have (3.7) y0=x0+ ∞  n=0 B(n)x(n)+ n  s=0 Q(n, s)x(s). Hence for suitable solution y(n) of (2.1’) we conclude that (2.7) holds. Since Condition 3 holds, we claim that the solution y(n) with the initial condition y0defined by (3.7) is unique in (2.7). 306 J. Morchalo From (2.5) for f(n) = 0 and (3.7) we have y0=x0I+ ∞  n=0 B(n)R(n, 0) + n  s=0 Q(n, s)R(s, 0) or (3.8) (I+P0)x0=y0 where P0= ∞  n=0 B(n)R(n, 0) + n  s=0 Q(n, s)R(s, 0). Assume that (I+P0)−1exists and x(n)= 0 for n∈N(x0= 0). Then, by (3.8), we have y0=0(y(n)= 0 for n∈N). Such a matrix exists if, for example, |P0|<1[12] (Banach Theorem’s). From Assumption 4 it follows that |P0|<1. Let the initial condition y0of solution y(n)=y(n, 0,y 0) be arbitrary. From (3.8) we obtain x0=(I+P0)−1y0. Hence for every solution y(n)=0onNthere exists a unique solution x(n)=0onNsuch that (2.7) holds and conversely. Remark. If all solutions of (2.1’) tends to zero as n→∞and Condition 2 hold, then all solutions of (2.1) tend to zero as n→∞. Now, we consider asymptotic equivalence between Equations (2.2) and (2.2’). Lemma 3.3. Suppose that the following conditions hold: 1. every solution of (2.2’) is bounded on N, 2. |F(n, x1)−F(n, x2)|≤g(n)||x1−x2|for n∈N,x1,x2∈U, 3. ∞  n=0 g(n)<∞and ∞  n=0 |F0(n)|<∞where F0(n)≡F(n, 0). Then every solution of (2.2) is bounded on Nand (3.9) |x(n)|≤LM(n),n∈N Asymptotic equivalence of Volterra difference systems 307 where Y0= sup N |Y(n)|, L= sup N y0(n)+ n−1  r=0 Y(n−r−1)F0(r) <∞, M(n) = exp Y0 n−1  r=0 g(r), y0(n)is a solution of (2.2’). Proof: By the formula (2.5) the solution of (2.2) can be written as (3.10) x(n, 0,x 0)=y0(n)+ n−1  r=0 Y(n−r−1)F(r, x(r)) where y0(n)=Y(n)x0+ n−1  r=0 Y(n−r−1)f(r),n∈N. Furthermore, it follows from (2.5) and in view 1 that all solutions of (2.1’) are bounded on N. Now, using the Relation (3.10) and the Condition 2, we get (3.11) |x(n)|≤L+Y0 n−1  r=0 g(r)|x(r)|, which implies, by Gronwall inequality |x(n)|≤Lexp Y0 n−1  r=0 g(r)=LM(n). Because the function M1(n) is bounded on N, we can conclude that the solution x(n) of (2.2) is also bounded on N. Remark. From (3.9), we have (3.12) |x(n)|≤[Y0|x0|+f0+Y0F1]M(n) where f0= sup N n−1  r=0 Y(n−r−1)f(r) <∞, F1= ∞  n=0 |F0(n)|<∞. 308 J. Morchalo Theorem 3.4. Let 1. all solutions of the system (2.1’) tend to finite limits as n→∞, 2. Conditions 2 and 3 of lemma 3.3 hold, 3. sup n∈Nn−1  r=0 |Z1(n, r)||g(r)|+ ∞  r=n |Z2(n, r)||g(r)|<1where Z1(n, r)=Y(n−r−1) −Y(n),Z 2(n, r)=−Y(n). Then for each solution x(n)of (2.2) there corresponds a solution y(n)of (2.2’) such that (2.7) holds. Moreover, suppose that Condition 3 of Theorem 3.2 holds. Then the solution y(n)of (2.2’) in Relation (2.7) is unique. Let Conditions 1-3 hold and 4. q1=Y0 ∞  n=0 g(n)M1(n)<1. Then for each solution of (2.2) with x0=0and |x0|>(1 −q1)−1[F2+q1(F1+Y−1 0f0)] where F2= ∞  n=0 F0(n) <∞ there corresponds a solution y(n)of (2.2’) with y0=0such that (2.7) holds and conversely. If, in addition the Condition 3 of Theorem 3.2 holds, then the solution x(n)of (2.2) in Relation (2.7) is unique. Proof: By 1 it follows that solutions of (2.2’) are bounded on N. Then, the bounded properties of solutions of (2.2’) imply that solutions of (2.1’) are bounded too. The first two parts of Theorem are easily verified (see Theorem 3.2 and Lemma 3.3). Since P= 0, then we can find a initial condition y0of the solution y(n) of (2.2’) such that (3.13) y0=x0+ ∞  n=0 F(n, x(n)), where x(n) is a given solution of (2.2). Asymptotic equivalence of Volterra difference systems 309 Let x0= 0, then from (3.13), (3.12) we have (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|+f0+Y0F1]−F2 ≥|x0|(1 −q1)−[(F1+Y−1 0f0)q1+F2]>0. Hence y0=0. Let the initial condition y0of the solution y(n) of (2.2’) be arbitrary selection. Then by Conditions 1, 2 and (3.13) the solution x(n) of (2.2) be defined for all n∈Nand (2.7) be hold. By this means we give some conditions for existence and uniqueness of the solution x(n) of (2.2) in Φ1which satisfied (3.13). Since the equation (2.2) with initial condition x0is equivalent to the equation (3.10), then substituting for x0from (3.13) into (3.10) (3.14) x(n)=y(n)+ n−1  r=0 Z1(n, r)F(r, x(r)) + ∞  r=n Z2(n, r)F(r, x(r)) where y(n)=Y(n)y0+ n−1  r=0 Y(n−r−1)f(r) is arbitrary solution of (2.2’), Z1(n, r)=Y(n−r−1) −Y(n) Z2(n, r)=−Y(n). Let Tbe the operator defined for each x∈Φ1by the equation Tx(n)= n−1  r=0 Z1(n, r)F(r, x(r)) + ∞  r=0 Z2(n, r)F(r, x(r)).