We show that a class of linear nonconvolution discrete Volterra equations has asymptotically periodic solutions. We also examine an example for which the calculations can be done explicitly. The results are established using theorems on the boundedness and convergence to a finite limit of solutions  of linear discrete Volterra equations

Győri, István

Reynolds, David W.

English

DCU Online Research Access Service

ON ASYMPTOTICALLY PERIODIC SOLUTIONS OFLINEAR DISCRETE VOLTERRA EQUATIONSISTVA´N GYO˝RI AND DAVID W. REYNOLDSAbstract. We show that a class of linear nonconvolution discreteVolterra equations has asymptotically periodic solutions. We alsoexamine an example for which the calculations can be done explic-itly. The results are established using theorems on the bounded-ness and convergence to a finite limit of solutions of linear discreteVolterra equations.1. IntroductionThe problem of finding periodic and asymptotically periodic solu-tions of linear discrete Volterra equations has been investigated in sev-eral papers, including [3, 4, 6, 7, 8]. Each of these considers the equationu(n+ 1) = A(n)u(n) +n∑j=0B(n, j)u(j) + e(n),and perturbations thereof, under the assumptions thatA(n+N) = A(n), e(n+N) = e(n),B(n+N,m+N) = B(n,m),}n ∈ Z+,for some integer N > 0.In this paper we study the asymptotic behaviour of solutions of theinitial-value problemx(n+ 1) = A(n)x(n) +n∑j=0K(n, j)x(j) + f(n), n ≥ 0, (1)assuming that the periodicity conditionA(n+N) = A(n), n ∈ Z+,holds for some integer N > 0. Here (1) is viewed as a perturbation ofu(n+ 1) = A(n)u(n).Date: 31 December 2009.2010 Mathematics Subject Classification. Primary 39A06, 39A23 Secondary39A22.Key words and phrases. Volterra difference equation, asymptotically periodicsolutions, asymptotic equilibria.David Reynolds thanks John Appleby for stimulating discussions on this topic.12 ISTVA´N GYO˝RI AND DAVID W. REYNOLDSOur method is to use a variation of constants to transform (1) into anequation of the formz(n+ 1) = h(n) +n∑j=0H(n, j)z(j), n ∈ Z+,and use results in [1, 5] on the boundedness and asymptotic constancyof solutions. Our main result establishes the existence of asymptoticallyperiodic solutionsIn the case of A(n) and K(n, j) being 2× 2 real matrices, a differentresult on the existence of asymptotically periodic solutions of (9) wasrecently proved in [2] using fixed point arguments. We analyse fully anexample found in [2] to show that our result can be less restrictive.2. PreliminariesFirstly we collect together some notation.Let Z+ = {0, 1, 2, . . .} denote the set of non-negative integers, andK either of the fields R or C. Kd×d is the space of all d × d matriceswith entries in K, and the zero and identity matrices are denoted by0 and I respectively. ‖A‖ = (∑dj=1∑di=1 |Aij|2)1/2 is the Euclideannorm of A = (Aij). Kd×d can be endowed with many norms, but theyare all equivalent. The absolute value of A is the matrix |A| defined by(|A|)ij = |Aij| for all 1 ≤ i ≤ d and 1 ≤ j ≤ d. The matrix A = (Aij) inRd×d is nonnegative if Aij ≥ 0, in which case we write A ≥ 0. A partialordering is defined on Kd×d by letting A ≤ B if and only if B−A ≥ 0,which is equivalent to Aij ≤ Bij for all 1 ≤ i ≤ d and 1 ≤ j ≤ d. Thespectral radius of a matrix A is defined by ρ(A) = limn→∞ ‖An‖1/n,and is independent of the norm used: it equals the maximum of theabsolute values of the eigenvalues of A. Note that ρ(A) ≤ ρ(|A|), andρ(A) ≤ ρ(B) if 0 ≤ A ≤ B.Let H : Z+ × Z+ → Kd×d be a double sequence of matrices withH(n, j) = 0 for all j > n, and h : Z+ → Kd a sequence of vectors. Foreach ξ ∈ Kd, there is a unique solution z : Z+ → Kd of the explicitdiscrete Volterra equationz(n+ 1) = h(n) +n∑j=0H(n, j)z(j), n ∈ Z+, (2)which satisfies the initial conditionz(0) = ξ. (3)Sufficient conditions for the solution of (2) and (3) to be boundedcan be found in [1, Theorem 5.1]. We state a variant of it.ASYMPTOTICALLY PERIODIC SOLUTIONS 3Lemma 1. Suppose thatWH := limm→∞lim supn→∞n∑j=m|H(n, j)| <∞, ρ(WH) < 1, (4)supn≥j|H(n, j)| <∞ for each j ≥ 0. (5)If h is bounded, then z is also bounded.Sufficient conditions for the solution of (2) and (3) to converge toa finite limit are given in [5, Theorem 3.1], a variant of which is nowgiven.Lemma 2. Suppose that (4) holds, and thatH(n, k)→ H∞(k) as n→∞. (6)Assume also that there is VH in Kd×d such thatlimm→∞lim supn→∞∣∣∣∣ n∑j=mH(n, j)− VH∣∣∣∣ = 0, (7)and h(n)→ h(∞) as n→∞. Then z(n)→ z(∞) as n→∞, and thislimit obeysz(∞) = (I − VH)−1[h(∞) +∞∑j=0H∞(j)z(j)]. (8)The hypotheses of this theorem ensure that all the terms in (8) arewell-defined. Since|VH | ≤∣∣∣∣VH − n∑j=mH(n, j)∣∣∣∣+ n∑j=m|H(n, j)|,we see that |VH | ≤ WH by firstly taking the limit superior as n→∞,followed by the limit superior as m → ∞. Hence ρ(VH) ≤ ρ(|VH |) ≤ρ(WH) < 1. Therefore I − VH is invertible and (I − VH)−1 ≥ 0.Also for large enough m ∈ Z+,lim supn→∞n∑j=0|H(n, j)| =m−1∑j=0|H∞(j)|+ lim supn→∞n∑j=m|H(n, j)|is certainly finite. Hencek∑j=0|H∞(j)| = lim supn→∞k∑j=0|H(n, j)| ≤ lim supn→∞n∑j=0|H(n, j)|,is uniformly bounded for k ∈ Z+, and the series ∑∞j=0 |H∞(j)| is sum-mable. Therefore if z is bounded,∣∣∣∣ ∞∑j=0H∞(j)z(j)∣∣∣∣ ≤ ∞∑j=0|H∞(j)| supj≥0|z(j)| <∞.4 ISTVA´N GYO˝RI AND DAVID W. REYNOLDS3. Formulation of ProblemWe study the asymptotic behaviour of the solution x(·; ξ) : Z+ →Kd×d of the initial-value problemx(n+ 1; ξ) = A(n)x(n; ξ) +n∑j=0K(n, j)x(j; ξ) + f(n), n ≥ 0, (9a)x(0; ξ) = ξ, (9b)assuming that the periodicity conditionA(n+N) = A(n), n ∈ Z+, (10)holds for some integer N > 0.We regard (9) as a perturbation ofu(n+ 1) = A(n)u(n),u(0) = ξ,}(11)which has solutionu(n) = Φ(n)ξ, n ≥ 0,where Φ : Z→ Kd×d is the principal matrix solution defined byΦ(n+ 1) = A(n)Φ(n), n ≥ 0,Φ(0) = I.}The matrix Φ(N) is termed a monodromy matrix, and its eigenvaluesare called Floquet multipliers.The following standing hypotheses are assumed to hold throughoutthe remainder of the paper.(A1) A : Z+ → Kd×d is a sequence of invertible matrices, whichsatisfies the periodicity condition (10) for some positive integerN .(A2) The monodromy matrix Φ(N) is assumed to satisfyΦ(N) = I (12)(A3) K : Z+ × Z+ → Kd×d is a double sequence of matrices withK(n, j) = 0 for all j > n, and∞∑n=0n∑j=0|K(n, j)| <∞. (13)(A4) f : Z+ → Kd is a sequence with∞∑j=0|f(j)| <∞. (14)ASYMPTOTICALLY PERIODIC SOLUTIONS 5Remark 1. (12) is equivalent toΦ(n+N) = Φ(n), n ∈ Z+. (15)Due to this periodicity,Φ(n) = Φ(n− [ nN]N), n ≥ 0, (16)where [·] is the greatest integer part function, though it is cumbersometo employ this expression in succeeding formula.4. Main Results and DiscussionOur main result clearly exhibits the asymptotic behaviour of all so-lutions of (9).The theorem refers to a matrix M and vector m, each of which isdefined to be the limit of the solution of a discrete Volterra equation.Let Y : Z+ → Kd×d be the solution ofY (n+ 1) = I +n∑j=0n∑k=jΦ(k + 1)−1K(k, j)Φ(j)Y (j),Y (0) = I.Then it is proved in Section 5 thatM = limn→∞Y (n) (17)exists. Clearly M is independent of the initial-value ξ. Also let w :Z+ → Kd be the solution ofw(n+ 1) =n∑j=0n∑k=jΦ(k + 1)−1K(k, j)Φ(j)w(j)+n∑k=0Φ(k + 1)−1f(k),which satisfies w(0) = 0. It is also shown that in Section 5 thatm = limn→∞w(n) (18)exists. Observe that because {w(n)} is independent of ξ, m is inde-pendent of ξ. In Section 6 an example is given in which M and m arecalculated explicitly.Theorem 1. Suppose that (A1)-(A4) hold, and let M be defined by(17) and m by (18). Then for every ξ ∈ Kd, the solution of (9) satisfiesx(n; ξ) = Φ(n)[Mξ +m] + v(n; ξ), n ∈ Z+, (19)where v(n; ξ)→ 0 as n→∞.6 ISTVA´N GYO˝RI AND DAVID W. REYNOLDSThe sequence {x(n; ξ)} is asymptotically periodic if there is a non-trivial periodic {p(n)} such thatx(n; ξ) = p(n) + o(1) as n→∞.Under the conditions of Theorem 1, x(·; ξ) is asymptotically periodicif and only if Mξ +m 6= 0. There are three cases:(a) M = 0 andm = 0, implying that {x(n; ξ)} is not asymptoticallyperiodic; the zero solution is a global attractor.(b) M = 0 but m 6= 0, implying that {x(n; ξ)} is asymptoticallyperiodic with respect to the {Φ(n)m} for all initial values.(c) If M 6= 0 and c − m is in the range of M , then {x(n; ξ)}is asymptotically periodic with respect to the {Φ(n)c} for allinitial-values obeying Mξ +m = c.It is clearly desirable to know what are the null space and range ofthe linear mapping associated with M .In order to compare our result with [2, Theorem 1], we state thefollowing.Corollary 1. Suppose that {a1(n)} and {a2(n)} are N-periodic se-quences in K \ {0}, satisfyingN−1∏j=0as(j) = 1, s = 1, 2. (20)LetA(n) =(a1(n) 00 a2(n)), n ∈ Z+, (21)and suppose that (A3)-(A4) hold with d = 2. Then the solution x(·, ξ)of (9) obeys (19).Note that in this caseΦ(n) =(∏n−1j=0 a1(j) 00∏n−1j=0 a2(j)), n ∈ Z+.There are important differences between [2, Theorem 1] and Corol-lary 1. Firstly the conclusion of [2, Theorem 1] says that for each c ina precisely defined region of R2, there is a solution of (9) such thatx(n; ξ) = Φ(n)c+ o(1) as n→∞. (22)Secondly, additional hypotheses to those in Corollary 1 are required in[2] to prove Theorem 1.In Section 6 an example is discussed for which M and m are explicitlycalculated. In the example M is invertible, and hence for each c ∈ R2,there is an initial-value ξ = M−1(c − m) for which (22) holds; alsox(·; ξ) is asymptotically periodic as long as ξ 6= −M−1m. This showsthat conditions on the vector c in [2, Theorem 1], though sufficient, arenot necessary for the existence of asymptotically periodic solutions.ASYMPTOTICALLY PERIODIC SOLUTIONS 75. Proof of Main TheoremWe define z(·; ξ) : Z+ → Kd byz(n; ξ) = Φ(n)−1x(n; ξ), n ≥ 0, (23)and investigate the auxiliary problem it satisfies.Lemma 3. x is a solution of (9) if and only if z solvesz(n+ 1; ξ) = ξ + f ∗(n) +n∑j=0H(n, j)z(j; ξ), (24a)z(0; ξ) = ξ, (24b)whereH(n, j) ={∑nk=j Φ(k + 1)−1K(k, j)Φ(j), 0 ≤ j ≤ n,0, j > n,(25)f ∗(n) =n∑k=0Φ(k + 1)−1f(k), n ≥ 0. (26)Proof. Clearly x(0; ξ) = ξ if and only if z(0; ξ) = Φ(0)−1x(0; ξ) = ξ.By substituting (23) into (9a) , we obtainz(n+1; ξ) = z(n; ξ)+n∑j=0Φ(n+1)−1K(n, j)Φ(j)z(j; ξ)+Φ(n+1)−1f(n).A variation of constants formula is easily obtained by observing thatz(n+ 1; ξ)− z(0; ξ) =n∑k=0{z(k + 1; ξ)− z(k; ξ)}=n∑j=0( n∑k=jΦ(k + 1)−1K(k, j)Φ(j))z(j; ξ)+ f ∗(n).Hence we obtain the equation in (24a). The converse is proved similarly.It is important to know what is the asymptotic behaviour of {f ∗(n)}.Lemma 4. f ∗(n)→ f ∗(∞) as n→∞, wheref ∗(∞) :=N−1∑j=0Φ(j)−1∞∑i=0f(iN + j − 1). (27)Proof. Recall that f : Z+ → Kd is in `1. Extending f to {−1, 0, 1, . . .}by setting f(−1) = 0, it follows from (26) thatf ∗(n) =n+1∑k=0Φ(k)−1f(k − 1),8 ISTVA´N GYO˝RI AND DAVID W. REYNOLDSWe write n + 1 = pN + r where p = [(n + 1)/N ], so that r is in{0, . . . , N − 1}, and p→∞ if n→∞. Thenf ∗(n) =pN−1∑k=0Φ(k)−1f(k − 1) +pN+r∑k=pNΦ(k)−1f(k − 1). (28)Using the periodicity of Φ, we can manipulate the first summation anddeduce thatpN−1∑k=0Φ(k)−1f(k − 1) =p−1∑i=0(i+1)N−1∑k=iNΦ(k)−1f(k − 1)=p−1∑i=0N−1∑j=0Φ(iN + j)−1f(iN + j − 1)=N−1∑j=0Φ(j)−1p−1∑i=0f(iN + j − 1)→N−1∑j=0Φ(j)−1∞∑i=0f(iN + j − 1) as p→∞,since f ∈ `1. To complete the proof, it suffices to notice that the secondterm in (28) satisfiespN+r∑k=pNΦ(k)−1f(k − 1) =r∑j=0Φ(j + pN)−1f(pN + j − 1)=r∑j=0Φ(j)−1f(pN + j − 1)→ 0 as p→∞,again since f ∈ `1. Our method is to apply Lemma 2 to (24). It must therefore beverified that H satisfies the hypotheses of that lemma.Lemma 5. The kernel H defined in (25) has the property (4) withWH = 0, (6) withH∞(j) =∞∑k=jΦ(k + 1)−1K(k, j)Φ(j), (29)and (7) with VH = 0.Proof. By the periodicity of n 7→ Φ(n) and n 7→ Φ(n)−1 there arepositive matrices C1 and C2 such that|Φ(n)| ≤ C1, |Φ(n)−1| ≤ C2, (30)ASYMPTOTICALLY PERIODIC SOLUTIONS 9for all n ≥ 0. Also by (13)∞∑n=0n∑j=0|K(n, j)| =∞∑j=0S(j) <∞,where S(j) =∑∞n=j |K(n, j)|.For an integer m > 0,lim supn→∞n∑j=m|H(n, j)| ≤ lim supn→∞n∑j=mn∑k=j|Φ(k + 1)−1||K(k, j)||Φ(j)|≤ C2 lim supn→∞n∑j=mn∑k=j|K(k, j)|C1≤ C2 lim supn→∞n∑j=m∞∑k=j|K(k, j)|C1≤ C2 lim supn→∞n∑j=mS(j)C1≤ C2∞∑j=mS(j)C1.Since∑∞j=0 S(j) is finite,∑∞j=m S(j)→ 0 as m→∞, andWH = limm→∞lim supn→∞n∑j=m|H(n, j)| = 0.Consequently (7) also holds with VH = 0.Lastly we demonstrate that (6) is true. We fix j ∈ Z+. For n ≥ j,H(n, j)→∞∑k=jΦ(k + 1)−1K(k, j)Φ(j) as n→∞.Because ∞∑k=j|Φ(k + 1)−1K(k, j)Φ(j)| ≤ C2S(j)C1,the limit is finite It is a consequence of Lemma 2 that z(n; ξ) converges to a limitz(∞; ξ) as n→∞, which satisfiesz(∞; ξ) = ξ + f ∗(∞) +∞∑j=0H∞(j)z(j; ξ), (31)where f ∗(∞) is given by (27) and H∞(j) by (29) .We now explicitly exhibit the dependence of the limit z(∞; ξ) onthe initial value ξ. For this reason, we examine (24) in the case that10 ISTVA´N GYO˝RI AND DAVID W. REYNOLDSf(n) ≡ 0, which corresponds to the homogeneous problem associatedwith (9):y(n+ 1; ξ) = ξ +n∑j=0H(n, j)y(j; ξ), (32a)y(0; ξ) = ξ, (32b)But ξ 7→ y(·; ξ) is linear; indeed y(n) = Y (n)ξ, whereY (n+ 1) = I +n∑j=0H(n, j)Y (j),Y (0) = I. (33)Lemma 2 says thatM = limn→∞Y (n) (34)exists, and y(n; ξ) → Mξ as n → ∞. Also the matrix M satisfies theimplicit limit formulaM = I +∞∑j=0H∞(j)Y (j).We summarise what has been discovered about (32).Lemma 6. The solution y of (32) satisfies y(n; ξ)→ Mξ as n→∞,where M is given by (34).Finally we introduce the sequence {w(n)} byw(n) = z(n; ξ)− Y (n)ξ, n ≥ 0,observing that it solvesw(n+ 1) = f ∗(n) +n∑j=0H(n, j)w(j),w(0) = 0. (35)We infer from Lemma 2 thatm = limn→∞w(n) (36)exists, and that m = f ∗(∞) +∑∞j=0H∞(j)w(j).In this section the following theorem has been proved.Theorem 2. Suppose that (A1)-(A4) hold. For every ξ ∈ Kd, thesolution z of (24) satisfieslimn→∞z(n; ξ) = Mξ +m, (37)where the matrix M is defined by (34) and the vector m by (36).Theorem 1 is now consequence of this and (23).ASYMPTOTICALLY PERIODIC SOLUTIONS 116. ExampleTo illustrate our results we examine an example from [2], whichconcerned (9) the case thatA(n) = −I, f(n) = 12n+13ne, K(n, j) =(−3)j2n3n+1Q,whereQ =(1 1−1 −1), e =(1−1).Note that Q2 = 0 and Qe = 0. Since A(0)A(1) = I, N = 2 andΦ(n) = (−1)nI. Also∞∑j=0|f(j)| = 35(11),∞∑n=0n∑j=0|K(n, j)| = 45(1 11 1),so that (A1)-(A4) hold with d = 2 and K = R.Let x(n; ξ) = (−1)nz(n; ξ) for all n ≥ 0. Then {z(n; ξ)} satisfiesz(n+ 1; ξ) = ξ +n∑j=0H(n, j)z(j; ξ) + f ∗(n),z(0; ξ) = ξ, (38)whereH(n, j) =27(−1)j+12j[1−(−16)n−j+1]Q,f ∗(n) = −37[1−(−16)n+1]e.These expressions can be written more concisely asH(n, j) = α(n− j)β(j)Q, f ∗(n) = γ(n)e,whereα(j) = 1−(−16)j+1, β(j) =27(−1)j+12j,γ(n) = −37[1−(−16)n+1].Here {α(n)} and {γ(n)} are convergent, and {β(n)} is summable withlimn→∞α(n) = 1,∞∑j=0β(j) = − 421, limn→∞γ(n) = −37.Note thatH∞(j) = limn→∞H(n, j) = β(j)Q, f ∗(∞) = limn→∞f ∗(n) = −37e.12 ISTVA´N GYO˝RI AND DAVID W. REYNOLDSThe solution of (38) can be split up asz(n; ξ) = Y (n)ξ + w(n),where Y is the solution ofY (n+ 1) = I +n∑j=0α(n− j)β(j)QY (j),Y (0) = I, (39)and w is the solution ofw(n+ 1) = γ(n)e+n∑j=0α(n− j)β(j)Qw(j),w(0) = 0. (40)We attempt to find a solution of (39) under the condition thatQY (n) = Q for all n ≥ 0. This is equivalent to Y having the formY (n) =(Y11(n) Y12(n)Y21(n) Y22(n))=(Y11(n) Y12(n)1− Y11(n) 1− Y12(n)).By substituting this into (39), we obtainY11(n+ 1) = 1 +n∑j=0α(n− j)β(j), Y12(n+ 1) =n∑j=0α(n− j)β(j),leading toY (n+ 1) = I +n∑j=0α(n− j)β(j)Q. (41)The solution of (40) is found in a similar fashion. We look for itssolution in the formw(n) =(w1(n)w2(n))=(w1(n)−w1(n))= w1(n)e,so that Qw(n) = 0 for all n ≥ 0. The equation becomes w1(n + 1) =γ(n), andw(n+ 1) = γ(n)e. (42)The exact solution of (38) is then z(0, ξ) = ξ andz(n+ 1; ξ) = ξ +n∑j=0α(n− j)β(j)Qξ + γ(n)e, n ≥ 0. (43)Sincelimn→∞n∑j=0α(n− j)β(j) = − 421, limn→∞γ(n)e = −37e = −37(1−1),ASYMPTOTICALLY PERIODIC SOLUTIONS 13it follows from (43) thatz(n+ 1; ξ)→(I − 421Q)ξ +(−37e)as n→∞. (44)We can calculate the matrix M defined by (34) from (41), and thevector m given by (36) from (42), and obtainM = I − 421Q =121(17 −44 25), m = −37e = −37(1−1).Observe that detM = 1 so that M is invertible. It follows from (44)that z(n + 1; ξ) = c + o(1) as n → ∞, where c = Mξ + m. Since Mis invertible, there is a solution with this property for every c ∈ R2.Therefore (9) has asymptotically periodic solutions of form x(n) =(−1)nc+ o(1) for every nonzero c in R2.We remark that if ξ =(2 1)T, thenc =121(17 −44 25)(21)+37(−11)=(12),which is the case examined in [2].References[1] J. A. D. Appleby, I. Gyo˝ri, and D. W. Reynolds, On exact convergence ratesfor solutions of linear systems of Volterra difference equations, J. DifferenceEqu. Appl. 12 (2006), no. 12, 1257–1275.[2] J. Dibl´ık, E. Schmeidel, and M. Ru˚zˇicˇkova´, Existence of asymptotically periodicsolutions of system of Volterra difference equations, J. Difference Equ. Appl.15 (2009), no. 11, 1165–1177.[3] S. Elaydi, Periodicity and stability of linear Volterra difference systems, J.Math. Anal. Appl. 181 (1994), no. 2, 483–492.[4] S. Elaydi and S. Zhang, Stability and periodicity of difference equations withfinite delay, Funkcial. Ekvac. 37 (1994), no. 3, 401–413.[5] I. Gyo˝ri and L. Horva´th, Asymptotic representation of the solutions of linearVolterra difference equations, Adv. Difference Equ. (2008), Art. ID 932831, 22.[6] Y. Song and C. T. H. Baker, Perturbations of Volterra difference equations, J.Difference Equ. Appl. 10 (2004), no. 4, 379–397.[7] , Perturbation theory for discrete Volterra equations, J. Difference Equ.Appl. 9 (2003), no. 10, 969–987.[8] S. Zhang, The unique existence of periodic solutions of linear Volterra differ-ence equations, J. Math. Anal. Appl. 193 (1995), no. 2, 419–430.Department of Mathematics, University of Pannonia, H-8200 Vezs-pre´m, Egyetem u. 10, Hungary.E-mail address: gyori@almos.uni-pannon.huURL: http://www.szt.vein.hu/~gyori/School of Mathematical Sciences, Dublin City University, Dublin9, IrelandE-mail address: david.reynolds@dcu.ieURL: http://webpages.dcu.ie/~reynoldd/

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