Trinity University From the SelectedWorks of William F. Trench 1985 Linear perturbations of general disconjugate equations William F. Trench, Trinity University Available at: https://works.bepress.com/william_trench/53/
c 1998 OPA (Overseas Publishing Association) N.V. Linear Asymptotic Equilibrium of Nilpotent Systems of Linear Difference Equations William F. Trench Trinity University San Antonio,TX 78212 J. Difference Equations Appl. 5 (1999), 549-556 Abstract We give sufficient conditions for a k k linear system of difference equations x n = A nx n, n = 0, 1,..., to have linear asymptotic equilibrium if A n = a 1(n)Q+ +a p(n)q p, where Q p+1 = 0 for some p {1,2,..., k 1} and {a 1(n)},..., {a p(n)} are sequences of scalars. The conditions involve convergence (perhaps conditional) of certain iterated sums involving these sequences. Key words: difference equations, nilpotent, linear asymptotic equilibrium, conditional convergence A k k linear system of difference equations x n = A n x n (1) is said to have linear asymptotic equilibrium if lim n x n exists and is nonzero whenever x 0 0. Since (1) can be written as its solution is x n = P n x 0, where x n+1 = (I + A n )x n, n = 0, 1,..., P n = (I + A m ) = (I + A ) (I + A 0 ). (2) Therefore (1) has linear asymptotic equilibrium if and only if I+A n is invertible for every n 0 and lim n P n exists and is invertible. If it is assumed that I + A n is invertible for every n then the most well known sufficient condition for (1) to have linear asymptotic equilibrium is that An <, where is any matrix norm such that AB A B for all square matrices A and B. (See [2, 4]). The following weaker condition for linear asymptotic equilibrium is given in [3]. For related results, see [1]. 1
2 William F. Trench Theorem 1 Suppose that I + A n is invertible for every n 0 and either A n < { or there is an integer R 1 such that the sequences (with B (0) n B (r) n = = I) are defined for 1 r R, and Then (1) has linear asymptotic equilibrium. B (r) n } given by B (r 1) m+1 A m, n = 0, 1, 2,... (3) B (R) n+1 A n <. (4) In [2] the author proved Theorem 1 with R = 1. In its full generality Theorem 1 is an analog of a result of Wintner [5] for a linear differential system y = A(t)y. In this paper we consider a class of systems that have linear asymptotic equilibrium under conditions that require only convergence which may be conditional of certain iterated series derived from {A n }. As we point out at the end of the paper, whether or not the conditions that we impose here actually imply the hypotheses of Theorem 1 for these systems is an open question which is in a way irrelevant, since the results given here provide a constructive way to obtain lim n P n for these systems, as opposed to Theorem 1, which is a nonconstructive existence theorem. We assume henceforth that k 2 and Q is a k k nonzero nilpotent matrix; that is, there is an integer p in {1, 2,..., k 1} such that Q p 0 and Q p+1 = 0. (5) For example, any strictly upper triangular matrix is nilpotent. We will say that (1) is a nilpotent system if A n = a 1 (n)q + + a p (n)q p, n = 0, 1, 2,..., (6) where Q is k k nilpotent matrix satisfying (5) and {a 1 (n)},..., {a p (n)} are sequences of scalars. We will show that the nilpotent system (1) has linear asymptotic equilibrium if certain iterated sums involving these sequences converge. The convergence may be conditional. Any matrix of the form Z = I + u 1 Q + + u p Q p
Nilpotent Linear Systems 3 is invertible, and any two matrices of this form commute. Therefore, the product P n in (2) is of the form with P n = (I + A m ) = I + b 1 (n)q + + b p (n)q p, b 1 (0) = b 2 (0) = = b p (0) = 0, and lim n P n exists if and only if the limits exist (finite), in which case lim b i(n) = b i, i = 1,..., p, n (I + A m ) = I + b 1 Q + + b p Q p. Hence, (1) has linear asymptotic equilbrium if and only if b 1,..., b p exist (finite). We will now introduce some notation which will enable us to give recursive formulas for b 1 (n),..., b p (n). If i 1, i 2,... are integers in {1, 2,..., r} and n is an arbitrary positive integer, let σ(n; i 1 ) = σ(n; i 2, i 1 ) =. σ(n; i s, i s 1,..., i 1 ) = a i1 (m), a i2 (m)σ(m; i 1 ), a is (m)σ(m; i s 1,..., i 1 ). We say that an s-tuple (i s, i s 1,..., i 1 ) of positive integers is an ordered partition of r if i s + i s 1 + + i 1 = r. We denote an ordered partition of r by ψ r. For each positive integer r let O r be the set of all ordered partitions of r. Lemma 1 If the sequence {A n } is as defined in (6) then P n = (I + A m ) = I + b 1 (n)q + + b p (n)q p,
4 William F. Trench with for 1 r p. b r (n) = ψ r O r σ(n; ψ r ) (7) Proof: We will establish (7) by finite induction on r. Since P n+1 = (I+A n )P n, P n+1 = [I + a 1 (n)q + + a p (n)q p ][I + b 1 (n)q + + b p (n)q p ]. Since Q p+1 = 0 we are interested in or so i 1 b i (n + 1) = b i (n) + a i (n) + a q (n)b i q (n), 1 i p, q=1 i 1 b i (n) = a i (n) + a q (n)b i q (n), 1 i p, i 1 b i (n) = σ(n; i) + Letting i = 1 here yields q=1 q=1 a q (m)b i q (m), 1 i p. b 1 (n) = σ(n; 1), which confirms (7) with r = 1. Now suppose that 2 i < p and (7) has been established for 1 r i 1. Then b i q (m) = σ(m; ψ i q ), 1 q i 1, ψ i q O i q so For a given q, i 1 b i (n) = σ(n, i) + ψ i q O i q q=1 ψ i q O i q a q (m)σ(m, ψ i q ) = a q (m)σ(m, ψ i q ). (8) ψ i O q i σ(n; ψ i ) where O q i is the set of all ordered partitions (i s, i s 1,..., i 1 ) of i for which i s = q. Therefore (8) implies that i 1 b i (n) = σ(n, i) + σ(n; ψ i ) q=1 ψ i O q i
Nilpotent Linear Systems 5 Since O i = i q=1 Oq i and Oi i = {(i)}, it now follows that b i (n) = i q=1 ψ i O q i σ(n; ψ i ) = ψ i O i σ(n; ψ i ), which implies (7) with r = i, completing the induction. The following theorem is our main result. Theorem 2 Let Q be a k k matrix such that Let Q p 0 and Q p+1 = 0. A n = a 1 (n)q + + a p (n)q p, where {a 1 (n)},..., {a p (n)} are sequences of scalars such that all the limits lim n σ(n; ψ r), ψ r O r, 1 r p, exist. (Convergence may be conditional.) Then the system has linear asymptotic equilibrium, and where x n = A n x n, n = 0, 1,..., (9) (I + A n ) = I + b 1 Q + + b p Q p, (10) b r = lim n ψ r O r σ(n; ψ r ), 1 r p. (11) The following example will motivate our discussion of the connection between Theorem 1 (as it applies to nilpotent systems) and Theorem 2. Example. Suppose that k 3 and let A n = ( 1) n α n Q + ( 1) n β n Q 2, where {α n } and {β n } are nonincreasing null sequences and α 2 n <. (12) Then the series S α = ( 1)n α n and S β = ( 1)n β n converge by the alternating series test. Now consider S αα = ( 1) n α n ( 1) m α m = Sα 2 ( 1) n α n ( 1) m α m. (13)
6 William F. Trench Since 0 < ( 1) n ( 1) m α m < α n, (12) implies that the last series in (13) converges. Therefore Theorem 2 implies that (9) has linear asymptotic equilibrium, and that (I + A n ) = I + S α Q + (S β + S αα )Q 2. (14) We note that S α, S β, and S αα may all converge conditionally. Theorem 1 is also applicable under the assumptions of this example, since (see (3)), ( ) ( ) B n (1) = ( 1) m α m Q + ( 1) m β m Q 2 is defined, as is B (2) n = ( ( 1) m α m l=m+1 ( 1) l α l ) Q 2. (Recall that Q 3 = 0.) Moreover, (4) obviously holds, since B (2) n+1 = 0. However, Theorem 1 does not imply (14). It is natural to ask whether the hypotheses of Theorem 2 in general imply those of Theorem 1. To establish this it would be sufficient to verify that the hypotheses of Theorem 2 imply that the sequences {B n (r) }, r = 1,..., p in (3) are all defined. However, to verify this if it is true would surely be at least as difficult as the proof of Lemma 1. If it is true then these sequences would necessarily be of the form B (r) n = p c rs (n)q s, 1 r p, s=r so B n (p) = 0 for all n, and (4) would hold automatically with R = p. Therefore, we could conclude that (9) has linear asymptotic equilibrium, but we would not have proved (10) and (11). References [1] Z. Benzaid and D. A. Lutz, Asymptotic representation of solutions of linear difference equations, Studies in Applied Mathematics 77 (1987), 195-221. [2] W. F. Trench, Invertibly convergent infinite products of matrices, with applications to difference equations, Computers Math. Applic. 30 (1995), 39-46.
Nilpotent Linear Systems 7 [3] W. F. Trench, Linear asymptotic equilibrium and uniform, exponential, and strict stability of linear difference systems, Computers Math. Applic. (to appear). [4] A. Trgo, Monodromy matrix for linear difference operators with almost constant coefficients, J. Math. Anal. Appl., 194 (1995), 697-719. [5] A. Wintner, On a theorem of Bòcher in the theory of ordinary linear differential equations, Amer. J. Math. 76 (1954), 183-190.