Note on nonstability of the linear recurrence (original) (raw)

Remarks on stability of linear recurrence of higher order

Applied Mathematics Letters, 2010

We prove some stability results for linear recurrences with constant coefficients in normed spaces. As a consequence we obtain a complete solution of the problem of the Hyers-Ulam stability for such recurrences. Throughout this note, N, N 0 , Z, R and C stand, as usual, for the sets of positive integers, nonnegative integers, integers, reals and complex numbers, respectively. Moreover T ∈ {N 0 , Z}, K is either the field R or C, X is a nontrivial normed space over K, S := {a ∈ C : |a| = 1}, p ∈ N, a 1 , . . . , a p ∈ K, (b n ) n∈T is a sequence in X , and r 1 , . . . , r p ∈ C denote all the roots of the equation r p

Hyers-Ulam stability of the linear recurrence with constant coefficients

Advances in Difference Equations, 2005

Let X be a Banach space over the field R or C, a 1 ,...,a p ∈ C, and (b n ) n≥0 a sequence in X. We investigate the Hyers-Ulam stability of the linear recurrence x n+p = a 1 x n+p−1 + ··· + a p−1 x n+1 + a p x n + b n , n ≥ 0, where x 0 ,x 1 ,...,x p−1 ∈ X.

Hyers–Ulam–Rassias stability of a linear recurrence

Journal of Mathematical Analysis and Applications, 2005

Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients

Discrete Dynamics in Nature and Society, 2015

We study the Hyers-Ulam stability in a Banach spaceXof the system of first order linear difference equations of the formxn+1=Axn+dnforn∈N0(nonnegative integers), whereAis a givenr×rmatrix with real or complex coefficients, respectively, and(dn)n∈N0is a fixed sequence inXr. That is, we investigate the sequences(yn)n∈N0inXrsuch thatδ∶=supn∈N0yn+1-Ayn-dn<∞(with the maximum norm inXr) and show that, in the case where all the eigenvalues ofAare not of modulus 1, there is a positive real constantc(dependent only onA) such that, for each such a sequence(yn)n∈N0, there is a solution(xn)n∈N0of the system withsupn∈N0yn-xn≤cδ.

On the stability of the first-order linear recurrence in topological vector spaces

Nonlinear Analysis: Theory, Methods & Applications, 2010

Suppose that X is a sequentially complete Hausdorff locally convex space over a scalar field K, V is a bounded subset of X , (a n ) n≥0 is a sequence in K \ {0} with the property lim inf n→∞ |a n | > 1 and (b n ) n≥0 is a sequence in X . We show that for every sequence (x n ) n≥0 in X satisfying

Hyers-Ulam stability of the first order difference equation generated by linear maps

Cornell University - arXiv, 2021

Hyers-Ulam stability of the difference equation z n+1 = a n z n + b n is investigated. If n j=1 |a j | has subexponential growth rate, then difference equation generated by linear maps has no Hyers-Ulam stability. Other complementary results are also found where lim n→∞ n j=1 |a j | 1 n is greater or less than one. These results contain Hyers-Ulam stability of the first order linear difference equation with periodic coefficients also.