On the best Ulam constant of a first order linear difference equation in Banach spaces
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ON THE BEST ULAM CONSTANT OF A FIRST ORDER LINEAR DIFFERENCE EQUATION IN BANACH SPACES A.-R. BAIAS∗, F. BLAGA and D. POPA Department of Mathematics, Technical University of Cluj-Napoca, G. Barit¸iu No. 25, 400027, Cluj-Napoca, Romania e-mails: [email protected], [email protected], [email protected] (Received April 15, 2020; revised May 13, 2020; accepted June 16, 2020)
Abstract. We obtain some results on Ulam stability for the linear difference equation xn+1 = an xn + bn , n ≥ 0, in a Banach space X. If there exists limn→∞ |an | = λ, then the equation is Ulam stable if and only if λ = 1. Moreover if (|an |)n≥0 is a monotone sequence, then the best Ulam constant of the equation 1 . is |λ−1|
1. Introduction The starting point of stability theory for functional equations was the problem of S.M. Ulam concerning approximate homomorphisms of groups, formulated in 1940 during a talk at Madison University, Wisconsin (see [23]). The first answer to Ulam’s problem was given a year later by D.H. Hyers who proved that Cauchy’s equation in Banach spaces is stable. Generally, we say that an equation is stable in Ulam sense if for every approximate solution of it there exists an exact solution of the equation near it. For more details and results on Ulam stability we refer the reader to [2,6,7,13]. As far as we know, the first result on generalized Ulam stability for a first order linear difference equation in a Banach space X was obtained in [19]. If (εn )n≥0 is a sequence of positive numbers, (an )n≥0 is a sequence of complex numbers, (bn )n≥0 is a sequence in X, and (1)
lim sup
εn |an+1 | 1, εn+1
∗ Corresponding
author. Key words and phrases: linear difference equation, Banach space, Ulam stability, best constant. Mathematics Subject Classification: 39A30, 39B82.
0236-5294/$20.00 © 2020 Akade ´miai Kiado ´, Budapest, Hungary
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A.-R. A.-R. BAIAS, BAIAS, F. BLAGA and D. POPA
then for every sequence (xn )n≥0 in X satisfying the relation (2)
xn+1 − an xn − bn ≤ εn ,
n ≥ 0,
there exists a sequence (yn )n≥0 in X, yn+1 = an yn + bn , n ≥ 0, such that (3)
L = sup n≥1
xn − yn < ∞. εn−1
n+1 | = 1 then the linear difference Moreover, in [8] is proved that if lim εnε|an+1 n→∞ equation xn+1 = an xn + bn is not Ulam stable. Finding the infimum of all constants L satisfying (3) means to obtain the best Ulam constant of the equation xn+1 = an xn + bn , n ≥ 0. For the case of a constant sequence (an )n≥0 , the best Ulam constant is obtained in [3]. It seems that until now there are no results on the best Ulam constant for the linear difference equations of order one with nonconstant coefficients. J. Brzdek, S.M. Jung and M.Th. Rassias obtained sharp estimates for the Ulam constant of some second order linear difference equations in [9,14]. Recently M. Onitsuka [15,16] and D.R. Anderson, M. Onitsuka [1] obtained results on Hyers–Ulam stability and on the best Ulam constant for a first order and a second order linear difference equation with constant stepsize. The best Ulam constant for a second or
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