Jung Advances in Difference Equations (2015) 2015:170 DOI 10.1186/s13662-015-0507-6 RESEARCH Open Access Hyers-Ulam stability of the first-order matrix difference equations Soon-Mo Jung* * Correspondence: smjung@hongik.ac.kr Mathematics Section, College of Science and Technology, Hongik University, Sejong, 339-701, Republic of Korea Abstract In this paper, we prove the Hyers-Ulam stability of the first-order linear homogeneous matrix difference equations xi = Axi–1 and xi–1 = Axi for all integers i ∈ Z MSC: Primary 39A45; 39B82; secondary 39A06; 39B42 Keywords: difference equation; matrix difference equation; recurrence; Hyers-Ulam stability; approximation Introduction Throughout this paper, let n be a fixed positive integer The nth order linear homogeneous difference equation with constant coefficients is of the form = α ai– + α ai– + · · · + αn ai–n , () where α , α , , αn are constants For example, the second-order difference equation with constant coefficients has the form = αai– + βai– () The solution of () is called the Fibonacci numbers when α = β = , a = , and a = , Lucas numbers when α = β = , a = , and a = , Pell numbers when α = , β = , a = , and a = , Pell-Lucas numbers when α = , β = , and a = a = , and Jacobsthal numbers if α = , β = , a = , and a = The polynomial p(x) = xn – α xn– – α xn– – · · · – αn– x – αn is called the characteristic polynomial of the difference equation () If the roots r , r , , rn of the characteristic polynomial are distinct, then the solution of the difference equation () is given by = k ri + k ri + · · · + kn rni , where the coefficients k , k , , kn are uniquely determined under the initial conditions of the difference equation © 2015 Jung This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made Jung Advances in Difference Equations (2015) 2015:170 Page of 13 If the characteristic polynomial has roots r , r , , rd with multiplicity m , m , , md , respectively, then the solution of the difference equation () is given by d mj cjk ik– rji , = j= k= where the cjk are constants and m + m + · · · + md = n (see [, ]) For the Hyers-Ulam stability of the linear difference equations, we may refer to [–] Let (Cn , · n ) be a complex normed space, each of whose elements is a column vector, and let Cn×n be a vector space consisting of all (n × n) complex matrices We choose a norm · n×n on Cn×n which is compatible with · n , i.e., both norms obey AB n×n ≤ A n×n B n×n and Ax n ≤ A n×n x n () for all A, B ∈ Cn×n and x ∈ Cn A matrix difference equation is a difference equation with matrix coefficients in which the value of vector of variables at one point is dependent on the values of preceding (succeeding) points In this paper, we prove the Hyers-Ulam stability of the first-order linear homogeneous matrix difference equations xi = Axi– and xi– = Axi for all integers i ∈ Z, where the transition matrix A is nonsingular More precisely, we prove that if a sequence {yi }i∈Z satisfies the inequality yi – Ayi– n ≤ ε for all i ∈ Z resp yi– – Ayi n ≤ ε for all i ∈ Z, then there exist a solution {xi }i∈Z ⊂ Cn of the first-order matrix difference equation () resp () and a constant K > such that yi – xi n ≤ Kε for all integers i ≥ (We refer the reader to [–] for the exact definition of Hyers-Ulam stability.) It should be remarked that many interesting theorems have been proved in [, ] concerning the linear (or nonlinear) recurrences Especially in , the Hyers-Ulam stability of the first-order matrix difference equations has been proved in [] in a general setting The substantial difference of this paper from [] lies in the fact that the stability problems for the ‘backward’ difference equations have been treated in Section of this paper Hyers-Ulam stability of xi = Axi–1 In this section, we investigate the Hyers-Ulam stability of the first-order linear homogeneous matrix difference equation xi = Axi– () for all integers i ∈ Z, where ⎞ xi ⎜x ⎟ ⎜ i ⎟ n ⎟ xi = ⎜ ⎜ ⎟ ∈ C ⎝ ⎠ xin ⎛ ⎛ and a ⎜a ⎜ A=⎜ ⎜ ⎝ an a a an ··· ··· ··· ⎞ an an ⎟ ⎟ n×n ⎟ ⎟∈C ⎠ ann Theorem . Given a fixed positive integer n, let (Cn , · n ) and (Cn×n , · n×n ) be complex normed spaces, whose elements are column vectors resp (n × n) complex matrices, with the Jung Advances in Difference Equations (2015) 2015:170 Page of 13 property () Assume that the transition matrix A ∈ Cn×n is nonsingular and {εi }i∈Z is a sequence of nonnegative real numbers If a sequence {yi }i∈Z ⊂ Cn satisfies the inequality yi – Ayi– n ≤ εi () for all i ∈ Z, then there exists a solution {xi }i∈Z ⊂ Cn of the first-order matrix difference equation () such that yi – xi i i–k i k= εk A n×n + A n×n y – x n –i – k – –i n×n + A n×n y – x n k= εk+i A ≤ n (for i ≥ ), (for i < ) Proof Assume that a sequence {yi }i∈Z ⊂ Cn satisfies the inequality () for all i ∈ Z First, we assume that i is a nonnegative integer It then follows from () and () that yi – Ai y n ≤ yi – Ayi– n + Ayi– – A yi– + A yi– – A yi– ≤ yi – Ayi– n×n + A ≤ εi + A n + A n + · · · + A y – Ai y n×n yi– – Ayi– n×n εi– n i– + A yi– – Ayi– n + ··· + A n×n εi– n n i– n×n + ··· + A y – Ay n i– n×n ε i = A i n×n εk A –k n×n () k= It is obvious that a sequence {xi }i∈Z ⊂ Cn satisfies the first-order matrix difference equation () if and only if xi = Ai x () for each i ∈ Z, where we set Ai = (A– )–i for all negative integers i Hence, by () and (), we have yi – xi ≤ yi – Ai y n n + Ai y – Ai x n + Ai x – xi n i ≤ εk A i–k n×n + A i n×n y – x n k= for any integer i ≥ On the other hand, we suppose i is a negative integer For this case, it follows from () and () that yi – Ai y n = yi – A– –i y ≤ yi – A yi+ + – A n – n + A– yi+ – A– yi+ – yi+ – A yi+ n + ··· + n A– –i– y– – A– –i y n Jung Advances in Difference Equations (2015) 2015:170 ≤ A– n×n + A– ≤ A– Ayi – yi+ n×n Page of 13 n Ayi+ – yi+ ε n×n i+ n Ayi+ – yi+ + · · · + A– ε n×n i+ + A– n×n + A– + A– –i n×n ε n×n i+ n Ay– – y n + · · · + A– –i ε n×n –i εk+i A– = k n×n () k= Moreover, by () and (), we have yi – xi n –i ≤ yi – A– + A – –i y n x – xi + –i A– y – A– –i x n n –i ≤ εk+i A– k n×n + A– –i n×n y – x n k= for all integers i < In view of (), if we assume the initial condition in the previous theorem, we can easily prove the uniqueness of the sequence {xi }i∈Z as we see in the following corollary Corollary . Given a fixed positive integer n, let (Cn , · n ) and (Cn×n , · n×n ) be complex normed spaces, whose elements are column vectors resp (n × n) complex matrices, with the property () Assume that the transition matrix A ∈ Cn×n is nonsingular and {εi }i∈Z is a sequence of nonnegative real numbers If a sequence {yi }i∈Z ⊂ Cn satisfies the inequality () for all i ∈ Z, then there exists a unique solution {xi }i∈Z ⊂ Cn of the first-order matrix difference equation () with the initial condition x = y such that yi – xi n i i–k k= εk A n×n –i – k n×n k= εk+i A ≤ (for i ≥ ), (for i < ) Some of the most important matrix norms are induced by p-norms For ≤ p ≤ ∞, the matrix norm induced by the p-norm, A p := sup x= Ax p , x p is called the matrix p-norm For example, we get n A ≤j≤n n |aij | = max and A ∞ i= |aij | = max ≤i≤n j= It is well known that the matrix p-norm, together with the p-norm, satisfies the conditions in (), where n x |xj | = j= for any x ∈ Cn and x ∞ = max |xj | ≤j≤n Jung Advances in Difference Equations (2015) 2015:170 Page of 13 In the following corollary, we prove the Hyers-Ulam stability of the second-order linear homogeneous difference equation with constant coefficients Corollary . Let (C , · ∞ ) and (C× , · ∞ ) be complex normed spaces and let α, β, γ be complex numbers satisfying the conditions α + β = , β = , γ = () Assume that ε > is an arbitrary constant If a sequence {ai }i∈Z of complex numbers satisfies the inequality |ai – αai– – βai– | ≤ ε () for all i ∈ Z, then there exists a sequence {ci }i∈Z of complex numbers such that c– = a– , c = a , ci = αci– + βci– , and i k= ε –i k= ε |ai – ci | ≤ where A ∞ (for i ≥ ), (for i < ), A i–k ∞ A– k∞ = max{|α| + |β/γ |, |γ |} and A– ∞ = max{|/γ |, |α/β| + |γ /β|} Proof If we define a sequence {bi }i∈Z of complex numbers by bi = γ ai– , it then follows from () that |ai – αai– – γβ bi– | ≤ ε, |bi – γ ai– | = for any i ∈ Z If we set yi := bi and A := α β γ γ , then we get yi – Ayi– ∞ ≤ε for each i ∈ Z According to Corollary ., there exists a unique solution {xi }i∈Z ⊂ C of the first-order a matrix difference equation () with the initial condition x = γ a– such that yi – xi ∞≤ i k= ε –i k= ε A i–k ∞ A– k∞ (for i ≥ ), (for i < ) In view of (), this last inequality implies that a – Ai γ ai– γ a– ≤ ∞ i k= ε –i k= ε A i–k ∞ A– k∞ (for i ≥ ), (for i < ) () Jung Advances in Difference Equations (2015) 2015:170 Page of 13 √ α– α +β Since the transition matrix A has two distinct eigenvalues λ = and λ = √ α+ α +β , which are the roots of the characteristic equation λ – αλ – β = , the matrix A can be expressed as A = CDC– () with λ γ C= λ , γ D= λ , λ C– = γ γ (λ – λ ) –γ –λ λ By (), we obtain Ai = CDi C– λi = λ γ (λ – λ ) γ = i+ γ (λi+ – λ ) i γ (λ – λ ) γ (λ – λi ) λ γ λi –λ λ γ –γ –λ λ (λi – λi ) i– –γ λ λ (λi– – λ ) for every integer i ≥ Using this equality, it follows from () that – λ i+ – λ i+ λ + aλ–a λ – aλ–a –λ –λ a –a– λ i – λ i γ ai– – γ aλ–a λ + γ λ λ –λ –λ i ≤ ∞ ε A i–k ∞ () k= for all integers i ≥ √ –α– α +β = On the other hand, the inverse matrix A– has two distinct eigenvalues ω = β √ +β α –α+ – β λ and ω = = – β λ , which are roots of the characteristic equation ω + βα ω – β = Hence, the matrix A– may be expressed as β A– = γ β γ – βα = γ ω γ ω ω–i ω–i ω γ ω ω γ ω – () Using (), we have Ai = A– = γ ω = –i γ ω γ ω – γ ω ω (ω–i– – ω–i– ) γ (ω – ω ) γ ω ω (ω–i – ω–i ) γ ω – ω–i – ω–i γ (ω–i – ω–i ) for all integers i < Thus, the inequality () yields ω –i ω –i ω + a–ω–a ω – a–ω–a –ω –ω a– –a ω –i a– –a ω –i γ ai– – γ ω –ω ω + γ ω –ω ω for any integer i < –i ≤ ∞ ε A– k= k ∞ () Jung Advances in Difference Equations (2015) 2015:170 Page of 13 Finally, considering (), (), and [], Theorem ., if we set ci := (for i ≥ ), (for i < ), a –a– λ i+ – λ i+ λ – aλ–a λ λ –λ –λ a– –a ω –i a– –a ω –i ω – ω –ω ω ω –ω then we get c– = a– , c = a , and it follows from () and () that i k= ε –i k= ε |ai – ci | ≤ A i–k ∞ A– k∞ (for i ≥ ), (for i < ) Furthermore, it is not difficult to show that the sequence {ci }i∈Z satisfies the second-order linear difference equation ci = αci– + βci– for any integer i √ If we set γ = lim A β→∞ ±α± ∞ α +β · A– ∞ For example, if we set γ = in Corollary ., then we get = α+ √ α +β and β > , then we have ⎧ √ √ ⎨ A = max α+ α +β , –α+ α +β , ∞ √ √ α+ α +β –α+ α +β ⎩ A– , , ∞ = max β β () and hence lim A β→∞ ∞ · A– ∞ = lim β→∞ β · √ = β √ √ √ α– α +β –α+ α +β –α– α +β , γ = , or γ = , we analogously obtain For the case when γ = limβ→∞ A ∞ · A– ∞ = If α and β are simultaneously small in absolute value, then the second-order difference equation () has the Hyers-Ulam stability as we see in the following example Example . Given an ε > , assume that a sequence {ai }i∈Z of complex numbers satisfies the inequality – ai– – ai– ≤ ε √ for √all i ∈ Z With α = and β = , it follows from () that A ∞ = + and A– ∞ = + Using these values, Corollary . implies that there exists a sequence {ci }i∈Z of complex numbers such that c– = a– , c = a , ci = ci– + ci– , and |ai – ci | ≤ √ + √ + √ + i ε + –i – –√ ε (for i ≥ ), (for i < ) Jung Advances in Difference Equations (2015) 2015:170 Page of 13 Hyers-Ulam stability of xi–1 = Axi In practical applications, we sometimes consider the first-order linear homogeneous matrix difference equation xi– = Axi () instead of (), where the transition matrix A is a nonsingular matrix of Cn×n We now investigate the Hyers-Ulam stability of the matrix difference equation () Theorem . Given a fixed positive integer n, let (Cn , · n ) and (Cn×n , · n×n ) be complex normed spaces, whose elements are column vectors resp (n × n) complex matrices, with the property () Assume that the transition matrix A ∈ Cn×n is nonsingular and {εi }i∈Z is a sequence of nonnegative real numbers If a sequence {yi }i∈Z ⊂ Cn satisfies the inequality yi– – Ayi ≤ εi n () for all i ∈ Z, then there exists a solution {xi }i∈Z ⊂ Cn of the first-order matrix difference equation () such that yi – xi i – k= εk A –i k= εk+i A n≤ i+–k – i n×n + A n×n k– –i + A n×n n×n y y – x – x n n (for i ≥ ), (for i < ) () Proof Assume that a sequence {yi }i∈Z ⊂ Cn satisfies the inequality () for all i ∈ Z First, we assume that i is a nonnegative integer Then, by () and (), we have yi – A–i y n ≤ yi – A– yi– – n + A– yi– – A– yi– – + A yi– – A yi– ≤ εi A – + ··· + A n –i+ n + εi– A– n×n n×n y – A–i y + εi– A– n×n n + · · · + ε A– i n×n i εk A– = i+–k n×n k= Obviously, a sequence {xi }i∈Z ⊂ Cn satisfies the first-order matrix difference equation () if and only if xi = A–i x () for all i ∈ Z, where we set A–i = (A– )i for each integer i ≥ Hence, we get yi – xi n ≤ yi – A–i y n + A–i y – A–i x n + A–i x – xi i ≤ εk A– k= for all integers i ≥ i+–k n×n + A– i n×n y – x n n Jung Advances in Difference Equations (2015) 2015:170 Page of 13 On the other hand, if i is a negative integer, then it follows from () and () that yi – A–i y n = yi – Ayi+ n + Ayi+ – A yi+ + A yi+ – A yi+ ≤ εi+ + εi+ A n×n n + ··· + A –i– n + εi+ A n×n y– – A–i y + · · · + ε A n –i– n×n –i = εk+i A k– n×n k= Thus, by () and the last inequality, we obtain yi – xi n ≤ yi – A–i y n + A–i y – A–i x n + A–i x – xi n –i ≤ εk+i A k– n×n + A –i n×n y – x n k= for any integer i < We now remark that if we apply Theorem . in place of the proof of Theorem ., then we would obtain an inequality () below, which seems not to be better than the inequality () given in Theorem ., as we see in the following remark, whose proof we omit Remark . Given a fixed positive integer n, let (Cn , · n ) and (Cn×n , · n×n ) be complex normed spaces, whose elements are column vectors resp (n × n) complex matrices, with the property () Assume that the transition matrix A ∈ Cn×n is nonsingular and {εi }i∈Z is a sequence of nonnegative real numbers If a sequence {yi }i∈Z ⊂ Cn satisfies the inequality () for all i ∈ Z, then there exists a solution {xi }i∈Z ⊂ Cn of the first-order matrix difference equation () such that yi – xi ≤ n εi+ + εi+ + i+ – i+–k n×n + k= εk A –i– k k= εk+i+ A n×n + A– i+ n×n Ay – x– n –i– A n×n Ay – x– n (for i ≥ ), (for i < ) () In view of (), assuming the initial condition in the previous theorem leads to the uniqueness of the sequence {xi }i∈Z , as we see in the following corollary Corollary . Given a fixed positive integer n, let (Cn , · n ) and (Cn×n , · n×n ) be complex normed spaces, whose elements are column vectors resp (n × n) complex matrices, with the property () Assume that the transition matrix A ∈ Cn×n is nonsingular and {εi }i∈Z is a sequence of nonnegative real numbers If a sequence {yi }i∈Z ⊂ Cn satisfies the inequality () for all i ∈ Z, then there exists a solution {xi }i∈Z ⊂ Cn of the first-order matrix difference equation () with the initial condition x = y such that yi – xi n ≤ i – k= εk A –i k= εk+i A i+–k n×n k– n×n (for i ≥ ), (for i < ) Jung Advances in Difference Equations (2015) 2015:170 Page 10 of 13 In the next corollary, we investigate the Hyers-Ulam stability of the second-order linear homogeneous difference equation with constant coefficients = αai+ + βai+ () Corollary . Let (C , · ∞ ) and (C× , · ∞ ) be complex normed spaces and let α, β, γ be complex numbers satisfying the conditions α + β = , β = , γ = () Assume that ε > is an arbitrary constant If a sequence {ai }i∈Z of complex numbers satisfies the inequality |ai – αai+ – βai+ | ≤ ε () for all i ∈ Z, then there exists a sequence {ci }i∈Z of complex numbers such that c = a , c = a , ci = αci+ + βci+ , and i k= ε –i k= ε |ai – ci | ≤ where A ∞ (for i ≥ ), (for i < ), A– i+–k ∞ A k– ∞ = max{|α| + |β/γ |, |γ |} and A– ∞ = max{|/γ |, |α/β| + |γ /β|} Proof If we define a sequence {bi }i∈Z of complex numbers by bi = γ ai+ , it then follows from () that |ai – αai+ – γβ bi+ | ≤ ε, |bi – γ ai+ | = for every i ∈ Z Hence, if we set yi := bi and A := α β γ γ , then we get yi – Ayi+ ∞ ≤ε for all i ∈ Z According to Corollary ., there exists a unique solution {xi }i∈Z of the first-order matrix a difference equation () with the initial condition x = γ a such that yi – xi ∞ ≤ i k= ε –i k= ε A– i+–k ∞ A k– ∞ (for i ≥ ), (for i < ) In view of () and the last inequality, we have a – A–i γ ai+ γ a ≤ ∞ i k= ε –i k= ε A– i+–k ∞ A k– ∞ (for i ≥ ), (for i < ) () Jung Advances in Difference Equations (2015) 2015:170 Page 11 of 13 √ α– α +β Since the matrix A has two distinct eigenvalues λ = and λ = did in the proof of Corollary ., if i is a negative integer, then we get A–i = –i γ (λ–i – λ ) –i γ (λ – λ ) γ (λ–i – λ ) α+ √ α +β , as we –i –λ λ (λ–i – λ ) –γ λ λ (λ–i– – λ–i– ) for all integers i < By using () and this equality, we have λ –i λ –i – aλ–a λ + aλ–a λ –λ –λ a –a λ –i a –a λ –i γ ai+ – γ λ –λ λ + γ λ –λ λ –i ≤ ∞ ε A k– ∞ () k= for any integer i < √ –α– α +β On the other hand, the inverse matrix A– has two distinct eigenvalues ω = = β √ +β –α+ α = – β λ In a similar way to the proof of Corollary ., if i is a – β λ and ω = β nonnegative integer, then we obtain – γ ω ω (ωi– – ωi– ) γ (ω – ω ) γ ω ω (ωi – ωi ) i A–i = A– = ωi – ωi γ (ωi+ – ωi+ ) for all integers i ≥ Thus, it follows from () and the last equality that ω i ω i – aω –a ω + aω –a ω –ω –ω a –a ω i+ a –a ω i+ γ ai+ – γ ω –ω ω + γ ω –ω ω i ≤ ∞ ε A– i+–k ∞ () k= for any integer i ≥ Finally, considering () and (), we define ci := a –a ω i ω i ω – aω –a ω ω –ω –ω a –a λ –i a –a λ –i λ – λ –λ λ λ –λ (for i ≥ ), (for i < ) We then have c = a , c = a , and it follows from () and () that i k= ε –i k= ε |ai – ci | ≤ A– i+–k ∞ A k– ∞ (for i ≥ ), (for i < ) Furthermore, it is easy to verify that the sequence {ci }i∈Z satisfies ci = αci+ + βci+ for any integer i If we set γ = α+ √ α +β and β > in Corollary ., then we obtain the equalities in (): ⎧ √ √ ⎨ A = max α+ α +β , –α+ α +β , ∞ √ √ α+ α +β –α+ α +β ⎩ A– = max , ∞ β β Jung Advances in Difference Equations (2015) 2015:170 Page 12 of 13 Thus, we get lim A β→∞ ∞ · A– = lim ∞ β→∞ β · √ = β If β is large in absolute value, then the second-order difference equation () has the Hyers-Ulam stability as we see in the next example Example . Given an ε > , assume that a sequence {ai }i∈Z of complex numbers satisfies the inequality |ai – ai+ – ai+ | ≤ ε () √ √ for all i ∈ Z With α = , β = , and γ = + , it follows from () that A ∞ = + and √ A– ∞ = + Using these values, Corollary . implies that there exists a sequence {ci }i∈Z of complex numbers such that c = a , c = a , ci = ci+ + ci+ , and |ai – ci | ≤ √ ( + ) – √ + √ + –i √ + i ε – ε (for i ≥ ), (for i < ) If we apply Corollary . with the inequality + ai– – ai– ≤ ε, () √ where we set α = – , β = , and γ = – , then there exists a sequence {ci }i∈Z of complex numbers such that c– = a– , c = a , ci = ci+ + ci+ , and √ |ai – ci | ≤ + √ + √ + i ε + –i – ε √ – (for i ≥ ), (for i < ) The inequalities () and () are equivalent In this case with inequality () or (), there is more efficiency with Corollary . than Corollary . for any integer i Competing interests The author declares that he has no competing interests Author’s contributions The author read and approved the final manuscript Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No 2013R1A1A2005557) Received: 26 February 2015 Accepted: 17 May 2015 References Cull, P, Flahive, M, Robson, R: Difference Equations: From Rabbits to Chaos Springer, Berlin (2005) Greene, DH, Knuth, DE: Mathematics for the Analysis of Algorithms, 2nd edn Birkhäuser, Basel (1982) Brzde¸k, J, Jung, S-M: A note on stability of a linear functional equation of second order connected with the Fibonacci numbers and Lucas sequences J Inequal Appl 2010, Article ID 793947 (2010) Brzde¸k, J, Jung, S-M: A note on stability of an operator linear equation of the second order Abstr Appl Anal 2011, Article ID 602713 (2011) Brzde¸k, J, Popa, D, Xu, B: Note on the nonstability of the linear recurrence Abh Math Semin Univ Hamb 76, 183-189 (2006) Jung Advances in Difference Equations (2015) 2015:170 Page 13 of 13 Brzde¸k, J, Popa, D, Xu, B: The Hyers-Ulam stability of linear equations of higher orders Acta Math Hung 120, 1-8 (2008) Popa, D: Hyers-Ulam-Rassias stability of a linear recurrence J Math Anal Appl 309, 591-597 (2005) Popa, D: Hyers-Ulam stability of the linear recurrence with constant coefficients Adv Differ Equ 2005, 101-107 (2005) Trif, T: Hyers-Ulam-Rassias stability of a linear functional equation with constant coefficients Nonlinear Funct Anal Appl 11, 881-889 (2006) 10 Czerwik, S: Functional Equations and Inequalities in Several Variables World Scientific, Singapore (2002) 11 Hyers, DH: On the stability of the linear functional equation Proc Natl Acad Sci USA 27, 222-224 (1941) 12 Hyers, DH, Isac, G, Rassias, TM: Stability of Functional Equations in Several Variables Birkhäuser, Boston (1998) 13 Jung, S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis Springer Optimization and Its Applications, vol 48 Springer, New York (2011) 14 Ulam, SM: A Collection of Mathematical Problems Interscience, New York (1960) 15 Brzde¸k, J, Popa, D, Xu, B: The Hyers-Ulam stability of nonlinear recurrences J Math Anal Appl 335, 443-449 (2007) 16 Brzde¸k, J, Popa, D, Xu, B: Remarks on stability of the linear recurrence of higher order Appl Math Lett 23, 1459-1463 (2010) 17 Xu, B, Brzde¸k, J: Hyers-Ulam stability of a system of first order linear recurrences with constant coefficients Discrete Dyn Nat Soc 2015, Article ID 269356 (2015) 18 Koshy, T: Fibonacci and Lucas Numbers with Applications Wiley, New York (2001)