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The z-transform method for the Ulam stability of linear difference equations with constant coefficients
Advances in Difference Equations volume 2018, Article number: 396 (2018)
Abstract
Applying the z-transform method, we study the Ulam stability of linear difference equations with constant coefficients. To a certain extent, our results can be viewed as an important complement to the existing methods and results.
Introduction
The notion of the Ulam stability was originated from a question on group homomorphisms posed by Ulam [24] in 1940. The essential problem of this type of stability is summarized as follows: “Under what conditions can a solution of a perturbed equation be close to a solution of the original equation?” In the following year, Hyers [9] gave a first affirmative partial answer to the Cauchy equation in a Banach space. Afterward, this work was generalized by Rassias [19] for linear mappings by considering unbounded Cauchy differences. It is worth mentioning that Rassias’ work has a great impact on the development of the Ulam stability of functional equations. Since then, almost all studies related to the Ulam stability have focused on different types of functional equations or abstract spaces. For more detailed results, we refer to the monographs [1, 7, 10, 12, 21] and references therein.
Let \((X,\|\cdot \|)\) be a normed space, n a positive integer, and \(F:X^{n}\rightarrow X\) a mapping. The nth-order difference equation
is said to have Hyers–Ulam stability or to be stable in the Hyers–Ulam sense if whenever a given \(\varepsilon >0\) and a sequence \(\{x_{k}\}\) satisfy the inequality
there exists a solution \(\{y_{k}\}\) of (1) such that
where \(K(\varepsilon)\) depends only on ε, and \(\lim_{\varepsilon \rightarrow 0}K(\varepsilon)=0\). More generally, we say that the difference equation (1) is Hyers–Ulam–Rassias stable or generalized Hyers–Ulam stable if ε and \(K(\varepsilon)\) are replaced by two control sequences \(\varphi_{k}\) and \(\Phi_{k}\), respectively.
In 2005, Popa [17] initiated the study of the Ulam stability of difference equations. Specifically, the author proved the Hyers–Ulam–Rassias stability of the linear difference equation of first order \(x_{k+1}=a_{k}x_{k}+b_{k}\) in a Banach space. At the same time, Popa [18] also established the Hyers–Ulam stability of higher-order linear difference equations with constant coefficients. These results showed that the Hyers–Ulam stability of a linear difference equation with constant coefficients depends strongly on the roots of the characteristic equation. Several examples showed that the difference equation is not Hyers–Ulam stable if the characteristic equation admits a root with modulus equal to 1. For this reason, Brzdȩk et al. [2] considered the nonstability of linear difference equations with constant coefficients. In 2007, Brzdȩk et al. [3] investigated the Ulam stability of the nonlinear difference equation \(x_{k+1}=a_{k}(x _{k})+b_{k}\) in an Abelian group with invariant metric d. Afterward, Brzdȩk et al. [4] presented the \((\varepsilon_{k})_{k\geq 0}\)-stability of difference equations that is weaker than the Hyers–Ulam stability, and they considered the \((\varepsilon_{k})_{k\geq 0}\)-stability of first-order linear difference equations. Based on the previous related works, they also gave a systematic answer to the Hyers–Ulam stability of higher-order linear difference equations with constants coefficients. In 2015, Xu and Brzdȩk [26] studied the Hyers–Ulam stability of systems of first-order linear difference equations with constant coefficients in a Banach space. Furthermore, Brzdȩk and Wójcik [6] established the Ulam stability of two kinds of difference equations in a metric space. As far as we know, this is the most general result associated with the Ulam stability of difference equations. In addition, Jung [13] also considered the Hyers–Ulam stability of first-order linear homogeneous matrix difference equations. Later, Jung and Nam [14] investigated the Hyers-stability of the Pielou logistic difference equation from a more general perspective. Recently, Onitsuka [16] established the Hyers-stability of first-order nonhomogeneous linear difference equations with constant stepsize, and the author further considered the best Hyers–Ulam stability constant of the corresponding difference equations.
As is well known, many different methods for solving differential equations have been used to study the Ulam stability of the corresponding equations, such as the integrating factor [22, 25], the power series method [11], the Laplace transform [20], the method of variation of constants [23], and so on. The Laplace transform method has a significant advantage in solving linear differential equations with constant coefficients, because it can turn a differential equation into an algebraic one. As a discrete case, the z-transform has a similar advantage in solving linear difference equations with constant coefficients. Inspired by the reference [20], the main purpose of this paper is to investigate the Ulam stability of linear difference equations with constant coefficients by using the z-transform method.
The z-transform and inverse z-transform
In this section, we recall some basic notions and results related to the z-transform and inverse z-transform. We denote by \(\mathbb{N}\), \(\mathbb{R}\), and \(\mathbb{C}\) the sets of nonnegative integers, real numbers, and complex numbers, respectively. Moreover, \(\mathbb{F}\) denotes either the real field \(\mathbb{R}\) or the complex field \(\mathbb{C}\).
Definition 2.1
(Kelley and Peterson [15])
Let \(\{x_{k}\}\) be a sequence in \(\mathbb{F}\). The z-transform of \(\{x_{k}\}\) is the function \(X(z)\) of a complex variable \(z\in \mathbb{C}\) defined by
and we say that the z-transform of \(\{x_{k}\}\) exists if there is a real number \(r>0\) such that the series \(\sum_{k=0}^{\infty }\frac{x _{k}}{z^{k}}\) converges for \(|z|>r\).
A sequence \(\{x_{k}\}\) is said to be exponentially bounded if there exist \(M>0\) and \(c>1\) such that
The following result is a sufficient condition for the existence of the z-transform of a sequence.
Theorem 2.1
(Kelley and Peterson [15])
If the sequence \(\{x_{k}\}\) is exponentially bounded, then the z-transform \(X(z)\) of \(\{x_{k}\}\) exists.
According to Definition 2.1, it is easy to show that the z-transform is linear, that is, if the two sequences \(\{x_{k}\}\) and \(\{y_{k}\}\) are exponentially bounded, then
for all z in the common domain of \(X(z)\) and \(Y(z)\).
Remark 1
Suppose that the sequence \(\{x_{k}\}\) is exponentially bounded. For a positive integer n, it is easy to verify that
Definition 2.2
(Gradshteyn and Ryzhik [8])
Suppose that \(X(z)\) is the z-transform of the sequence \(\{x_{k}\}\) with the domain of convergence \(D=\{z\in \mathbb{C}: |z|>r\}\). The inverse z-transform \(\{x_{k}\}\) of \(X(z)\) is given by
where C is a counterclockwise simple closed contour encircling all poles of \(X(z)\) and lying entirely within the domain of convergence D.
Now we define the unit step sequence \(\{u_{k}(n)\}\), \(n\geq 1\), and the unit impulse sequence \(\{\delta_{k}(m)\}\), \(m\geq 0\), respectively, by
It follows immediately from Definition 2.1 that
Notice that the convolution of two sequences \(\{x_{k}\}\) and \(\{y_{k}\}\) is defined by
Theorem 2.2
(Kelley and Peterson [15])
Suppose that \(X(z)\) and \(Y(z)\) exist for \(|z|>r_{1}\) and \(|z|>r_{2}\), respectively. Then
for \(|z|>\max \{r_{1},r_{2}\}\).
Theorem 2.3
(Kelley and Peterson [15])
Let \(\{y_{k}\}\) be a sequence such that the z-transform \(Y(z)=Z(y_{k})\) exists for \(|z|>r\). Then
where \(k^{(n)}=k(k+1)(k+2)\cdots (k+n-1)\).
Lemma 2.4
Let \(a\in \mathbb{C}\). Then
for \(|z|>|a|\).
Proof
Since \(Z(a^{k})=\frac{z}{z-a}\) for \(|z|>|a|\), from Theorem 2.3 we can infer that
□
The Ulam stability of linear difference equations with constant coefficients
Theorem 3.1
(Kelley and Peterson [15])
Suppose that the sequence \(\{f_{k}\}\) is exponentially bounded. Then each solution of the nth-order linear difference equation
is exponentially bounded, and hence its z-transform exists.
First of all, we consider the Ulam stability of first-order linear difference equations with constant coefficients.
Theorem 3.2
Let \(\{f_{k}\}\) and \(\{\varphi_{k}\}\) be two exponentially bounded sequences, and let \(\{\varphi_{k}\}\) be a positive sequence. Assume that \(p\in \mathbb{F}\setminus \{0\}\). If a sequence \(\{x_{k}\}\) satisfies the inequality
for all \(k\in \mathbb{N}\), then there exists a solution \(\{y_{k}\}\) of the first-order linear difference equation
such that
for all \(k\in \mathbb{N}\).
Proof
Set \(v_{k}=x_{k+1}-p x_{k}-f_{k}\), \(k\in \mathbb{N}\). Obviously, \(v_{k}\) is exponentially bounded. Taking into account Theorem 2.1, Theorem 3.1, and Remark 1 and applying the z-transform to \(v_{k}\), we have
where \(V(z)\) and \(F(z)\) denote the z-transforms of the sequences \(\{v_{k}\}\) and \(\{f_{k}\}\), respectively. Then, we obtain
Set
It is easily seen that \(y_{0}=x_{0}\). By Theorem 2.2, using the identity \(p^{k-1}u_{k}(1)=p^{k}\ast \delta_{k}(1)\), we get
Then we obtain
Notice that the inverse z-transform gives \(y_{k+1}=py_{k}+f_{k}\). This shows that \(y_{k}\) is a solution of the difference equation (3).
From (5) and (6) it follows that
Again using the inverse z-transform, we obtain
From (2) we can infer that
which completes the proof. □
As a direct consequence of Theorem 3.2, for \(|p|<1\), we obtain the Hyers–Ulam stability of the linear difference equation (3).
Corollary 3.3
Let \(\{f_{k}\}\) be given as in Theorem 3.2, and let \(0<|p|<1\). For a given \(\varepsilon >0\), if a sequence \(\{x_{k}\}\) satisfies the inequality
for all \(k\in \mathbb{N}\), then there exists a solution \(\{y_{k}\}\) of the first-order linear difference equation (3) such that
for all \(k\in \mathbb{N}\).
Corollary 3.4
Let \(\{f_{k}\}\) be given as in Theorem 3.2, and let \(|p|=1\). Assume that \(\lambda >1\). For a given \(\varepsilon >0\), if a sequence \(\{x_{k}\}\) satisfies the inequality
for all \(k\in \mathbb{N}\), then there exists a solution \(y_{k}\) of the first-order linear difference equation (3) such that
for all \(k\in \mathbb{N}\).
Lemma 3.5
Let
and
where \(m,n\in \mathbb{N}\) with \(m< n\), and \(\alpha_{j}\) and \(\beta_{j}\) are scalars. Then there exists a sequence \(\{g_{k}\}\) such that
for \(|z|>r_{p}\), where \(r_{p}=\max \{|z_{1}|,|z_{2}|,\ldots,|z_{l}| \}\), and \(z_{i}\), \(i=1,2,\ldots,l\), are mutually different roots of the polynomial equation \(P(z)=0\).
Proof
Using the polynomial decomposition theorem, the polynomial \(P(z)\) can be decomposed as
for some complex numbers \(z_{i}\), \(n_{i}\in \mathbb{N}\), \(i=1,2, \ldots,l\), such that \(n_{1}+n_{2}+\cdots +n_{l}=n\).
Furthermore, applying the partial fraction decomposition, we obtain
where \(\lambda_{ij}\) is a scalar for all \(i=1,2,\ldots,l\), \(j=1,2,\ldots,n_{i}\).
Now we set
where \(i=1,2,\ldots,l\) and \(j=1,2,\ldots,n_{i}\).
Define
From Lemma 2.4 and the linearity of the z-transform it follows that
for \(|z|>r_{p}\), where \(r_{p}=\max \{|z_{1}|,|z_{2}|,\ldots,|z_{l}| \}\). □
Lemma 3.6
Let \(\{f_{k}\}\) be an exponentially bounded sequence, and let \(P(z)\) be a complex polynomial of degree \(n\geq 1\). Then there exists a sequence \(\{h_{k}\}\) such that
for \(|z|>\max \{r_{f},r_{p}\}\), where \(r_{p}=\max \{|z^{\ast }|: P(z ^{\ast })=0\}\), and \(r_{f}\) is a nonnegative real number such that the z-transform of \(\{f_{k}\}\) exists for \(|z|>r_{f}\).
Proof
Set \(Q(z)=1\). Note that \(P(z)\) is a complex polynomial of degree \(n\geq 1\). By Lemma 3.5 there exists a sequence \(\{g_{k}\}\) such that
for \(|z|>r_{p}\). Define \(h_{k}=g_{k}\ast f_{k}\). For any \(|z|>\max \{r _{f},r_{p}\}\), by Theorem 2.2 we have
□
Now we discuss the Ulam stability of linear difference equations of nth order (\(n>1\)).
Theorem 3.7
Let \(\{f_{k}\}\) and \(\{\varphi_{k}\}\) be two exponentially bounded sequences, and let \(\{\varphi_{k}\}\) be a positive sequence. Assume that \(p_{0},p_{1},\ldots,p_{n-1}\) are scalars, \(n\in \mathbb{N}\setminus \{0,1\}\). If a sequence \(\{x_{k}\}\) satisfies the inequality
for all \(k\in \mathbb{N}\), then there exists a solution \(\{y_{k}\}\) of the nth-order linear difference equation
such that
for all \(k\in \mathbb{N}\), where \(\widetilde{p_{k}}=Z^{-1} (\frac{1}{P _{n,0}(z)} )\), and \(P_{n,0}(z)\) is the characteristic polynomial of the corresponding homogeneous equation of (10).
Proof
By Remark 1, for any \(n\in \mathbb{N}\setminus \{0,1\}\), we have
For notational convenience, we let \(p_{n}=1\). Then, we note that a sequence \(\{\widetilde{y}_{k}\}\) is a solution of (10) if and only if
where the polynomial \(P_{n,j}(z)\) is given by
Set
Similarly, applying the z-transform to both sides of (13), we obtain
Then we have
For convenience, we assume that the z-transform of \(\{f_{k}\}\) exists for \(|z|>r_{f}\). Let \(z_{1},z_{2},\ldots,z_{n}\) be the roots of the polynomial \(P_{n,0}(z)\), and let \(r_{p}=\max \{|z_{i}|:i=1,2,\ldots n \}\).
For \(|z|>\max \{r_{f},r_{p}\}\), we let
By Lemma 3.6 there exists a sequence \(\{\widetilde{f}_{k}\}\) such that
for \(|z|>\max \{r_{f},r_{p}\}\). Moreover, we also notice that
for \(j=1,2,\ldots,n\) and \(|z|>r_{p}\).
Let \(Q(z)=p_{0}+p_{1}z+\cdots +p_{j-1}z^{j-1}\) and \(P(z)=z^{j-1}P_{n,0}(z)\). By Lemma 3.5 there exists a sequence \(\{g_{k}\}\) such that
Let us define
Furthermore, we define
Then it follows from (15)–(20) that
for \(|z|>\max \{r_{f},r_{p}\}\). From (14) and (21) we know that the sequence \(\{y_{k}\}\) given in (20) is a solution of (10). Meantime, we can infer from (14) and (20) that
Therefore, using the inverse z-transform, we get
for each \(k\in \mathbb{N}\), where \(\widetilde{p_{k}}=Z^{-1} (\frac{1}{P _{n,0}(z)} )\). □
Remark 2
In fact, \(P_{n,0}(z)\) is the characteristic polynomial of the corresponding homogeneous equation of (10). Assume that \(z_{1},z_{2}, \ldots,z_{l}\) are distinct roots of the polynomial equation \(P_{n,0}(z)=0\) with multiplicities \(n_{1},n_{2},\ldots,n_{l}\), respectively, such that \(n_{1}+n_{2}+\cdots +n_{l}=n\). Then we obtain
Using the partial fraction decomposition, we have
where \(\mu_{ij}\) are scalars, \(i=1,2,\ldots,l\), \(j=1,2,\ldots,n_{i}\).
Similar to the proof of Lemma 3.5, we define
where \(i=1,2,\ldots,l\) and \(j=1,2,\ldots,n_{i}\). Furthermore, we set
Hence we can infer that
Then the inverse z-transform implies that
From the previous statement we can obtain the following results associated with the Ulam stability of the linear difference equation (10).
Corollary 3.8
Let \(\{f_{k}\}\), \(\{\varphi_{k}\}\) be given as in Theorem 3.7, and let \(p_{0},p_{1},\ldots,p_{n}\) be scalars with \(p_{n}=1\), \(n\in \mathbb{N}\setminus \{0,1\}\). Assume that \(z_{1},z_{2},\ldots,z_{l}\) are distinct roots of the polynomial equation \(P_{n,0}(z)=0\) with multiplicities \(n_{1},n_{2},\ldots,n_{l}\), respectively, such that \(n_{1}+n_{2}+\cdots +n_{l}=n\). If a sequence \(\{x_{k}\}\) satisfies inequality (9) for all \(k\in \mathbb{N}\), then there exists a solution \(\{y_{k}\}\) of the nth-order linear difference equation (10) such that
for each \(k\in \mathbb{N}\), where \(\widetilde{p_{m}}\) is defined by (22).
Corollary 3.9
Let \(\{f_{k}\}\) be an exponentially bounded sequence, and let \(p_{0},p_{1},\ldots,p_{n}\) be scalars with \(p_{n}=1\), \(n\in \mathbb{N}\setminus \{0,1\}\). Assume that \(z_{1},z_{2},\ldots,z_{n}\) are n distinct roots of the polynomial equation \(P_{n,0}(z)=0\) with \(\max \{|z_{i}|:i=1,2,\ldots,n\}<1\). For a given \(\varepsilon >0\), if a sequence \(\{x_{k}\}\) satisfies the inequality
for all \(k\in \mathbb{N}\), then there exists a solution \(\{y_{k}\}\) of the nth-order linear difference equation (10) such that
for all \(k\in \mathbb{N}\), where \(\mu_{i}\) can be determined by the partial fraction decomposition of \(\frac{1}{P_{n,0}(z)}\).
Proof
By Corollary 3.8 there exists a solution \(\{y_{k}\}\) of (10) such that
However, if \(z_{i}\neq 0\) for each i, then by (22) we obtain
which implies the desired result. Moreover, if there exists \(i_{0}\in \{1,2,\ldots,n\}\) such that \(z_{i_{0}}=0\), then the foregoing result remains true. □
Corollary 3.10
Let \(\{f_{k}\}\) be an exponentially bounded sequence, and let \(p_{0},p_{1},\ldots,p_{n}\) be scalars with \(p_{n}=1\), \(n\in \mathbb{N}\setminus \{0,1\}\). Assume that \(z_{1},z_{2},\ldots,z_{l}\) are roots of the polynomial equation \(P_{n,0}(z)=0\) with multiplicities \(n_{1},n_{2},\ldots,n_{l}\), respectively, such that \(n_{1}+n_{2}+ \cdots +n_{l}=n\), \(\max \{|z_{i}|:i=1,2,\ldots,l\}<1\). For a given \(\varepsilon >0\), if a sequence \(\{x_{k}\}\) satisfies inequality (23) for all \(k\in \mathbb{N}\), then there exists a solution \(\{y_{k}\}\) of the nth-order linear difference equation (10) such that
where \(\mu_{ij}\) can be determined by the partial fraction decomposition of \(\frac{1}{P_{n,0}(z)}\).
Proof
Without loss of generality, we assume that \(z_{1}=0\) with multiplicity \(n_{1}< l\), and \(z_{2},z_{3},\ldots,z_{t}\) are simple roots, that is, \(n_{2}=n_{3}=\cdots =n_{t}=1\), \(2\leq t< l\). Just notice that
□
Conclusion
By using the z-transform method we have established the Ulam stability of linear difference equations with constant coefficients. In fact, the results obtained in this paper can be regarded as a discrete analogue of the stability results for linear differential equations in [20]. To a certain extent, our results constitute an important complement to the stability results obtained in [4, 5]. Additionally, this paper also provides another method to study the Ulam stability of difference equations. Obviously, this paper shows that the z-transform method is more convenient to study the Ulam stability of linear difference equations with constant coefficients.
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Acknowledgements
The authors would like to express their sincere thanks to the editor and three anonymous referees for their useful comments and valuable suggestions, which helped to improve the quality of the paper.
Funding
This work was supported by the National Natural Science Foundation of China (No. 11701425). The second author acknowledges the support from the National Natural Science Foundation of China (No. 11571378).
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Shen, Y., Li, Y. The z-transform method for the Ulam stability of linear difference equations with constant coefficients. Adv Differ Equ 2018, 396 (2018). https://doi.org/10.1186/s13662-018-1843-0
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DOI: https://doi.org/10.1186/s13662-018-1843-0
Keywords
- Ulam stability
- z-transform
- Inverse z-transform
- Linear difference equations