Invariance of Hyers-Ulam stability of linear differential equations and its applications
- Ginkyu Choi^{1} and
- Soon-Mo Jung^{2}Email author
https://doi.org/10.1186/s13662-015-0617-1
© Choi and Jung 2015
Received: 16 June 2015
Accepted: 25 August 2015
Published: 5 September 2015
Abstract
We prove that the generalized Hyers-Ulam stability of linear differential equations of nth order (defined on I) is invariant under any monotone one-to-one correspondence \(\tau: I \to J\) which is n times continuously differentiable. Moreover, using this result, we investigate the generalized Hyers-Ulam stability of the linear differential equation of second order and the Cauchy-Euler equation.
Keywords
MSC
1 Introduction
Obłoza seems to be the first author who investigated the Hyers-Ulam stability of linear differential equations (see [5, 6]): Given real-valued constants a and b, let \(g, r : (a,b) \to\mathbf{R}\) be continuous functions with \(\int_{a}^{b} | g(x) | \,dx < \infty\). Assume that \(\varepsilon> 0\) is an arbitrary real number. Obłoza proved that if a differentiable function \(y : (a,b) \to\mathbf{R}\) satisfies the inequality \(| y'(x) + g(x) y(x) - r(x) | \leq\varepsilon\) for all \(x \in(a,b)\) and if a function \(y_{0} : (a,b) \to\mathbf{R}\) satisfies \(y'_{0}(x) + g(x) y_{0}(x) = r(x)\) for all \(x \in(a,b)\) and \(y(\tau) = y_{0}(\tau)\) for some \(\tau\in(a,b)\), then there exists a constant \(\delta> 0\) such that \(| y(x) - y_{0}(x) | \leq\delta\) for all \(x \in(a,b)\).
Thereafter, Alsina and Ger [7] proved that if a differentiable function \(y : (a,b) \to\mathbf{R}\) satisfies the differential inequality \(| y'(x) - y(x) | \leq\varepsilon\), then there exists a function \(y_{0} : (a,b) \to\mathbf{R}\) such that \(y_{0}'(x) = y_{0}(x)\) and \(| y(x) - y_{0}(x) | \leq3\varepsilon\) for all \(x \in(a,b)\). This result of Alsina and Ger was generalized by Takahasi et al. [8]. Indeed, they proved the Hyers-Ulam stability of the Banach space valued differential equation \(y'(x) = \lambda y(x)\) (see also [9–19]).
The main goal of this paper is to prove that the (generalized) Hyers-Ulam stability of the linear differential equations is invariant under any monotone one-to-one correspondence which is n times continuously differentiable. In other words, if the differential equation (1) has the (generalized) Hyers-Ulam stability, then the reduced differential equation (4) also has the (generalized) Hyers-Ulam stability, and vice versa.
Moreover, we investigate the generalized Hyers-Ulam stability of the linear differential equation of second order and the Cauchy-Euler equation.
2 Hyers-Ulam stability is invariant
In the following main theorem, we prove that the (generalized) Hyers-Ulam stability of the linear differential equation of nth order is invariant.
Theorem 2.1
Assume that the linear differential equation (1) defined on I can be reduced to another differential equation (4) defined on J via a monotone one-to-one correspondence \(\tau: I \to J\) which is n times continuously differentiable. If the differential equation (1) has the (generalized) Hyers-Ulam stability, so does the reduced differential equation (4).
Proof
If the differential equation (1) has the Hyers-Ulam stability and if an n times continuously differentiable function \(y : I \to\mathbf{R}\) satisfies the inequality (2) for all \(x \in I\) and for some \(\varepsilon> 0\), then there exists a solution \(y_{0} : I \to\mathbf{R}\) of the differential equation (1) such that the inequality (3) holds for any \(x \in I\), where \(K(\varepsilon)\) depends on ε only and satisfies \(\lim_{\varepsilon\to0} K(\varepsilon) = 0\).
Finally, it is obvious that \(z_{0}\) is a solution of the differential equation (4) by considering the last part of the Introduction.
The rest of the proof runs analogously to the first part of this proof. □
By exchanging the roles of the monotone one-to-one correspondence \(\tau: I \to J\) and its inverse \(\sigma: J \to I\), we can prove a corollary to Theorem 2.1.
3 Stability of linear differential equation of second order
The proof of the following lemma can be found in [25], Section 2.16.
Lemma 3.1
We now investigate the generalized Hyers-Ulam stability of the linear inhomogeneous differential equation of the second order (7) in the class of twice continuously differentiable functions.
Theorem 3.2
Proof
If we set \(c := a_{1} = a_{2}\) in Theorem 3.2 and use the equality (14), then we obtain the following corollary.
Corollary 3.3
4 Hyers-Ulam stability of Cauchy-Euler equation
By using Theorem 2.1 and Corollary 3.3, we prove the generalized Hyers-Ulam stability of the Cauchy-Euler equation (15) for the case of \((\alpha- 1)^{2} - 4 \beta> 0\).
Theorem 4.1
Proof
Furthermore, by Corollary 3.3, the linear differential equation (18) has the generalized Hyers-Ulam stability. Therefore, due to Theorem 2.1, the Cauchy-Euler equation (15) has the generalized Hyers-Ulam stability.
If we set \(\varphi(x) = \varepsilon\) in Theorem 4.1, then we get the following corollary.
Corollary 4.2
Proof
We now consider the case when \((\alpha-1)^{2} - 4\beta= 0\) and use Theorem 2.1 and Corollary 3.3 to prove the generalized Hyers-Ulam stability of the inhomogeneous Cauchy-Euler equation (15).
Theorem 4.3
Proof
Analogously to the proof of Theorem 4.1, we define a monotone one-to-one correspondence \(\tau: (0,\infty) \to\mathbf{R}\) and a twice continuously differentiable function \(z : \mathbf{R} \to\mathbf{R}\) by \(\tau(x) = \ln x = t\) and \(z(t) = y(x) = y(e^{t})\), respectively. In a similar way to the first part of the proof of Theorem 4.1, the Cauchy-Euler equation (15) has the generalized Hyers-Ulam stability.
If we set \(\varphi(x) = \varepsilon\) in Theorem 4.3, then we obtain the following corollary.
Corollary 4.4
Proof
We apply Theorem 2.1 and Corollary 3.3 to prove the generalized Hyers-Ulam stability of the Cauchy-Euler equation (15) for the case of \((\alpha-1)^{2} - 4\beta< 0\).
Theorem 4.5
Proof
In a similar way to the proofs of Theorems 4.1 and 4.3, we conclude that the Cauchy-Euler equation (15) has the generalized Hyers-Ulam stability.
If we set \(\varphi(x) = \varepsilon\) in Theorem 4.5, then we can easily prove the following corollary.
Corollary 4.6
Proof
Declarations
Acknowledgements
The authors are very grateful to anonymous referees for their kind comments and suggestions. 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). This work was supported by 2015 Hongik University Research Fund.
Open Access 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.
Authors’ Affiliations
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