Optimal Hyers-Ulam’s constant for the linear differential equations
- Soon-Mo Jung^{1},
- Jaiok Roh^{2}Email author and
- Juri Lee^{3}
https://doi.org/10.1186/s13660-016-1145-6
© Jung et al. 2016
Received: 20 February 2016
Accepted: 11 August 2016
Published: 22 August 2016
Abstract
In this paper, we will obtain the optimal Hyers-Ulam’s constant for the first-order linear differential equations \(p(t)y'(t) - q(t)y(t) - r(t) = 0\).
Keywords
MSC
1 Introduction
The question concerning the stability of functional equations has been originally raised by Ulam [1]: Given a metric group \((G, \cdot, d)\), a positive number ε, and a function \(f : G \to G\) that satisfies the inequality \(d ( f(xy), f(x)f(y) ) \leq\varepsilon\) for all \(x, y \in G\), do there exist an homomorphism \(a : G \to G\) and a constant δ depending only on G and ε such that \(d ( a(x), f(x) ) \leq\delta\) for all \(x \in G\)?
If the answer to this question is affirmative, then the functional equation \(a(xy) = a(x)a(y)\) is said to be stable. A first answer to this question was given by Hyers [2] in 1941, who proved that the Cauchy additive equation is stable in Banach spaces. In general, a functional equation is said to be stable in the sense of Hyers and Ulam (or the equation has the Hyers-Ulam stability) if for each solution to the perturbed equation, there exists a solution to the equation that differs from the solution to the perturbed equation with a small error. We recommend the reader to refer [3] for the exact definition of Hyers-Ulam stability and an excellent survey on that subject.
After Hyers’ result, mathematicians extended and generalized Hyers’ theorem in various directions. Aoki [4] and Rassias [5] provided a generalization of the Hyers theorem for additive and linear mappings, respectively. Găvruţa [6] generalized it by considering a general control function instead of ε.
Obloza [7, 8] investigated the Hyers-Ulam stability for the linear differential equations. Thereafter, Alsina and Ger [9] considered a differentiable function \(f : I \to{\mathbf{R}}\) that satisfies the differential inequality \(| y'(t) - y(t) | \leq\varepsilon\) and proved that there exists a solution \(f_{0} : I \to{\mathbf{R}}\) of the differential equation \(y'(t) = y(t)\) such that \(| f(t) - f_{0}(t) | \leq3\varepsilon\) for any \(t \in I\). In this paper, we prove that \(| f(t) - f_{0}(t) | \leq\varepsilon\) for any \(t \in I\). So, our result is a better estimation. Miura et al. [10, 11], Takahasi et al. [12, 13], and Jung [14] also generalized their results.
Furthermore, the Hyers-Ulam stability for the nonhomogeneous linear differential equations of the form \(y'(t) + g(t)y(t) + h(t) = 0\) has been investigated by many mathematicians (see [13, 15–18]).
Proposition 1.1
([18])
In the proof of the proposition, Wang et al. assumed that \(|q(t)| \geq1\) by saying without the loss of generality. Indeed, their proof is just for the case of \(|q(t)| \geq1\), and they said nothing about general cases. In this paper, we clearly generalize their assumption for the proof.
It seems that in Proposition 1.1, Wang et al. denoted a Hyers-Ulam constant K as \(2\exp \{ |\int_{a}^{b}\frac{q(s)}{p(s)}\,ds | \}-1\). In this case, the constant K is very big when \(b-a\) is a big number. Here, we find a Hyers-Ulam constant K that is independent of a and b.
After we introduce two examples, we will prove that, under the same assumptions of Proposition 1.1, we have a Hyers-Ulam constant K in (3) equal to \(\frac{1}{\mu}\), where μ is a lower bound of the function \(|q(t)|\).
2 Motivation
In this section, we assume that a and b are given real numbers with \(a < b\). Before we prove our main theorem, in the following two examples, we will see the reason why we try to obtain our main theorem.
Example 2.1
Example 2.2
3 Main theorem
In this section, let \(I = (a,b)\) be an arbitrary interval, where \(-\infty\leq a < b \leq\infty\). Now, we are in the position to introduce our main theorem.
Theorem 3.1
Proof
We use the method of integrating factor of Wang et al. [18].
Case 3: For the cases \(p(t) < 0\), \(q(t) \geq\mu> 0\) and \(p(t) < 0\), \(q(t) \leq-\mu< 0\), we can prove our assertion by similar calculations. □
Remark 3.2
Declarations
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01059467). Soon-Mo Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2015R1D1A1A02061826).
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
References
- Ulam, SM: A Collection of Mathematical Problems. Interscience, New York (1960) MATHGoogle Scholar
- Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222-224 (1941) MathSciNetView ArticleMATHGoogle Scholar
- Brillouët-Belluot, N, Brzdȩk, J, Ciepliński, K: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012, Article ID 716936 (2012) View ArticleMATHGoogle Scholar
- Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64-66 (1950) MathSciNetView ArticleMATHGoogle Scholar
- Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297-300 (1978) MathSciNetView ArticleMATHGoogle Scholar
- Găvruţa, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431-436 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Obloza, M: Hyers stability of the linear differential equation. Rocz. Nauk.-Dydakt., Pr. Mat. 13, 259-270 (1993) MathSciNetMATHGoogle Scholar
- Obloza, M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocz. Nauk.-Dydakt., Pr. Mat. 14, 141-146 (1997) MathSciNetMATHGoogle Scholar
- Alsina, C, Ger, R: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 2, 373-380 (1998) MathSciNetMATHGoogle Scholar
- Miura, T, Takahasi, SE, Choda, H: On the Hyers-Ulam stability of real continuous function valued differential map. Tokyo J. Math. 24, 467-476 (2001) MathSciNetView ArticleMATHGoogle Scholar
- Miura, T, Miyajima, S, Takahasi, SE: A characterization of Hyers-Ulam stability of first order linear differential operators. J. Math. Anal. Appl. 286, 136-146 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Takahasi, SE, Miura, T, Miyajima, S: On the Hyers-Ulam stability of the Banach space valued differential equation \(y' = \lambda y\). Bull. Korean Math. Soc. 39, 309-315 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Takahasi, SE, Takagi, H, Miura, T, Miyajima, S: The Hyers-Ulam stability constant of first order linear differential operators. J. Math. Anal. Appl. 296, 403-409 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Jung, S-M: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17, 1135-1140 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Miura, T, Hirasawa, G, Takahasi, SE: Note on the Hyers-Ulam-Rassias stability for the first order linear differential equation \(y'(t) + p(t)y(t) + q(t) = 0\). Int. Math. Sci. 22, 1151-1158 (2004) MathSciNetView ArticleGoogle Scholar
- Jung, S-M: Hyers-Ulam stability of linear differential equations of first order (II). Appl. Math. Lett. 19, 854-858 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Jung, S-M: Hyers-Ulam stability of linear differential equations of first order (III). J. Math. Anal. Appl. 311, 139-146 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Wang, G, Zhou, M, Sun, L: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024-1028 (2008) MathSciNetView ArticleMATHGoogle Scholar