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
linear differential equation Hyers-Ulam stability integrating factor perturbationMSC
34A40 34D10 34A30 39B821 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
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