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# Approximate perfect differential equations of second order

*Advances in Difference Equations*
**volume 2012**, Article number: 225 (2012)

## Abstract

In this paper we prove the Hyers-Ulam stability of the perfect linear differential equation f(t){y}^{\u2033}(t)+{f}_{1}(t){y}^{\prime}(t)+{f}_{2}(t)y(t)=Q(t), where f,y\in {C}^{2}[a,b], Q\in C[a,b], {f}_{2}(t)={f}_{1}^{\prime}(t)-{f}^{\u2033}(t) and -\mathrm{\infty}<a<b<+\mathrm{\infty}.

**MSC:**34K20, 26D10, 39B82, 34K06, 39B72.

## 1 Introduction

The question concerning the stability of group homomorphisms was posed by Ulam [1]. Hyers [2] solved the case of approximately additive mappings in Banach spaces and T.M. Rassias generalized the result of Hyers [3].

**Definition 1.1** Let *X* be a normed space over a scalar field and let *I* be an open interval. Assume that {a}_{0},{a}_{1},\dots ,{a}_{n}, h:I\to \mathbb{K} are continuous functions. We say that the differential equation

has the Hyers-Ulam stability if, for any function f:I\to X satisfying the differential inequality

for all t\in I and some \epsilon \ge 0, there exists a solution g:I\to X of (1.1) such that \parallel f(t)-g(t)\parallel \le K(\epsilon ) for all t\in I, where K(\epsilon ) is a function depending only on *ε*.

Obłoza [4, 5] was the first author who investigated the Hyers-Ulam stability of differential equations (also see [6]).

Jung [7] solved the inhomogeneous differential equation of the form {y}^{\u2033}+2x{y}^{\prime}-2ny={\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}, where *n* is a positive integer, and he used this result to prove the Hyers-Ulam stability of the differential equation {y}^{\u2033}+2x{y}^{\prime}-2ny=0 in a special class of analytic functions.

Li and Shen [8] proved that if the characteristic equation {\lambda}^{2}+\alpha \lambda +\beta =0 has two different positive roots, then the linear differential equation of second order with constant coefficients {y}^{\u2033}(x)+\alpha {y}^{\prime}(x)+\beta y(x)=f(x) has the Hyers-Ulam stability where y\in {C}^{2}[a,b], f\in C[a,b] and -\mathrm{\infty}<a<b<+\mathrm{\infty} (see also [9, 10]). Abdollahpour and Najati [11] proved that the third-order differential equation {y}^{(3)}(t)+\alpha {y}^{\u2033}(t)+\beta {y}^{\prime}(t)+\gamma y(t)=f(t) has the Hyers-Ulam stability. Ghaemi *et al.* [12] proved the Hyers-Ulam stability of the exact second-order linear differential equation

with {p}_{0}^{\u2033}(x)-{p}_{1}^{\prime}(x)+{p}_{2}(x)=0. Here {p}_{0}, {p}_{1}, {p}_{2}, f:(a,b)\to \mathbb{R} are continuous functions. For more results about the Hyers-Ulam stability of differential equations, we can refer to [13–21].

**Definition 1.2** We say that the differential equation

is perfect if it can be written as \frac{d}{dt}[f(t){y}^{\prime}(t)+({f}_{1}(t)-{f}^{\prime}(t))y(t)]=Q(t).

It is clear that the differential equation (1.2) is perfect if and only if {f}_{2}(t)={f}_{1}^{\prime}(t)-{f}^{\u2033}(t). The aim of this paper is to investigate the Hyers-Ulam stability of the perfect differential equation (1.2), where f,y\in {C}^{2}[a,b], Q\in C[a,b], {f}_{1}\in {C}^{1}[a,b], {f}_{2}(t)={f}_{1}^{\prime}(t)-{f}^{\u2033}(t) and -\mathrm{\infty}<a<b<+\mathrm{\infty}. More precisely, we prove that the equation (1.2) has the Hyers-Ulam stability.

## 2 Hyers-Ulam stability of the perfect differential equation f(t){y}^{\u2033}(t)+{f}_{1}(t){y}^{\prime}(t)+{f}_{2}(t)y(t)=Q(t)

In the following theorem, we prove the Hyers-Ulam stability of the differential equation (1.2).

Throughout this section, *a* and *b* are real numbers with -\mathrm{\infty}<a<b<+\mathrm{\infty}.

**Theorem 2.1**
*The perfect differential equation*

*has the Hyers*-*Ulam stability*, *where* f,y\in {C}^{2}[a,b], {f}_{1}\in {C}^{1}[a,b], Q\in C[a,b] *and* f(t)\ne 0 *for all* t\in [a,b].

*Proof* Let \epsilon >0 and y\in {C}^{2}[a,b] with

Let g(t)=f(t){y}^{\prime}+({f}_{1}(t)-{f}^{\prime}(t))y for all t\in [a,b]. It is clear that

We define

Then

Also, we have

for all x\in [a,b]. Now we define

for all x\in [a,b]. It is clear that u\in {C}^{2}[a,b] and

Therefore,

Hence, (2.1) implies that

Also, we have

for all x\in [a,b]. Since \frac{{f}_{1}}{f}\in C[a,b], there exist constants {m}^{\prime} and {M}^{\prime} such that {m}^{\prime}\u2a7d\frac{{f}_{1}(x)}{f(x)}\u2a7d{M}^{\prime}. Thus

for all x\in [a,b]. Since f\in C[a,b] and |f|>0, there exist constants 0<m\u2a7dM such that m\u2a7d|f(x)|\u2a7dM for all x\in [a,b]. Hence, (2.4) implies that

for all x\in [a,b]. It follows from (2.3) that

for all x\in [a,b]. □

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## Acknowledgements

CP was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299) and DYS was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Abdollahpour, M.R., Najati, A., Park, C. *et al.* Approximate perfect differential equations of second order.
*Adv Differ Equ* **2012, **225 (2012). https://doi.org/10.1186/1687-1847-2012-225

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DOI: https://doi.org/10.1186/1687-1847-2012-225

### Keywords

- Hyers-Ulam stability
- differential equation