#### Abstract

The aim of this paper is to prove the stability in the sense of Hyers-Ulam of differential equation of second order . That is, if is an approximate solution of the equation , then there exists an exact solution of the equation near to .

#### 1. Introduction and Preliminaries

In 1940, Ulam [1] posed the following problem concerning the stability of functional equations: give conditions in order for a linear mapping near an approximately linear mapping to exist. The problem for the case of approximately additive mappings was solved by Hyers [2] when and are Banach spaces, and the result of Hyers was generalized by Rassias (see [3]). Since then, the stability problems of functional equations have been extensively investigated by several mathematicians (cf. [3–5]).

In connection with the stability of exponential functions, Alsina and Ger [6] remarked that the differential equation has the Hyers-Ulam stability. More explicitly, they proved that if a differentiable function satisfies for all , then there exists a differentiable function satisfying for any such that for every.

The above result of Alsina and Ger has been generalized by Miura et al. [7], by Miura [8], and also by Takahasi et al. [9]. Indeed, they dealt with the Hyers-Ulam stability of the differential equation , while Alsina and Ger investigated the differential equation .

Furthermore, the result of Hyers-Ulam stability for first-order linear differential equations has been generalized by Miura et al. [10], by Takahasi et al. [11], and also by Jung [12]. They dealt with the nonhomogeneous linear differential equation of first order

Jung [13] proved the generalized Hyers-Ulam stability of differential equations of the form

and also applied this result to the investigation of the Hyers-Ulam stability of the differential equation

Recently, Wang et al. [14] discussed the Hyers-Ulam stability of the first-order nonhomogeneous linear differential equation

They proved the following theorem.

Theorem 1.1 (see [14]). *Let and be continuous real functions on the interval such that and for all and some independent of . Then (1.4) has the Hyers-Ulam stability.*

The aim of this paper is to investigate the Hyers-Ulam stability of the following linear differential equations of second order under some special conditions:

where .

For the sake of convenience, all the integrals in the rest of the work will be viewed as existing. We say that (1.5) has the Hyers-Ulam stability if there exists a constant with the following property: for every , if

then there exists some satisfying

such that . We call such a Hyers-Ulam stability constant for (1.5).

#### 2. Main Results

Now, the main results of this work are given in the following theorems.

Theorem 2.1. *If a twice continuously differentiable function satisfies the differential inequality**
for all and for some , and the Riccati equation has a particular solution, then there exists a solution of (1.5) such that
**
where is a constant.*

*Proof. *Let and be a continuously differentiable function such that

We will show that there exists a constant independent of and such that for some satisfying .

Assume that is a particular solution of Riccati equation ; if we set

then
thus

Using the similar technique in [14], we can prove

satisfying
and there exists an such that

By , we get

Using the same technique as above, we know that

satisfying
and there exists a such that
The desired conclusion is proved.

Theorem 2.2. *Let and be continuous real functions on the interval such that and is a nonzero bounded particular solution . If is a twice continuously differentiable function, which satisfies the differential inequality
**
for all and for some , then there exists a solution such that
**
where is a constant, and satisfies .*

*Proof. *Setting , we obtain

and also

By a simple calculation, we see that

Without loss of generality, we may assume that . Using the similar technique in [14], we know that

satisfies
and also
for some .

From the inequalities and the similar technique in [14], we further get that

satisfies
and also
for some .

Consequently, we have

for some positive constant .

Define . It then follows from the above inequality that holds for every . We can easily verify that satisfies . This completes the proof of our theorem.

We can prove the following corollaries by using an analogous argument. Hence, we omit the proofs.

Corollary 2.3. *Let and be continuous real functions on the interval such that and . If is a twice continuously differentiable function, which satisfies the differential inequality**
for all and for some , then there exists a solution such that
**
where is a constant, and satisfies .*

Corollary 2.4. *Let and be continuous real functions on the interval such that and is a nonzero bounded particular solution . If is a twice continuously differentiable function, which satisfies the differential inequality**
for all and for some , then there exists a solution such that
**
where is a constant, and satisfies .*

#### Acknowledgments

The work of this paper was supported by the National Natural Science Foundation of China (10871213). The authors would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper.