#### Abstract

By using of the Gronwall inequality, we prove the Hyers-Ulam stability of differential equations of second order with initial conditions.

#### 1. Introduction

In 1940, Ulam [1] posed a problem concerning the stability of functional equations: “Give conditions in order for a linear function near an approximately linear function to exist.”

A year later, Hyers [2] gave an answer to the problem of Ulam for additive functions defined on Banach spaces: let and be real Banach spaces and . Then for every function satisfying for all , there exists a unique additive function with the property for all .

After Hyers’s result, many mathematicians have extended Ulam’s problem to other functional equations and generalized Hyers’s result in various directions (see [3–6]). A generalization of Ulam’s problem was recently proposed by replacing functional equations with differential equations: the differential equation has the Hyers-Ulam stability if for a given and a function such that there exists a solution of the differential equation such that and .

Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [7, 8]). Thereafter, Alsina and Ger published their paper [9], which handles the Hyers-Ulam stability of the linear differential equation : if a differentiable function is a solution of the inequality for any , then there exists a constant such that for all .

Those previous results were extended to the Hyers-Ulam stability of linear differential equations of first order and higher order with constant coefficients in [10–13], respectively. Furthermore, Jung has also proved the Hyers-Ulam stability of linear differential equations (see [14–17]). Rus investigated the Hyers-Ulam stability of differential and integral equations using the Gronwall lemma and the technique of weakly Picard operators (see [18, 19]). Recently, the Hyers-Ulam stability problems of linear differential equations of first order and second order with constant coefficients were studied by using the method of integral factors (see [20, 21]). The results given in [10, 15, 20] have been generalized by Cimpean and Popa [22] and by Popa and Raşa [23, 24] for the linear differential equations of th order with constant coefficients. Furthermore, the Laplace transform method was recently applied to the proof of the Hyers-Ulam stability of linear differential equations (see [25]).

This paper consists of two main sections. In Section 2, we introduce some sufficient conditions under which each solution of the linear differential equation (11) is bounded. In Section 3, we prove the Hyers-Ulam stability of the linear differential equations of the form (11) as well as the nonlinear differential equations of the form (55) by using the Gronwall lemma that was recently introduced by Rus [18, 19] in studying the Hyers-Ulam stability of differential equations.

One of the advantages of this paper is that the authors have applied the Gronwall lemma, which is now recognized as a powerful method, for proving the Hyers-Ulam stability of various differential equations of second order.

#### 2. Preliminaries

In this section, we first introduce and prove a lemma which is a kind of the Gronwall inequality.

Lemma 1. *Let be integrable functions, let be a constant, and let be given. If satisfies the inequality
**
for all , then
**
for all .*

*Proof. *It follows from (5) that
for all . Integrating both sides of the last inequality from to , we obtain
or
for each , which together with (5) implies that
for all .

In the following theorem, using Lemma 1, we investigate sufficient conditions under which every solution of the differential equation is bounded.

Theorem 2. *Let be a differentiable function. Every solution of the linear differential equation (11) is bounded provided that and as .*

*Proof. *First, we choose to be large enough so that for all . Multiplying (11) by and integrating it from to , we obtain
for all . Integrating by parts, this yields
for any . Then it follows from (13) that
for all . Thus, it holds that
for any .

In view of Lemma 1, (15), and our hypothesis, there exists a constant such that
for all . On the other hand, since is continuous, there exists a constant such that for all , which completes the proof.

Corollary 3. *Let be a differentiable function satisfying as . Every solution of the linear differential equation
**
is bounded provided that .*

#### 3. Main Results on Hyers-Ulam Stability

Given constants and , let denote the set of all functions with the following properties:(i)is twice continuously differentiable;(ii);(iii).

We now prove the Hyers-Ulam stability of the linear differential equation (11) by using the Gronwall inequality.

Theorem 4. *Given constants and , assume that is a differentiable function with and . If a function satisfies the inequality
**
for all and for some , then there exist a solution of the differential equation (11) and a constant such that
**
for any , where
*

*Proof. *We multiply (18) with to get
for all . If we integrate each term of the last inequalities from to , then it follows from that
for any .

Integrating by parts and using , we have
for all .

Since holds for all , it follows from (23) that
or
for any .

Applying Lemma 1, we obtain
for all . Hence, it holds that
for any . Obviously, satisfies (11) and the conditions , , and such that
for all , where .

If we set , then the following corollary is an immediate consequence of Theorem 4.

Corollary 5. *Given constants and , assume that is a differentiable function with and . If a function satisfies the inequality
**
for all and for some , then there exist a solution of the differential equation (17) and a constant such that
**
for any , where .*

*Example 6. *Let be a constant function defined by for all and for a constant . Then, we have and . Assume that a twice continuously differentiable function satisfies , , and
for all and for some and . According to Corollary 5, there exists a solution of the differential equation, , such that
for any .

Indeed, if we define a function by
where we set , then satisfies the conditions stated in the first part of this example, as we see in the following. It follows from the definition of that
and, hence, we get . Moreover, we obtain
For any given , if we choose the constant such that , then we can easily see that
for any .

Theorem 7. *Given constants and , assume that is a monotone increasing and differentiable function. If a function satisfies the inequality (18) for all and for some , then there exists a solution of the differential equation (11) such that
**
for any .*

*Proof. *We multiply (18) with to get
for all . If we integrate each term of the last inequalities from to , then it follows from that
for any .

Integrating by parts, the last inequalities together with yield
for all . Then we have
for any .

Applying Lemma 1, we obtain
for all , since is a monotone increasing function. Hence, it holds that
for any . Obviously, satisfies (11), , and the inequality (37) for all .

Corollary 8. *Given constants and , assume that is a monotone increasing and differentiable function with . If a function satisfies the inequality
**
for all and for some , then there exists a solution of the differential equation (17) such that
**
for any .*

If we set , then the following corollary is an immediate consequence of Theorem 7.

Corollary 9. *Given constants and , assume that is a monotone decreasing and differentiable function with . If a function satisfies the inequality
**
for all and for some , then there exists a solution of the differential equation
**
such that
**
for any .*

*Example 10. *Let be a monotone decreasing function defined by for all . Then, we have . Assume that a twice continuously differentiable function satisfies , , and
for all and for some and . According to Corollary 9, there exists a solution of the differential equation, , such that
for any .

Indeed, if we define a function by
where is a real number with , then satisfies the conditions stated in the first part of this example, as we see in the following. It follows from the definition of that
and, hence, we get . Moreover, we obtain
We can see that
for any .

Now, we investigate the Hyers-Ulam stability of the nonlinear differential equation

Theorem 11. *Given constants and , assume that is a function satisfying and for all and . If a function satisfies and the inequality
**
for all and for some , then there exists a solution of the differential equation (55) such that
**
for any .*

*Proof. *We multiply (56) with to get
for all . If we integrate each term of the last inequalities from to , then it follows from that
for any .

Integrating by parts and using , the last inequalities yield
for all . Then we have
for any .

Applying Lemma 1, we obtain
for all . Hence, it holds that for any . Obviously, satisfies (55) and such that
for all .

In the following theorem, we investigate the Hyers-Ulam stability of the Emden-Fowler nonlinear differential equation of second order for the case where is a positive odd integer.

Theorem 12. *Given constants and , assume that is a differentiable function. Let be an odd integer larger than . If a function satisfies and the inequality
**
for all and for some , then there exists a solution of the differential equation (64) such that
**
for any , where .*

*Proof. *We multiply (65) with to get
for all . If we integrate each term of the last inequalities from to , then it follows from that
for any .

Integrating by parts and using , the last inequalities yield
for all . Then we have
for any .

Applying Lemma 1, we obtain
for all , from which we have
for all . Hence, it holds that
for any , where we set . Obviously, satisfies (64) and . Moreover, we get
for all .

Given constants , , and , let denote the set of all functions with the following properties: is twice continuously differentiable;; for all ; for all .

We now investigate the Hyers-Ulam stability of the differential equation of the form where is a positive odd integer.

Theorem 13. *Given constants , , and , assume that is a function satisfying . Let be an odd integer larger than . If a function satisfies the inequality
**
for all and for some , then there exists a solution of the differential equation (75) such that
**
for any .*

*Proof. *We multiply (76) with to get
for all . If we integrate each term of the last inequalities from to , then it follows from that
for any .

Integrating by parts and using and , the last inequalities yield
for all . Then it follows from that
for any .

Applying Lemma 1, we obtain
for all . Hence, it holds that
for any . Obviously, satisfies (75) and . Furthermore, we get
for all .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1 A1A2005557).