Journal of Function Spaces
Volume 2016 (2016), Article ID 2406158, 7 pages
http://dx.doi.org/10.1155/2016/2406158

Research Article

Approximation by First-Order Linear Differential Equations with an Initial Condition

Jaiok Roh¹ and Soon-Mo Jung²

¹Department of Mathematics, Hallym University, Chuncheon, Gangwon-do 24252, Republic of Korea
²Mathematics Section, College of Science and Technology, Hongik University, Sejong 30016, Republic of Korea

Received 10 May 2016; Accepted 25 July 2016

Academic Editor: Seppo Hassi

Copyright © 2016 Jaiok Roh and Soon-Mo Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We will consider a continuously differentiable function satisfying the inequality for all and for some and some . Then we will approximate by a solution of the linear equation with .

1. Introduction

The question concerning the stability of functional equations has been originally raised by Ulam [1]: given a metric group , a positive number , and a function which satisfies inequality for all , does there exist homomorphism and a constant depending only on and such that for all ?

If the answer to this question is affirmative, the functional equation 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 refer the reader to [3–8] for the exact definition of Hyers-Ulam stability.

Usually the experiment (or the observed) data do not exactly coincide with theoretical ones. We may express natural phenomena by use of equations but because of the errors due to measurement or observance the actual experiment data can almost always be a little bit off the expectations. If we would use inequalities instead of equalities to explain natural phenomena, then these errors could be absorbed into the solutions of inequalities; that is, those errors would be no more errors.

There is another way to explain the Hyers-Ulam stability. Let us consider a closed system which can be explained by the first-order linear differential equation, namely, . The past, present, and future of this system are completely determined if we know the general solution and an initial condition of that differential equation. So we can say that this system is “predictable.” Sometimes, because of the disturbances (or noises) of the outside, the system may not be determined by but can only be explained by an inequality like . Then it is impossible to predict the exact future of the disturbed system.

Even though the system is not predictable exactly because of outside disturbances, we say the differential equation has the Hyers-Ulam stability if the “real” future of the system follows the solution of with a bounded error. But if the error bound is “too big,” we say that differential equation does not have the Hyers-Ulam stability. Resonance is the case. Considering this point of view, the Hyers-Ulam stability (of differential equations) is fundamental (see [9]).

A generalization of Ulam’s problem was recently proposed by replacing functional equations with differential equations. Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [10, 11]). Thereafter, Alsina and Ger proved the Hyers-Ulam stability of the linear differential equations: indeed, they proved in [12] that if a differentiable function is a solution of the differential inequality , where is an open subinterval of , then there exists a solution of the differential equation such that for any .

The above result by Alsina and Ger was generalized by Miura et al. [13], by Miura et al. [14], and also by Takahasi et al. [15, 16]. They proved that the Hyers-Ulam stability holds true for the Banach space-valued differential equation .

Miura et al. [14] proved the Hyers-Ulam stability of the first-order linear differential equation , where is a continuous function, while Jung [17] proved the Hyers-Ulam stability of the differential equation of the form . Furthermore, results of Hyers-Ulam stability of first-order linear differential equation have been generalized by Miura et al. [18] and also by Takahasi et al. [16]. They dealt with the nonhomogeneous linear differential equation of the form .

Wang et al. [19] used the method of integrating factor to prove the Hyers-Ulam stability of the nonhomogeneous linear differential equation of the form

where with .

Theorem 1 (see [19]). Let be continuous functions such that and

for all and some constant . Given a constant , if a continuously differentiable function satisfies the differential inequality

for all , then there exists a solution of the differential equation (1) such that

for all , where is a function defined by

We wondered if a stability of (1) can be proven without the condition in Theorem 1 and whether the error estimation of Theorem 1 can be improved or not.

In this paper, we are going to prove the Hyers-Ulam stability of the differential equation (1) with an initial condition and then compare the stability results of (1) with and without the condition . And also we are going to generalize the stability result of (1) by replacing the bounded difference with a general control function.

Throughout this paper, let denote an open interval with .

2. Hyers-Ulam Stability without (2)

In the following theorem, we shall prove the Hyers-Ulam stability of the differential equation (1) with an initial condition when condition (2) is excluded, while Wang et al. proved the Hyers-Ulam stability of (1) with condition (2) (see Theorem 1 or [19]).

Theorem 2. Let be continuous functions such that

(i) for all ;(ii) exists for every .

Given constants and , if a continuously differentiable function satisfies the differential inequality

for all with

for some and , then there exists a solution of the differential equation (1) with such that

for all .

Proof. Assume that for all . In view of (6), we have

for any . Multiplying the formula by , we get

for all .
Integrating each term of (10) from to , we have

for all . Then

for all .
It follows from (7) and (12) that

for all . Multiplying (13) by , we have

for all .
If we define a function by

for all , then is a solution of the differential equation (1) with . Moreover, in view of (14), we get

for all .
By an argument similar to the above, for the case when for all , we get the same result. This completes the proof of this theorem.

Corollary 3. Let be continuous functions such that

(i) for all ;(ii) for all ;(iii) exists for all .

Moreover, assume that there exists such that

(iv)⁡ .

Given constants and , if a continuously differentiable function satisfies the differential inequality (6) with (7) for all and for some , then there exists a solution of the differential equation (1) with such that

for all .

Proof. In view of (ii), (iii), and (iv), it is not difficult to show that

for all . Hence, it follows from Theorem 2 that

for all .

In the next theorem, we are going to generalize the stability result of the differential equation (1) by a general control function .

Theorem 4. Let be continuous functions such that

(i) for all ;(ii) exists for any .

Suppose is a continuous function. Given a constant , if a continuously differentiable function satisfies the differential inequality

for all with

for some and , then there exists a solution of the differential equation (1) with such that

for all .

Proof. Similarly to Theorem 2, we can prove this theorem.

Corollary 5. Let be continuous functions and let be given such that

(i) for all ;(ii) for all ;(iii) exists for all ;(iv).

Suppose is a continuous function. Given a constant , if a continuously differentiable function satisfies the differential inequality (20) with (21) for all and for some , then there exists a solution of the differential equation (1) with such that

for all .

Proof. As in the proof of Corollary 3, by (ii), (iii), and (iv), we can easily show that

for all . Therefore, it follows from Theorem 4 that

for all .

3. Hyers-Ulam Stability with (2)

In this section, we also assume that is an open interval for and we prove, in the following theorem, the Hyers-Ulam stability of the differential equation (1) with an initial condition when condition (2) is included.

Theorem 6. Let be continuous functions such that

(i) for all ;(ii) for all ;(iii) exists for any .

Moreover, assume that there exist and a constant such that

(iv)⁡ ;(v)⁡ for all .

Given constants and , if a continuously differentiable function satisfies the differential inequality (6) with (7) for all and for some , then there exists a solution of the differential equation (1) with such that

for all .

Proof. Assume that and for all . Since for each , it follows from (10) that

for all . Since the function is continuous, we know that is finite. And, for any , by integrating (27) from to , we have

Then

for each .
It follows from (7) and (29) that

for all . Multiplying (30) by , we get

for all .
It obviously follows from (ii), (iii), and (iv) that

for all . Hence, by (31), we obtain

for all . So, we have

where

for all . Obviously, is solution of (1) with .
By an argument similar to the above, for the case when for every , we can prove our assertion.

We remark that if , for all , and , and if , then it seems more feasible that our result (Theorem 6) is better than the result of Wang et al. (Theorem 1) as we see in the following example.

Example 7. For any and for some , let , , , , , , and . If a continuously differentiable function satisfies the differential inequality

for all and for some real number , then Theorem 6 implies that there exists a continuously differentiable function such that

for all , where is uniquely determined as

According to Theorem 1, we have

instead of (37), which implies that Theorem 6 is better than Theorem 1 for small values of .

Corollary 8. Let be continuous functions such that

(i) for all ;(ii) for all ;(iii) exists for any .

Moreover, assume that there exist and a constant such that

(iv)⁡ ;(v)⁡ for all .

Given constants and , if a continuously differentiable function satisfies the differential inequality (6) with (7) for all and for some , then there exists a solution of the differential equation (1) with such that

for all .

4. Conclusion

In this paper, we proved the Hyers-Ulam stability of the linear differential equation with an initial condition, , when the condition, , for all and some constant , is excluded or included. Due to the hypotheses of Corollary 3 or Theorem 6, we have

that is,

We now define a subinterval , , and by

If , then we get

This relation implies that if , then the result of Corollary 3 is better than that of Theorem 6, while if , then the result of Corollary 3 is worse than that of Theorem 6. Thus, roughly speaking, if is so “large” that holds, then the result of Corollary 3 is better than that of Theorem 6.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors read and approved the final paper.

Acknowledgments

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) and Hallym University Research Fund, 2014 (HRF-201409-017). 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).

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