Journal of Nonlinear Dynamics
Volume 2014 (2014), Article ID 487257, 6 pages
http://dx.doi.org/10.1155/2014/487257

Research Article

Hyers-Ulam Stability of Third Order Euler’s Differential Equations

A. K. Tripathy and A. Satapathy

Department of Mathematics, Sambalpur University, Sambalpur 768019, India

Received 1 May 2014; Accepted 8 October 2014; Published 4 November 2014

Academic Editor: Ivo Petras

Copyright © 2014 A. K. Tripathy and A. Satapathy. 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 investigate the Hyers-Ulam stability of third order Euler's differential equations of the form on any open interval , or , where , and are complex constants.

1. Introduction

In 1940, Ulam gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems [1]. Among such problems is a problem concerning the stability of functional equations: give conditions in order for a linear function near an approximately linear function to exist.

In the following year, Hyers [2] gave an answer to the problem of Ulam for additive functions defined on Banach spaces.

Let and be two real Banach spaces and let . Then, for every function satisfying

there exists a unique additive function with the property

Furthermore, the result of Hyers has been generalized by Rassias [3, 4]. Since then, the stability problems of various functional equations have been investigated by many authors (see, e.g., [1, 5–8]). A generalization of Ulam’s problem was recently proposed by replacing functional equations with differential equations.

The differential equation

has Hyers-Ulam stability; if for given and a function such that

then there exists a solution of the differential equation such that and .

If the preceding statement is also true when we replace and by and , respectively, where and are appropriate functions not depending on and explicitly, then we say that the corresponding differential equation has the generalized Hyers-Ulam stability.

Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see, e.g., [9, 10]). Thereafter, Alsina and Ger published their work [11], 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 .

In [1], Rezaei et al. have discussed the Hyers-Ulam stability of linear differential equations of first and th order by applying Laplace transform which is comparable with the other methods available in the literature.

It is Jung et al. who have investigated the Hyers-Ulam stability of linear differential equations of different classes including the stability of the delay differential equation , where is a constant (see, e.g., [5–7, 12–14]). Among the works, we are motivated by the results of [13], where he has studied the Hyers-Ulam stability of the following Euler’s differential equations:

where , , and are complex constants. We may note that the Hyers-Ulam stability of (6) depends on the Hyers-Ulam stability of (5) for every fixed . We mentioned here the important results of [13].

Theorem 1. Let be a complex Banach space and let be an open interval such that or . Assume that a function is given, that , , and are complex constants, and that is a fixed element of . Furthermore, suppose a continuously differentiable function satisfies

If both and are integrable on , for any with , then there exists a unique solution of the differential equation (5) such that

for any .

Indeed, the unique solution of (5) in the complex Banach space is given by

Theorem 2. If a twice continuously differentiable function satisfies the differential inequality

for all and for some , then there exists a solution of (6) such that

for any , where is a negative real number. In particular, if , , then

In this theorem, we may note that the unique solution of (6) is given by

where

are real constants and is any arbitrary constant.

Theorem 3. Let be a complex Banach space and let be an open interval such that or . Assume that a function is given and that is a continuous function. Furthermore, suppose a continuously differentiable function satisfies

If both and are integrable on , for any with , then there exists a unique solution of the differential equation

such that (8) holds for any , where is a complex constant.

Proof. The proof of the theorem follows from the proof of Theorem 1, if we set

where the unique solution is given by

The details are omitted.

Remark 4. We may note that (5) is a particular case of (16). Hence, Theorems 1 and 3 are comparable.

The objective of this work is to investigate the generalized Hyers-Ulam stability of the following Euler’s differential equations of the form

where , , , and are complex constants. Intuitively, we shall prove that if a twice continuously differentiable function satisfies the differential inequality

for all , where is a fixed element of the complex Banach space and with or , then there exists a twice continuously differentiable solution of (19) such that

for any , where

We also apply this result to the investigation of the Hyers-Ulam stability of (20).

2. Hyers-Ulam Stability of (19)

Throughout this section, we let with or . For fixed , the general solution of (19) in the class of real valued functions defined on is given by

where and , are arbitrary constants.

Remark 5. Indeed, . If , then either or . Therefore, implies that and implies that . Hence, the second solution could be any one of the following:

Theorem 6. Let be a complex Banach space and let be an open interval with either or . Assume that a function is given, that , , , and are complex constants, and that is a fixed element of . Furthermore, suppose a twice continuously differentiable function satisfies the differential inequality (21). If , , , , and are integrable on , for any with , then there exists a unique solution (which is twice continuously differentiable) of (19) such that (22) holds for any , where and are the roots of .

Proof. We prove the theorem for five possible cases; namely,

(i), ;(ii), ; (iii), ; (iv), ;(v), .

Case (i). Suppose that is a complex Banach space and is a twice continuously differentiable function satisfying the differential inequality (21). Define such that . Then it is easy to verify that

Hence, by Theorem 1, it follows that there exists a unique solution of the differential equation such that

provided that and are integrable on , for any with . Indeed,

where is a limit point in . If we denote

then (27) becomes

Clearly, . By Theorem 3, there exists a unique solution of the differential equation such that

provided that and are integrable on , for any with . Clearly, is integrable if and only if and are integrable. According to Theorem 3,

where is limit point in . It is easy to verify that

Consequently,

which is a solution of (19).
Case (ii). Proceeding as in Case   (i), we find a unique solution of the equation such that

where is a limit point in . Clearly,

implies that

which is a solution of (19).
Case (iii). Interchanging the roles of and in Case  (ii), we find the unique solution as

where .
Case (iv). In this case, we proceed as in Case  (i). Indeed, for ,

and the unique solution is given by

where .
Case (v). In this case,

Using the same type of argument as in Case  (i), we find the unique solution as

where . This completes the proof of the theorem.

3. Hyers-Ulam Stability of (20)

In this section, we shall investigate the Hyers-Ulam stability of (20) on any open interval with either or . We may note that , , and , when , , and are the characteristic roots of the associated characteristic equation of (20).

Theorem 7. Let be a complex Banach space. Assume that is given. Furthermore, assume that , , , , , and are integrable on with , where

Suppose that and satisfies

for all . Then, there exists a unique solution of (20) such that

Proof. Let be such that (44) holds, for . Define such that

Indeed,

Hence, using Theorem 3, it follows that there exists a unique such that

where and is a limit point in . Ultimately, (48) becomes

Now, we can apply Theorem 6 to the above differential inequality for five possible cases. Consequently,

are the unique solutions with respect to the five possible cases, respectively. In fact, they are the possible solutions of (20). Hence, the theorem is proved.

Corollary 8. Let be a real Banach space. Assume that , , and are integrable on with or , where , , and are positive roots of the associated characteristic equation of (20). Suppose that and satisfies

for all . Then, there exists a unique solution of (20) such that

Proof. Let be such that (51) holds, for all . Since all the conditions of Theorem 7 are satisfied, then there exists a unique solution of (20) such that

when . Then, for ,

Hence,

Consequently,

where can be chosen from Theorem 7.

Example 9. Let be a Banach space and let , . Consider Euler’s equation

with , , and . Suppose that satisfies the differential inequality

for any . By Corollary 8, there exists a unique solution of (57) such that

for any and for unique ,

Conflict of Interests

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

Acknowledgment

The authors are thankful to the referee for his helpful suggestions and necessary corrections in the completion of this paper.

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