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

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.