Journal of Function Spaces
Volume 2016 (2016), Article ID 7874061, 5 pages
http://dx.doi.org/10.1155/2016/7874061

Research Article

Approximation of Analytic Functions by Solutions of Cauchy-Euler Equation

Soon-Mo Jung

Mathematics Section, College of Science and Technology, Hongik University, Sejong 30016, Republic of Korea

Received 2 November 2015; Accepted 15 May 2016

Academic Editor: Gestur Ólafsson

Copyright © 2016 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 investigate the approximation properties of a special class of twice continuously differentiable functions by solutions of the Cauchy-Euler equation.

1. Introduction

Throughout this paper, let be a positive integer, let be a nondegenerate interval of , and let denote either or . We will consider the (linear) differential equation of th order

defined on , where is an times continuously differentiable function.

For arbitrary , assume that an times continuously differentiable function satisfies the differential inequality

for all . If for each function satisfying inequality (2) there exists a solution of the differential equation (1) such that

for any , where depends on only and satisfies , then we say that the differential equation (1) satisfies (or has) the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain is not the whole space ). If the above statement also holds when we replace and with some appropriate and , respectively, then we say that the differential equation (1) satisfies the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability). We may apply these terminologies for other differential equations. For more detailed definition of the Hyers-Ulam stability and recent papers on this subject, refer to [1–4].

Obłoza seems to be the first author who investigated the Hyers-Ulam stability of linear differential equations (see [5, 6]). Let be continuous functions with , where and are real constants. Assume that is an arbitrary real number. Obłoza proved that if a differentiable function satisfies the inequality for all and if a function satisfies for all and for some , then there exists a constant such that for all .

Thereafter, Alsina and Ger [7] proved that if a differentiable function satisfies the differential inequality , then there exists a solution of the differential equation such that for any . This result of Alsina and Ger was generalized by Takahasi et al. They proved in [8] that the Hyers-Ulam stability holds for the Banach space valued differential equation (see also [9–13]). For a recent result on the Hyers-Ulam stability for second-order linear differential equations, we refer to [14, 15].

In this paper, we consider the (inhomogeneous) Cauchy-Euler equation

where and are real-valued constants and is a continuous function, and we investigate the approximation properties of twice continuously differentiable functions by solutions of the Cauchy-Euler equation

which is associated with (4).

2. Preliminaries

Recently, Choi and Jung [16, Corollary  4.2] proved the Hyers-Ulam stability of the Cauchy-Euler equation (4) for the case of .

Theorem 1. Assume that the real-valued constants , are given with and is an arbitrarily given positive constant. Let be a positive real-valued constant and let , be given as

If is a differentiable function and is a twice continuously differentiable function such that the inequality

holds for any , then there exists a solution of the inhomogeneous Cauchy-Euler equation (4) such that

for all .

For the case of , the Hyers-Ulam stability of the inhomogeneous Cauchy-Euler equation (4) was proved in [16, Corollary  4.4].

Theorem 2. Assume that the real-valued constants and are given with , and is an arbitrarily given positive constant. Let be a positive real-valued constant and let . If is a differentiable function and is a twice continuously differentiable function such that the inequality

holds for all , then there exists a solution of the inhomogeneous Cauchy-Euler equation (4) with such that

for all .

Finally, the Hyers-Ulam stability of the Cauchy-Euler equation (4) was also proven in [16, Theorem  4.5] for the case of .

Theorem 3. Assume that the real-valued constants and are given with and is an arbitrarily given positive constant. Let be a given real-valued constant and let

If a differentiable function and a twice continuously differentiable function satisfy inequality (7) for all , then there exists a solution of the inhomogeneous Cauchy-Euler equation (4) such that

for all .

Remark 4. Cîmpean and Popa [14] proved the Hyers-Ulam stability of the linear differential equations of th order with constant coefficients. Indeed, they proved a general theorem for the Hyers-Ulam stability which includes Theorems 1, 2, and 3 as its corollaries with the inequality

However, Theorems 1, 2, and 3 have the advantage of more exact local approximation over the result of Cîmpean and Popa as we see in Theorems 5, 6, and 7.

3. Approximation Properties

We denote by the set of all twice continuously differentiable functions for which there exists a constant such that

for all , where and are real-valued constants.

If we define

for all and , then is a vector space over . This fact implies that the set is large enough to be a vector space.

In the following theorems, we investigate approximation properties of functions of by solutions of the Cauchy-Euler equation (5).

Theorem 5. Let be a given real number and let be given with . If , then there exists a solution of the Cauchy-Euler equation (5) such that

as .

Proof. We define and by the formulas given in Theorem 1; that is, and are the distinct roots of the indicial equation . Since , we have . Since , there exists a constant such that inequality (14) holds for all .
According to Theorem 1 with , there exists a solution of the Cauchy-Euler equation (5) such that

for any .
We will only estimate the following limit for the case of by applying L’Hospital’s rule:

which implies the validity of this theorem.

We now consider the case of and use Theorem 2 to prove the following theorem.

Theorem 6. Let and be real numbers and let . If , then there exists a solution of the Cauchy-Euler equation (5) with such that

as .

Proof. Since , there exists a constant such that inequality (14) holds for all . According to Theorem 2 with , there exists a solution of the Cauchy-Euler equation (5) with such that

for all .
Therefore, we estimate the limit by applying L’Hospital’s rule:

which implies the validity of this theorem.

Finally, we investigate the approximation property of each function of by a solution of the differential equation (5) when .

Theorem 7. Let be a given real number and let be given with . If , then there exists a solution of the Cauchy-Euler equation (5) such that

as .

Proof. Let us define

Since , there exists a constant such that inequality (14) holds for all . According to Theorem 3 with , there exists a solution of the Cauchy-Euler equation (5) such that

for all .
If we substitute , then we have

Hence, we further apply L’Hospital’s rule to obtain

which implies the validity of this theorem.

Competing Interests

The author declares that there are no competing interests.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557). This work was supported by 2014 Hongik University Research Fund.

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