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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Power series method and approximate linear differential equations of second order

Soon-Mo Jung1 and Hamdullah Şevli2*

Author Affiliations

1 Mathematics Section, College of Science and Technology, Hongik University, Sejong, 339-701, Republic of Korea

2 Department of Mathematics, Faculty of Sciences and Arts, Istanbul Commerce University, Uskudar, Istanbul, 34672, Turkey

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Advances in Difference Equations 2013, 2013:76  doi:10.1186/1687-1847-2013-76

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/76


Received:14 December 2012
Accepted:4 March 2013
Published:26 March 2013

© 2013 Jung and Şevli; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we will establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

MSC: 34A05, 39B82, 26D10, 34A40.

Keywords:
power series method; approximate linear differential equation; simple harmonic oscillator equation; Hyers-Ulam stability; approximation

1 Introduction

Let X be a normed space over a scalar field <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M1">View MathML</a>, and let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M2">View MathML</a> be an open interval, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M1">View MathML</a> denotes either ℝ or ℂ. Assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M4">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M5">View MathML</a> are given continuous functions. If for every n times continuously differentiable function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M6">View MathML</a> satisfying the inequality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M7">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M8">View MathML</a> and for a given <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M9">View MathML</a>, there exists an n times continuously differentiable solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M10">View MathML</a> of the differential equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M11">View MathML</a>

such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M12">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M8">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M14">View MathML</a> is an expression of ε with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M15">View MathML</a>, then we say that the above differential equation has the Hyers-Ulam stability. For more detailed definitions of the Hyers-Ulam stability, we refer the reader to [1-8].

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [9,10]). Thereafter, Alsina and Ger [11] proved the Hyers-Ulam stability of the differential equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M16">View MathML</a>. It was further proved by Takahasi et al. that the Hyers-Ulam stability holds for the Banach space valued differential equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M17">View MathML</a> (see [12] and also [13-15]).

Moreover, Miura et al.[16] investigated the Hyers-Ulam stability of an nth-order linear differential equation. The first author also proved the Hyers-Ulam stability of various linear differential equations of first order (ref. [17-25]).

Recently, the first author applied the power series method to studying the Hyers-Ulam stability of several types of linear differential equations of second order (see [26-34]). However, it was inconvenient that he had to alter and apply the power series method with respect to each differential equation in order to study the Hyers-Ulam stability. Thus, it is inevitable to develop a power series method that can be comprehensively applied to different types of differential equations.

In Sections 2 and 3 of this paper, we establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

Throughout this paper, we assume that the linear differential equation of second order of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M18">View MathML</a>

(1)

for which <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M19">View MathML</a> is an ordinary point, has the general solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M20">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M21">View MathML</a> is a constant with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M22">View MathML</a> and the coefficients <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M23">View MathML</a> are analytic at 0 and have power series expansions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M24">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M25">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M19">View MathML</a> is an ordinary point of (1), we remark that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M27">View MathML</a>.

2 Inhomogeneous differential equation

In the following theorem, we solve the linear inhomogeneous differential equation of second order of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M28">View MathML</a>

(2)

under the assumption that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M19">View MathML</a> is an ordinary point of the associated homogeneous linear differential equation (1).

Theorem 2.1Assume that the radius of convergence of power series<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M30">View MathML</a>is<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M31">View MathML</a>and that there exists a sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32">View MathML</a>satisfying the recurrence relation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M33">View MathML</a>

(3)

for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M35">View MathML</a>be the radius of convergence of power series<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36">View MathML</a>and let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M37">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M38">View MathML</a>is the domain of the general solution to (1). Then every solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M39">View MathML</a>of the linear inhomogeneous differential equation (2) can be expressed by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M40">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M41">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M42">View MathML</a>is a solution of the linear homogeneous differential equation (1).

Proof Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M19">View MathML</a> is an ordinary point, we can substitute <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M44">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M45">View MathML</a> in (2) and use the formal multiplication of power series and consider (3) to get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M46">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M41">View MathML</a>. That is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36">View MathML</a> is a particular solution of the linear inhomogeneous differential equation (2), and hence every solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M39">View MathML</a> of (2) can be expressed by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M50">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M51">View MathML</a> is a solution of the linear homogeneous differential equation (1). □

For the most common case in applications, the coefficient functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M52">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M53">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M54">View MathML</a> of the linear differential equation (1) are simple polynomials. In such a case, we have the following corollary.

Corollary 2.2Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M52">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M53">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M54">View MathML</a>be polynomials of degree at most<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M58">View MathML</a>. In particular, let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M59">View MathML</a>be the degree of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M52">View MathML</a>. Assume that the radius of convergence of power series<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M30">View MathML</a>is<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M31">View MathML</a>and that there exists a sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32">View MathML</a>satisfying the recurrence formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M64">View MathML</a>

(4)

for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M66">View MathML</a>. If the sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32">View MathML</a>satisfies the following conditions:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M68">View MathML</a>,

(ii) there exists a complex numberLsuch that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M69">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M70">View MathML</a>,

then every solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M39">View MathML</a>of the linear inhomogeneous differential equation (2) can be expressed by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M72">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M41">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M74">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M42">View MathML</a>is a solution of the linear homogeneous differential equation (1).

Proof Let m be any sufficiently large integer. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M76">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M77">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M78">View MathML</a>, if we substitute <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M79">View MathML</a> for k in (4), then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M80">View MathML</a>

By (i) and (ii), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M81">View MathML</a>

which implies that the radius of convergence of the power series <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36">View MathML</a> is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M83">View MathML</a>. The rest of this corollary immediately follows from Theorem 2.1. □

In many cases, it occurs that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M84">View MathML</a> in (1). For this case, we obtain the following corollary.

Corollary 2.3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M85">View MathML</a>be a distance between the origin 0 and the closest one among singular points of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M86">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M87">View MathML</a>, or<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M88">View MathML</a>in a complex variablez. If there exists a sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32">View MathML</a>satisfying the recurrence relation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M90">View MathML</a>

(5)

for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34">View MathML</a>, then every solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M39">View MathML</a>of the linear inhomogeneous differential equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M93">View MathML</a>

(6)

can be expressed by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M94">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M41">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M42">View MathML</a>is a solution of the linear homogeneous differential equation (1) with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M84">View MathML</a>.

Proof If we put <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M98">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M99">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M100">View MathML</a>, then the recurrence relation (3) reduces to (5). As we did in the proof of Theorem 2.1, we can show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36">View MathML</a> is a particular solution of the linear inhomogeneous differential equation (6).

According to [[35], Theorem 7.4] or [[36], Theorem 5.2.1], there is a particular solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M102">View MathML</a> of (6) in a form of power series in x whose radius of convergence is at least <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M85">View MathML</a>. Moreover, since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36">View MathML</a> is a solution of (6), it can be expressed as a sum of both <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M102">View MathML</a> and a solution of the homogeneous equation (1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M84">View MathML</a>. Hence, the radius of convergence of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36">View MathML</a> is at least <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M85">View MathML</a>.

Now, every solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M39">View MathML</a> of (6) can be expressed by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M110">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M51">View MathML</a> is a solution of the linear differential equation (1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M84">View MathML</a>. □

3 Approximate differential equation

In this section, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M31">View MathML</a> be a constant. We denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M114">View MathML</a> the set of all functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M115">View MathML</a> with the following properties:

(a) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M45">View MathML</a> is expressible by a power series <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M117">View MathML</a> whose radius of convergence is at least <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M83">View MathML</a>;

(b) There exists a constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M119">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M120">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M121">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M122">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M27">View MathML</a>.

Lemma 3.1Given a sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M125">View MathML</a>, let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32">View MathML</a>be a sequence satisfying the recurrence formula (3) for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34">View MathML</a>. If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M27">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M129">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M130">View MathML</a>is a linear combination of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M131">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M132">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M133">View MathML</a>.

Proof We apply induction on n. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M27">View MathML</a>, if we set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M135">View MathML</a> in (3), then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M136">View MathML</a>

i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M137">View MathML</a> is a linear combination of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M138">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M132">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M133">View MathML</a>. Assume now that n is an integer not less than 2 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M141">View MathML</a> is a linear combination of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M142">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M132">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M133">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M145">View MathML</a>, namely,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M146">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M147">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M148">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M149">View MathML</a> are complex numbers. If we replace m in (3) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M150">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M151">View MathML</a>

which implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M152">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M153">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M154">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M155">View MathML</a> are complex numbers. That is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M156">View MathML</a> is a linear combination of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M157">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M132">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M133">View MathML</a>, which ends the proof. □

In the following theorem, we investigate a kind of Hyers-Ulam stability of the linear differential equation (1). In other words, we answer the question whether there exists an exact solution near every approximate solution of (1). Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M19">View MathML</a> is an ordinary point of (1), we remark that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M27">View MathML</a>.

Theorem 3.2Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M32">View MathML</a>be a sequence of complex numbers satisfying the recurrence relation (3) for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M34">View MathML</a>, where (b) is referred for the value of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M164">View MathML</a>, and let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M35">View MathML</a>be the radius of convergence of the power series<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36">View MathML</a>. Define<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M37">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M38">View MathML</a>is the domain of the general solution to (1). Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M115">View MathML</a>is an arbitrary function belonging to<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M114">View MathML</a>and satisfying the differential inequality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M171">View MathML</a>

(7)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172">View MathML</a>and for some<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M9">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M174">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M175">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M176">View MathML</a>be the complex numbers satisfying

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M177">View MathML</a>

(8)

for any integer<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M129">View MathML</a>. If there exists a constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M179">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M180">View MathML</a>

(9)

for all integers<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M129">View MathML</a>, then there exists a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M182">View MathML</a>of the linear homogeneous differential equation (1) such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M183">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172">View MathML</a>, whereKis the constant determined in (b).

Proof By the same argument presented in the proof of Theorem 2.1 with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M117">View MathML</a> instead of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M36">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M187">View MathML</a>

(10)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172">View MathML</a>. In view of (b), there exists a constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M119">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M190">View MathML</a>

(11)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M191">View MathML</a>.

Moreover, by using (7), (10), and (11), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M192">View MathML</a>

for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172">View MathML</a>. (That is, the radius of convergence of power series <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M30">View MathML</a> is at least <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M85">View MathML</a>.)

According to Theorem 2.1 and (10), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M45">View MathML</a> can be written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M197">View MathML</a>

(12)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M42">View MathML</a> is a solution of the homogeneous differential equation (1). In view of Lemma 3.1, the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M130">View MathML</a> can be expressed by a linear combination of the form (8) for each integer <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M129">View MathML</a>.

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M202">View MathML</a> is a particular solution of (2), if we set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M203">View MathML</a>, then it follows from (8), (9), and (12) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M204">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M172">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that this paper is their original paper. All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research was completed with the support of The Scientific and Technological Research Council of Turkey while the first author was a visiting scholar at Istanbul Commerce University, Istanbul, Turkey.

References

  1. Brillouet-Belluot, N, Brzdȩk, J, Cieplinski, K: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal.. 2012, Article ID 716936 (2012)

  2. Czerwik, S: Functional Equations and Inequalities in Several Variables, World Scientific, River Edge (2002)

  3. Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA. 27, 222–224 (1941). PubMed Abstract | Publisher Full Text | PubMed Central Full Text OpenURL

  4. Hyers, DH, Isac, G, Rassias, TM: Stability of Functional Equations in Several Variables, Birkhäuser, Boston (1998)

  5. Hyers, DH, Rassias, TM: Approximate homomorphisms. Aequ. Math.. 44, 125–153 (1992). Publisher Full Text OpenURL

  6. Jung, S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York (2011)

  7. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc.. 72, 297–300 (1978). Publisher Full Text OpenURL

  8. Ulam, SM: Problems in Modern Mathematics, Wiley, New York (1964)

  9. Obłoza, M: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat.. 13, 259–270 (1993)

  10. Obłoza, M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk.-Dydakt. Prace Mat.. 14, 141–146 (1997)

  11. Alsina, C, Ger, R: On some inequalities and stability results related to the exponential function. J. Inequal. Appl.. 2, 373–380 (1998)

  12. Takahasi, S-E, Miura, T, Miyajima, S: On the Hyers-Ulam stability of the Banach space-valued differential equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M206">View MathML</a>. Bull. Korean Math. Soc.. 39, 309–315 (2002). Publisher Full Text OpenURL

  13. Miura, T: On the Hyers-Ulam stability of a differentiable map. Sci. Math. Jpn.. 55, 17–24 (2002)

  14. Miura, T, Jung, S-M, Takahasi, S-E: Hyers-Ulam-Rassias stability of the Banach space valued differential equations <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M208">View MathML</a>. J. Korean Math. Soc.. 41, 995–1005 (2004). Publisher Full Text OpenURL

  15. Miura, T, Miyajima, S, Takahasi, S-E: A characterization of Hyers-Ulam stability of first order linear differential operators. J. Math. Anal. Appl.. 286, 136–146 (2003). Publisher Full Text OpenURL

  16. Miura, T, Miyajima, S, Takahasi, S-E: Hyers-Ulam stability of linear differential operator with constant coefficients. Math. Nachr.. 258, 90–96 (2003). Publisher Full Text OpenURL

  17. Cimpean, DS, Popa, D: On the stability of the linear differential equation of higher order with constant coefficients. Appl. Math. Comput.. 217(8), 4141–4146 (2010). Publisher Full Text OpenURL

  18. Cimpean, DS, Popa, D: Hyers-Ulam stability of Euler’s equation. Appl. Math. Lett.. 24(9), 1539–1543 (2011). Publisher Full Text OpenURL

  19. Jung, S-M: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett.. 17, 1135–1140 (2004). Publisher Full Text OpenURL

  20. Jung, S-M: Hyers-Ulam stability of linear differential equations of first order, II. Appl. Math. Lett.. 19, 854–858 (2006). Publisher Full Text OpenURL

  21. Jung, S-M: Hyers-Ulam stability of linear differential equations of first order, III. J. Math. Anal. Appl.. 311, 139–146 (2005). Publisher Full Text OpenURL

  22. Jung, S-M: Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients. J. Math. Anal. Appl.. 320, 549–561 (2006). PubMed Abstract | Publisher Full Text | PubMed Central Full Text OpenURL

  23. Lungu, N, Popa, D: On the Hyers-Ulam stability of a first order partial differential equation. Carpath. J. Math.. 28(1), 77–82 (2012)

  24. Lungu, N, Popa, D: Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl.. 385(1), 86–91 (2012). Publisher Full Text OpenURL

  25. Popa, D, Rasa, I: The Frechet functional equation with application to the stability of certain operators. J. Approx. Theory. 164(1), 138–144 (2012). Publisher Full Text OpenURL

  26. Jung, S-M: Legendre’s differential equation and its Hyers-Ulam stability. Abstr. Appl. Anal.. 2007, Article ID 56419. doi:10.1155/2007/56419 (2007)

  27. Jung, S-M: Approximation of analytic functions by Airy functions. Integral Transforms Spec. Funct.. 19(12), 885–891 (2008). Publisher Full Text OpenURL

  28. Jung, S-M: Approximation of analytic functions by Hermite functions. Bull. Sci. Math.. 133, 756–764 (2009). Publisher Full Text OpenURL

  29. Jung, S-M: Approximation of analytic functions by Legendre functions. Nonlinear Anal.. 71(12), e103–e108 (2009). Publisher Full Text OpenURL

  30. Jung, S-M: Hyers-Ulam stability of differential equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/76/mathml/M210">View MathML</a>. J. Inequal. Appl.. 2010, Article ID 793197. doi:10.1155/2010/793197 (2010)

  31. Jung, S-M: Approximation of analytic functions by Kummer functions. J. Inequal. Appl.. 2010, Article ID 898274. doi:10.1155/2010/898274 (2010)

  32. Jung, S-M: Approximation of analytic functions by Laguerre functions. Appl. Math. Comput.. 218(3), 832–835 doi:10.1016/j.amc.2011.01.086 (2011)

    doi:10.1016/j.amc.2011.01.086

    Publisher Full Text OpenURL

  33. Jung, S-M, Rassias, TM: Approximation of analytic functions by Chebyshev functions. Abstr. Appl. Anal.. 2011, Article ID 432961. doi:10.1155/2011/432961 (2011)

  34. Kim, B, Jung, S-M: Bessel’s differential equation and its Hyers-Ulam stability. J. Inequal. Appl.. 2007, Article ID 21640. doi:10.1155/2007/21640 (2007)

  35. Ross, CC: Differential Equations - An Introduction with Mathematica, Springer, New York (1995)

  36. Kreyszig, E: Advanced Engineering Mathematics, Wiley, New York (2006)