On Approximate Euler Differential Equations

Abstract

We solve the inhomogeneous Euler differential equations of the form 𝑥²𝑦^′′(𝑥)+𝛼𝑥𝑦^′(𝑥)+𝛽𝑦(𝑥)=∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚 and apply this result to the approximation of analytic functions of a special type by the solutions of Euler differential equations.

1. Introduction

The stability problem for functional equations starts from the famous talk of Ulam and the partial solution of Hyers to the Ulam's problem (see [1, 2]). Thereafter, Rassias [3] attempted to solve the stability problem of the Cauchy additive functional equation in a more general setting.

The stability concept introduced by Rassias’ theorem significantly influenced a number of mathematicians to investigate the stability problems for various functional equations (see [4–10] and the references therein).

Assume that 𝑋 and 𝑌 are a topological vector space and a normed space, respectively, and that 𝐼 is an open subset of 𝑋. If for any function 𝑓∶𝐼→𝑌 satisfying the differential inequality

‖𝑎_𝑛(𝑥)𝑦^(𝑛)(𝑥)+𝑎_𝑛−1(𝑥)𝑦^(𝑛−1)(𝑥)+⋯+𝑎₁(𝑥)𝑦^′(𝑥)+𝑎₀(𝑥)𝑦(𝑥)+ℎ(𝑥)‖≤𝜀 (1.1) for all 𝑥∈𝐼 and for some 𝜀≥0, there exists a solution 𝑓₀∶𝐼→𝑌 of the differential equation

𝑎_𝑛(𝑥)𝑦^(𝑛)(𝑥)+𝑎_𝑛−1(𝑥)𝑦^(𝑛−1)(𝑥)+⋯+𝑎₁(𝑥)𝑦^′(𝑥)+𝑎₀(𝑥)𝑦(𝑥)+ℎ(𝑥)=0 (1.2) such that ‖𝑓(𝑥)−𝑓₀(𝑥)‖≤𝐾(𝜀) for any 𝑥∈𝐼, where 𝐾(𝜀) is an expression of 𝜀 only, then we say that the above differential equation satisfies the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain 𝐼 is not the whole space 𝑋). We may apply these terminologies for other differential equations. For more detailed definition of the Hyers-Ulam stability, refer to [1, 3, 5, 6, 8–11].

Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [12, 13]). Here, we will introduce a result of Alsina and Ger (see [14]): if a differentiable function 𝑓∶𝐼→ℝ is a solution of the differential inequality |𝑦^′(𝑥)−𝑦(𝑥)|≤𝜀, where 𝐼 is an open subinterval of ℝ, then there exists a solution 𝑓₀∶𝐼→ℝ of the differential equation 𝑦^′(𝑥)=𝑦(𝑥) such that |𝑓(𝑥)−𝑓₀(𝑥)|≤3𝜀 for any 𝑥∈𝐼.

This result of Alsina and Ger has been generalized by Takahasi, Miura, and Miyajima: they proved in [15] that the Hyers-Ulam stability holds for the Banach space-valued differential equation 𝑦^′(𝑥)=𝜆𝑦(𝑥) (see also [16]).

Using the conventional power series method, the first author investigated the general solution of the inhomogeneous Hermite differential equation of the form

𝑦^′′(𝑥)−2𝑥𝑦^′(𝑥)+2𝜆𝑦(𝑥)=^∞_𝑚=0𝑎_𝑚𝑥^𝑚 (1.3) under some specific condition, where 𝜆 is a real number and the convergence radius of the power series is positive. This result was applied to prove that every analytic function can be approximated in a neighborhood of 0 by a Hermite function with an error bound expressed by 𝐶𝑥²𝑒^𝑥2 (see [17–20]).

In Section 2 of this paper, using power series method, we will investigate the general solution of the inhomogeneous Euler (or Cauchy) differential equation

𝑥²𝑦^′′(𝑥)+𝛼𝑥𝑦^′(𝑥)+𝛽𝑦(𝑥)=^∞_𝑚=0𝑎_𝑚𝑥^𝑚, (1.4) where 𝛼 and 𝛽 are fixed complex numbers and the coefficients 𝑎_𝑚 of the power series are given such that the radius of convergence is 𝜌>0. Moreover, using the idea from [17–19], we will approximate some analytic functions by the solutions of Euler differential equations.

In this paper, ℕ₀ denotes the set of all nonnegative integers.

2. General Solution of Inhomogeneous Euler Equations

The second-order Euler differential equation

𝑥²𝑦^′′(𝑥)+𝛼𝑥𝑦^′(𝑥)+𝛽𝑦(𝑥)=0, (2.1) which is sometimes called the second-order Cauchy differential equation, is one of the most famous differential equations and frequently appears in applications.

The quadratic equation

𝑚²+(𝛼−1)𝑚+𝛽=0 (2.2) is called the auxiliary equation of the Euler differential equation (2.1), and every solution of (2.1) is of the form

𝑦_ℎ𝑐(𝑥)=¹𝑥^𝑚1+𝑐₂𝑥^𝑚2if𝑚₁and𝑚₂aredistinctrootsof(2.2),(𝑐₁+𝑐₂ln𝑥)𝑥^(1−𝛼)/2if(1−𝛼)2isadoublerootof(2.2), (2.3) where 𝑐₁ and 𝑐₂ are complex constants (see [21, Section 2.7]).

Theorem 2.1. Let 𝛼 and 𝛽 be complex constants such that no root of the auxiliary equation (2.2) is a nonnegative integer. If the radius of convergence of power series ∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚 is at least 𝜌>0, then every solution 𝑦∶(0,𝜌)→ℂ of the inhomogeneous Euler differential equation (1.4) can be expressed by 𝑦(𝑥)=𝑦_ℎ(𝑥)+^∞_𝑚=0𝑎_𝑚𝑥^𝑚𝑚²+(𝛼−1)𝑚+𝛽 (2.4) for all 𝑥∈(0,𝜌), where 𝑦_ℎ(𝑥) is a solution of the Euler differential equation (2.1).

Proof. Assume that 𝑦∶(0,𝜌)→ℂ is a function given by (2.4), where 𝑦_ℎ(𝑥) is a solution of the homogeneous Euler differential equation (2.1). We first prove that the function 𝑦_𝑝(𝑥), defined by 𝑦(𝑥)−𝑦_ℎ(𝑥), satisfies the inhomogeneous equation (1.4).
Indeed, we have 𝑥²𝑦_𝑝^′′(𝑥)+𝛼𝑥𝑦^′_𝑝(𝑥)+𝛽𝑦_𝑝(𝑥)=^∞_𝑚=2𝑚(𝑚−1)𝑎_𝑚𝑥^𝑚𝑚²++(𝛼−1)𝑚+𝛽^∞_𝑚=1𝛼𝑚𝑎_𝑚𝑥^𝑚𝑚²++(𝛼−1)𝑚+𝛽_∞𝑚=0𝛽𝑎_𝑚𝑥^𝑚𝑚²=+(𝛼−1)𝑚+𝛽_∞𝑚=0𝑎_𝑚𝑥^𝑚, (2.5) which proves that 𝑦_𝑝(𝑥) is a particular solution of the inhomogeneous equation (1.4). Moreover, each power series appearing in the above equalities has the same radius of convergence as ∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚 (which can be verified by using the ratio test). Since every solution to (1.4) can be expressed as a sum of a solution 𝑦_ℎ(𝑥) of the homogeneous equation and a particular solution 𝑦_𝑝(𝑥) of the inhomogeneous equation, every solution of (1.4) is certainly of the form (2.4).
We will now apply the ratio test to the power series in (2.4). Indeed, by setting 𝑐_𝑚=𝑎_𝑚/[𝑚²+(𝛼−1)𝑚+𝛽], we get |𝑐_𝑚+1𝑐_𝑚𝑚|=|²+(𝛼−1)𝑚+𝛽(𝑚+1)²𝑎+(𝛼−1)(𝑚+1)+𝛽||^𝑚+1𝑎_𝑚|, (2.6) and hence we have limsup_𝑚→∞|𝑐_𝑚+1𝑐_𝑚|=limsup_𝑚→∞|𝑎_𝑚+1𝑎_𝑚|, (2.7) which implies that the power series given in (2.4) has the same radius of convergence as power series ∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚, which is at least 𝜌. That is, 𝑦(𝑥) given in (2.4) is well defined on its domain (0,𝜌).

3. Approximate Euler Differential Equations

In this section, assume that 𝛼 and 𝛽 are complex constants and 𝜌 is a positive constant. For a given 𝐾≥0, we denote by 𝒞_𝐾 the set of all functions 𝑦∶(0,𝜌)→ℂ with the properties (a) and (b):

(a)𝑦(𝑥) is expressible by a power series ∑^∞_𝑚=0𝑏_𝑚𝑥^𝑚 whose radius of convergence is at least 𝜌; (b)∑^∞_𝑚=0|𝑎_𝑚𝑥^𝑚|≤𝐾|∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚| for any 𝑥∈(0,𝜌), where we set 𝑎_𝑚=[𝑚²+(𝛼−1)𝑚+𝛽]𝑏_𝑚 for 𝑚≥0.

Let {𝑏_𝑚} be a sequence of positive real numbers such that the radius of convergence of the series ∑^∞_𝑚=0𝑏_𝑚𝑥^𝑚 is at least 𝜌, and let 𝛼 and 𝛽 satisfy either 𝛼≥1 and 𝛽>0 or 𝛽≥(𝛼−1)²/4. If a function 𝑦∶(0,𝜌)→ℝ is defined by 𝑦(𝑥)=∑^∞_𝑚=0𝑏_𝑚𝑥^𝑚, then 𝑦 certainly belongs to 𝒞_𝐾 with 𝐾≥1. So, the set 𝒞_𝐾 is not empty if 𝐾≥1. In particular, if 𝜌 is small and 𝐾 is large, then 𝒞_𝐾 is a large class of analytic functions 𝑦∶(0,𝜌)→ℂ.

We will now solve the approximate Euler differential equations in a special class of analytic functions, 𝒞_𝐾.

Theorem 3.1. Let 𝛼 and 𝛽 be complex constants such that no root of the auxiliary equation (2.2) is a nonnegative integer. If a function 𝑦∈𝒞_𝐾 satisfies the differential inequality |𝑥²𝑦^′′(𝑥)+𝛼𝑥𝑦^′(𝑥)+𝛽𝑦(𝑥)|≤𝜀 (3.1) for all 𝑥∈(0,𝜌) and for some 𝜀≥0, then there exists a solution 𝑦_ℎ∶(0,𝜌)→ℂ of the Euler differential equation (2.1) such that |𝑦(𝑥)−𝑦_ℎ(𝑥)|≤(sup_𝑘∈ℕ01|𝑘²)+(𝛼−1)𝑘+𝛽|𝐾𝜌𝜀𝜌−𝑥 (3.2) for any 𝑥∈(0,𝜌).

Proof. Since 𝑦 belongs to 𝒞_𝐾, it follows from (a) and (b) that 𝑥²𝑦^′′(𝑥)+𝛼𝑥𝑦^′(𝑥)+𝛽𝑦(𝑥)=^∞_𝑚=0[𝑚²+(𝛼−1)𝑚+𝛽]𝑏_𝑚𝑥^𝑚=^∞_𝑚=0𝑎_𝑚𝑥^𝑚 (3.3) for all 𝑥∈(0,𝜌). By considering (3.1) and (3.3), we get |^∞_𝑚=0𝑎_𝑚𝑥^𝑚|≤𝜀 (3.4) for any 𝑥∈(0,𝜌). This inequality, together with (b), yields that ^∞_𝑚=0|𝑎_𝑚𝑥^𝑚|≤𝐾|^∞_𝑚=0𝑎_𝑚𝑥^𝑚|≤𝐾𝜀 (3.5) for each 𝑥∈(0,𝜌).
Now, suppose that an arbitrary 𝑥∈(0,𝜌) is given. Then we can choose an arbitrary constant 𝜌₀∈(𝑥,𝜌). By Abel's formula (see [22, Theorem 6.30] ), we have ^𝑛_𝑚=0|𝑎_𝑚𝜌^𝑚₀||𝑥/𝜌₀|^𝑚|𝑚²+(𝛼−1)𝑚+𝛽|=(^𝑛_𝑚=0|𝑎_𝑚𝜌^𝑚₀|)|𝑥/𝜌₀|^𝑛+1|(𝑛+1)²++(𝛼−1)(𝑛+1)+𝛽|_𝑛𝑘=0(^𝑘_𝑚=0|𝑎_𝑚𝜌^𝑚₀|)|𝑥/𝜌₀|^𝑘|𝑘²+(𝛼−1)𝑘+𝛽|𝑀(𝑘;𝜌₀;𝑥), (3.6) where we set 𝑀(𝑘;𝜌₀𝑘;𝑥)=1−|²+(𝛼−1)𝑘+𝛽(𝑘+1)²𝑥+(𝛼−1)(𝑘+1)+𝛽||𝜌₀|. (3.7) Since 𝑀(𝑘;𝜌₀;𝑥)≤1 for any 𝑘∈ℕ₀ and 𝜌₀∈(𝑥,𝜌), it follows from (3.5) and (3.6) that ^𝑛_𝑚=0|𝑎_𝑚𝜌^𝑚₀||𝑥/𝜌₀|^𝑚|𝑚²+(𝛼−1)𝑚+𝛽|≤𝐾𝜀(|𝑥/𝜌₀|^𝑛+1|(𝑛+1)²++(𝛼−1)(𝑛+1)+𝛽|^𝑛_𝑘=0|𝑥/𝜌₀|^𝑘|𝑘²)+(𝛼−1)𝑘+𝛽| (3.8) If we let 𝑛→∞ in the above inequality, then we obtain ^∞_𝑚=0|𝑎_𝑚𝑥^𝑚||𝑚²+(𝛼−1)𝑚+𝛽|≤𝐾𝜀^∞_𝑘=01|𝑘²|𝑥+(𝛼−1)𝑘+𝛽|𝜌₀|^𝑘≤(sup_𝑘∈ℕ01|𝑘²)+(𝛼−1)𝑘+𝛽|𝐾𝜌₀𝜌₀𝜀−|𝑥| (3.9) for all 𝑥∈(0,𝜌) and for any 𝜌₀∈(𝑥,𝜌). Since 𝜌₀/(𝜌₀−𝑥)↓𝜌/(𝜌−𝑥) as 𝜌₀→𝜌, we get ^∞_𝑚=0|𝑎_𝑚𝑥^𝑚||𝑚²+(𝛼−1)𝑚+𝛽|≤(sup_𝑘∈ℕ01|𝑘²)+(𝛼−1)𝑘+𝛽|𝐾𝜌𝜀𝜌−𝑥 (3.10) for every 𝑥∈(0,𝜌).
Finally, it follows from (3.3), Theorem 2.1, and (3.10) that there exists a solution 𝑦_ℎ of the Euler differential equation (2.1) such that |𝑦(𝑥)−𝑦_ℎ(𝑥)|≤|^∞_𝑚=0𝑎_𝑚𝑥^𝑚𝑚²|≤+(𝛼−1)𝑚+𝛽_∞𝑚=0|𝑎_𝑚𝑥^𝑚||𝑚²+(𝛼−1)𝑚+𝛽|≤(sup_𝑘∈ℕ01|𝑘²)+(𝛼−1)𝑘+𝛽|𝐾𝜌𝜀𝜌−𝑥 (3.11) for any 𝑥∈(0,𝜌).

4. An Example

We fix 𝛼=0, 𝛽=1/4 and suppose that a small 𝜀>0 is given. We can choose a constant 0<𝜌<1 such that

𝜀=^∞_𝑚=11(𝑚−1+4𝑚)𝜌^𝑚. (4.1)

We will consider the function 𝑦(𝑥)=ln(1−𝑥) which can be expressed by the power series ∑^∞_𝑚=1(−𝑥^𝑚/𝑚), whose radius of convergence is 1.

If we set 𝑏₀=0 and 𝑏_𝑚=−1/𝑚 for any 𝑚∈ℕ, then it follows from (b) that

𝑎_𝑚=10for𝑚=0,−𝑚+1−4𝑚for𝑚∈ℕ. (4.2) Thus, we have

^∞_𝑚=0|𝑎_𝑚𝑥^𝑚|=^∞_𝑚=11(𝑚−1+4𝑚)𝑥^𝑚=|^∞_𝑚=0𝑎_𝑚𝑥^𝑚| (4.3) for any 𝑥∈(0,𝜌), which enables us to choose 𝐾=1. Thus, it holds that 𝑦∈𝐶₁.

Moreover, we have

|𝑥²𝑦^′′1(𝑥)+4𝑦(𝑥)|=^∞_𝑚=11(𝑚−1+4𝑚)𝑥^𝑚≤^∞_𝑚=11(𝑚−1+4𝑚)𝜌^𝑚=𝜀 (4.4) for all 𝑥∈(0,𝜌). It now follows from Theorem 3.1 that there exist complex numbers 𝐴 and 𝐵 with

|ln(1−𝑥)−(𝐴+𝐵ln𝑥)√𝑥|≤4𝜌𝜌−𝑥^∞_𝑚=11(𝑚−1+4𝑚)𝜌^𝑚 (4.5) for any 𝑥∈(0,𝜌).

Acknowledgments

The authors would like to express their cordial thanks to the referees for their useful remarks and constructive comments which have improved the first version of this paper. This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0071206).