We solve the inhomogeneous Euler differential equations of the form x2y′′(x)+αxy′(x)+βy(x)=∑m=0∞amxm 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

∥an(x)y(n)(x)+an-1(x)y(n-1)(x)+⋯+a1(x)y′(x)+a0(x)y(x)+h(x)∥≤ε
for all x∈I and for some ε≥0, there exists a solution f0:I→Y of the differential equation

an(x)y(n)(x)+an-1(x)y(n-1)(x)+⋯+a1(x)y′(x)+a0(x)y(x)+h(x)=0
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=0∞amxm
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 Cx2ex2 (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=0∞amxm,
where α and β are fixed complex numbers and the coefficients am 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,
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
is called the auxiliary equation of the Euler differential equation (2.1), and every solution of (2.1) is of the form

yh(x)={c1xm1+c2xm2ifm1andm2aredistinctrootsof(2.2),(c1+c2lnx)x(1-α)/2if(1-α)2isadoublerootof(2.2),
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=0∞amxm 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)=yh(x)+∑m=0∞amxmm2+(α-1)m+β
for all x∈(0,ρ), where yh(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 yh(x) is a solution of the homogeneous Euler differential equation (2.1). We first prove that the function yp(x), defined by y(x)-yh(x), satisfies the inhomogeneous equation (1.4).

Indeed, we have
x2yp′′(x)+αxyp′(x)+βyp(x)=∑m=2∞m(m-1)amxmm2+(α-1)m+β+∑m=1∞αmamxmm2+(α-1)m+β+∑m=0∞βamxmm2+(α-1)m+β=∑m=0∞amxm,
which proves that yp(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=0∞amxm (which can be verified by using the ratio test). Since every solution to (1.4) can be expressed as a sum of a solution yh(x) of the homogeneous equation and a particular solution yp(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 cm=am/[m2+(α-1)m+β], we get
|cm+1cm|=|m2+(α-1)m+β(m+1)2+(α-1)(m+1)+β||am+1am|,
and hence we have
limsupm→∞|cm+1cm|=limsupm→∞|am+1am|,
which implies that the power series given in (2.4) has the same radius of convergence as power series ∑m=0∞amxm, 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 𝒞K the set of all functions y:(0,ρ)→ℂ with the properties (a) and (b):

y(x) is expressible by a power series ∑m=0∞bmxm whose radius of convergence is at least ρ;

∑m=0∞|amxm|≤K|∑m=0∞amxm| for any x∈(0,ρ), where we set am=[m2+(α-1)m+β]bm for m≥0.

Let {bm} be a sequence of positive real numbers such that the radius of convergence of the series ∑m=0∞bmxm 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=0∞bmxm, then y certainly belongs to 𝒞K with K≥1. So, the set 𝒞K is not empty if K≥1. In particular, if ρ is small and K is large, then 𝒞K 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, 𝒞K.

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∈𝒞K satisfies the differential inequality
|x2y′′(x)+αxy′(x)+βy(x)|≤ε
for all x∈(0,ρ) and for some ε≥0, then there exists a solution yh:(0,ρ)→ℂ of the Euler differential equation (2.1) such that
|y(x)-yh(x)|≤(supk∈ℕ01|k2+(α-1)k+β|)Kρρ-xε
for any x∈(0,ρ).

Proof.

Since y belongs to 𝒞K, it follows from (a) and (b) that
x2y′′(x)+αxy′(x)+βy(x)=∑m=0∞[m2+(α-1)m+β]bmxm=∑m=0∞amxm
for all x∈(0,ρ). By considering (3.1) and (3.3), we get
|∑m=0∞amxm|≤ε
for any x∈(0,ρ). This inequality, together with (b), yields that
∑m=0∞|amxm|≤K|∑m=0∞amxm|≤Kε
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
∑m=0n|amρ0m||x/ρ0|m|m2+(α-1)m+β|=(∑m=0n|amρ0m|)|x/ρ0|n+1|(n+1)2+(α-1)(n+1)+β|+∑k=0n(∑m=0k|amρ0m|)|x/ρ0|k|k2+(α-1)k+β|M(k;ρ0;x),
where we set
M(k;ρ0;x)=1-|k2+(α-1)k+β(k+1)2+(α-1)(k+1)+β||xρ0|.
Since M(k;ρ0;x)≤1 for any k∈ℕ0 and ρ0∈(x,ρ), it follows from (3.5) and (3.6) that
∑m=0n|amρ0m||x/ρ0|m|m2+(α-1)m+β|≤Kε(|x/ρ0|n+1|(n+1)2+(α-1)(n+1)+β|+∑k=0n|x/ρ0|k|k2+(α-1)k+β|)
If we let n→∞ in the above inequality, then we obtain
∑m=0∞|amxm||m2+(α-1)m+β|≤Kε∑k=0∞1|k2+(α-1)k+β||xρ0|k≤(supk∈ℕ01|k2+(α-1)k+β|)Kρ0ρ0-|x|ε
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∞|amxm||m2+(α-1)m+β|≤(supk∈ℕ01|k2+(α-1)k+β|)Kρρ-xε
for every x∈(0,ρ).

Finally, it follows from (3.3), Theorem 2.1, and (3.10) that there exists a solution yh of the Euler differential equation (2.1) such that
|y(x)-yh(x)|≤|∑m=0∞amxmm2+(α-1)m+β|≤∑m=0∞|amxm||m2+(α-1)m+β|≤(supk∈ℕ01|k2+(α-1)k+β|)Kρρ-xε
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=1∞(m-1+14m)ρm.

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

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

am={0form=0,-m+1-14mform∈ℕ.
Thus, we have

∑m=0∞|amxm|=∑m=1∞(m-1+14m)xm=|∑m=0∞amxm|
for any x∈(0,ρ), which enables us to choose K=1. Thus, it holds that y∈C1.

Moreover, we have

|x2y′′(x)+14y(x)|=∑m=1∞(m-1+14m)xm≤∑m=1∞(m-1+14m)ρm=ε
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=1∞(m-1+14m)ρm
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).

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