# Approximation of Analytic Functions by Solutions of Cauchy-Euler Equation

Authors: Soon-Mo Jung
Journal: Journal of Function Spaces
 (2016)

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## Abstract

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


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## Body

## 1. Introduction

Throughout this paper, letn be a positive integer, let I be a nondegenerate interval of R, and let K denote either R or C. We will consider the (linear) differential equation of nth order(1)Fyn,yn-1,…,y′,y,x=0defined on I, where y:I→K is an n times continuously differentiable function.For arbitraryε>0, assume that an n times continuously differentiable function y:I→K satisfies the differential inequality(2)Fyn,yn-1,…,y′,y,x≤εfor all x∈I. If for each function y:I→K satisfying inequality (2) there exists a solution y0:I→K of the differential equation (1) such that(3)yx-y0x≤Kεfor any x∈I, where K(ε) depends on ε only and satisfies limε→0⁡K(ε)=0, then we say that the differential equation (1) satisfies (or has) the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain I is not the whole space R). If the above statement also holds when we replace ε and K(ε) with some appropriate φ(x) and Φ(x), 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 g,r:(a,b)→R be continuous functions with ∫ab‍gxdx<∞, where a and b are real constants. Assume that ε>0 is an arbitrary real number. Obłoza proved that if a differentiable function y:(a,b)→R satisfies the inequality y′x+gxyx-rx≤ε for all x∈(a,b) and if a function y0:(a,b)→R satisfies y0′(x)+g(x)y0(x)=r(x) for all x∈(a,b) and y(τ)=y0(τ) for some τ∈(a,b), then there exists a constant δ>0 such that yx-y0x≤δ for all x∈(a,b).Thereafter, Alsina and Ger [7] proved that if a differentiable function f:(a,b)→R satisfies the differential inequality y′x-yx≤ε, then there exists a solution f0:(a,b)→R of the differential equation y′(x)=y(x) such that fx-f0x≤3ε for any x∈(a,b). 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 y′(x)=λy(x) (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(4)x2y′′x+αxy′x+βyx=rx,where α and β are real-valued constants and r:R→R is a continuous function, and we investigate the approximation properties of twice continuously differentiable functions by solutions of the Cauchy-Euler equation(5)x2y′′x+αxy′x+βyx=0,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 (α-1)2-4β>0.Theorem 1.
Assume that the real-valued constantsα, β are given with (α-1)2-4β>0 and ε is an arbitrarily given positive constant. Let c be a positive real-valued constant and let m1, m2 be given as (6)m1=-α-1-α-12-4β2,m2=-α-1+α-12-4β2.If r:(0,∞)→R is a differentiable function and y:(0,∞)→R is a twice continuously differentiable function such that the inequality(7)x2y′′x+αxy′x+βyx-rx≤εholds for any x∈(0,∞), then there exists a solution yc:(0,∞)→R of the inhomogeneous Cauchy-Euler equation (4) such that (8)
y
x
-
y
c
x
≤
ε
m
1
m
2
+
ε
m
2
-
m
1
1
m
2
x
c
m
2
-
1
m
1
x
c
m
1
f
o
r
m
1
≠
0
≠
m
2
,
ε
m
2
2
x
c
m
2
-
1
-
ε
m
2
ln
⁡
x
c
f
o
r
m
1
=
0
,
ε
m
1
2
x
c
m
1
-
1
-
ε
m
1
ln
⁡
x
c
f
o
r
m
2
=
0for all x∈(0,∞).For the case of(α-1)2-4β=0, 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 constantsα and β are given with α≠1, β=(α-1)2/4 and ε is an arbitrarily given positive constant. Let c be a positive real-valued constant and let λ=-α-1/2. If r:(0,∞)→R is a differentiable function and y:(0,∞)→R is a twice continuously differentiable function such that the inequality (9)x2y′′x+αxy′x+α-124yx-rx≤εholds for all x∈(0,∞), then there exists a solution yc:(0,∞)→R of the inhomogeneous Cauchy-Euler equation (4) with β=(α-1)2/4 such that (10)yx-ycx≤ελ2+ελxcλln⁡xc-1λfor all x∈(0,∞).Finally, the Hyers-Ulam stability of the Cauchy-Euler equation (4) was also proven in [16, Theorem  4.5] for the case of (α-1)2-4β<0.Theorem 3.
Assume that the real-valued constantsα and β are given with (α-1)2-4β<0 and ε is an arbitrarily given positive constant. Let c>0 be a given real-valued constant and let (11)λ=-α-12,μ=124β-α-12.If a differentiable function r:(0,∞)→R and a twice continuously differentiable function y:(0,∞)→R satisfy inequality (7) for all x∈(0,∞), then there exists a solution yc:(0,∞)→R of the inhomogeneous Cauchy-Euler equation (4) such that (12)yx-ycx≤εμ∫cxxλζλ+1sin⁡μln⁡xζdζfor all x∈(0,∞).Remark 4.
Cîmpean and Popa [14] proved the Hyers-Ulam stability of the linear differential equations of nth 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 (13)
y
x
-
y
c
x
≤
ε
β
for  either
α
2
-
4
β
>
0
,
β
≠
0
or
α
2
-
4
β
=
0
,
α
≠
0
,
4
ε
α
2
for
α
2
-
4
β
<
0
,
α
≠
0
.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 byB(α;β) the set of all twice continuously differentiable functions y:(0,∞)→R for which there exists a constant ε>0 such that(14)x2y′′x+αxy′x+βyx≤εfor all x∈(0,∞), where α and β are real-valued constants.If we define(15)y1+y2x≔y1x+y2x,γy1x≔γy1xfor all y1,y2∈B(α;β) and γ∈R, then B(α;β) is a vector space over R. This fact implies that the set B(α;β) is large enough to be a vector space.In the following theorems, we investigate approximation properties of functions ofB(α;β) by solutions of the Cauchy-Euler equation (5).Theorem 5.
Letc>0 be a given real number and let α,β∈R be given with (α-1)2-4β>0. If y∈B(α;β), then there exists a solution yc:(0,∞)→R of the Cauchy-Euler equation (5) such that (16)yx-ycx=ox-cas x→c.Proof.
We definem1 and m2 by the formulas given in Theorem 1; that is, m1 and m2 are the distinct roots of the indicial equation m2+(α-1)m+β=0. Since (α-1)2-4β>0, we have m1<m2. Since y∈B(α;β), there exists a constant ε>0 such that inequality (14) holds for all x∈(0,∞).
According to Theorem1 with r(x)≡0, there exists a solution yc:(0,∞)→R of the Cauchy-Euler equation (5) such that (17)
y
x
-
y
c
x
≤
ε
m
1
m
2
+
ε
m
2
-
m
1
1
m
2
x
c
m
2
-
1
m
1
x
c
m
1
for
m
1
≠
0
≠
m
2
,
ε
m
2
2
x
c
m
2
-
1
-
ε
m
2
ln
⁡
x
c
for
m
1
=
0
,
ε
m
1
2
x
c
m
1
-
1
-
ε
m
1
ln
⁡
x
c
for
m
2
=
0for any x∈(0,∞).
We will only estimate the following limit for the case ofm1=0 by applying L’Hospital’s rule: (18)limx→c⁡yx-ycxx-c≤limx→c⁡ε/m22x/cm2-1-ε/m2ln⁡x/cx-c=limx→c⁡εm2cxcm2-1-εm21x=0,which implies the validity of this theorem.We now consider the case of(α-1)2-4β=0 and use Theorem 2 to prove the following theorem.Theorem 6.
Letc>0 and α≠1 be real numbers and let λ=-α-1/2. If y∈B(α;λ2), then there exists a solution yc:(0,∞)→R of the Cauchy-Euler equation (5) with β=λ2 such that (19)yx-ycx=ox-cas x→c.Proof.
Sincey∈B(α;λ2), there exists a constant ε>0 such that inequality (14) holds for all x∈(0,∞). According to Theorem 2 with r(x)≡0, there exists a solution yc:(0,∞)→R of the Cauchy-Euler equation (5) with β=λ2 such that (20)yx-ycx≤ελ2+ελxcλln⁡xc-1λfor all x∈(0,∞).
Therefore, we estimate the limit by applying L’Hospital’s rule:(21)limx→c⁡yx-ycxx-c≤limx→c⁡ε/λ2+ε/λx/cλln⁡x/c-1/λx-c=limx→c⁡εxλ-1cλln⁡xc=0,which implies the validity of this theorem.Finally, we investigate the approximation property of each function ofB(α;β) by a solution of the differential equation (5) when (α-1)2-4β<0.Theorem 7.
Letc>0 be a given real number and let α,β∈R be given with (α-1)2-4β<0. If y∈B(α;β), then there exists a solution yc:(0,∞)→R of the Cauchy-Euler equation (5) such that (22)yx-ycx=ox-cas x→c.Proof.
Let us define(23)λ=-α-12,μ=124β-α-12.Since y∈B(α;β), there exists a constant ε>0 such that inequality (14) holds for all x∈(0,∞). According to Theorem 3 with r(x)≡0, there exists a solution yc:(0,∞)→R of the Cauchy-Euler equation (5) such that (24)yx-ycx≤εμ∫cxxλζλ+1sin⁡μln⁡xζdζfor all x∈(0,∞).
If we substituteη=cx/ζ, then we have (25)∫cxxλζλ+1sin⁡μln⁡xζdζ=1cλ∫cxηλ-1sin⁡μln⁡ηcdη.Hence, we further apply L’Hospital’s rule to obtain (26)limx→c⁡yx-ycxx-c≤εμlimx→c⁡1x-c∫cxxλζλ+1sin⁡μln⁡xζdζ=εμcλlimx→c⁡1x-c∫cxηλ-1sin⁡μln⁡ηcdη=εμcλlimx→c⁡xλ-1sin⁡μln⁡xc=0,which implies the validity of this theorem.



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*Source: 7874061-2016-06-06.xml*