We solve the inhomogeneous Chebyshev's differential equation and apply
this result for approximating analytic functions by the Chebyshev functions.

1. Introduction

Let X be a normed space over a scalar field 𝕂, and let I⊂ℝ be an open interval, where 𝕂 denotes either ℝ or ℂ. Assume that a0,a1,…,an:I→𝕂, and g:I→X are given continuous functions and that y:I→X is an n times continuously differentiable function satisfying the inequality ‖an(t)y(n)(t)+an-1(t)y(n-1)(t)+⋯+a1(t)y′(t)+a0(t)y(t)+g(t)‖≤ɛ
for all t∈I and for a given ɛ>0. If there exists an n times continuously differentiable function y0:I→X satisfying an(t)y0(n)(t)+an-1(t)y0(n-1)(t)+⋯+a1(t)y0′(t)+a0(t)y0(t)+g(t)=0
and ∥y(t)-y0(t)∥≤K(ɛ) for any t∈I, where K(ɛ) is an expression of ɛ with limɛ→0K(ɛ)=0, 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 [1–7].

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [8, 9]). Here, we will introduce a result of Alsina and Ger [10]. They proved that if a differentiable function f:I→ℝ satisfies the inequality |y′(t)-y(t)|≤ɛ, where I is an open subinterval of ℝ, then there exists a constant c such that |f(t)-cet|≤3ɛ for any t∈I. Their result was generalized by Takahasi et al. Indeed, it was proved in [11] that the Hyers-Ulam stability holds true for the Banach space valued differential equation y′(t)=λy(t) (see also [12, 13]).

Moreover, Miura et al. [14] investigated the Hyers-Ulam stability of nth order linear differential equation with complex coefficients. They [15] also proved the Hyers-Ulam stability of linear differential equations of first order, y′(t)+g(t)y(t)=0, where g(t) is a continuous function.

Jung also proved the Hyers-Ulam stability of various linear differential equations of first order [16–19]. Moreover, he applied the power series method to the study of the Hyers-Ulam stability of Legendre's differential equation (see [20, 21]). Recently, Jung and Kim tried to prove the Hyers-Ulam stability of the Chebyshev's differential equation(1-x2)y′′(x)-xy′(x)+n2y(x)=0
for all x∈(-1,1). However, the obtained theorem unfortunately does not describe the Hyers-Ulam stability of the Chebyshev's differential equation in a strict sense (see [22]).

In Section 2 of this paper, by using the ideas from [20–26], we investigate the general solution of the inhomogeneous Chebyshev's differential equation of the form(1-x2)y′′(x)-xy′(x)+n2y(x)=∑m=0∞amxm,
where n is a given positive integer. Section 3 will be devoted to the investigation of the Hyers-Ulam stability and an approximation property of the Chebyshev functions.

2. Inhomogeneous Chebyshev’s Equation

Every solution of the Chebyshev's differential equation (1.3) is called a Chebyshev function. The Chebyshev's differential equation has regular singular points at −1, 1, and ∞, and it plays a great role in physics and engineering. In particular, this equation is most useful for treating the boundary value problems exhibiting certain symmetries.

In this section, we set c0=c1=0 and define, for all m∈ℕ,c2m=12m∑i=0m-1a2i2i+1∏j=i+1m-1(2j)2-n22j(2j+1),c2m+1=12m+1∑i=0m-1a2i+12i+2∏j=i+1m-1(2j+1)2-n2(2j+1)(2j+2),
where we refer to (1.4) for the am's and we follow the convention ∏j=mm-1[⋯]=1. We can easily check that cm's satisfy the following relation:(m+2)(m+1)cm+2-(m2-n2)cm=am
for any m∈{0,1,2,…}.

Theorem 2.1.

Assume that n is a positive integer and the radius of convergence of the power series ∑m=0∞amxm is ρ>0. Let ρ0=min{1,ρ}. Then, every solution y:(-ρ0,ρ0)→ℂ of the Chebyshev's differential equation (1.4) can be expressed by
y(x)=yh(x)+∑m=2∞cmxm,
where yh(x) is a Chebyshev function and the cm's are given in (2.1).

Proof.

It is not difficult to see that, if j∈ℕ and |(2j)2-n2|>2j(2j+1), then
j<-1+1+8n28<8n28(for2j<n).
Hence, we have 1≤j≤ne with ne=[n/8]. If m>ne, then it follows from (2.1) that
|c2m|≤12m∑i=0ne-1|a2i|2i+1(∏j=i+1ne|(2j)2-n2|2j(2j+1))(∏j=ne+1m-1|(2j)2-n2|2j(2j+1))+12m∑i=nem-1|a2i|2i+1∏j=i+1m-1|(2j)2-n2|2j(2j+1)≤12m∑i=0ne-1|a2i|2i+1∏j=i+1nen2-42j(2j+1)+12m∑i=nem-1|a2i|2i+1≤12m∑i=0ne-1(2i)!(n2-4)ne-i(2ne+1)!|a2i|+12m∑i=nem-1|a2i|2ne+1≤max0≤i≤ne((2i)!/(2ne+1)!)(n2-4)ne-i2m∑i=0m-1|a2i|.
We now suppose 1≤m≤ne. Then it holds true that n≥3, and we have
|c2m|≤12m∑i=0m-1|a2i|2i+1∏j=i+1m-1|(2j)2-n2|2j(2j+1)≤12m∑i=0m-1|a2i|2i+1∏j=i+1m-1n2-42j(2j+1)=12m∑i=0m-1(2i)!(n2-4)m-1-i(2m-1)!|a2i|≤max0≤i≤m-1((2i)!/(2m-1)!)(n2-4)m-1-i2m∑i=0m-1|a2i|.
Hence, we conclude from the above two inequalities that
|c2m|≤Me2m∑i=0m-1|a2i|
for all m∈ℕ, where we set
Me=max0≤i≤l≤ne(2i)!(2l+1)!(n2-4)l-i.

On the other hand, if j∈ℕ and |(2j+1)2-n2|>(2j+1)(2j+2), then
j<8n2+1-58<8n2-48<n2-12(for2j+1<n).
Hence, we get 1≤j≤no with no=[n/8-1/2]. If m>no, then it follows from (2.1) that
|c2m+1|≤12m+1∑i=0no-1|a2i+1|2i+2(∏j=i+1no|(2j+1)2-n2|(2j+1)(2j+2))(∏j=no+1m-1|(2j+1)2-n2|(2j+1)(2j+2))+12m+1∑i=nom-1|a2i+1|2i+2∏j=i+1m-1|(2j+1)2-n2|(2j+1)(2j+2)≤12m+1∑i=0no-1|a2i+1|2i+2∏j=i+1non2-9(2j+1)(2j+2)+12m+1∑i=nom-1|a2i+1|2i+2≤12m+1∑i=0no-1(2i+1)!(n2-9)no-i(2no+2)!|a2i+1|+12m+1∑i=nom-1|a2i+1|2no+2≤max0≤i≤no((2i+1)!/(2no+2)!)(n2-9)no-i2m+1∑i=0m-1|a2i+1|.
If 1≤m≤no, then we have n≥5, and it follows from (2.1) that
|c2m+1|≤12m+1∑i=0m-1|a2i+1|2i+2∏j=i+1m-1|(2j+1)2-n2|(2j+1)(2j+2)≤12m+1∑i=0m-1|a2i+1|2i+2∏j=i+1m-1n2-9(2j+1)(2j+2)
since j<no and hence 2j+1<2n/8<n. Furthermore, we have
|c2m+1|≤12m+1∑i=0m-1(2i+1)!(n2-9)m-1-i(2m)!|a2i+1|≤max0≤i≤m-1((2i+1)!/(2m)!)(n2-9)m-1-i2m+1∑i=0m-1|a2i+1|.
Thus, we may conclude from the last two inequalities that
|c2m+1|≤Mo2m+1∑i=0m-1|a2i+1|
for any m∈ℕ, where
Mo=max0≤i≤l≤no(2i+1)!(2l+2)!(n2-9)l-i.
Let ρ1 be an arbitrary positive number less than ρ0. Then it follows from (2.7) and (2.13) that
|∑m=2∞cmxm|≤∑m=1∞|c2m||x|2m+∑m=1∞|c2m+1||x|2m+1≤Me∑m=1∞|x|2m2m∑i=0m-1|a2i|+Mo∑m=1∞|x|2m+12m+1∑i=0m-1|a2i+1|=Me|a0|(|x|22+|x|44+|x|66+|x|88+|x|1010+⋯)+Me|a2||x|2(|x|24+|x|46+|x|68+|x|810+|x|1012+⋯)+Me|a4||x|4(|x|26+|x|48+|x|610+|x|812+|x|1014+⋯)+⋯+Mo|a1||x|(|x|23+|x|45+|x|67+|x|89+|x|1011+⋯)+Mo|a3||x|3(|x|25+|x|47+|x|69+|x|811+|x|1013+⋯)+Mo|a5||x|5(|x|27+|x|49+|x|611+|x|813+|x|1015+⋯)+⋯=Me∑m=0∞|a2m||x|2m∑i=1∞|x|2i2(m+i)+Mo∑m=0∞|a2m+1||x|2m+1∑i=1∞|x|2i2(m+i)+1
for any x∈[-ρ1,ρ1].

Because of 0<ρ1<ρ0≤1, we obtain
∑i=1∞|x|2i2(m+i)≤12m+2|x|21-|x|2,∑i=1∞|x|2i2(m+i)+1≤12m+3|x|21-|x|2
for all x∈[-ρ1,ρ1]. Thus, we have
|∑m=2∞cmxm|≤Me∑m=0∞|a2mx2m|2m+2|x|21-|x|2+Mo∑m=0∞|a2m+1x2m+1|2m+3|x|21-|x|2≤Me|x|21-|x|2∑m=0∞|amxm|m+2
for all x∈[-ρ1,ρ1]. Since ρ1 is arbitrarily given with 0<ρ1<ρ0, inequality (2.17) holds true for all x∈(-ρ0,ρ0). Moreover, the power series ∑m=0∞amxm is absolutely convergent on (-ρ,ρ). Hence, we conclude that
|∑m=2∞cmxm|<∞
for all x∈(-ρ0,ρ0). That is, the power series ∑m=2∞cmxm is convergent for each x∈(-ρ0,ρ0).

We will now prove that ∑m=2∞cmxm satisfies the inhomogeneous Chebyshev's differential equation (1.4) for all x∈(-ρ0,ρ0). If we substitute ∑m=2∞cmxm=∑m=1∞c2mx2m+∑m=1∞c2m+1x2m+1 for y(x) in (1.4), then it follows from (2.2) that
(1-x2)y′′(x)-xy′(x)+n2y(x)=∑m=0∞(2m+2)(2m+1)c2m+2x2m+∑m=0∞(2m+3)(2m+2)c2m+3x2m+1-∑m=1∞2m(2m-1)c2mx2m-∑m=1∞(2m+1)(2m)c2m+1x2m+1-∑m=1∞2mc2mx2m-∑m=1∞(2m+1)c2m+1x2m+1+∑m=1∞n2c2mx2m+∑m=1∞n2c2m+1x2m+1=2c2+6c3x+∑m=1∞[(2m+2)(2m+1)c2m+2+(n2-(2m)2)c2m]x2m+∑m=1∞[(2m+3)(2m+2)c2m+3+(n2-(2m+1)2)c2m+1]x2m+1=2c2+6c3x+∑m=1∞a2mx2m+∑m=1∞a2m+1x2m+1=∑m=0∞amxm
for all x∈(-ρ0,ρ0). That is, ∑m=2∞cmxm is a particular solution of the inhomogeneous Chebyshev's differential equation (1.4), and hence every solution y:(-ρ0,ρ0)→ℂ of (1.4) can be expressed by
y(x)=yh(x)+∑m=2∞cmxm,
where yh(x) is a Chebyshev function.

3. Approximate Chebyshev Differential Equation

In this section, let K≥0 and ρ>0 be constants. We denote by 𝒞K the set of all functions y:(-ρ,ρ)→ℂ with the following properties:

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∈(-ρ,ρ), where
am=(m+2)(m+1)bm+2-(m2-n2)bm
for all m∈ℕ0 and set b0=b1=0.

We now investigate the (local) Hyers-Ulam stability problem of the Chebyshev differential equation. More precisely, we try to answer the question, whether there exists a Chebyshev function near any approximate Chebyshev function.

Theorem 3.1.

Let n be a positive integer, and assume that a function y∈𝒞K satisfies the differential inequality
|(1-x2)y′′(x)-xy′(x)+n2y(x)|≤ɛ
for all x∈(-ρ,ρ) and for some ɛ>0. Let ρ0=min{1,ρ}. Then there exists a Chebyshev function yh:(-ρ0,ρ0)→ℂ such that
|y(x)-yh(x)|≤KMeɛ2x21-x2
for all x∈(-ρ0,ρ0), where the constant Me is defined in (2.8).

Proof.

It follows from (a) and (b) that
(1-x2)y′′(x)-xy′(x)+n2y(x)=∑m=0∞amxm
for all x∈(-ρ,ρ) (cf. (2.19)). Moreover, by using (b) and (3.2), we get
∑m=0∞|amxm|≤K|∑m=0∞amxm|≤Kɛ
for any x∈(-ρ,ρ).

According to Theorem 2.1 and (3.4), y(x) can be written as yh(x)+∑m=2∞cmxm for all x∈(-ρ0,ρ0), where yh is some Chebyshev function and cm's are given in (2.1). It moreover follows from (2.17) and (3.5) that
|y(x)-yh(x)|=|∑m=2∞cmxm|≤Mex21-x2K2ɛ
for all x∈(-ρ0,ρ0).

If ρ is assumed to be less than 1, then ρ0=ρ<1 and Theorem 3.1 implies the Hyers-Ulam stability of the Chebyshev's differential equation (1.3).

Remark 3.2.

We give some values for ne, no, Me, and Mo in Table 1.

n

ne

no

Me

Mo

1

0

−1

1

-∞

2

0

0

1

1/2

3

1

0

1

1/2

4

1

0

2

1/2

5

1

1

7/2

2/3

6

2

1

128/15

9/8

Corollary 3.3.

Let n be a positive integer, and assume that a function y∈𝒞K satisfies the differential inequality (3.2) for all x∈(-ρ,ρ) and for some ɛ>0. Let ρ0=min{1,ρ}. Then there exists a Chebyshev function yh:(-ρ0,ρ0)→ℂ such that
|y(x)-yh(x)|=O(x2)
as x→0.

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0004919).

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