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Siafarikas We solve the inhomogeneous Bessel differential equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the Bessel differential equati

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2007, Article ID 21640, 8 pages

doi:10.1155/2007/21640

Research Article

Bessel’s Differential Equation and Its Hyers-Ulam Stability

Byungbae Kim and Soon-Mo Jung

Received 23 August 2007; Accepted 25 October 2007

Recommended by Panayiotis D Siafarikas

We solve the inhomogeneous Bessel differential equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the Bessel differential equation Copyright © 2007 B Kim and S.-M Jung This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

In 1940, Ulam gave a wide ranging talk before the Mathematics Club of the University of Wisconsin, in which he discussed a number of important unsolved problems (see [1]) Among those was the question concerning the stability of homomorphisms: letG1be a group and letG2be a metric group with a metricd( ·,·) Given anyδ > 0, does there exist

anε > 0 such that if a function h : G1→ G2satisfies the inequalityd(h(xy), h(x)h(y)) < ε

for allx, y ∈ G1, then there exists a homomorphismH : G1→ G2withd(h(x), H(x)) < δ

for allx ∈ G1?

In the following year, Hyers [2] partially solved the Ulam problem for the case where

G1 andG2are Banach spaces Furthermore, the result of Hyers has been generalized by Rassias (see [3]) Since then, the stability problems of various functional equations have been investigated by many authors (see [4–6])

We will now consider the Hyers-Ulam stability problem for the differential equations: assume that X is a normed space over a scalar field Kand that I is an open interval,

whereKdenotes eitherRorC Leta0,a1, , a n:I →K be given continuous functions, letg : I → X be a given continuous function, and let y : I → X be an n times continuously

differentiable function satisfying the inequality

a n(t)y(n)(t) + a n −1(t)y(n −1)(t) + ···+a1(t)y (t) + a0(t)y(t) + g(t)  ≤ ε (1.1)

for allt ∈ I and for a given ε > 0 If there exists an n times continuously differentiable functiony0:I → X satisfying

a n(t)y(0n)(t) + a n −1(t)y(0n −1)(t) + ···+a1(t)y 0(t) + a0(t)y0(t) + g(t) =0 (1.2)

and y(t) − y0(t) ≤ K(ε) for any t ∈ I, where K(ε) is an expression of ε with lim ε →0K(ε) =

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 [4–8]

Alsina and Ger were the first authors who investigated the Hyers-Ulam stability of differential equations They proved in [9] that if a differentiable function f : I→Ris a solution of the differential inequality| y (t) − y(t) | ≤ ε, where I is an open subinterval of

R, then there exists a solution f0:I →Rof the differential equation y (t) = y(t) such that

| f (t) − f0(t) | ≤3ε for any t ∈ I.

This result of Alsina and Ger has been generalized by Takahasi et al They proved in [10] that the Hyers-Ulam stability holds true for the Banach space valued differential equationy (t) = λy(t) (see also [11,12])

Moreover, Miura et al [13] investigated the Hyers-Ulam stability of nth order

lin-ear differential equation with complex coefficients They [14] 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 Indeed, they dealt with the differential inequality y (t) + g(t)y(t)  ≤ ε for some ε > 0.

Recently, Jung proved the Hyers-Ulam stability of various linear differential equations

of first order (see [15–18]) and further investigated the general solution of the inhomo-geneous Legendre differential equation and its Hyers-Ulam stability (see [14,19])

In Section 2 of this paper, by using the ideas from [19], we investigate the general solution of the inhomogeneous Bessel differential equation of the form

x2y (x) + xy (x) +

x2− ν2 

y(x) =

∞



m =0

where the parameterν is a given positive nonintegral number.Section 3will be devoted to

a partial solution of the Hyers-Ulam stability problem for the Bessel differential equation (2.1) in a subclass of analytic functions

2 Inhomogeneous Bessel equation

A function is called a Bessel function if it satisfies the Bessel differential equation

x2y (x) + xy (x) +

x2− ν2 

The Bessel equation plays a great role in physics and engineering In particular, this equation is most useful for treating the boundary-value problems exhibiting cylindrical symmetries

B Kim and S.-M Jung 3

In this section, we define

c m = −

[m/2]

i =0

a m −2i i



j =0

1

for eachm ∈ {0, 1, 2, }, where [m/2] denotes the largest integer not exceeding m/2, and

we refer to (1.3) for thea m’s We can easily check thatc m’s satisfy

a0= − ν2c0, a1= −(ν2−1)c1,

a m+2 = c m −ν2−(m + 2)2

for anym ∈ {0, 1, 2, }

Lemma 1 (a) If the power series∞

m =0a m x m converges for all x ∈(− ρ, ρ) with ρ > 1, then the power series∞

m =0c m x m with c m ’s given in ( 2.2 ) satisfies the inequality |∞ m =0c m x m | ≤

C1/(1 − | x | ) for some positive constant C1and for any x ∈(− 1, 1).

(b) If the power series∞

m =0a m x m converges for all x ∈(− ρ, ρ) with ρ ≤ 1, then for any positive ρ0< ρ, the power series∞

m =0c m x m with c m ’s given in ( 2.2 ) satisfies the inequality

|∞ m =0c m x m | ≤ C2for any x ∈(− ρ0,ρ0) and for some positive constant C2which depends

on ρ0 Since ρ0 is arbitrarily close to ρ, this means that ∞

m =0c m x m is convergent for all

x ∈(− ρ, ρ).

Proof (a) Since the power series∞

m =0a m x m is absolutely convergent on its interval of convergence, withx =1,∞

m =0a mconverges absolutely, that is,∞

m =0| a m | < M1by some numberM1 Suppose that p < ν < p + 1 for some integer p Then for any nonnegative

integerq, 1/ | ν2− q2| =1/ | ν + q |1/ | ν − q |is less than 1 except, possibly, forq = p and

q = p + 1 Therefore,

i



j =0

1

ν2−(m −2j)2 ≤max

1

ν2− p2 ,ν2−(1p + 1)2 = M2 (2.4) for anym and i Now,

c m  ≤[m/2]

i =0

a m −2i i

j =0

1

ν2−(m −2j)2 ≤

[m/2]

i =0

a m −2iM2≤ M1M2= C1 (2.5)

and, therefore,







∞



m =0

c m x m



 ≤

∞



m =0

c mx m  ≤ C1

∞



m =0

x m  ≤ C1

1− | x | (2.6)

forx ∈(−1, 1)

(b) The power series∞

m =0a m x m is absolutely convergent on its interval of conver-gence, and, therefore, for any givenρ0<ρ, the series∞

m =0| a m x m |is convergent on [− ρ0,ρ0] and

∞



m =0

a m | x | m ≤

∞



m =0

a mρ m

for anyx ∈[− ρ0,ρ0]

Also form ≥ p + 2, if we let M2 =max{1,M2}, then

i



j =0

1

ν2−(m −2j)2 ≤ν2−1m2 M2 ≤ 1

(m − p −1)2M



Now,







∞



m = p+2

c m x m



 =





 −

∞



m = p+2

x m

[m/2]

i =0

a m −2i i



j =0

1

ν2−(m −2j)2







≤

∞



m = p+2

[m/2]

i =0

a m −2iρ m

0

1 (m − p −1)2M



2

≤

∞



m = p+2

1 (m − p −1)2

[m/2]

i =0

a m −2iρ m −2i

0 M2

≤

∞



m = p+2

1 (m − p −1)2M3M



2

=∞

k =1

1

k2M3M 2≤2M3M 2,

(2.9)

and, therefore, if|m p+1 =0c m x m | ≤m p+1 =0

 [m/2]

i =0 | a m −2i | ρ m −2i

0 M2≤(p + 2)M3M2, then







∞



m =0

c m x m



 ≤(p + 2)M2M3+ 2M2 M3=(p + 2)M2+ 2M2

M3= C2 (2.10)

Lemma 2 Suppose that the power series∞

m =0a m x m converges for all x ∈(− ρ, ρ) with some positive ρ Let ρ1=min{1,ρ } Then the power series∞

m =0c m x m with c m ’s given in ( 2.2 ) is convergent for all x ∈(− ρ1,ρ1) Further, for any positive ρ0< ρ1, |∞ m =0c m x m | ≤ C for any

x ∈(− ρ0,ρ0) and for some positive constant C which depends on ρ0.

Proof The first statement follows from the latter statement Therefore, let us prove the

latter statement If ρ ≤1, then ρ1= ρ By Lemma 1(b), for any positive ρ0< ρ = ρ1,

|∞ m =0c m x m | ≤ C2 for x ∈(− ρ0,ρ0) and for some positive constant C2 which depends

onρ

B Kim and S.-M Jung 5

Ifρ > 1, then byLemma 1(a), for any positiveρ0< 1 = ρ1,







∞



m =0

c m x m



 ≤1− | C1x | <1C −1ρ0 = C (2.11)

forx ∈(− ρ0,ρ0) and for some positive constantC which depends on ρ0 

Using these definitions and the lemmas above, we will show that∞

m =0c m x mis a par-ticular solution of the inhomogeneous Bessel equation (1.3)

Theorem 2.1 Assume that ν is a given positive nonintegral number and the radius of convergence of the power series∞

m =0a m x m is ρ Let ρ1=min{1,ρ } Then, every solution

y : ( − ρ1,ρ1)→C of the differential equation ( 1.3 ) can be expressed by

y(x) = y h(x) +

∞



m =0

where y h(x) is a Bessel function and c m ’s are given by( 2.2 ).

Proof We show that∞

m =0c m x msatisfies (1.3) ByLemma 2, the power series∞

m =0c m x m

is convergent for eachx ∈(− ρ1,ρ1)

Substituting∞

m =0c m x mfory(x) in (1.3) and collecting like powers together, we have

x2y (x) + xy (x) +

x2− ν2 

y(x)

= − ν2c0−ν2−1

c1x +

∞



m =0

c m −ν2−(m + 2)2

c m+2

x m+2

= a0+a1x +

∞



m =0

a m+2 x m+2 =

∞



m =0

a m x m

(2.13)

for allx ∈(− ρ1,ρ1) by (2.3)

Therefore, every solutiony : ( − ρ1,ρ1)→Cof the differential equation (1.3) can be ex-pressed by

y(x) = y h(x) +

∞



m =0

3 Partial solution to Hyers-Ulam stability problem

In this section, we will investigate a property of the Bessel differential equation (2.1) con-cerning the Hyers-Ulam stability problem That is, we will try to answer the question whether there exists a Bessel function near any approximate Bessel function

Theorem 3.1 Let y : ( − ρ, ρ) →C be a given analytic function which can be represented by

a power-series expansion centered at x = 0 Suppose there exists a constant ε > 0 such that

x2y (x) + xy (x) +

x2− ν2 

for all x ∈(− ρ, ρ) and for some positive nonintegral number ν Let ρ1=min{1,ρ } Suppose, further, that x2y (x) + xy (x) + (x2− ν2)y(x) =∞ m =0a m x m satisfies

∞



m =0

a m x m  ≤ K







∞



m =0

a m x m





for all x ∈(− ρ, ρ) and for some constant K Then there exists a Bessel function y h: (− ρ1,ρ1)→

Csuch that

y(x) − y h(x)  ≤ Cε (3.3)

for all x ∈(− ρ0,ρ0), where ρ0< ρ1is any positive number and C is some constant which depends on ρ0.

Proof We assumed that y(x) can be represented by a power series and

x2y (x) + xy (x) +

x2− ν2 

y(x) =

∞



m =0

also satisfies

∞



m =0

a m x m  ≤ K





∞



m =0

a m x m



for allx ∈(− ρ, ρ) from (3.1)

According toTheorem 2.1,y can be written as y h+∞

m =0c m x mforx ∈(− ρ1,ρ1), where

y his some Bessel function andc m’s are given by (2.2) Then by Lemmas1and2and their proofs (replaceM1andM3withKε inLemma 1),

y(x) − y h(x)  =∞

m =0

c m x m

for allx ∈(− ρ0,ρ0), whereρ0< ρ1is any positive number andC is some constant which

depends onρ0 This completes the proof of our theorem 

4 Example

In this section, our task is to show that there certainly exist functionsy(x) which satisfy

all the conditions given inTheorem 3.1

Example 1 Let y : ( −1, 1)→Rbe an analytic function given by

y(x) = J1/2(x) + b

x2+x4+···+x2n

whereJ1/2(x) is the Bessel function of the first kind of order 1/2, n is a given positive

integer, andb is a constant satisfying

0≤ b ≤ 2

3n



2n2+ 3n +17

8

−1

B Kim and S.-M Jung 7 for some ε ≥0 Since J1/2(x) is a particular solution of the Bessel differential equation (2.1) withν =1/2, we then have

x2y (x) + xy (x) +



x2−1

4



y(x) = bx2n+2+

n



m =2

2m 2

+3 4



bx2m+15

4 bx

2. (4.3)

If we set

a m =

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪



m2+3 4



b form ∈ {4, 6, , 2n },



15 4



(4.4)

then we obtain

x2y (x) + xy (x) +



x2−1

4



y(x) =

∞



m =0

for allx ∈(−1, 1) It further follows from (4.2) and (4.4) that

∞



m =0

a m x m  =∞

m =0

a m x m



for anyx ∈(−1, 1)

Indeed, if we choose theJ1/2(x) as a Bessel function, then we have

y(x) − J1/2(x)  = bx2+x4+···+x2n  ≤ nb ≤ n 2

3n



2n2+ 3n +17

8

−1

ε (4.7) for allx ∈(−1, 1), which is consistent with the assertion ofTheorem 3.1

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and Applied Analysis.

Byungbae Kim: Mathematics Section, College of Science and Technology, Hong-Ik University, Chochiwon 339-701, South Korea

Email address:bkim@hongik.ac.kr

Soon-Mo Jung: Mathematics Section, College of Science and Technology, Hong-Ik University, Chochiwon 339-701, South Korea

Email address:smjung@hongik.ac.kr

and Applied Analysis.

Byungbae Kim: Mathematics Section, College of Science and. .. data-page="8">

[8] E Liz and M Pituk, “Exponential stability in a scalar functional differential equation, ” Journal

of Inequalities and Applications, vol 2006, Article ID 37195,... his some Bessel function and< i>c m’s are given by (2.2) Then by Lemmas 1and2 and their proofs (replaceM1and< i>M3withKε