Hyers-Ulam Stability of Differential Equation y

Volume 2010, Article ID 793197,12pages doi:10.1155/2010/793197

Research Article

Hyers-Ulam Stability of Differential Equation y

2xy

− 2ny 0

Soon-Mo Jung

Mathematics Section, College of Science and Technology, Hongik University, Jochiwon 339-701, South Korea

Correspondence should be addressed to Soon-Mo Jung,[email protected] Received 13 October 2009; Accepted 24 November 2009

Academic Editor: Yeol Je Cho

Copyrightq2010 Soon-Mo 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.

We solve the inhomogeneous differential equation of the formy2xy−2ny_∞

m0amx^m, where nis a nonnegative integer, and apply this result to the proof of a local Hyers-Ulam stability of the differential equationy2xy−2ny0 in a special class of analytic functions.

1. Introduction

Assume thatX andY are a topological vector space and a normed space, respectively, and thatIis an open subset ofX. If for any functionf:I → Ysatisfying the differential inequality anxyⁿx an−1xyⁿ⁻¹x · · ·a1xyx a0xyx hx≤ε 1.1

for allx∈Iand for someε≥0, there exists a solutionf0:I → Y of the differential equation anxyⁿx an−1xyⁿ⁻¹x · · ·a1xyx a0xyx hx 0 1.2 such thatfx−f0x ≤Kεfor anyx∈I, whereKεis an expression ofεonly, then we say that the above differential equation satisfies the Hyers-Ulam stabilityor the local Hyers- Ulam stability if the domainIis not the whole spaceX. We may apply this terminology for other differential equations. For more detailed definition of the Hyers-Ulam stability, refer to 1–6.

Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equationssee7,8. Here, we will introduce a result of Alsina and Ger see9: If a differentiable functionf : I → Ris a solution of the differential inequality

|yx−yx| ≤ε, whereIis an open subinterval ofR, then there exists a solutionf0 :I → R of the differential equationyx yxsuch that|fx−f0x| ≤3εfor anyx∈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 equationyx λyx see also11.

Using the conventional power series method, the author in 12 investigated the general solution of the inhomogeneous Legendre differential equation of the form

1−x²

yx−2xyx p p1

yx ^∞

m0

amx^m 1.3

under some specific conditions, wherepis a real number and the convergence radius of the power series is positive. Moreover, he applied this result to prove that every analytic function can be approximated in a neighborhood of 0 by the Legendre function with an error bound expressed byCx²/1−x² see13–15.

Let us consider the error function and the complementary error function defined by

erfx 2

√π

x 0

e^−t2dt, erfcx 2

√π

∞ x

e^−t2dt1−erfx, 1.4

respectively. We recursively define the integrals of the error function as follows:

i⁻¹erfcx 2

√πe^−x2, i⁰erfcxerfcx, i^merfcx ^∞

x

i^m−1erfctdt 1.5

for any m ∈ N0. Suppose that we are given a nonnegative integern, and we introduce a differential equation

yx 2xyx−2nyx 0, 1.6

whose general solution is given by

yx AiⁿerfcxBiⁿerfc−x 1.7 see16,§7.2.2.

InSection 2of this paper, using power series method, we will investigate the general solution of the inhomogeneous differential equation:

yx 2xyx−2nyx ^∞

m0

amx^m, 1.8

where the radius of convergence of the power series_∞

m0amx^misρ >0, whose value is in general permitted to have infinity. Moreover, using the idea from12–14, we will prove the Hyers-Ulam stability of the differential equation1.6in a class of special analytic functions see the classCKinSection 3.

In this paper,N0denotes the set of all nonnegative integers.

2. General Solution of 1.8

In the following theorem, we solve the inhomogeneous differential equation1.8.

Theorem 2.1. Assume thatnis a nonnegative integer, the radius of convergence of the power series _∞

m0amx^misρ >0, and that there exists a real numberμ≥0 with

|a2m| ≤μ²m2m22m1|α2m2| if nis odd,

|a2m1| ≤μ²m2m32m2β2m3 ifnis even 2.1

for all sufficiently large integersm, where

α2m 2^m−1 2m!

m−2

k0

2k!

2^k a2k

m−1 ik1

n−2i,

β_2m1 2^m−1 2m1!

m−2

k0

2k1!

2^k a_2k1

m−1

ik1

n−2i1

2.2

for anym∈ {2,3, . . .}. Let us defineρ0min{ρ,1/μ}and 1/0∞. Every solutiony:−ρ0, ρ0 → Cof the inhomogeneous differential equation1.8can be represented by

yx yhx ^∞

m2

a_m−2

mm−1x^m∞

m2

α2mx^2m∞

m2

β_2m1x^2m1, 2.3

whereyhxis a solution of the homogeneous differential equation1.6.

Proof. Assume that a functiony : −ρ0, ρ0 → Cis given by2.3. We first prove that the function ypx, defined byyx−yhx, satisfies the inhomogeneous differential equation 1.8. Since

y_px ^∞

m2

am−2

m−1x^m−1∞

m2

2mα2mx^2m−1∞

m2

2m1β_2m1x^2m,

y_px ^∞

m0

amx^m∞

m1

2m22m1α2m2x^2m

^∞

m1

2m32m2β2m3x^2m1,

2.4

we have

y_px 2xy_px−2nypx ^∞

m0

amx^m∞

m1

2m22m1α_2m2x^2m

^∞

m1

2m32m2β_2m3x^2m1∞

m2

2a_m−2 m−1x^{m ∞}

m2

4mα2mx^2m∞

m2

22m1β2m1x^2m1

−^∞

m2

2nam−2

mm−1x^m−^∞

m2

2nα2mx^2m−^∞

m2

2nβ2m1x^2m1

^∞

m0amx^m12α4x^2∞

m2

2m22m1α_2m2x^2m

20β5x^3∞

m2

2m32m2β_2m3x^2m1

^∞

m2

2m−n

mm−1am−2x^m−^∞

m2

2n−2mα2mx^2m

−^∞

m2

2n−2m1β2m1x^2m1

2.5

for allx∈−ρ0, ρ0.

It is not difficult to see that

2m22m1α_2m22n−2mα2m n−2m

m2m−1a_2m−2, 2m32m2β_2m32n−2m1β_2m1n−2m1

2m1m a_2m−1

2.6

for anym∈N. Hence, we obtain

y_px 2xy_px−2nypx ^∞

m0

amx^m12α4x^2∞

m2

n−2m

m2m−1a_2m−2x^2m20β5x^{3 ∞}

m2

n−2m1

m2m1 a_2m−1x^2m1∞

m2

2m−n

mm−1a_m−2x^{m ∞}

m0

amx^m,

2.7

which proves thatypxis a particular solution of the inhomogeneous equation1.8.

We now apply the ratio test to the power series expression ofypx. Ifn is an odd integer not less than 0, then β_n2 β_n4 β_n6 · · · 0. Hence, the power series _∞

m2β2m1x^2m1 is a polynomial. And it follows from the first conditions of2.1and2.6 that

mlim→ ∞

α_2m2 α2m

lim

m→ ∞

n−2m

m12m1 n−2m

m2m22m12m−1 a_2m−2

α2m

lim

m→ ∞

|n−2m|

m2m22m12m−1 a_2m−2

α2m

≤ lim

m→ ∞

|n−2m|

m2m22m12m−1μ²m−12m2m−1 μ².

2.8

Ifn ≥0 is an even integer, then we haveαn2 αn4 αn6 · · · 0. Thus, for each even integern≥0, the power series_∞

m2α2mx^2mis a polynomial. By the second conditions in2.1and2.6, we get

mlim→ ∞

β2m3

β2m1

lim

m→ ∞

n−2m1

2m3m1 n−2m1 2m32m22m1m

a2m−1

β2m1

lim

m→ ∞

|n−2m1|

2m32m22m1m a2m−1

β2m1

≤ lim

m→ ∞

|n−2m1|

2m32m22m1mμ²m−12m12m μ².

2.9

Therefore, the power series expression ofypxconverges for allx∈−ρ0, ρ0.

Moreover, the convergence region of the power series forypxis the same as those of power series fory_pxand y_px. In this paper, the convergence region will denote the maximum open set where the relevant power series converges. Hence, the power series expression fory_px 2xy_px−2nypxhas the same convergence region as that ofypx.

This implies thatypxis well defined on−ρ0, ρ0and so does foryxin2.3becauseyhx converges for allx∈Runder our hypotheses.

Since every solution to 1.8 can be expressed as a sum of a solution yhx of the homogeneous equation and a particular solutionypxof the inhomogeneous equation, every solution of1.8is certainly in the form of2.3.

Remark 2.2. We might have thought that the conditions presented in2.1were too strong.

However, we can show that some familiar sequences{am}satisfy the conditions in2.1. For example, letn 0 and a0 a1 1,a2m a2m1 1/m−1! for allm∈ Nand choose an arbitrary μ > 0. Then, by some manipulations, we can show that the coefficients sequence

{am}satisfies the second condition of2.1for all sufficiently large integersmas we see in the following:

μ²m2m32m2β2m3 μ² m−1!

1^m−1

k1

−1^kk

≥ 1

m−1!

|a2m1|.

2.10

3. Hyers-Ulam Stability of 1.6

In this section, letnbe a nonnegative integer and letρbe a constant with 0 < ρ≤ ∞. For a givenK ≥0, let us denote byCKthe set of all functionsy:−ρ, ρ → Cwith the properties aandb:

ayxis represented by a power series_∞

m0bmx^mwhose radius of convergence is at leastρ;

bit holds true that _∞

m0|amx^m| ≤ K|_∞

m0amx^m| for allx ∈ −ρ, ρ, where am m2m1bm22m−nbmfor eachm∈N0.

It should be remarked that the power series_∞

m0amx^minbhas the same radius of convergence as that of_∞

m0bmx^mgiven ina.

In the following theorem, we prove that if an analytic function satisfies some given conditions, then it can be approximated by a combination of integrals of the error function see the last part ofSection 1or16,§7.2.2.

Theorem 3.1. Letnbe a nonnegative integer. For given constantsKand ρwith K ≥ 0 and 0 <

ρ ≤ ∞, suppose thaty : −ρ, ρ → Cis a function which belongs toCK. Assume that there exist constantsμ, ν≥0 satisfying

ν²|α2m2| ≤ |a2m| ≤μ²m2m22m1|α2m2|, 3.1 ν²β_2m3≤ |a2m1| ≤μ²m2m32m2β_2m3 3.2 for allm∈N.See the definitions ofα2mandβ_2m1given inTheorem 2.1. Indeed, it is sufficient for the second inequalities in3.1and3.2to hold true for all sufficiently large integersm.Let us define ρ0min{ρ,1/μ}, where 1/0∞. If the functionysatisfies the differential inequality

yx 2xyx−2nyx≤ε 3.3

for allx ∈−ρ0, ρ0and for someε≥0, then there exists a solutionyh :R → Cof the differential equation1.6such that

yx−yhx≤ 1

2 1 ν²

Kεx² 3.4

for anyx∈−ρ0, ρ0.

Proof. Sincey∈ CK, it follows fromaandbthat yx 2xyx−2nyx ^∞

m0

m2m1bm22m−nbmx^m

^∞

m0

amx^m

3.5

for allx∈−ρ, ρ. It follows from the last equality and3.3that

∞ m0

amx^m

≤ε 3.6

for anyx∈−ρ0, ρ0. This inequality, together withb, yields that ∞

m0

|amx^m| ≤K

∞ m0

amx^m

≤Kε 3.7 for eachx∈−ρ0, ρ0.

By Abel’s formulasee17, Theorem 6.30, we have p

m0

|amx^m| 1

m2m1 p m0

|amx^m| 1 p2

p1

−^p−1

m0

_m

i0

aixⁱ 1

m3m2− 1 m2m1

1 p2

p1 p m0

|amx^m|

^p−1

m0

_m

i0

aixⁱ

2

m1m2m3

≤^∞

m0

Kε 2

m1m2m3 K

2ε

3.8

for anyx∈−ρ0, ρ0andp∈N, since ∞

m0

2

m1m2m3 ^∞

m0

1

m1− 1 m2

1

m3− 1 m2

1

2. 3.9

Hence, it follows from3.7and3.8that ∞

m2

am−2

mm−1x^m x²

∞ m0

|amx^m| 1

m2m1 ≤ K

2εx² 3.10

for eachx∈−ρ0, ρ0.

Finally, it follows from Theorem 2.1, 3.1,3.2,3.7, and3.10 that there exists a solution functionyh:R → Cof the homogeneous differential equation1.6such that

yx−yhx≤^∞

m2

a_m−2 mm−1x^m

^∞

m2

|α2m|x^2m∞

m2

β_2m1x^2m1

≤ K

2εx² 1 ν²

∞ m2

|a2m−2|x^2m 1 ν²

∞ m2

|a2m−1|x^2m1

≤ 1

2 1 ν²

Kεx²

3.11

for allx∈−ρ0, ρ0.

If ρ is finite, then the local Hyers-Ulam stability of the differential equation 1.6 immediately follows fromTheorem 3.1.

Corollary 3.2. Letnbe a nonnegative integer. For given constantsK and ρwithK ≥ 0 and 0 <

ρ < ∞, suppose thaty : −ρ, ρ → Cis a function which belongs toCK. Assume that there exist constantsμ, ν ≥0 satisfying the conditions in3.1and3.2for allm∈N.It is sufficient for the second inequalities in3.1and3.2to hold true for all sufficiently large integersm.Let us define ρ0 min{ρ,1/μ}and 1/0 ∞. If the functionysatisfies the differential inequality3.3for all x∈−ρ0, ρ0and for someε≥0, then there exists a solutionyh:R → Cof the differential equation 1.6such that

yx−yhx≤ 1

2 1 ν²

Kρ²ε 3.12

for anyx∈−ρ0, ρ0.

We now deal with an asymptotic behavior of functions in CK under the additional conditions3.1and3.2.

Corollary 3.3. Letnbe a nonnegative integer. For given constantsK,ρ, andρ1 with K ≥ 0 and 0< ρ1< ρ≤ ∞, suppose thaty:−ρ, ρ → Cis a function belonging toCK. Assume that there exist constantsμ≥0 andν >0 satisfying the conditions in3.1and3.2for anym∈N.It is sufficient

for the second inequalities in3.1and3.2to hold true for all sufficiently large integersm.Then there exists a solutionyh:R → Cof the differential equation1.6such that

yx−yhxO x²

3.13

asx → 0.

Proof. Sincey∈ CK, it follows from the first 4 lines of the proof ofTheorem 3.1that

yx 2xyx−2nyx ^∞

m0

amx^m 3.14

for allx ∈−ρ, ρ. As was remarked in the first part ofSection 3, the radius of convergence of the power series_∞

m0amx^mis same as that of_∞

m0bmx^myx, that is, it is at leastρ.

Since 0< ρ1< ρ, if we setρ0min{ρ1,1/μ}, then there exists a constantδ >0 such that yx 2xyx−2nyx

∞ m0

amx^m

≤δ 3.15

for anyx∈−ρ0, ρ0.

According to Theorem 3.1, there exists a solution yh : R → C of the differential equation1.6satisfying

yx−yhx≤ 1

2 1 ν²

Kδx² 3.16

for anyx∈−ρ0, ρ0. Hence, we have

yx−yhxO x²

3.17

asx → 0.

4. An Example

The conditions in3.1and 3.2may seem too strong to construct some examples for the coefficients am’s. In this section, however, we will show that the sequence {am} given in Remark 2.2satisfies these conditions: let n 0 anda0 a1 1,a2m a2m1 1/m−1!

for allm∈Nand choose some constantsμ >0 andν√

2. The second inequality in3.2has been verified inRemark 2.2.

The first inequality in3.2is also true for allm∈Nas we see in the following:

ν²β2m32

−1^m 2m32m2

1 m!

m−1

k0

−1^kk!a2k1

≤ 2

2m32m2m|a2m1| 1^m−1

k1

−1^kk

≤ 2

2m32m2m|a_2m1| m1

2

≤ m1

2m32m2m|a2m1|

≤ |a2m1|,

4.1

wherexdenotes the largest integer not exceedingx.

It is not difficult to show that

1

4k ≤ 2k2!

4^k1k1!k!− 2k!

4^kk!k−1! ≤ 2k−1

4k , 4.2

1

2 ≤ 2k!

4^kk!k−1! ≤k−1

2 4.3

for allk∈N.

By using4.2, we will now prove that

1^m−1

k1

−1^k 4^k

2k!

k!k−1!

−→ ∞ 4.4

asm → ∞: ifm2for some∈N, then

1^m−1

k1

−1^k 4^k

2k!

k!k−1! 1 2⁻¹

i1

1 4²ⁱ

4i!

2i!2i−1!− 1 4²ⁱ¹

4i2!

2i1!2i!

≤ 1 2⁻¹

i1

−1 8i

−→ −∞ asm−→ ∞.

4.5

Ifm21 for some∈N, then

1^m−1

k1

−1^k 4^k

2k!

k!k−1! 1

i1

1 4²ⁱ

4i!

2i!2i−1!− 1 4²ⁱ⁻¹

4i−2!

2i−1!2i−2!

≥1

i1

1 8i−4

−→ ∞ asm−→ ∞.

4.6

It then follows from4.3and4.4that

μ²m2m22m1|α_2m2|μ² 4^mm!

22m−1!

m−1

k0

−1^k 4^k

2k!

k! a2k

μ²m4^mm!m−1!

2m! |a2m| 1^m−1

k1

−1^k 4^k

2k!

k!k−1!

≥μ²m1 m|a2m|

1^m−1

k1

−1^k 4^k

2k!

k!k−1!

≥ |a2m|

4.7

for all sufficiently large integersm, which proves that the sequence{am}satisfies the second inequality in3.1.

Finally, we will show that the sequence{am}satisfies the first inequality in3.1. It follows from4.3that

ν²|α2m2|2 4^mm!

2m2!

1^m−1

k1

−1^k 4^k

2k!

k!k−1!

24^mm!m−1!

2m2! |a2m| 1^m−1

k1

−1^k 4^k

2k!

k!k−1!

2

2m22m1

4^mm!m−1!

2m! |a2m| 1^m−1

k1

−1^k 4^k

2k!

k!k−1!

≤ 4

2m22m1|a2m| 1^m−1

k1

−1^k 4^k

2k!

k!k−1!

≤ 4

2m22m1|a2m| 1^m−1

k1

2k!

4^kk!k−1!

≤ 4

2m22m1|a2m| 1^m−1

k1

k−1

2

2m²−4m6 2m22m1|a2m|

<|a2m|

4.8 for eachm∈N.

Acknowledgments

The author would like to express his cordial thanks to the referee for useful remarks which have improved the first version of this paper. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Governmentno. 2009-0071206.

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