# Generalized Hyers-Ulam Stability of the Second-Order Linear Differential Equations

Authors: A. Javadian; E. Sorouri; G. H. Kim; M. Eshaghi Gordji
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

We prove the generalized Hyers-Ulam stability of the 2nd-order linear differential
equation of the formy′′+p(x)y′+q(x)y=f(x), with condition that there exists a nonzero y1:I→X in C2(I) such that y1′′+p(x)y1′+q(x)y1=0 and I is an open
interval. As a consequence of our main theorem, we prove the generalized Hyers-Ulam
stability of several important well-known differential equations.


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## Body

## 1. Introduction

Thestability problem of functional equations started with the question concerning stability of group homomorphisms proposed by Ulam [1] during a talk before a Mathematical Colloquium at the University of Wisconsin, Madison. In 1941, Hyers [2] gave a partial solution of Ulam’s problem for the case of approximate additive mappings in the context of Banach spaces. In 1978, Rassias [3] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences ∥f(x+y)-f(x)-f(y)∥≤ϵ(∥x∥p+∥y∥p),(ϵ>0,p∈[0,1)). This phenomenon of stability that was introduced by Rassias [3] is called the Hyers-Ulam-Rassias stability (or the generalized Hyers-Ulam stability).LetX be a normed space over a scalar field 𝕂, and let I be an open interval. Assume that for any function f:I→X satisfying the differential inequality
(1.1)‖an(t)y(n)(t)+an-1(t)y(n-1)(t)+⋯+a1(t)y′(t)+a0(t)y(t)+h(t)‖≤ϵ
for all t∈I and for some ϵ≥0, there exists a function f0:I→X satisfying
(1.2)an(t)y(n)(t)+an-1(t)y(n-1)(t)+⋯+a1(t)y′(t)+a0(t)y(t)+h(t)=0,‖f(t)-f0(t)‖≤K(ϵ)
for all t∈I; here K(t) is an expression for ϵ with limϵ→0K(ϵ)=0. Then, we say that the above differential equation has the Hyers-Ulam stability.If the above statement is also true when we replaceϵ and K(ϵ) by φ(t) and ϕ(t), where φ,ϕ:I→[0,∞) are functions not depending on f and f0 explicitly, then we say that the corresponding differential equation has the Hyers-Ulam-Rassias stability (or the generalized Hyers-Ulam stability).The Hyers-Ulam stability of differential equationy′=y was first investigated by Alsina and Ger [4]. This result has been generalized by Takahasi et al. [5] for the Banach space-valued differential equation y′=λy. In [6], Miura et al. also proved the Hyers-Ulam-Rassias stability of linear differential of first order, y′+g(t)y(t)=0, where g(t) is a continuous function, while the author [7] proved the Hyers-Ulam-Rassias stability of linear differential of the form c(t)y′(t)=y(t). Jung [8] proved the Hyers-Ulam-Rassias stability of linear differential of first order of the form c(t)y′(t)+g(t)y(t)+h(t)=0.In this paper, we investigate the generalized Hyers-Ulam stability of differential equations of the form(1.3)y′′+p(x)y′+q(x)y=f(x).
We assume that X is a complex Banach space, I=(a,b) is an arbitrary interval, and y1:I→X is a nonzero solution of corresponding homogeneous equation of (1.3), where
(1.4)y1′′+p(x)y1′+q(x)y1=0.


## 2. Main Results

Taking some idea from [8], we are going to investigate the stability of the 2nd-order linear differential equations. For the sake of convenience, all the integrals and derivations will be viewed as existing and ℜ(ω) denotes the real part of complex number ω. Moreover, let I=(a,b) be an open interval, where a,b∈ℝ⋃{±∞} are arbitrarily given with a<b.Theorem 2.1.
LetX be a complex Banach space. Assume that  p,q:I→ℂ and  f:I→X are continuous functions and  y1:I→X is a nonzero twice continuously differentiable function which satisfies the differential equation (1.4). If a twice continuously differentiable function  y:I→X satisfies
(2.1)‖y′′+p(x)y′+q(x)y-f(x)‖≤φ(x)
for all x∈I, where k=y(a)/y1(a)∈X and φ:I→(0,∞) is a continuous function, then there exists a unique x0∈X such that
(2.2)‖y(x)-y1(x)⋅(∫ax(exp⁡{-∫as(2y1(u)′y1(u)+p(u))du}⋅[x0+∫asf(v)y1(v)exp⁡{∫av(2y1(u)′y1(u)+p(u))du}dv])ds+k)‖≤‖y1(x)‖⋅∫ax((∫at(2y1(u)′y1(u)+p(u)du))exp⁡{-R∫as(2y1(u)′y1(u)+p(u))du}⋅|∫sbexp⁡{R(∫at(2y1(u)′y1(u)+p(u)du))⋅φ(t)‖y1(t)‖dt}|)ds.Proof.
We assume that(2.3)v(x)=y(x)y1(x)
for all x∈I. It follows from (1.4), (2.1), and (2.3) that
(2.4)‖(v(x)y1(x))′′+p(x)(v(x)y1(x))′+q(x)(v(x)y1(x))-f(x)‖=‖(v(x)′y1(x)+v(x)y1(x)′)′+p(x)(v(x)′y1(x)+v(x)y1(x)′)+q(x)v(x)y1(x)-f(x)‖=‖v(x)′′y1(x)+v(x)′(2y1(x)′+p(x)y1(x))+v(x)(y1(x)′′+p(x)y1(x)′+q(x)y1(x))-f(x)‖=‖v(x)′′y1(x)+v(x)′(2y1(x)′+p(x)y1(x))-f(x)‖=‖y1(x)‖‖v(x)′′+v(x)′(2y1(x)′y1(x)+p(x))-f(x)y1(x)‖≤φ(x),
so, we have
(2.5)‖v(x)′′+v(x)′(2y1(x)′y1(x)+p(x))-f(x)y1(x)‖≤φ(x)‖y1(x)‖.
For simplicity, we use the following notation:
(2.6)z(s):=exp⁡{∫as(2y1(u)′y1(u)+p(u))du}⋅(y(s)y1(s))′-∫as(f(v)y1(v)exp⁡{∫av(2y1(u)′y1(u)+p(u))du})dv
for all s∈I. By making use of this notation and by (2.5), we get
(2.7)‖z(s)-z(l)‖=‖exp⁡{∫as(2y1(u)′y1(u)+p(u))du}⋅(y(s)y1(s))′-∫as(f(v)y1(v)exp⁡{∫av(2y1(u)′y1(u)+p(u))du})dv-exp⁡{∫al(2y1(u)′y1(u)+p(u))du}⋅(y(l)y1(l))′+∫al(f(v)y1(v)exp⁡∫av(2y1(u)′y1(u)+p(u))dudv)‖=‖∫lsdt(exp⁡∫at(2y1(u)′y1(u)+p(u))du⋅(y(t)y1(t))′-∫at(f(v)y1(v)exp⁡{∫av(2y1(u)′y1(u)+p(u))du})dv)‖=‖∫ls((2y1(t)′y1(t)+p(t))⋅exp⁡{∫at(2y1(u)′y1(u)+p(u))du}⋅(y(t)y1(t))′+exp⁡{∫at(2y1(u)′y1(u)+p(u))du}⋅(y(t)y1(t))′′)+(-f(t)y1(t)exp⁡{∫at(2y1(u)′y1(u)+p(u))du})dt‖=‖∫lsexp⁡{∫at(2y1(u)′y1(u)+p(u))du}⋅((2y1(t)′y1(t)+p(t))⋅(y(t)y1(t))′+(y(t)y1(t))′′-f(t)y1(t))dt‖≤|∫lsexp⁡{R(∫at(2y1(u)′y1(u)+p(u)du))⋅φ(t)‖y1(t)‖dt}|
for all l,x∈I. Since exp{ℜ(∫at((2y1(u)′/y1(u))+p(u)du))·(φ(t)/∥y1(t)∥) is assumed to be integrable on I, we may select l0∈I, for any given ϵ>0, such that l,x≥l0 implies ∥z(x)-z(l)∥<ϵ. That is, {z(l)}l∈I is a Cauchy net. By completeness of X, there exists an x0∈X such that z(l) converges to x0 as l→b. It follows from (2.7) and the previous argument that, for any x∈I,
(2.8)‖y(x)-y1(x)(∫ax(exp⁡{-∫as(2y1(u)′y1(u)+p(u))du}×[x0+∫asf(v)y1(v)exp⁡{∫av(2y1(u)′y1(u)+p(u))du}dv])ds+k)‖=‖y1(x)⋅(∫ax(exp⁡{-∫as(2y1(u)′y1(u)+p(u))du}⋅(z(s)-x0))ds)‖≤‖y1(x)‖⋅∫ax(exp⁡{-R∫as(2y1(u)′y1(u)+p(u))du}⋅‖z(s)-z(l)‖)ds+‖y1(x)‖⋅∫ax(exp⁡{-R∫as(2y1(u)′y1(u)+p(u))du}⋅‖z(l)-x0‖)ds≤‖y1(x)‖⋅∫ax(exp⁡{-R∫as(2y1(u)′y1(u)+p(u))du}⋅|∫lsexp⁡{R(∫at(2y1(u)′y1(u)+p(u)du))⋅φ(t)‖y1(t)‖dt|)ds+‖y1(x)‖⋅∫ax(exp⁡{-R∫as(2y1(u)′y1(u)+p(u))du}⋅‖z(l)-x0‖)ds⟶‖y1(x)‖⋅∫ax(exp⁡{-R∫as(2y1(u)′y1(u)+p(u))du}⋅|∫sbexp⁡{R(∫at(2y1(u)′y1(u)+p(u)du))⋅φ(t)‖y1(t)‖dt|)ds
as l→b. Moreover,
(2.9)y0(x)=y1(x)⋅(∫ax(exp⁡{∫as(2y1(u)′y1(u)+p(u))du}⋅[x0+∫asf(v)y1(v)exp⁡{∫av(2y1(u)′y1(u)+p(u))du}dv])ds+k)
is a solution of (1.3).
Now, we prove the uniqueness property ofx0. Assume that x1,x2∈X satisfy inequality (2.2) in place of x0. Then, we have
(2.10)‖y1(x)⋅∫ax(exp⁡{-∫as(2y1(u)′y1(u)+p(u))du}⋅(x2-x1))ds‖≤2‖y1(x)‖⋅∫ax(exp⁡{-R∫as(2y1(u)′y1(u)+p(u))du}⋅|∫sbexp⁡{R(∫at(2y1(u)′y1(u)+p(u)du))⋅φ(t)‖y1(t)‖dt|)ds,
thus,
(2.11)‖x2-x1‖≤2⋅∫ax(exp⁡{-R∫asAdu}⋅|∫sbexp⁡{R(∫at(Adu))⋅(φ(t)/‖y1(t)‖)dt|)ds|∫ax(exp⁡{-R∫asAdu})ds|,
where 𝒜 denotes ((2y1(u)′/y1(u))+p(u)).
It follows from the integrability hypothesis that(2.12)|∫sbexp⁡{R(∫at(2y1(u)′y1(u)+p(u)du))⋅φ(t)‖y1(t)‖dt|⟶0
as s→b. This implies that x1=x2 and the proof is complete.Remark 2.2.
It follows from Theorem2.1 that
(2.13)y(x)=y1(x)⋅(∫ax(exp⁡{∫as(2y1(u)′y1(u)+p(u))du}⋅[c1+∫asf(v)y1(v)exp⁡{∫av(2y1(u)′y1(u)+p(u))du}dv])ds+c2)
is the general solution of the differential equation (1.3), where c1,c2 are arbitrary elements of X and y1(x) is a nonzero solution of the corresponding homogeneous equation (1.3).Remark 2.3.
If we replaceℂ by ℝ in the proof of Theorem 2.1 and we assume that p,q are real-valued continuous functions, then we can see that Theorem 2.1 is true for a real Banach space X.Hence, every 2nd-order linear differential equation has the generalized Hyers-Ulam stability with the condition that there exists a solution of corresponding homogeneous equation or there exists a general solution in the ordinary differential equations.Example 2.4.
Consider the second-order linear differential equation with constant coefficients(2.14)y′′+by′+cy=f(x).
Let b2-4c≥0, m=(-b±b2-4c)/2, and let f:I→ℝ,φ:I→[0,∞) be continuous functions. Assume that y:I→ℝ is a twice continuously differential function satisfying the differential inequality
(2.15)|y′′+by′+cy-f(x)|≤φ(x)
for all x∈I. On the other hand, by ordinary differential equations, we know that y1(x)=exp(mx) is a solution of corresponding homogeneous equation of (2.14). It follows from Theorem 2.1, Remark 2.3, and (2.14) that there exists a solution y0:I→ℝ of (2.14) such that
(2.16)y0(x)=exp⁡(mx)⋅(∫ax(exp⁡(-(2m+b)(s-a))⋅[x0+∫asf(v)⋅exp⁡(v(m+b)-a(2m+b))dv])ds+k)
for all x∈I and that
(2.17)|y(x)-y0(x)|≤|exp⁡(mx)|⋅∫ax(φ(t)|exp⁡(mx)|exp⁡(-(2m+b)(s-a))⋅|∫sbexp⁡((2m+b)(t-a))⋅φ(t)|exp⁡(mx)|dt|)ds.Example 2.5.
Consider (2.14). Let b2-4c<0, m=(-b±b2-4c)/2=α±iβ, and let f:I→ℝ,φ:I→[0,∞) be continuous functions. Let y:I→ℝ be a twice continuously differential function satisfying the differential inequality of (2.15) for all x∈I. It follows from the ordinary differential equations that y1(x)=exp(αx)cos(βx). Then it follows from Theorem 2.1, Remark 2.3, and (2.15) that there exists a solution y0:I→ℝ of (2.14) such that
(2.18)y0(x)=exp⁡(αx)cos⁡(βx)⋅(x0cos⁡2(βa)∫axexp⁡((2α+b)(a-s))cos⁡2(βs)ds)+exp⁡(αx)cos⁡(βx)⋅(∫axexp⁡(-(2α+b)s)cos⁡2(βs)⋅(∫asf(v)⋅exp⁡v(α+b)⋅cos⁡(βv)dv∫asf(v)⋅exp⁡v(α+b))ds+k)
for all x∈I, where k=y(a)/(exp(αa)cos(βa)) and x0∈ℝ is unique and
(2.19)|y(x)-y0(x)|≤|exp⁡(αx)cos⁡(βx)|⋅∫ax(exp⁡(-(2α+b)s)cos⁡2(βs)⋅|∫sbcos⁡2(βt)⋅exp⁡((α+b)t)⋅φ(t)dt|)ds.Example 2.6.
Consider the equation(2.20)y′′-2x1+x2y′+21+x2y=6(1+x2).
Let I=(a,b) be an open interval, where a,b∈[1,+∞] are arbitrarily given with a<b, f:I→ℝ and φ:I→[0,∞) are continuous functions. Assume that y:I→ℝ is a twice continuously differential function satisfying the differential inequality
(2.21)|y′′-2x1+x2y′+21+x2y-6(1+x2)|≤φ(x)
for all x∈I. By the trial of y0(x)=x, we see that it is a solution of corresponding homogeneous equation of (2.20). Then it follows from Theorem 2.1, Remark 2.3, and (2.21) that there exists a solution y0:I→ℝ of (2.20) such that
(2.22)y0(x)=x(x0a(1-a21+a2)+k-6a+2a3)+(x2-1)(x0a21+a2-3a2)+x4+3x2
for all x∈I, where k=y(a)/a and x0∈ℝ is unique and
(2.23)|y(x)-y0(x)|≤x⋅∫ax((1+s2s2)⋅|∫sbt1+t2⋅φ(t)dt|)ds.Remark 2.7.
We know that Eulars differential equation of second order has the general solution in ordinary differential equations, then we can use Theorem2.1 and Remark 2.3 for the Hyers-Ulam-Rassias stability in this case.Letp be a real constant, and I=[-1,1]. We know that Legender’s differential equation(2.24)(1-x2)y′′-2xy′+p(p+1)y=0has the general solution(2.25)y=a0y1(x)+a1y2(x),where(2.26)y1(x)=1-p(p+1)2x2+(p-2)p(p+1)(p+3)4!x4-⋯,y2(x)=x-(P-1)(p+2)3!x3+(p-3)(p-1)(p+2)(p+4)5!x5-⋯
and a0,a1 are arbitrary constants. By Theorem 2.1 and Remark 2.3, Legender’s differential equation has Hyers-Ulam-Rassias stability.Hermite’s differential equation(2.27)y′′-2xy′+2py=0,
where p is a real constant, has the general solution(2.28)y=a0y1(x)+a1y2(x)
that
(2.29)y1(x)=1+∑n=1∞(-1)n2np(p-2)⋯(p-2n+2)(2n)!x2n,y2(x)=x+(-1)n2n(p-1)(p-3)⋯(p-2n+1)(2n+1)!x2n+1
for all x∈ℝ, and a0,a1 are arbitrary constants. Thus Hermites differential equation has generalized Hyers-Ulam stability.It is well known from the ordinary differential equations that(2.30)y1(x)=Jp(x)=∑n=0∞(-1)nn!Γ(n+p+1)(x2)2n+p,
for all x∈ℝ, is a solution of Bessel’s differential equation(2.31)x2y′′+xy′+(x2-p2)y=0
that p≥0.Then Bessel’s differential equation has Hyers-Ulam-Rassias stability.We know from the ordinary differential equations that Laguerre, Chebyshev, and Gauss hypergeometric differential equations have the general solution. Then we can show that those have generalized Hyers-Ulam stability.



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