# Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order

Authors: Yongjin Li; Yan Shen
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

The aim of this paper is to prove the stability in the sense of Hyers-Ulam of differential equation of second ordery′′+p(x)y′+q(x)y+r(x)=0. That is, if f is an approximate solution of the equation y′′+p(x)y′+q(x)y+r(x)=0, then there exists an exact solution of the equation near to f.


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## Body

## 1. Introduction and Preliminaries

In 1940, Ulam [1] posed the following problem concerning the stability of functional equations: give conditions in order for a linear mapping near an approximately linear mapping to exist. The problem for the case of approximately additive mappings was solved by Hyers [2] when G1 and G2 are Banach spaces, and the result of Hyers was generalized by Rassias (see [3]). Since then, the stability problems of functional equations have been extensively investigated by several mathematicians (cf. [3–5]).In connection with the stability of exponential functions, Alsina and Ger [6] remarked that the differential equation y′=y has the Hyers-Ulam stability. More explicitly, they proved that if a differentiable function y:I→R satisfies |y′(t)-y(t)|≤ε for all t∈I, then there exists a differentiable function g:I→R satisfying g′(t)=g(t) for any t∈I such that |y(t)-g(t)|≤3ε for everyt∈I.The above result of Alsina and Ger has been generalized by Miura et al. [7], by Miura [8], and also by Takahasi et al. [9]. Indeed, they dealt with the Hyers-Ulam stability of the differential equation y′(t)=λy(t), while Alsina and Ger investigated the differential equation y′(t)=y(t).Furthermore, the result of Hyers-Ulam stability for first-order linear differential equations has been generalized by Miura et al. [10], by Takahasi et al. [11], and also by Jung [12]. They dealt with the nonhomogeneous linear differential equation of first order(1.1)y′+p(t)y+q(t)=0.Jung [13] proved the generalized Hyers-Ulam stability of differential equations of the form(1.2)ty′(t)+αy(t)+βtrx0=0
and also applied this result to the investigation of the Hyers-Ulam stability of the differential equation(1.3)t2y′′(t)+αty′(t)+βy(t)=0.
Recently, Wang et al. [14] discussed the Hyers-Ulam stability of the first-order nonhomogeneous linear differential equation(1.4)p(x)y′+q(x)y+r(x)=0.
They proved the following theorem.Theorem 1.1 (see [14]).
Letp(x),q(x), and r(x) be continuous real functions on the interval I=(a,b) such that p(x)≠0 and |q(x)|≥δ for all x∈I and some δ>0 independent of x∈I. Then (1.4) has the Hyers-Ulam stability.The aim of this paper is to investigate the Hyers-Ulam stability of the following linear differential equations of second order under some special conditions:(1.5)y′′+p(x)y′+q(x)y+r(x)=0,
where y∈C2(I)=C2(a,b),-∞<a<b<+∞.For the sake of convenience, all the integrals in the rest of the work will be viewed as existing. We say that (1.5) has the Hyers-Ulam stability if there exists a constant K>0 with the following property: for every ε>0,y∈C2(I), if(1.6)|y′′+p(x)y′+q(x)y+r(x)|≤ε,
then there exists some z∈C2(I) satisfying(1.7)z′′+p(x)z′+q(x)z+r(x)=0
such that |y(x)-z(x)|<Kε. We call such K a Hyers-Ulam stability constant for (1.5).


## 2. Main Results

Now, the main results of this work are given in the following theorems.Theorem 2.1.
If a twice continuously differentiable functiony:I→R satisfies the differential inequality
(2.1)|y′′+p(x)y′+q(x)y+r(x)|≤ε
for all t∈I and for some ε>0, and the Riccati equation u′(x)+p(x)u(x)-u2(x)=q(x) has a particular solution, then there exists a solution v:I→R of (1.5) such that
(2.2)|y(x)-v(x)|≤Kε,
where K>0 is a constant.Proof.
Letε>0 and y:I→R be a continuously differentiable function such that
(2.3)|y′′+p(x)y′+q(x)y+r(x)|≤ε.
We will show that there exists a constantK independent of ε and v such that |y-v|<Kε for some v∈C2(I) satisfying v′′+p(x)v′+q(x)v+r(x)=0.
Assume thatc(x) is a particular solution of Riccati equation u′(x)+p(x)u(x)-u2(x)=q(x); if we set
(2.4)g(x)=y′(x)+c(x)y(x),d(x)=p(x)-c(x),
then
(2.5)g′(x)=y′′(x)+c(x)y′(x)+c′(x)y(x),
thus
(2.6)|g′(x)+d(x)g(x)+r(x)|=|y′′(x)+(c(x)+d(x))y′(x)+(c′(x)+d(x)c(x))y(x)+r(x)|=|y′′+p(x)y′+q(x)y+r(x)|≤ε.
Using the similar technique in [14], we can prove
(2.7)w(x)=exp{∫ax(-d(s))ds}[(g(b)-ε)exp{-∫ab(-d(x))ds}-∫xb-r(s)exp{-∫as(-d(t))dt}ds]=exp{∫ax(-d(s))ds}[(g(b)-ε)exp{∫ab(d(x))ds}+∫xbr(s)exp{∫as(d(t))dt}ds]
satisfying
(2.8)w′(x)+d(x)w(x)+r(x)=0,
and there exists an L>0 such that
(2.9)|g(x)-w(x)|≤Lε.
Byg(x)=y′(x)+c(x)y(x), we get
(2.10)|y′(x)+c(x)y(x)-w(x)|≤ε.
Using the same technique as above, we know that
(2.11)v(x)=exp{∫ax(-c(s))ds}[(y(b)-ε)exp{-∫ab(-c(x))ds}-∫xbw(s)exp{-∫as(-c(t))dt}ds]=exp{∫ax(-c(s))ds}[(y(b)-ε)exp{∫ab(c(s))ds}-∫xbw(s)exp{∫as(c(t))dt}ds]
satisfying
(2.12)v′(x)+c(x)v(x)-w(x)=0,
and there exists a K>0 such that
(2.13)|y(x)-v(x)|≤Kε.
The desired conclusion is proved.Theorem 2.2.
Letp(x),q(x), and r(x) be continuous real functions on the interval I=(a,b) such that p(x)≠0 and y0(x) is a nonzero bounded particular solution p(x)y′′+q(x)y′+r(x)y=0. If y:I→R is a twice continuously differentiable function, which satisfies the differential inequality
(2.14)|p(x)y′′+q(x)y′+r(x)y|≤ε
for all t∈I and for some ε>0, then there exists a solution v:I→R such that
(2.15)|y(x)-v(x)|≤Kε,
where K>0 is a constant, and v satisfies p(x)v′′+q(x)v′+r(x)v=0.Proof.
Settingy(x)=y0(x)∫axz(s)ds, we obtain
(2.16)y′(x)=y0′(x)∫axz(s)ds+y0(x)z(x)
and also
(2.17)y′′(x)=y0′′(x)∫axz(s)ds+2y0′(x)z(x)+y0(x)z′(x)
By a simple calculation, we see that
(2.18)|p(x)y0(x)z′(x)+[2p(x)y0′(x)+q(x)y0(x)]z(x)|=|p(x)y′′(x)+q(x)y′(x)+r(x)y(x)|≤ε.
Without loss of generality, we may assume thatp(x)y0(x)>0. Using the similar technique in [14], we know that
(2.19)z1(x)=exp{-∫ax2p(s)y0′(s)+q(s)y0(s)ds}[(z(b)-ε)exp{∫ab2p(s)y0′(s)+q(s)y0(s)ds}]
satisfies
(2.20)p(x)y1(x)z1′(x)+[2p(x)y1′(x)+q(x)y1(x)]z1(x)=0
and also
(2.21)|z(x)-z1(x)|≤Lε
for some L>0.
From the inequalities-Lε≤z(x)-z1(x)≤Lε and the similar technique in [14], we further get that
(2.22)z2(x)=(y(b)y1(b)-ε)-∫xbz1(s)ds
satisfies
(2.23)z2(x)-z1(x)=0
and also
(2.24)|z(x)-z2(x)|≤Qε
for some Q>0.
Consequently, we have
(2.25)|y(x)-z2(x)y0(x)|≤Mε
for some positive constant M.
Definev(x)=z2(x)y0(x). It then follows from the above inequality that |z(x)-v(x)|≤Mε holds for every x∈I. We can easily verify that v satisfies p(x)v′′+q(x)v′+r(x)v=0. This completes the proof of our theorem.We can prove the following corollaries by using an analogous argument. Hence, we omit the proofs.Corollary 2.3.
Letp(x),q(x), and r(x) be continuous real functions on the interval I=(a,b) such that p(x)≠0 and r2+p(x)r+q(x)=0. If y:I→R is a twice continuously differentiable function, which satisfies the differential inequality
(2.26)|p(x)y′′+q(x)y′+r(x)y|≤ε
for all t∈I and for some ε>0, then there exists a solution v:I→R such that
(2.27)|y(x)-v(x)|≤Kε,
where K>0 is a constant, and v satisfies p(x)v′′+q(x)v′+r(x)v=0.Corollary 2.4.
Letp(x),q(x),r(x), and s(x) be continuous real functions on the interval I=(a,b) such that p(x)≠0 and y0(x) is a nonzero bounded particular solution p(x)y′′′+q(x)y′′+r(x)y′+s(x)y=0. If y:I→R is a twice continuously differentiable function, which satisfies the differential inequality
(2.28)|p(x)y′′′+q(x)y′′+r(x)y′+s(x)y|≤ε
for all t∈I and for some ε>0, then there exists a solution v:I→R such that
(2.29)|y(x)-v(x)|≤Kε,
where K>0 is a constant, and v satisfies p(x)v′′′+q(x)v′′+r(x)v′+s(x)v=0.



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*Source: 576852-2009-10-19.xml*