# Hyers-Ulam Stability of Third Order Euler’s Differential Equations

Authors: A. K. Tripathy; A. Satapathy
Journal: Journal of Nonlinear Dynamics
 (2014)

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## Abstract

We investigate the Hyers-Ulam stability of third order Euler's differential equations of the formt3y'''+αt2y''+βty'+γy=0 on any open interval I=(a,b), 0<a<b≤∞ or -∞<a<b<0, where α,β, and γ are complex constants.


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## Body

## 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 [1]. Among such problems is a problem concerning the stability of functional equations:give conditions in order for a linear function near an approximately linear function to exist.In the following year, Hyers [2] gave an answer to the problem of Ulam for additive functions defined on Banach spaces.LetX1 and X2 be two real Banach spaces and let ϵ>0. Then, for every function f:X→Y satisfying
(1)fx+y-fx-fy≤ϵ,x,y∈X1
there exists a unique additive function A:X1→X2 with the property
(2)fx-Ax≤ϵ,x∈X1.
Furthermore, the result of Hyers has been generalized by Rassias [3, 4]. Since then, the stability problems of various functional equations have been investigated by many authors (see, e.g., [1, 5–8]). A generalization of Ulam’s problem was recently proposed by replacing functional equations with differential equations.The differential equation(3)ϕt,y,y′,…,y(n)=0
has Hyers-Ulam stability; if for given ϵ>0 and a function y such that
(4)ϕt,y,y′,…,y(n)≤ϵ,
then there exists a solution ya of the differential equation such that y(t)-ya(t)≤K(ϵ) and lim⁡ϵ→0⁡K(ϵ)=0.If the preceding statement is also true when we replaceϵ and K(ϵ) by ϕ(t) and ψ(t), respectively, where ϕ and ψ are appropriate functions not depending on y and ya explicitly, then we say that the corresponding differential equation has the generalized Hyers-Ulam stability.Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see, e.g., [9, 10]). Thereafter, Alsina and Ger published their work [11], which handles the Hyers-Ulam stability of the linear differential equation y′(t)=y(t).If a differentiable functiony(t) is a solution of the inequality y′(t)-y(t)≤ϵ for any t∈(a,∞), then there exists a constant c such that y′(t)-cet≤3ϵ, for all t∈(a,∞).In [1], Rezaei et al. have discussed the Hyers-Ulam stability of linear differential equations of first and nth order by applying Laplace transform which is comparable with the other methods available in the literature.It is Jung et al. who have investigated the Hyers-Ulam stability of linear differential equations of different classes including the stability of the delay differential equationy′(t)=λy(t-τ), where τ>0 is a constant (see, e.g., [5–7, 12–14]). Among the works, we are motivated by the results of [13], where he has studied the Hyers-Ulam stability of the following Euler’s differential equations:
(5)ty′t+αyt+βtrx0=0,(6)t2y′′t+αty′t+βyt=0,
where α, β, and γ are complex constants. We may note that the Hyers-Ulam stability of (6) depends on the Hyers-Ulam stability of (5) for every fixed x0≠0. We mentioned here the important results of [13].Theorem 1.
LetX be a complex Banach space and let I=(a,b) be an open interval such that 0<a<b≤∞ or -∞<a<b<0. Assume that a function ϕ:I→0,∞ is given, that α, β, and γ are complex constants, and that x0 is a fixed element of X. Furthermore, suppose a continuously differentiable function f:I→X satisfies
(7)ty′t+αyt+βtrx0≤ϕt,t∈I.
If both tα+r-1 and tα-1ϕ(t) are integrable on (a,c), for any c with a<c≤b, then there exists a unique solution f0:I→X of the differential equation (5) such that
(8)ft-f0t≤t-α∫tb‍vα-1ϕvdv
for any t∈I.Indeed, the unique solution of (5) in the complex Banach space X is given by
(9)f0(t)=atαx-βα+r1tαtα+r-aα+rx0,α+r≠0,atαx-βtαln⁡t-ln⁡ax0.α+r=0.Theorem 2.
If a twice continuously differentiable functionf:I→R satisfies the differential inequality
(10)t2f′′t+αtf′t+βft≤ϵ
for all t∈I and for some ϵ>0, then there exists a solution f0:I→R of (6) such that
(11)ft-f0t≤ϵβbtc-1
for any t∈I, where c is a negative real number. In particular, if I=(a,∞), a>0, then
(12)ft-f0t≤ϵβ,∀t>a.In this theorem, we may note that the unique solution of (6) is given by
(13)f0t=c3+c0dc-datc-c0dc-datd,
where
(14)c=(α-1)-1-α2-4β1/22,d=(α-1)+1-α2-4β1/22
are real constants and c3 is any arbitrary constant.Theorem 3.
LetX be a complex Banach space and let I=(a,b) be an open interval such that 0<a<b≤∞ or -∞<a<b<0. Assume that a function ϕ:I→0,∞ is given and that h:I→X is a continuous function. Furthermore, suppose a continuously differentiable function f:I→X satisfies
(15)ty′t+αyt+h(t)≤ϕt,t∈I.
If both tα-1ϕ(t) and tα-1h(t) are integrable on (a,c), for any c with a<c≤b, then there exists a unique solution f0:I→X of the differential equation
(16)ty′t+αyt+ht=0
such that (8) holds for any t∈I, where α is a complex constant.Proof.
The proof of the theorem follows from the proof of Theorem1, if we set
(17)zt=taαft+∫at‍uα-1aαhudu,
where the unique solution is given by
(18)f0t=atα-1tα∫at‍uα-1hudu.
The details are omitted.Remark 4.
We may note that (5) is a particular case of (16). Hence, Theorems 1 and 3 are comparable.The objective of this work is to investigate the generalized Hyers-Ulam stability of the following Euler’s differential equations of the form(19)t2y′′(t)+αty′(t)+βy(t)+γtrx0=0,(20)t3y′′′t+αt2y′′t+βty′t+γyt=0,
where α, β, γ, and r are complex constants. Intuitively, we shall prove that if a twice continuously differentiable function f:I→X satisfies the differential inequality
(21)t2y′′t+αty′t+βyt+γtrx0≤ϕt
for all t∈I, where x0 is a fixed element of the complex Banach space X and I=(a,b) with 0<a<b≤∞ or -∞<a<b<0, then there exists a twice continuously differentiable solution f0(t) of (19) such that
(22)ft-f0t≤tl∫tb‍u-l-1ψudu
for any t∈I, where
(23)ψt=tm∫tb‍u-m-1ϕudu.
We also apply this result to the investigation of the Hyers-Ulam stability of (20).


## 2. Hyers-Ulam Stability of (19)

Throughout this section, we letI=(a,b) with 0<a<b≤∞ or -∞<a<b<0. For fixed x0≠0, the general solution of (19) in the class of real valued functions defined on I is given by
(24)yt=c1tl+c2tm-γx0trgr,gr≠0,l≠m,c1tl+c2tm-γx0trln⁡tg′r,gr=0≠g′r,l≠m,c1+c2ln⁡ttl-γx0trgr,gr≠0,l=m,yt=c1+c2ln⁡ttl-γx0trln⁡t22,gr=0=g′r,l=m,
where g(r)=r2+(α-1)r+β and c1, c2 are arbitrary constants.Remark 5.
Indeed,g′(r)=2r+α-1. If g(r)=0, then either r-l=0 or r-m=0. Therefore, r-l=0 implies that 2r+α-1=2r-l-m=r-m≠0 and r-m=0 implies that 2r+α-1=r-l≠0. Hence, the second solution could be any one of the following:
(25)c1tl+c2tm-γx0trln⁡tr-m;r-l=0,r-m≠0,l≠m,c1tl+c2tm-γx0trln⁡tr-l;r-l≠0,r-m=0,l≠m.Theorem 6.
LetX be a complex Banach space and let I=(a,b) be an open interval with either 0<a<b≤∞ or -∞<a<b<0. Assume that a function ϕ:I→0,∞ is given, that α, β, γ, and r are complex constants, and that x0 is a fixed element of X. Furthermore, suppose a twice continuously differentiable function f:I→X satisfies the differential inequality (21). If tr-l-1, tr-m-1, tm-l-1, t-m-1ϕ(t), and t-l-1ψ(t) are integrable on (a,c), for any c with a<c≤b, then there exists a unique solution f0:I→X (which is twice continuously differentiable) of (19) such that (22) holds for any t∈I, where l and m are the roots of g(r)=0.Proof.
We prove the theorem for five possible cases; namely,(i)
(r-l)(r-m)≠0, l≠m;(ii)
r-l=0≠r-m,
l≠m;(iii)
r-l≠0=r-m,
l≠m;(iv)
r-l≠0, l=m;(v)
r-l=0, l=m.
Case (i). Suppose that X is a complex Banach space and f:I→X is a twice continuously differentiable function satisfying the differential inequality (21). Define h:I→X such that h(t)=tf′(t)-lf(t). Then it is easy to verify that
(26)th′t-mht+γtrx0=t2f′′t+1-l-mtf′t+lmft+γtrx0=t2f′′t+αtf′t+βft+γtrx0≤ϕt.
Hence, by Theorem 1, it follows that there exists a unique solution h0:I→X of the differential equation ty′-my+γtrx0=0 such that
(27)ht-h0t≤tm∫tb‍v-m-1ϕvdv
provided that tr-m-1 and t-m-1ϕ(t) are integrable on (a,c), for any c with a<c≤b. Indeed,
(28)h0t=at-mx-γtmr-mtr-m-ar-mx0,
where x is a limit point in X. If we denote
(29)ψt=tm∫tb‍v-m-1ϕvdv,
then (27) becomes
(30)tf′t-lft-h0t≤ψt.
Clearly, ψ:I→0,∞. By Theorem 3, there exists a unique solution f1:I→X of the differential equation ty′(t)-ly(t)-h0(t)=0 such that
(31)ft-f1t≤tl∫tb‍u-l-1ψudu
provided that t-l-1ψ(t) and t-l-1h0(t) are integrable on (a,c), for any c with a<c≤b. Clearly, t-l-1h0(t) is integrable if and only if tm-l-1 and tr-l-1 are integrable. According to Theorem 3,
(32)f1t=talx1+tl∫at‍u-l-1h0udu,
where x1 is limit point in X. It is easy to verify that
(33)∫at‍u-l-1h0udu=xam+γx0ar-mr-mtm-lm-l-xalm-l-γx0tr-lr-lr-m-γx0ar-lr-lm-l.
Consequently,
(34)f1t=xal-γx0ar-lr-lm-l-xalm-ltl+xamm-l+γx0ar-mr-mm-ltm-γx0trgr
which is a solution of (19).
Case (ii). Proceeding as in Case   (i), we find a unique solution f2:I→X of the equation ty′(t)-ly(t)-h0(t)=0 such that
(35)ft-f2t≤tl∫tb‍u-l-1ψudu,f2t=talx2+tl∫at‍u-l-1h0udu,
where x2 is a limit point in X. Clearly,
(36)∫at‍u-l-1h0udu=∫at‍xamum-l-1-γx0u-1r-m+γx0ar-mr-mum-l-1du=xam+γx0ar-mr-mtm-lm-l+γx0r-ml-m-xalm-l-γx0r-mln⁡ta
implies that
(37)f2t=x2al+γx0r-m2+xalr-mtl+xamm-l-γx0ar-mr-m2tm-γx0trr-mln⁡ta
which is a solution of (19).
Case (iii). Interchanging the roles of l and m in Case  (ii), we find the unique solution f3(t) as
(38)f3t=x3al-xalm-l-γx0am-lm-l2tl+xamm-l+γx0m-l2tm-γx0trr-lln⁡ta,
where x3∈X.
Case (iv). In this case, we proceed as in Case  (i). Indeed, for l=m,
(39)∫at‍u-l-1h0udu=xal+γx0ar-lr-lln⁡ta+γx0r-l2ar-l-tr-l
and the unique solution f4(t) is given by
(40)f4t=x4al+γx0ar-lr-l2+xal+γx0ar-lr-lln⁡tatl-γx0trr-l2,
where x4∈X.
Case (v). In this case,
(41)h0t=talx-γx0tlln⁡ta.
Using the same type of argument as in Case  (i), we find the unique solution f5(t) as
(42)f5t=talx5+tl∫at‍u-l-1h0udu=talx5+xaltlln⁡ta-γx02tlln⁡ta2=x5al+xalln⁡tatl-γx02trln⁡ta2,
where x5∈X. This completes the proof of the theorem.


## 3. Hyers-Ulam Stability of (20)

In this section, we shall investigate the Hyers-Ulam stability of (20) on any open interval I=(a,b) with either 0<a<b≤∞ or -∞<a<b<0. We may note that α=3-(l+m+n), β=lm+mn+nl-l-m-n+1, and γ=-lmn, when l, m, and n are the characteristic roots of the associated characteristic equation of (20).Theorem 7.
LetX be a complex Banach space. Assume that θ:I→[0,∞) is given. Furthermore, assume that tn-l-1, tn-m-1, tm-l-1, t-n-1θ(t), t-m-1λ(t), and t-l-1η(t) are integrable on (a,c) with a<c≤b, where
(43)λt=tn∫tb‍u-n-1θudu,ηt=tm∫tb‍v-m-1λvdv.
Suppose that f∈C3(I,X) and satisfies
(44)t3f′′′t+αt2f′′t+βtf′t+γft≤θt,
for all t∈I. Then, there exists a unique solution f0∈C3(I,X) of (20) such that
(45)ft-f0t≤tl∫tb‍s-l-1ηsds.Proof.
Letf:I→X be such that (44) holds, for t∈I. Define w:I→X such that
(46)wt=t2f′′t+1-l-mtf′t+lmft.
Indeed,
(47)tw′t-nwt≤θt,t∈I.
Hence, using Theorem 3, it follows that there exists a unique w0:I→X such that
(48)wt-w0t≤tn∫tb‍u-n-1θudu=λt,
where w0(t)=t/anx and x is a limit point in X. Ultimately, (48) becomes
(49)t2f′′t+1-l-mtf′t+lmft-a-ntnx≤λt,t∈I.
Now, we can apply Theorem 6 to the above differential inequality for five possible cases. Consequently,
(50)f1t=x1-x2m-l+xn-ln-mtal+x2m-l+xn-ml-mtam+xn-ln-mtan,f2t=x3al+x2aln-m-xann-m2+xann-mln⁡tatl+x2amm-l+xamn-m2tm,f3t=x4al-x2alm-l+xalm-l2tl+x2amm-l-xamm-l2+xamm-lln⁡tatm,f4t=x4al-xaln-l2+x2al-xaln-lln⁡tatl+xn-l2tan,f5t=x5al+x2alln⁡ta+x2anln⁡ta2tl
are the unique solutions with respect to the five possible cases, respectively. In fact, they are the possible solutions of (20). Hence, the theorem is proved.Corollary 8.
LetX=R be a real Banach space. Assume that tn-l-1, tn-m-1, and tm-l-1 are integrable on I=(a,b) with 0<a<b≤∞ or -∞<a<b<0, where l, m, and n are positive roots of the associated characteristic equation of (20). Suppose that f∈C3(I,X) and satisfies
(51)t3f′′′t+αt2f′′t+βtf′t+γft≤ϵ,
for all t∈I. Then, there exists a unique solution f0∈C3(I,X) of (20) such that
(52)f(t)-f0t≤ϵlmn.Proof.
Letf:I→X be such that (51) holds, for all t∈I. Since all the conditions of Theorem 7 are satisfied, then there exists a unique solution f0:I→X of (20) such that
(53)λt=tn∫tb‍u-n-1ϵdu=ϵtnu-n-ntb=ϵn
when I=-∞,∞. Then, for I=-∞,∞,
(54)ηt=tm∫tb‍v-m-1λvdv=ϵmntbm-1=ϵmn.
Hence,
(55)ft-f0t≤tl∫tb‍s-l-1ηsds=ϵlmntbl-1.
Consequently,
(56)ft-f0t≤ϵlmnasb⟶∞,
where f0(t) can be chosen from Theorem 7.Example 9.
LetX=R be a Banach space and let I=(a,∞), a>0. Consider Euler’s equation
(57)t3y′′′t-6t2y′′t+18ty′t-24yt=0
with l=4, m=3, and n=2. Suppose that f:I→X satisfies the differential inequality
(58)t3f′′′t-6t2f′′t+18tf′t-24ft≤ϵ,
for any t∈I. By Corollary 8, there exists a unique solution f0:I→X of (57) such that
(59)ft-f0t≤ϵ24,
for any t∈I and for unique x,x1,x2∈X,
(60)f0(t)=x2+x1+x2ta4+x-x2ta3+x2ta2.



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