# Fourier Transformation and Stability of a Differential Equation onL1ℝ

Authors: Hamid Rezaei; Zahra Zafarasa; Lotfollah Karimi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

In the present paper, by the Fourier transform, we show that every linear differential equation with constant coefficients ofn-th order has a solution in L1ℝ which is infinitely differentiable in ℝ∖0. Moreover the Hyers–Ulam stability of such equations on L1ℝ is investigated.


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## Body

## 1. Introduction

One of the main mathematical problems in the study of functional equations is the Hyers–Ulam stability of equations. S. M. Ulam was the first one who investigated this problem by the question under what conditions does there exist an additive mapping near an approximately additive mapping? Hyers in [1] gave an answer to the problem of Ulam for additive functions defined on Banach spaces. Let X1,X2 be two real Banach spaces, and ε>0 is given. Then, for every mapping, f:X1⟶X2 satisfying(1)fx+y−fx−fy<εfor allx;y∈X1.There exists a unique additive mappingg:X1⟶X2 with the property(2)fx−gx<ε,for allx∈X1.After Hyers result, these problems have been extended to other functional equations [2]. This may be the most important extension in the Hyers–Ulam stability of the differential equations.The differential equationφy,y′,…,yn=f has Hyers–Ulam stability on normed space X if, for given ε>0 and a function y such that φy,y′,…,yn−f<ε, there is a solution ya∈X of the differential equation such that y−ya<Kε and limε⟶0Kε=0.In the theory of differential equations, we search for a classical symmetric solution or a weak or generalized solution. Usually, these solutions at least satisfy the equation almost everywhere. It seems that theϵ-solution provides us a wider notion of a solution, and for some physical applications, it models the underlying physics more appropriate. We hope this paper is a beginning for further research about the ϵ-solution of a differential equation. In the theory of differential equations, it is particularly important for problems arising from physical applications that we would prefer our solution changes only a little when the data of the problem change a little. It is called the continuity of the equation with respect to data. See [3] and section 3.2 in [4]. It is straightforward to check that the Hyers–Ulam stability of our problems is equivalent to the continuity of the equations φy,y′,…,yn=f with respect to the right-hand side f.Alsina and Ger [5] were the first authors who investigated the Hyers–Ulam stability of a differential equation. The result of the work of Alsina and Ger was extended to the Hyers–Ulam stability of a higher-order differential equation with constant coefficients in [6]. We mention here only the recent contributions in this subject of some mathematicians [6–15].Rus has proved some results on the stability of ordinary differential and integral equations using Gronwall lemma and the technique of weakly Picard operators [13, 14]. Using the method of integral factors, the Hyers–Ulam stability of some ordinary differential equations of first and second order with constant coefficients has been proved in [9, 15].We recall that the Laplace transform is a powerful integral transform used to switch a function from the time domain to thes-domain. It maps a function to a new function on the complex plane. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions. Applying the Laplace transform method, Rezaei et al. [16] investigated the Hyers–Ulam stability of the linear differential equations of functions defined on 0,+∞.Fourier transforms can also be applied to the solution of differential equations of functions with domain−∞,+∞. They can convert a function to a new function on the real line. Since the Laplace transform cannot be used for the functions defined on −∞,+∞, in the present paper, we apply the Fourier transforms to show that every n-order linear differential equation with constant coefficients has a solution in L1ℝ which is infinitely differentiable in ℝ∖0. Moreover, it proves the Hyers–Ulam stability of the equation on L1ℝ.It seems that authors of [17], Theorem 3.1, by using the Fourier transform, proved the Hyers–Ulam stability of the linear differential equation of functions on 0,+∞. But, this is not a new result because the restriction of the Fourier transform on the space of functions with domain in 0,+∞ is the Laplace transform, and it has already been proven that the Laplace transform established the Hyers–Ulam stability of the linear differential equation of functions on 0,+∞ (see [16]).


## 2. Fourier Transform and Inversion Formula

Throughout this paper,F will denote either the real field, ℝ, or the complex field, ℂ. Assume that f:−∞,+∞⟶F is absolutely integrable. Then, the Fourier transform associated to f is a mapping ℱf:ℝ⟶ℂ defined by(3)ℱfw=∫−∞∞e−iwtftdt.Also, the inverse Fourier transform associated to a functionf∈L1ℝ is defined by(4)ℱ−1ft=∫−∞∞eiwtfwdw.By the Fourier Inversion Theorem (see Theorem A.14 in [18]) if f∈L1ℝ and f and f′ are piecewise continuous on ℝ, that is, both are continuous in any finite interval except possibly for a finite number of jump discontinuities, then at each point t where f is continuous,(5)ℱ−1ℱft=ft.In particular, iff,ℱf∈L1ℝ, then the abovementioned relation holds almost everywhere on ℝ (see Theorem 9.11 in [19]).In the following, some required properties of the Fourier transform are presented.Proposition 1.
Letf∈L1ℝ, F=ℱf, and u be the step function defined by ut=1 for t≥0 and ut=0 for t<0. Then,(i)
ℱe−ztutw=1/iw+z provided that Rez>0(ii)
ℱft−t0/kw=ke−iwt0Fkw(iii)
ℱfteiw0tw=Fw−w0(iv)
ℱ−itnftw=Fnw(v)
ℱyntw=iwnℱyw.Proof.
Parts (i)–(iii) are obtained by the definition of Fourier transform. For part (iv), see Theorem 1.6, page 136, in [20]. For part (v), see Theorem 3.3.1, part (f), in [21].We recall that the convolution of two functionsf,g∈L1ℝ is defined by(6)f∗gt=∫−∞+∞ft−xgxdx.Then,f∗g∈L1ℝ, f∗g1≤f1g1, and(7)ℱf∗g=ℱf⋅ℱg.Moreover, we have the following theorem.Theorem 1.
Letf,g∈L1ℝ; then,(i)
ℱf∗g=ℱfℱg.(ii)
ℱ−1fg=ℱ−1f∗ℱ−1g.(iii)
If eitherf or g is differentiable, then f∗g is differentiable. If f′ exists and is continuous, then f∗g′=f′∗g.Proof.
For part (i), see Theorems 1.3 and 1.5, page 135, in [20]. Part (ii) follows from the definition of ℱ−1 and the Fourier Inversion Theorem. For part (iii), see Proposition 2.1, page 68, in [22].Note that ifu is the step function, then(8)f∗ut=u∗ft=∫−∞+∞ut−xfxdx=∫−∞tfxdx.The following corollary is deduced from Theorem 1.5, page 135, in [20].Corollary 1.
Letf∈L1ℝ; then,(9)ℱf∗uw=ℱfwiw,w≠0.


## 3. Hyers–Ulam Stability of the Linear Differential Equation

Before stating the main theorem, we need the following important proposition.Proposition 2.
Letf∈L1ℝ and p be a polynomial with the complex roots w0,w1,…,wk−1, k≥1. Then, there is a function y0∈L1ℝ which is infinitely differentiable in ℝ∖0, k−1-times differentiable at zero, and satisfying(10)ℱy0w=ℱfwpw,w≠w0,w1,…,wk−1.Proof.
First assume thatpw=w−w0k and Imw0≠0. Put z1:=−iw0 when Imw0>0 and z2:=iw0 when Imw0<0. Since Rezi>0,i=1,2, Proposition 1, part (i), implies that(11)ℱe−z1tutw=1iw+z1=1iw−iw0=1iw−w0,ℱe−z2tutw=1iw+z2=1iw+iw0=1iw+w0.
In the second case, using part (ii) of Proposition1 for t0=0 and k=−1, we get(12)ℱez2tu−tw=1−iw+iw0=−1iw−w0.
According to the abovementioned computations, the functionf0:−∞,+∞⟶F is defined by(13)f0t=ieiw0tut,ifImw0>0,−ieiw0tu−t,ifImw0<0.
It satisfies the equation(14)ℱf0w=1w−w0,and by part (iv) of Proposition 1, we see that(15)ℱ−itk−1f0w=1w−w0k−1=−1k−1k−1!w−w0k.
Now, we put(16)y0≔−itk−1k−1!f0∗f.
Then,y0∈L1ℝ, by Theorem 1 part (ii), y0 is infinitely differentiable in ℝ∖0, y0k−10 exists, and(17)ℱy0w=ℱ−itk−1k−1!fwℱyw=ℱyww−w0m,for w≠w0. In the next case, we suppose that Imw0=0, namely, w0 is a real number. In this case, the argument is different. Indeed, let f0=f and(18)fi+1t=eiw0tfie−iw0t∗ut,for i=0,1,2,…,k. Then, by Corollary 1, we have(19)ℱf∗uw=ℱfwiw,w≠0,and so, by (iii) of Proposition 1,(20)ℱfte−iw0t∗uw=ℱfte−iw0twiw=ℱyw+w0iw.
Hence,(21)ℱf1w=ℱeiw0tfte−iw0t∗utw=ℱfwiw−w0.
Continuing in this way, we get(22)ℱfi+1w=ℱfiwiw−w0.
Therefore,(23)ℱfkw=ℱwyikw−w0k.
Finally, we puty0:=ikfk. Then, y0 has the requested properties and satisfies in (10) for pw=w−w0k, and in this case, the proof is completed.
Now, we suppose thatpw expresses as a product of linear factors(24)pw=w−w1k1,w−w2k2,…,w−wknk,for some complex number wi,i=1,…,k and some integer ki,i=1,2,…,k. Applying the partial fraction decomposition of 1/Pw, we obtain(25)1pw=∑i=1k∑j=1niλijw−wij,where λij is a complex number for i=1,2,…,k and j=1,2,…,ni. Considering the first part of proof, there exists a yij∈L1ℝ which is k-times differentiable on ℝ∖0 and(26)ℱyijw=ℱyw−wij,w≠wi,for every integer 1≤i≤k and 1≤j≤ni. Then, we put(27)y0w=∑i=1k∑j=1niλijyijw,w≠w1,w2,…,wkand use the linearity of Fourier transform and (26) to obtain(28)ℱy0w=ℱ∑i=1k∑j=1niλijyijw=∑i=1k∑j=1niλijℱyijw,=∑i=1k∑j=1niλijℱyw−wij=ℱywpw.Theorem 2.
consider the differential equation(29)y′t+a0yt=ft,t∈ℝ,where a0 is a nonzero scalar such that Rea0≠0 and f∈L1ℝ. Then, there exists a constant M with the following property: for every y∈L1ℝ and for given ε>0 satisfying(30)y′+a0y−f1≤ε,there exists a differentiable solution ya∈L1ℝ of the equation such that ya−y1≤Mε.Proof.
Let(31)ht=y′t+a0yt−ft.
Then,(32)ℱhw=ℱy′w+a0ℱyw−ℱfw,=iwℱyw+a0ℱyw−ℱfw,=iw+a0ℱyw−ℱfw.
Hence,(33)ℱyw−ℱfwiw+a0=ℱhwiw+a0.
By the preceding proposition, there exists a functionya∈L1ℝ such that(34)ℱyaw=ℱfwiw+a0.
According to (33), for h=0, we find that ya, in fact, is a solution of the equation. Without loss of generality, we suppose that Rea0>0. Then, by considering (33) and part (i) of Proposition 1, we obtain(35)ℱy−yaw=ℱyw−ℱyaw,=ℱyw−ℱfwiw+a0,=ℱhtwiw+a0=ℱe−a0tut∗htw.
Consequently,yt−yat=e−a0tut∗ht and(36)y−ya1=∫−∞+∞yt−yatdt,=∫−∞+∞e−a0tut∗htdt≤∫−∞+∞∫−∞+∞e−xa0uxht−xdxdt≤∫−∞+∞∫−∞+∞e−xa0|uxht−xdxdt≤∫∞+∞∫0+∞e−xRea0ht−xdxdt≤∫0+∞∫−∞+∞e−xRea0ht−xdtdx≤1Rea0h1<εRea0.
This completes the proof.We denote byCnℝ the space of all n-times differentiable continuous functions on ℝ.Theorem 3.
consider the differential equation(37)ynt+∑k=0n−1akykt=ft,where f∈L1ℝ, n>1, and a0,a1,…,an−1 are given scalars such that Rean−1≠0. Then, there exists a constant M with the following property: for every y∈L1ℝ and for given ε>0 satisfying(38)yn+∑k=0n−1akyk−f1≤ε,there exists a solution ya of equation (37) such that ya∈Cnℝ∩L1ℝ and(39)ya−y1≤Mε,t∈ℝ.Proof.
Let(40)ht=ynt+∑k=0n−1akykt−ft.
Applying Proposition1, part (v), we may derive(41)ℱhw=ℱynw+∑k=0n−1akℱykw−ℱfw,=iwnℱy+∑k=0n−1akiwkℱy−ℱfw,=piwℱyw−ℱfw,where p is a complex polynomial determined by(42)pz=zn+∑k=0n−1akzk.
Applying relations (40) and (41) for h=0, we find that y is a solution of (37) if and only if(43)ℱyw=ℱfwpiw.
Now, from (41), we deduce that(44)ℱhwpiw=ℱyw−ℱfwpiw,iw≠w0,w1,…,wn−1,where w0,w1,…,wn−1 are the roots of the polynomial pw. By Proposition 2, there exists a function ya∈L1ℝ such that(45)ℱyaw=ℱfwpiw.
Since every function satisfying (43) is a solution of (37), we find that ya is a solution of equation (37), and from (44), we obtain(46)ℱyw−ℱyaw=ℱhwpiw,and consequently,(47)yt−yat=ℱ−1ℱhwpiwt.
By the definition ofh and the inequality (38), h1≤ε, so(48)ℱhw≤∫−∞+∞e−iwthtdt≤ε.
Now, by considering part (i) of Theorem1, we have(49)y−ya1=∫−∞+∞yt−yatdt,=∫−∞+∞ℱ−1ℱhwpiwdt,=∫−∞+∞ℱ−1ℱhwt∗ℱ−11piwtdt,=∫−∞+∞h∗ℱ−11piwtdt,=h∗ℱ−11piw1,≤h1ℱ−11piw1=Mε,where(50)M=ℱ−11piw1<+∞,becausedegp>1.
To see the last relation, looking at (42) and the fact that w0,w1,…,wn−1 are the roots of polynomial p, we get(51)w0+w1+⋯wn−1=−an−1,piw=iw−w0iw−w1…iw−wn−1.
SinceRean−1≠0, there exist some 0≤j≤n−1 such that Rewj≠0. Then, by Proposition 1, part (i), there exist some yj∈L1ℝ such that(52)ℱyjw=1iw−wj.
Letqw=∏k≠jiw−wk. Then,(53)1piw=1/iw−wjqw=ℱyjwqw.
Sinceyj∈L1ℝ, by Proposition 2, there exist some yj∈L1ℝ such that(54)ℱyjw=ℱyjwqw.
Comparing the abovementioned relations, we see that(55)ℱyjw=1piw,and hence,(56)1piw=ℱ−1yjw∈L1ℝ.
Therefore, (37) has Hyers–Ulam stability, and the proof is completed.Remark 1.
Note that Proposition2, in fact, states that equation (37) has a solution in L1ℝ which is infinitely differentiable in ℝ∖0, k−1-times differentiable at zero.Remark 2.
In Theorem2, it is easy to see that M=1/Rea0 is the minimal Hyers–Ulam constant. To see this, let the assumptions of Theorem 2 hold with the Hyers–Ulam constant M. Suppose that Rea0>0 and δ=Rea0ε. Putting fδt:=ft+δe−a0tut, we see that fδ∈L1ℝ, and by Remark 1, the equation y′t+a0yt−fδt=0 has a solution yδ∈L1ℝ satisfying(57)yδ′+a0yδ−f1=yδ′+a0yδ−fδ+fδ−f1,=fδ−f1=δe−2a0tut1=δRea0≤ε.
In this case, the functionh in the proof of Theorem 2 is obtained by the following equation:(58)ht=yδ′t+a0yδt−ft=yδ′t+a0yδt−fδt+δe−a0tut=δe−a0tut.
Applying Theorem2, there is a solution ya of the equation y′+a0y−f=0 such that yδ−ya1≤Mε. Considering the proof of Theorem 2 for h=δe−a0tut and yδ instead of y, we obtain(59)Mε≥yδ−ya1=∫−∞+∞yδt−yatdt≥∫0+∞yδt−yatdt,=∫0+∞e−a0tut∗htdt,=∫0+∞∫−∞+∞e−xa0uxδe−t−xRea0ut−xdxdt,=∫0+∞∫0tδe−ta0dxdt,=∫0+∞δte−ta0dt,=δ∫0+∞te−tRea0dt=δRea02.
Hence,δ/Rea02≤Mε, and considering δ=Rea0ε, we obtain 1/Rea0≤M.The following example shows that the conditionRean−1≠0 in the abovementioned theorem is necessary and cannot be removed.Example 1.
Every nonzero solution of equationy′t−iyt=0 has the form y=λeit which is not in L1ℝ. Hence, y=0 is the only solution of equation in L1ℝ. Now, for ϵ>0, consider the function(60)yεt=ei−1tut+ε2e−tut,which is in L1ℝ. To see this, note that(61)∫−∞+∞yεtdt=∫−∞+∞ei−1tut+ε2e−tutdt≤∫−∞+∞ei−1tutdt+ε2∫−∞+∞e−tutdt,=∫0+∞eite−tdt+ε2∫0+∞e−tdt,=∫0+∞e−tdt+ε2∫0+∞e−tdt,<+∞.
On the other hand, fort<0, yε′t=yεt=0. Thus,(62)yε′−iyε1=∫−∞+∞yε′t−iyεtdt,=∫−∞+∞ieit−ε2e−tut−iMeit−ε2ie−tut,=ε2∫0+∞1+ie−tdt=ε.
If the equation,y′t−iyt=0 has Hyers–Ulam stability, and there must be a solution ya∈L1ℝ of the differential equation such that yε−ya1<Kε and limε⟶0Kε=0. Since the only solution of equation in L1ℝ is zero function, ya=0, and on the other hand,(63)yε−ya1=yε1=∫0+∞ei−1t+ε2e−t≥∫0+∞ei−1tdt−ε2∫0+∞e−tdt=1−ε2.
Therefore,(64)1−ε2≤Kε,which is a contradiction with the fact that limε⟶0Kε=0.


## 4. Conclusions

The main conclusions of this paper are as follows:(1)
Every differential equationynt+∑k=0n−1akykt=ft, where f∈L1ℝ, n>1, and Rean−1≠0 has a n-times differentiable solution in Cnℝ∩L1ℝ(2)
Equation (37) is stable on L1ℝ in the sense of Hyers–Ulam



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