In the present paper, by the Fourier transform, we show that every linear differential equation with constant coefficients of n-th order has a solution in L¹(ℝ) which is infinitely differentiable in ℝ∖{0}. Moreover the Hyers–Ulam stability of such equations on L¹(ℝ) is investigated.

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)

There exists a unique additive mapping g : X₁⟶X₂ with the property

(2)

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 equation φ(y, y ^′, …, y⁽ⁿ⁾) = f has Hyers–Ulam stability on normed space X if, for given ε > 0 and a function y such that ‖φ(y, y ^′, …, y⁽ⁿ⁾) − f‖ < ε, there is a solution y_a ∈ X of the differential equation such that ‖y − y_a‖ < K(ε) and lim_ε⟶0K(ε) = 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 the ϵ-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 φ(y, y′, …, y⁽ⁿ⁾) = 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 the s-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 L¹(ℝ) which is infinitely differentiable in ℝ∖{0}. Moreover, it proves the Hyers–Ulam stability of the equation on L¹(ℝ).

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,

will denote either the real field, ℝ, or the complex field, ℂ. Assume that

is absolutely integrable. Then, the Fourier transform associated to f is a mapping ℱ(f) : ℝ⟶ℂ defined by

(3)

Also, the inverse Fourier transform associated to a function f ∈ L¹(ℝ) is defined by

(4)

By the Fourier Inversion Theorem (see Theorem A.14 in [18]) if f ∈ L¹(ℝ) 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)

In particular, if f, ℱ(f) ∈ L¹(ℝ), 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. Let f ∈ L¹(ℝ), F = ℱ(f), and u be the step function defined by u(t) = 1 for t ≥ 0 and u(t) = 0 for t < 0. Then,

(i)
ℱ(e^−ztu(t))(w) = (1/iw + z) provided that Re(z) > 0
(ii)
(iii)
(iv)
ℱ((−it)ⁿf(t))(w) = F⁽ⁿ⁾(w)
(v)
ℱ(y⁽ⁿ⁾(t))(w) = (iw)ⁿℱ(y)(w).

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 functions f, g ∈ L¹(ℝ) is defined by

(6)

Then, f∗g ∈ L¹(ℝ), ‖f∗g‖₁ ≤ ‖f‖₁‖g‖₁, and

(7)

Moreover, we have the following theorem.

Theorem 1. Let f, g ∈ L¹(ℝ); then,

(i)
ℱ(f∗g) = ℱ(f)ℱ(g).
(ii)
ℱ⁻¹(fg) = ℱ⁻¹(f)∗ℱ⁻¹(g).
(iii)
If either f 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 if u is the step function, then

(8)

The following corollary is deduced from Theorem 1.5, page 135, in [20].

Corollary 1. Let f ∈ L¹(ℝ); then,

(9)

3. Hyers–Ulam Stability of the Linear Differential Equation

Before stating the main theorem, we need the following important proposition.

Proposition 2. Let f ∈ L¹(ℝ) and p be a polynomial with the complex roots w₀, w₁, …, w_k−1, k ≥ 1. Then, there is a function y₀ ∈ L¹(ℝ) which is infinitely differentiable in ℝ∖{0}, (k − 1)-times differentiable at zero, and satisfying

(10)

Proof. First assume that and Im(w₀) ≠ 0. Put z₁ : = −iw₀ when Im(w₀) > 0 and z₂ : = iw₀ when Im(w₀) < 0. Since Re(z_i) > 0, i = 1,2, Proposition 1, part (i), implies that

(11)

In the second case, using part (ii) of Proposition 1 for t₀ = 0 and k = −1, we get

(12)

According to the abovementioned computations, the function is defined by

(13)

It satisfies the equation

(14)

and by part (iv) of Proposition 1, we see that

(15)

Now, we put

(16)

Then, y₀ ∈ L¹(ℝ), by Theorem 1 part (ii), y₀ is infinitely differentiable in ℝ∖{0}, exists, and

(17)

for w ≠ w₀. In the next case, we suppose that Im(w₀) = 0, namely, w₀ is a real number. In this case, the argument is different. Indeed, let f₀ = f and

(18)

for i = 0,1,2, …, k. Then, by Corollary 1, we have

(19)

and so, by (iii) of Proposition 1,

(20)

Hence,

(21)

Continuing in this way, we get

(22)

Therefore,

(23)

Finally, we put y₀ : = i^kf_k. Then, y₀ has the requested properties and satisfies in (10) for , and in this case, the proof is completed.

Now, we suppose that p(w) expresses as a product of linear factors

(24)

for some complex number w_i, i = 1, …, k and some integer k_i, i = 1,2, …, k. Applying the partial fraction decomposition of 1/P(w), we obtain

(25)

where λ_ij is a complex number for i = 1,2, …, k and j = 1,2, …, n_i. Considering the first part of proof, there exists a y_ij ∈ L¹(ℝ) which is k-times differentiable on ℝ∖{0} and

(26)

for every integer 1 ≤ i ≤ k and 1 ≤ j ≤ n_i. Then, we put

(27)

and use the linearity of Fourier transform and (26) to obtain

(28)

Theorem 2. consider the differential equation

(29)

where a₀ is a nonzero scalar such that Re(a₀) ≠ 0 and f ∈ L¹(ℝ). Then, there exists a constant M with the following property: for every y ∈ L¹(ℝ) and for given ε > 0 satisfying

(30)

there exists a differentiable solution y_a ∈ L¹(ℝ) of the equation such that

Proof. Let

(31)

Then,

(32)

Hence,

(33)

By the preceding proposition, there exists a function y_a ∈ L¹(ℝ) such that

(34)

According to (33), for h = 0, we find that y_a, in fact, is a solution of the equation. Without loss of generality, we suppose that Re(a₀) > 0. Then, by considering (33) and part (i) of Proposition 1, we obtain

(35)

Consequently, and

(36)

This completes the proof.

We denote by Cⁿ(ℝ) the space of all n-times differentiable continuous functions on ℝ.

Theorem 3. consider the differential equation

(37)

where f ∈ L¹(ℝ), n > 1, and a₀, a₁, …, a_n−1 are given scalars such that Re(a_n−1) ≠ 0. Then, there exists a constant M with the following property: for every y ∈ L¹(ℝ) and for given ε > 0 satisfying

(38)

there exists a solution y_a of equation (37) such that y_a ∈ Cⁿ(ℝ)∩L¹(ℝ) and

(39)

Proof. Let

(40)

Applying Proposition 1, part (v), we may derive

(41)

where p is a complex polynomial determined by

(42)

Applying relations (40) and (41) for h = 0, we find that y is a solution of (37) if and only if

(43)

Now, from (41), we deduce that

(44)

where w₀, w₁, …, w_n−1 are the roots of the polynomial p(w). By Proposition 2, there exists a function y_a ∈ L¹(ℝ) such that

(45)

Since every function satisfying (43) is a solution of (37), we find that y_a is a solution of equation (37), and from (44), we obtain

(46)

and consequently,

(47)

By the definition of h and the inequality (38), ‖h‖₁ ≤ ε, so

(48)

Now, by considering part (i) of Theorem 1, we have

(49)

where

(50)

To see the last relation, looking at (42) and the fact that w₀, w₁, …, w_n−1 are the roots of polynomial p, we get

(51)

Since Re(a_n−1) ≠ 0, there exist some 0 ≤ j ≤ n − 1 such that Re(w_j) ≠ 0. Then, by Proposition 1, part (i), there exist some y_j ∈ L¹(ℝ) such that

(52)

Let q(w) = ∏_k≠j(iw − w_k). Then,

(53)

Since y_j ∈ L¹(ℝ), by Proposition 2, there exist some y_j ∈ L¹(ℝ) such that

(54)

Comparing the abovementioned relations, we see that

(55)

and hence,

(56)

Therefore, (37) has Hyers–Ulam stability, and the proof is completed.

Remark 1. Note that Proposition 2, in fact, states that equation (37) has a solution in L¹(ℝ) which is infinitely differentiable in ℝ∖{0}, (k − 1)-times differentiable at zero.

Remark 2. In Theorem 2, it is easy to see that M = 1/|Re(a₀)| is the minimal Hyers–Ulam constant. To see this, let the assumptions of Theorem 2 hold with the Hyers–Ulam constant M. Suppose that Re(a₀) > 0 and δ = Re(a₀)ε. Putting , we see that f_δ ∈ L¹(ℝ), and by Remark 1, the equation y ^′(t) + a₀y(t) − f_δ(t) = 0 has a solution y_δ ∈ L¹(ℝ) satisfying

(57)

In this case, the function h in the proof of Theorem 2 is obtained by the following equation:

(58)

Applying Theorem 2, there is a solution y_a of the equation y ^′ + a₀y − f = 0 such that . Considering the proof of Theorem 2 for and y_δ instead of y, we obtain

(59)

Hence, , and considering δ = Re(a₀)ε, we obtain (1/Re(a₀)) ≤ M.

The following example shows that the condition Re(a_n−1) ≠ 0 in the abovementioned theorem is necessary and cannot be removed.

Example 1. Every nonzero solution of equation y′(t) − iy(t) = 0 has the form y = λe^it which is not in L¹(ℝ). Hence, y = 0 is the only solution of equation in L¹(ℝ). Now, for ϵ > 0, consider the function

(60)

which is in L¹(ℝ). To see this, note that

(61)

On the other hand, for t < 0, . Thus,

(62)

If the equation, y′(t) − iy(t) = 0 has Hyers–Ulam stability, and there must be a solution y_a ∈ L¹(ℝ) of the differential equation such that and lim_ε⟶0K(ε) = 0. Since the only solution of equation in L¹(ℝ) is zero function, y_a = 0, and on the other hand,

(63)

Therefore,

(64)

which is a contradiction with the fact that lim_ε⟶0K(ε) = 0.

4. Conclusions

The main conclusions of this paper are as follows:

(1)
Every differential equation , where f ∈ L¹(ℝ), n > 1, and Re(a_n−1) ≠ 0 has a n-times differentiable solution in Cⁿ(ℝ)∩L¹(ℝ)
(2)
Equation (37) is stable on L¹(ℝ) in the sense of Hyers–Ulam

Disclosure

This manuscript has been submitted as a preprint in the following link: http://export.arxiv.org/pdf/2005.03296.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Open Research

Data Availability

No data were used to support this study.

References

1 Hyers D. H., On the stability of the linear functional equation, Proceedings of the National Academy of Sciences. (1941) 27, no. 4, 222–224, https://doi.org/10.1073/pnas.27.4.222.
10.1073/pnas.27.4.222
CAS PubMed Web of Science® Google Scholar
2 Jung S. M., Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, 2011, Springer, New York, NY, USA.
10.1007/978-1-4419-9637-4
Google Scholar
3 Mynt-U T. and Debenath L., Linear Partial Differential Equations for Scientists and Engineers, 2007, fourth edition, Birkhauser, Boston, MA, USA.
Google Scholar
4 Lia C. P. and Zhang F. R., A survey on the stability of fractional differential equations, The European Physical Journal Special Topics. (2011) 193, no. 1, 27–47, https://doi.org/10.1140/epjst/e2011-01379-1, 2-s2.0-79953701362.
10.1140/epjst/e2011-01379-1
Web of Science® Google Scholar
5 Alsina C. and Ger R., On some inequalities and stability results related to the exponential function, Journal of Inequalities and Applications. (1998) 2, 373–380, https://doi.org/10.1155/s102558349800023x.
10.1155/s102558349800023x
Web of Science® Google Scholar
6 Takahasi S.-E., Takagi H., Miura T., and Miyajima S., The Hyers-Ulam stability constants of first order linear differential operators, Journal of Mathematical Analysis and Applications. (2004) 296, no. 2, 403–409, https://doi.org/10.1016/j.jmaa.2003.12.044, 2-s2.0-3442896171.
10.1016/j.jmaa.2003.12.044
Web of Science® Google Scholar
7 Cîmpean D. S. and Popa D., On the stability of the linear differential equation of higher order with constant coefficients, Applied Mathematics and Computation. (2010) 217, no. 8, 4141–4146, https://doi.org/10.1016/j.amc.2010.09.062, 2-s2.0-78650028247.
10.1016/j.amc.2010.09.062
Web of Science® Google Scholar
8 Guo Y., Chen M., Shu X.-B., and Xu F., The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm, Stochastic Analysis and Applications. (2020) 39, no. 4, 643–666, https://doi.org/10.1080/07362994.2020.1824677.
10.1080/07362994.2020.1824677
Web of Science® Google Scholar
9 Li Y. and Shen Y., Hyers-Ulam stability of linear differential equations of second order, Applied Mathematics Letters. (2010) 23, no. 3, 306–309, https://doi.org/10.1016/j.aml.2009.09.020, 2-s2.0-70449672333.
10.1016/j.aml.2009.09.020
Web of Science® Google Scholar
10 Li S., Shu L., Shu X.-B., and Xu F., Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays, Stochastics. (2018) 91, no. 6, 857–872, https://doi.org/10.1080/17442508.2018.1551400, 2-s2.0-85057582226.
10.1080/17442508.2018.1551400
Google Scholar
11 Muthaiah S., Thangaraj D. B. N. G., Baleanu D., and Gopal Thangaraj N., Existence and Hyers-Ulam type stability results for nonlinear coupled system of Caputo-Hadamard type fractional differential equations, Aims Mathematics. (2021) 6, no. 1, 168–194, https://doi.org/10.3934/math.2021012.
10.3934/math.2021012
Web of Science® Google Scholar
12 Miura T., Miyajima S., and Takahasi S.-E., A characterization of Hyers-Ulam stability of first order linear differential operators, Journal of Mathematical Analysis and Applications. (2003) 286, no. 1, 136–146, https://doi.org/10.1016/s0022-247x(03)00458-x, 2-s2.0-0142087794.
10.1016/S0022-247X(03)00458-X
Web of Science® Google Scholar
13 Rus I. A., Remarks on Ulam stability of the operatorial equations, Fixed Point Theory. (2009) 10, 305–320.
Web of Science® Google Scholar
14 Rus I. A., Ulam stability of ordinary differential equations, Stud. Univ. Babes-Bolyai Math. (2009) 54, 125–134.
Google Scholar
15 Wang G., Zhou M., and Sun L., Hyers-Ulam stability of linear differential equations of first order, Applied Mathematics Letters. (2008) 21, no. 10, 1024–1028, https://doi.org/10.1016/j.aml.2007.10.020, 2-s2.0-48849092137.
10.1016/j.aml.2007.10.020
Web of Science® Google Scholar
16 Rezaei H., Jung S.-M., and Rassias T. M., Laplace transform and Hyers-Ulam stability of linear differential equations, Journal of Mathematical Analysis and Applications. (2013) 403, no. 1, 244–251, https://doi.org/10.1016/j.jmaa.2013.02.034, 2-s2.0-84875605874.
10.1016/j.jmaa.2013.02.034
Web of Science® Google Scholar
17 Ramdoss M., Selvan-Arumugam P., and Park C., Ulam stability of linear differential equations using Fourier transform, AIMS Mathematics. (2020) 5, no. 2, 766–780, https://doi.org/10.3934/math.2020052.
10.3934/math.2020052
Web of Science® Google Scholar
18 Schif J. L., The Laplace Transform, Theory and its Application, 1999, Springer, New York, NY, USA.
10.1007/978-0-387-22757-3
Google Scholar
19 Rudin W., Real and Complex Analysis, 1987, McGraw-Hill, New York, NY, USA.
Google Scholar
20 Katzenelson Y., An Introduction to Harmonic Analysis, 2002, Cambrige University Press, Cambridge, UK.
Google Scholar
21 Deitmar A., A First Course in Harmonic Analysis, 2005, Springer, Berlin, Germany.
Google Scholar
22 Conway J. B., Functions of One Complex Variable, 1978, Springer, New York, NY, USA.
10.1007/978-1-4612-6313-5
Google Scholar

Citing Literature

All articles

Fourier Transformation and Stability of a Differential Equation on L¹(ℝ)

Abstract

1. Introduction

2. Fourier Transform and Inversion Formula

3. Hyers–Ulam Stability of the Linear Differential Equation

4. Conclusions

Disclosure

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Fourier Transformation and Stability of a Differential Equation on L1(ℝ)

Abstract

1. Introduction

2. Fourier Transform and Inversion Formula

3. Hyers–Ulam Stability of the Linear Differential Equation

4. Conclusions

Disclosure

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

References

Related

Information

Fourier Transformation and Stability of a Differential Equation on L¹(ℝ)