Fourier Transformation and Stability of a Differential Equation on

Abstract

In the present paper, by the Fourier transform, we show that every linear differential equation with constant coefficients of -th order has a solution in which is infinitely differentiable in . Moreover the Hyers–Ulam stability of such equations on 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 be two real Banach spaces, and is given. Then, for every mapping, satisfying

There exists a unique additive mapping with the property

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 has Hyers–Ulam stability on normed space if, for given and a function such that , there is a solution of the differential equation such that and .

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 -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 -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 with respect to the right-hand side .

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 -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 .

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 -order linear differential equation with constant coefficients has a solution in which is infinitely differentiable in . Moreover, it proves the Hyers–Ulam stability of the equation on .

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 . But, this is not a new result because the restriction of the Fourier transform on the space of functions with domain in 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 (see [16]).

2. Fourier Transform and Inversion Formula

Throughout this paper, will denote either the real field, , or the complex field, . Assume that is absolutely integrable. Then, the Fourier transform associated to is a mapping defined by

Also, the inverse Fourier transform associated to a function is defined by

By the Fourier Inversion Theorem (see Theorem A.14 in [18]) if and and are piecewise continuous on , that is, both are continuous in any finite interval except possibly for a finite number of jump discontinuities, then at each point where is continuous,

In particular, if , then the abovementioned relation holds almost everywhere on (see Theorem 9.11 in [19]).

In the following, some required properties of the Fourier transform are presented.

Proposition 1. Let , , and be the step function defined by for and for . Then,(i) provided that (ii)(iii)(iv)(v)

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 is defined by

Then, , , and

Moreover, we have the following theorem.

Theorem 1. Let ; then,(i).(ii).(iii)If either or is differentiable, then is differentiable. If exists and is continuous, then .

Proof. For part (i), see Theorems 1.3 and 1.5, page 135, in [20]. Part (ii) follows from the definition of and the Fourier Inversion Theorem. For part (iii), see Proposition 2.1, page 68, in [22].

Note that if is the step function, then

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

Corollary 1. Let ; then,

3. Hyers–Ulam Stability of the Linear Differential Equation

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

Proposition 2. Let and be a polynomial with the complex roots , . Then, there is a function which is infinitely differentiable in , -times differentiable at zero, and satisfying

Proof. First assume that and . Put when and when . Since , Proposition 1, part (i), implies thatIn the second case, using part (ii) of Proposition 1 for and , we getAccording to the abovementioned computations, the function is defined byIt satisfies the equationand by part (iv) of Proposition 1, we see thatNow, we putThen, , by Theorem 1 part (ii), is infinitely differentiable in , exists, andfor . In the next case, we suppose that , namely, is a real number. In this case, the argument is different. Indeed, let andfor . Then, by Corollary 1, we haveand so, by (iii) of Proposition 1,Hence,Continuing in this way, we getTherefore,Finally, we put . Then, has the requested properties and satisfies in (10) for , and in this case, the proof is completed.
Now, we suppose that expresses as a product of linear factorsfor some complex number and some integer . Applying the partial fraction decomposition of , we obtainwhere is a complex number for and . Considering the first part of proof, there exists a which is -times differentiable on andfor every integer and . Then, we putand use the linearity of Fourier transform and (26) to obtain

Theorem 2. consider the differential equationwhere is a nonzero scalar such that and . Then, there exists a constant with the following property: for every and for given satisfyingthere exists a differentiable solution of the equation such that .

Proof. LetThen,Hence,By the preceding proposition, there exists a function such thatAccording to (33), for , we find that , in fact, is a solution of the equation. Without loss of generality, we suppose that . Then, by considering (33) and part (i) of Proposition 1, we obtainConsequently, andThis completes the proof.

We denote by the space of all -times differentiable continuous functions on .

Theorem 3. consider the differential equationwhere , , and are given scalars such that . Then, there exists a constant with the following property: for every and for given satisfyingthere exists a solution of equation (37) such that and

Proof. LetApplying Proposition 1, part (v), we may derivewhere is a complex polynomial determined byApplying relations (40) and (41) for , we find that is a solution of (37) if and only ifNow, from (41), we deduce thatwhere are the roots of the polynomial . By Proposition 2, there exists a function such thatSince every function satisfying (43) is a solution of (37), we find that is a solution of equation (37), and from (44), we obtainand consequently,By the definition of and the inequality (38), , soNow, by considering part (i) of Theorem 1, we havewhereTo see the last relation, looking at (42) and the fact that are the roots of polynomial , we getSince , there exist some such that . Then, by Proposition 1, part (i), there exist some such thatLet . Then,Since , by Proposition 2, there exist some such thatComparing the abovementioned relations, we see thatand hence,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 which is infinitely differentiable in , -times differentiable at zero.

Remark 2. In Theorem 2, it is easy to see that is the minimal Hyers–Ulam constant. To see this, let the assumptions of Theorem 2 hold with the Hyers–Ulam constant . Suppose that and . Putting , we see that , and by Remark 1, the equation has a solution satisfyingIn this case, the function in the proof of Theorem 2 is obtained by the following equation:Applying Theorem 2, there is a solution of the equation such that . Considering the proof of Theorem 2 for and instead of , we obtainHence, , and considering , we obtain .

The following example shows that the condition in the abovementioned theorem is necessary and cannot be removed.

Example 1. Every nonzero solution of equation has the form which is not in . Hence, is the only solution of equation in . Now, for , consider the functionwhich is in . To see this, note thatOn the other hand, for , . Thus,If the equation, has Hyers–Ulam stability, and there must be a solution of the differential equation such that and . Since the only solution of equation in is zero function, , and on the other hand,Therefore,which is a contradiction with the fact that .

4. Conclusions

The main conclusions of this paper are as follows:(1)Every differential equation , where , , and has a -times differentiable solution in (2)Equation (37) is stable on in the sense of Hyers–Ulam

Data Availability

No data were used to support this study.

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.