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

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. &en, for every mapping, f: X1⟶ X2 satisfying ‖f(x + y) − f(x) − f(y)‖< ε for all x; y ∈ X1. (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 X 1 , X 2 be two real Banach spaces, and ε > 0 is given. en, for every mapping, f: X 1 ⟶ X 2 satisfying ere exists a unique additive mapping g: X 1 ⟶ X 2 with the property ‖f(x) − g(x)‖ < ε, for all x ∈ X 1 .
(2) After Hyers result, these problems have been extended to other functional equations [2]. is may be the most important extension in the Hyers-Ulam stability of the differential equations. e 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⟶0 K(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 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. e 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 (− ∞, +∞). ey 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 L 1 (R) which is infinitely differentiable in R∖ 0 { }. Moreover, it proves the Hyers-Ulam stability of the equation on L 1 (R).
It seems that authors of [17], eorem 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]).

Fourier Transform and Inversion Formula
roughout this paper, F will denote either the real field, R, or the complex field, C. Assume that f: (− ∞, +∞) ⟶ F is absolutely integrable. en, the Fourier transform associated to f is a mapping F(f): R ⟶ C defined by Also, the inverse Fourier transform associated to a function f ∈ L 1 (R) is defined by By the Fourier Inversion eorem (see eorem A.14 in [18]) if f ∈ L 1 (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, In particular, if f, F(f) ∈ L 1 (R), then the abovementioned relation holds almost everywhere on R (see eorem 9.11 in [19]). In the following, some required properties of the Fourier transform are presented. en, Proof. Parts (i)-(iii) are obtained by the definition of Fourier transform. For part (iv), see eorem 1.6, page 136, in [20]. For part (v), see eorem 3.3.1, part (f ), in [21]. □ We recall that the convolution of two functions Moreover, we have the following theorem.

□
Note that if u is the step function, then e following corollary is deduced from eorem 1.5, page 135, in [20].

Hyers-Ulam Stability of the Linear Differential Equation
Before stating the main theorem, we need the following important proposition.

International Journal of Mathematics and Mathematical Sciences
Proposition 2. Let f ∈ L 1 (R) and p be a polynomial with the complex roots w 0 , w 1 , . . . , w k− 1 , k ≥ 1. en, there is a function y 0 ∈ L 1 (R) which is infinitely differentiable in R∖ 0 { }, (k − 1)-times differentiable at zero, and satisfying Proof. First assume that p(w) � (w − w 0 ) k and Im(w 0 ) ≠ 0. Put z 1 : � − iw 0 when Im(w 0 ) > 0 and z 2 : � iw 0 when Im(w 0 ) < 0. Since Re(z i ) > 0, i � 1, 2, Proposition 1, part (i), implies that In the second case, using part (ii) of Proposition 1 for t 0 � 0 and k � − 1, we get According to the abovementioned computations, the It satisfies the equation and by part (iv) of Proposition 1, we see that Now, we put en, y 0 ∈ L 1 (R), by eorem 1 part (ii), y 0 is infinitely differentiable in R∖ 0 { }, y (k− 1) 0 (0) exists, and for w ≠ w 0 . In the next case, we suppose that Im(w 0 ) � 0, namely, w 0 is a real number. In this case, the argument is different. Indeed, let f 0 � f and for i � 0, 1, 2, . . . , k. en, by Corollary 1, we have and so, by (iii) of Proposition 1, (21) Continuing in this way, we get erefore, Finally, we put y 0 : � i k f k . en, y 0 has the requested properties and satisfies in (10) for p(w) � (w − w 0 ) k , and in this case, the proof is completed. Now, we suppose that p(w) expresses as a product of linear factors 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 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 ∈ L 1 (R) which is k-times differentiable on R∖ 0 { } and for every integer 1 ≤ i ≤ k and 1 ≤ j ≤ n i . en, we put and use the linearity of Fourier transform and (26) to obtain □ International Journal of Mathematics and Mathematical Sciences Theorem 2. consider the differential equation where a 0 is a nonzero scalar such that Re(a 0 ) ≠ 0 and f ∈ L 1 (R). en, there exists a constant M with the following property: for every y ∈ L 1 (R) and for given ε > 0 satisfying there exists a differentiable solution y a ∈ L 1 (R) of the equation such that ‖y a − y‖ 1 ≤ Mε.

Proof.
Let en, Hence, By the preceding proposition, there exists a function y a ∈ L 1 (R) such that 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(a 0 ) > 0. en, by considering (33) and part (i) of Proposition 1, we obtain F y − y a (w) � F(y)(w) − F y a (w), Consequently, y(t) − y a (t) � e − a 0 t u(t) * h(t) and is completes the proof.

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We denote by C n (R) the space of all n-times differentiable continuous functions on R.
Theorem 3. consider the differential equation where f ∈ L 1 (R), n > 1, and a 0 , a 1 , . . . , a n− 1 are given scalars such that Re(a n− 1 ) ≠ 0. en, there exists a constant M with the following property: for every y ∈ L 1 (R) and for given ε > 0 satisfying there exists a solution y a of equation (37) such that y a ∈ C n (R) ∩ L 1 (R) and Proof. Let Applying Proposition 1, part (v), we may derive 4 International Journal of Mathematics and Mathematical Sciences where p is a complex polynomial determined by Applying relations (40) and (41) for h � 0, we find that y is a solution of (37) if and only if Now, from (41), we deduce that where w 0 , w 1 , . . . , w n− 1 are the roots of the polynomial p(w). By Proposition 2, there exists a function y a ∈ L 1 (R) such that 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 and consequently, By the definition of h and the inequality (38), ‖h‖ 1 ≤ ε, so Now, by considering part (i) of eorem 1, we have where To see the last relation, looking at (42) and the fact that w 0 , w 1 , . . . , w n− 1 are the roots of polynomial p, we get w 0 + w 1 + · · · w n− 1 � − a n− 1 , Since Re(a n− 1 ) ≠ 0, there exist some 0 ≤ j ≤ n − 1 such that Re(w j ) ≠ 0. en, by Proposition 1, part (i), there exist some y j ∈ L 1 (R) such that Let q(w) � k≠j (iw − w k ). en, Since y j ∈ L 1 (R), by Proposition 2, there exist some y j ∈ L 1 (R) such that F y j (w) � F y j (w) q(w) .
Comparing the abovementioned relations, we see that International Journal of Mathematics and Mathematical Sciences 5

Conclusions
e main conclusions of this paper are as follows: (1) Every differential equation y (n) (t) + n− 1 k�0 a k y (k) (t) � f(t), where f ∈ L 1 (R), n > 1, and Re(a n− 1 ) ≠ 0 has a n-times differentiable solution in C n (R) ∩ L 1 (R) (2) Equation (37) is stable on L 1 (R) in the sense of Hyers-Ulam

Data Availability
No data were used to support this study.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.