Laplace Transform and Semi-Hyers–Ulam–Rassias Stability of Some Delay Differential Equations

: In this paper, we study semi-Hyers–Ulam–Rassias stability and generalized semi-Hyers– Ulam–Rassias stability of differential equations x (cid:48) ( t ) + x ( t − 1 ) = f ( t ) and x (cid:48)(cid:48) ( t ) + x (cid:48) ( t − 1 ) = f ( t ) , x ( t ) = 0 if t ≤ 0, using the Laplace transform. Our results complete those obtained by S. M. Jung and J. Brzdek for the equation x (cid:48) ( t ) + x ( t − 1 ) = 0. Laplace 44A10; 34K20


Introduction
The study of Ulam stability began in 1940, when Ulam posed a problem concerning the stability of homomorphisms (see [1]). In 1941, Hyers [2] gave an answer, in the case of the additive Cauchy equation in Banach spaces, to the problem posed by Ulam [1].
In 1993, Obloza [3] started the study of Hyers-Ulam stability of differential equations. Later, in 1998, Alsina and Ger [4] studied the equation y (x) − y(x) = 0. Many mathematicians have further studied the stability of various equations. For a collection of results regarding this problematic, see [5] or [6].
There are many methods for studying Hyers-Ulam stability of differential equations, such as the direct method, the Gronwall inequality method, the fixed point method, the integral transform method, etc.
We mention that the Laplace transform method was used by H. Rezaei, S. M. Jung and Th. M. Rassias [7] and by Q. H. Alqifiary and S. M. Jung [8] to study the differential equation This method was also used in [9], where Laguerre differential equation xy + (1 − x)y + ny = 0, n positive integer and Bessel differential equation xy + y + xy = 0, was studied. In [10], Mittag-Leffler-Hyers-Ulam stability of the following linear differential equation of first order was studied with this method: u (t) + lu(t) = r(t), t ∈ I, u, r ∈ C(I), I = [a, b].
In [11], the semi-Hyers-Ulam-Rassias stability of a Volterra integro-differential equation of order I with a convolution type kernel was studied via Laplace transform: f , g, y : (0, ∞) → F functions of exponential order and continuous and F the real field R or the complex field C.
In [13], the delay equation was studied, using direct method.
In the following, we will study semi-Hyers-Ulam-Rassias stability and generalized semi-Hyers-Ulam-Rassias stability of some equations, with delay of order one and two, with Laplace transform. We complete the results obtained in [13]. Delay differential equations have many applications in various areas of engineering science, biology, physics, etc. The monograph [14] contains some modeling examples from mechanics, chemistry, ecology, biology, psychology, etc. For other applications, see also [15].
We first recall some notions and results regarding the Laplace transform.

Definition 1.
A function x : R → R is called an original function if the following conditions are satisfied: 1.
∃M > 0 and σ 0 ≥ 0 such that We denote by O the set of original functions. We denote by M(x) the set of all numbers that satisfy the condition 3.
The number σ x = inf{σ 0 | σ 0 ∈ M(x)} is called abscissa of convergence of x. The functions that appear below are considered original functions. Hence, since in definition of Laplace transform are involved only the values of x on [0, ∞), we may suppose that is the unit step function of Heaviside. We write x (n) (0) instead the lateral limit x (n) (0 + ) for n ≥ 0. We denote by L(x) the Laplace transform of the function x, defined by It is well known that the Laplace transform is linear and one-to-one if the functions involved are continuous. The inverse Laplace transform will be denoted by L −1 (X) or by L −1 (L(x)).
The following properties are used in the paper: dτ is the convolution product of f and g. In the following, we consider the original functions x, f : R → R. The following Gronwall Lemma is also used in the paper ( [16], p. 6):

Semi-Hyers-Ulam-Rassias Stability of a Delay Differential Equation of Order One
Let f ∈ O. In what follows, we consider the equation x continuous, piecewise differentiable. Let ε > 0. We also consider the inequality According to [17], we give the following definition:

Remark 1.
A function x : (0, ∞) → R is a solution of (2) if and only if there exists a function p

Lemma 2.
For s > 1 we have Proof. As in [18] (p. 15), for s > 1 we have e −s s < 1, hence where [t] denotes the integer part of the real number t.
Applying a method used in [19], we prove now that the Laplace transform exist for the functions satisfying (1) and (2).
Then the Laplace transform of x, which is the exact solution of (1) and of x exist Proof. Integrating the relation (1) from 0 to t, we obtain Changing the variable v = u − 1 in the first integral, we have Applying now Gronwall Lemma 1, we obtain that is the function x is of exponential order.
Applying now Gronwall Lemma, we obtain that is the function x is of exponential order.
From (1), we have Hence, x is of exponential order.
Then the Laplace transform of x (which is a solution of (2) and of x exist for all Integrating from 0 to t, we obtain Applying now Gronwall Lemma, we obtain Applying now Gronwall Lemma, we obtain that is the function x is of exponential order.
From (2), we have We have We remark that x 0 (0) = 0. Hence, we obtain Since L is one-to-one, it follows that that is x 0 is a solution of (1). We have From Lemma 2, we obtain For t > 1, we have We obtain

Definition 3. The Equation
Remark 2. A function x : (0, ∞) → R is a solution of (6) if and only if there exists a function p Lemma 3. For s > 1, we have Proof. For s > 1, we have e −s s < 1, hence Proof. We can apply Theorem 3.1 from [19].
Theorem 5. Let f ∈ O. Let σ f be abscissa of convergence of f and M f > 0 such that | f (t)| ≤ M f · e σ f t , ∀t > 0. Then the Laplace transform of x, which is a solution of (6) and of x , x exist for a certain σ > σ f , for all s > σ.
Proof. The proof is similar to that of Theorem 2.

that is the Equation (5) is semi-Hyers-Ulam-Rassias stable.
Proof. Let p : (0, ∞) → R, We have We remark that x 0 (0) = 0 and x 0 (0) = 0 Hence, we obtain Since L is one-to-one, it follows that that is x 0 is a solution of (5). We have

From Lemma 3, we obtain
We obtain

Generalized Semi-Hyers-Ulam-Rassias Stability of a Delay Differential Equation of Order One
We continue to study generalized semi-Hyers-Ulam-Rassias stability of the Equation (1). Let ϕ ∈ O. We consider the inequality Remark 3. A function x : (0, ∞) → R is a solution of (9) if, and only if, there exists a function p Theorem 7. If a function x : (0, ∞) → R satisfies the inequality (9), where f , ϕ ∈ O, then there exists a solution x 0 : (0, ∞) → R of (1) such that that is the Equation (1) is generalized semi-Hyers-Ulam-Rassias stable.
Proof. Let p : (0, ∞) → R, As in Theorem 6, for x that is a solution of (13) and Laplace transform of x, x , x exists, we have is a solution of (5).

Conclusions
The use of the Laplace transform in the study of Hyers-Ulam stability of differential equations is relatively recent (2013, see [7]). This method was not used to study the stability of equations with delay. In this paper, we have studied semi-Hyers-Ulam-Rassias stability and generalized semi-Hyers-Ulam-Rassias stability of Equations (1) and (5) using the Laplace transform. Some examples were given. The results obtained complete those of S. M. Jung and J. Brzdek from [13]. This method can be used successfully in the case of other equations with delay, integro-differential equations, partial differential equations or for fractional calculus. In [11], we have already studied a Volterra integro-differential equation of order I with a convolution type kernel and, in [12], the convection partial differential equation. In [20], the Poisson partial differential equation was studied via the double Laplace transform method. We intend to further study other equations.