Hyers–Ulam stability of second-order differential equations using Mahgoub transform

The aim of this research is investigating the Hyers–Ulam stability of second-order differential equations. We introduce a new method of investigation for the stability of differential equations by using the Mahgoub transform. This is the first attempt of the investigation of Hyers–Ulam stability by using Mahgoub transform. We deal with both homogeneous and nonhomogeneous second-order differential equations.


Preliminaries
In this paper, F denotes the real field R or complex field C. A function f : (0, ∞) → F is said to be of exponential order if there exist constants A, B ∈ R such that |f (t)| ≤ Ae tB for all t > 0.
For each function f : (0, ∞) → F of exponential order, consider the set where the constant M is finite, whereas k 1 and k 2 may be infinite. The Mahgoub transform is defined by where the variable v in the Mahgoub transform is used to factor the variable t in the argument of the function f , especially, for f ∈ A.
Definition 2.1 (Convolution of two functions) Let f and g be Lebesgue-integrable functions on (-∞, +∞). Let S denote the set of x for which the Lebesgue integral exists. This integral defines the function h on S called the convolution of f and g and denoted by h = f * g.
Now we give some definitions related to the Hyers-Ulam stability of the differential equations (1.1) and (1.2).
Let I, J ⊆ R be intervals. We denote the space of k continuously differentiable functions from I to J by C k (I, J) and denote C k (I, I) by C k (I). Further, C(I, J) = C 0 (I, J) denotes the space of continuous functions from I to J. In addition, R + := [0, ∞). From now on, we assume that I = [τ 1 , τ 2 ], where -∞ < τ 1 < τ 2 < ∞.

Definition 2.2
We say that the differential equation (1.1) has the Hyers-Ulam stability if there exists a constant L > 0 satisfying the following condition: If for every > 0, there exists x ∈ C 2 (I) satisfying the inequality for all t ∈ I, then there exists a solution y ∈ C 2 (I) satisfying the differential equation y (t) + μ 2 y(t) = 0 such that for all t ∈ I. We call such L the Hyers-Ulam stability constant for (1.1).

Definition 2.3
We say that the differential equation (1.1) has the generalized Hyers-Ulam stability with respect to φ ∈ C(R + , R + ) if there exists a constant L φ > 0 with the following property: If for every > 0, there exists x ∈ C 2 (I) satisfying the inequality for all t ∈ I, then there exists a solution y ∈ C 2 (I) satisfying the differential equation y (t) + μ 2 y(t) = 0 such that for all t ∈ I. We call such L the generalized Hyers-Ulam stability constant for (1.1).

Definition 2.4
We say that the differential equation (1.2) has the Hyers-Ulam stability if there exists a constant L > 0 satisfying the following condition: If for every > 0, there exists x ∈ C 2 (I) satisfying the inequality for all t ∈ I, then there exists y ∈ C 2 (I) satisfying y (t) + μ 2 y(t) = q(t) such that for all t ∈ I. We call such L the Hyers-Ulam stability constant for (1.2).

Definition 2.5
We say that the differential equation (1.2) has the generalized Hyers-Ulam stability with respect to φ ∈ C(R + , R + ) if there exists a constant L φ > 0 such that for every > 0 and for each solution x ∈ C 2 (I) satisfying the inequality for all t ∈ I, there exists y ∈ C 2 (I) satisfying the differential equation such that for all t ∈ I. We call such L the generalized Hyers-Ulam stability constant for (1.2).

Hyers-Ulam stability for (1.1)
In this section, we prove the Hyers-Ulam stability and generalized Hyers-Ulam stability of the differential equation (1.1) by using the Mahgoub transform.
Proof Let > 0. Suppose that x ∈ C 2 (I) satisfies for all t ∈ I. We will prove that there exists a real number L > 0 such that and thus Then we have y(0) = x(0) and y (0) = x (0). Taking the Mahgoub transform of y, we obtain On the other hand, Since M is a one-to-one linear operator, we have y (t) + μ 2 y(t) = 0. This means that y is a solution of (1.1). It follows from (3.3) and (3.4) that , These equalities show that Taking the modulus on both sides and using |p(t)| ≤ , we get which exists. Hence |x(t)-y(t)| ≤ L . By Definition 2.2 the linear differential equation (1.1) has the Hyers-Ulam stability. This finishes the proof.
By using the same technique as in Theorem 3.1, we can also prove the following theorem, which shows the generalized Hyers-Ulam stability of the differential equation (1.1). The method of the proof is similar, but we include it for completeness.

Hyers-Ulam stability for (1.2)
In this section, we investigate the Hyers-Ulam stability and generalized Hyers-Ulam stability of the differential equation (1.2). Firstly, we prove the Hyers-Ulam stability of the nonhomogeneous linear differential equation (1.2).

Theorem 4.1 The differential equation (1.2) has the Hyers-Ulam stability.
Proof For every > 0 and for each solution x ∈ C 2 (I) satisfying for all t ∈ I, we will prove that there exists L > 0 such that |x(t)y(t)| ≤ L for some y ∈ C 2 (I) satisfying y (t) + μ 2 y(t) = q(t) for all t ∈ I. The function p : (0, ∞) → R defined by p(t) =: x (t) + μ 2 x(t)q(t) satisfies |p(t)| ≤ . Taking the Mahgoub transform of p, we have Equality (4.2) shows that a function x 0 : (0, ∞) → F is a solution of (1.2) if and only if If there exist constants a and b in F such that v 2 + μ 2 = (va)(vb) with a + b = 0 and ab = μ 2 , then (4.2) becomes Set r(t) = e at -e bt a-b and y(t) = x(0)( le lt -me mt l-m ) + x (0)r(t) + [(r * q)(t)]. So, y(0) = x(0) and y (0) = x (0). Once more, taking the Mahgoub transform of y, we have On the other hand, M{y (t) + μ 2 y(t)} = (v 2 + μ 2 )M{y(t)}v 3 y(0)v 2 y (0). Using (4.4), the last equality becomes Since M is a one-to-one linear operator, we have y (t) + μ 2 y(t) = q(t), which shows that y is a solution of (1.2). Now relations (4.3) and (4.4) necessitate that and hence xy = p * r. Taking the modulus of both sides of the last equality and using |p(t)| ≤ , we get which exists for all t > 0. Therefore the linear differential equation (1.2) has the Hyers-Ulam stability.
Analogously to Theorem 4.1, we have the following result, which shows the generalized Hyers-Ulam stability of the differential equation (1.2).

Conclusion
In this paper, we first initiated and proposed a new method for the investigation of Hyers-Ulam stability of differential equations by using the Mahgoub transform. Also, using this new idea, we investigated the Hyers-Ulam stability of second-order homogeneous and nonhomogeneous differential equations by using the Mahgoub transform.