MITTAG-LEFFLER-HYERS-ULAM STABILITY OF A LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER USING LAPLACE TRANSFORM

1Department of Mathematics, Kings Engineering College, Irungattukottai, Sri Perumbudur, Chennai – 602 117, Tamil Nadu, India 2Department of Mathematics, Sri Sankara Arts & Science College, Enathur, Kanchipuram – 631 561, Tamil Nadu, India 3Department of Mathematics, Panimalar Engineering College, Chennai, Tamil Nadu, India 4Department of Mathematics, Sree Abiraami Arts and Science College for Women, Gudiyattam, Vellore Dt. 635 803, Tamil Nadu, India


INTRODUCTION
A simulating and famous talk presented by Ulam [38] in 1940, motivated the study of stability problems for various functional equations. He gave a wide range of talk before a Mathematical Colloquium at the University of Wisconsin in which he presented a list of unsolved problems.
One of his question was that when is it true that a mapping that approximately satisfies a functional equation must be close to an exact solution of the equation? If the answer is affirmative, we say that the functional equation for homomorphisms is stable. In 1941, Hyers [8] was the first Mathematician to present the result concerning the stability of functional equations. He brilliantly answered the question of Ulam, the problem for the case of approximately additive mappings on Banach spaces. In course of time, the Theorem formulated by Hyers was generalized by Rassias [34], Aoki [3] and Bourgin [4] for additive mappings (see also [32,39]).
The generalization of Ulam's question has been relatively recently proposed by replacing functional equations with differential equations: Let I be a subinterval of R, let K denote either R or C, and let n be a positive integer. The differential equation ψ f , x, x , x , . . . , x (n) = 0 has the Hyers-Ulam stability if there exists a constant K > 0 such that the following statement is true for any ε > 0: If an n times continuously differentiable function z : I → K satisfies the inequality ψ f , z, z , z , . . . , z (n) ≤ ε for all t ∈ I, then there exists a solution y : I → K of the differential equation that satisfies the inequality |z(t) − y(t)| ≤ Kε for all t ∈ I.
Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [28,29]). Then, in 1998, Alsina and Ger continued the study of Obłoza's Hyers-Ulam stability of differential equations. Indeed, they proved in [2] the following theorem.
0 be an open subinterval of R. If a differentiable function x : I → R satisfies the differential inequality x (t) − x(t) ≤ ε for any t ∈ I and for some ε > 0, then there exists a differentiable function y : I → R satisfying y (t) = y(t) and x(t) − y(t) ≤ 3ε for any t ∈ I.
This result of Alsina and Ger has been generalized by Takahashi et al. They proved in [37] that the Hyers-Ulam stability holds true for the Banach space valued differential equation x (t) = λ x(t). Indeed, the Hyers-Ulam stability has been proved for the first-order linear differential equations in more general settings (see [9,10,11,12,17]).
In 2006, Jung [12] investigated the Hyers-Ulam stability of a system of first-order linear differential equations with constant coefficients by using matrix method. Then, in 2008, Wang et al. [40] studied the Hyers-Ulam stability of linear differential equations of first order using the integral factor method. Meanwhile, Rus [36] discussed various types of Hyers-Ulam stability of the ordinary differential equation x (t) = Ax(t) + f (t, x(t)). In 2014, Alqifiary and Jung [1] proved the generalized Hyers-Ulam stability of linear differential equation of the form by using the Laplace transform method, where α k are scalars and x(t) is an n times continuously differentiable function and of the exponential order (see also [35]).
In recent years, many authors are studying the Hyers-Ulam stability of differential equations, and a number of mathematicians are paying attention to the new results of the Hyers-Ulam stability of differential equations (see [5,6,16,19,20,24,30,31]).
Note that, in particular, during these days most of the mathematicians are studied only the Hyers-Ulam stability of the second order differential equations by various directions (See [7,14,15,18,21,22]).
In recent days, few authors have investigated the Ulam stability of the linear differential equations using various integral transform techniques, like, Fourier transform, Mahgoub transform and Aboodh transform in [21,25,26,27,33].
Based on the above discussions, by applying the Laplace Transform Method, we study the Mittag-Leffler-Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam-Rassias stability of a general homogeneous and non-homogeneous linear differential equations of second order and for all t ∈ I, l, m are constants in F, u(t) ∈ C 2 (I) and r(t) ∈ C(I) where I = [a, b] ⊂ R.

PRELIMINARIES
In this section, we introduce some notations, definitions and preliminaries which are used throughout this paper.
Throughout this paper, F denotes the real field R or the complex field C. A function f : (0, ∞) → F of exponential order if there exists a constants M(> 0) ∈ R such that | f (t)| ≤ Me at for all t > 0. For each function f : (0, ∞) → F of exponential order, we define the Laplace Transform of f by The Laplace transform of f is sometimes denoted by L ( f ). It is also well known that L is linear and one-to-one. Then, at points of continuity of f , we have this is called the inverse Laplace transforms.
exists. This integral defines a function h on S called the convolution of f and g. We also write h = f * g to denote this function.
Theorem 3. The Laplace transform of the convolution of f (x) and g(x) is the product of the Laplace transform of f (x) and g(x). That is, where F(s) and G(s) are Laplace transform of f (x) and g(x) respectively.

Definition 4. [13]
The Mittag-Leffler function of one parameter is denoted by E α (z) and defined Let I, J ⊆ R. Throughout this paper, 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). Furthermore, 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 firstly give some definitions of various forms of Mittag-Leffler-Hyers-Ulam stability of the second order differential equations (1) and (2).
Definition 6. We say that the differential equation (1) has the Mittag-Leffler-Hyers-Ulam stability, if there exists a positive constant K satisfies the following conditions: For every ε > 0 and there exists u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I. Then there exists a solution v ∈ C 2 (I) , for all t ∈ I. We call such K as the Mittag-Leffler-Hyers-Ulam stability constant for (1).

Definition 7.
We say that the differential equation (2) has the Mittag-Leffler-Hyers-Ulam stability, if there exists a positive constant K satisfies the following conditions: For every ε > 0 and there exists u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I. Then there exists a solution v ∈ C 2 (I) satisfying the linear differential equation , for all t ∈ I. We call such K as the Mittag-Leffler-Hyers-Ulam stability constant for (2).
Definition 8. We say that the differential equation (1) has the Mittag-Leffler-Hyers-Ulam- , if there exists a positive constant K satisfies the following conditions: For every ε > 0 and there exists u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I. We call such K as the Mittag-Leffler-Hyers-Ulam-Rassias stability constant for (1).
Definition 9. We say that the differential equation (2) for all t ∈ I. Then there exists a solution v ∈ C 2 (I) satisfies the linear differential equation , for all t ∈ I. We call such K as the Mittag-Leffler-Hyers-Ulam-Rassias stability constant for (2).
Proof. Given ε > 0. Suppose that u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I. We wish to prove that there exists real number K > 0 which is independent of ε and u such that |u(t) − v(t)| ≤ KεE α (t), for some v ∈ C 2 (I) satisfies the differential equation v (t) + l v (t) + m v(t) = 0 for all t ∈ I. Define a function p : (0, ∞) → R such that for all t > 0. In view of (3), we have |p(t)| ≤ εE α (t). Taking Laplace transform to p(t), then and thus In view of the (4) On the other hand, Using (7), we get L {v (t) + l v (t) + m v(t)} = 0. Since L is one-to-one operator and linear, then we get v (t) + l v (t) + m v(t) = 0. This means that v(t) is a solution of (1). It follows from (5) and (7) that The above equalities show that Taking modulus on both sides and using |p(t)| ≤ εE α (t), we get for all t > 0, where This finishes the proof.
By applying the same idea as Theorem 10, we can prove the following corollary which shows the Hyers-Ulam stability of the differential equation (1).
Corollary 11. Given ε > 0. Suppose that u(t) ∈ C 2 (I) satisfying the inequality u (t) + l u (t) + m u(t) ≤ ε for all t ∈ I. Then there exists real number K > 0 which is independent of ε and u such that Proof. Substituting εE α (t) as ε in the inequality (3) and by applying same methodology of the Theorem 10, we can arrive that the corresponding differential equation (1) has the Hyers-Ulam stability.
By using the same technique in Theorem 10, we can also prove that the following theorem which shows the Mittag-Leffler-Hyers-Ulam-Rassias stability of the differential equation (1).
The method of proof is similar, but we include it for the sake of completeness. Proof. Given ε > 0. Suppose that u(t) ∈ C 2 (I) and φ (t) : (0, ∞) → (0, ∞) satisfying for all t ∈ I. We wish to prove that there exists real number K > 0 which is independent of ε and u such that for all t ∈ I. Define a function p : (0, ∞) → R such that p(t) =: u (t) + l u (t) + m u(t) for all t > 0. In view of (8), we have |p(t)| ≤ φ (t)εE α (t). Taking Laplace transform to p(t), we have On the other hand, Using (12), we get L {v (t) + l v (t) + m v(t)} = 0. Since L is one-to-one operator and linear, then we get v (t) + l v (t) + m v(t) = 0. This means that v(t) is a solution of (1). It follows from (10) and (12) that The above equalities show that u(t) − v(t) = p(t) * q(t). Taking modulus on both sides and using for all t > 0, where where the integrals Hence, by the virtue of Definition 8, the linear differential equation (1) has the Mittag-Leffler-Hyers-Ulam-Rassias stability. This completes the proof.
By using the same technique as applied in the Theorem 12, we can prove the following corollary which shows the Hyers-Ulam-Rassias stability of the differential equation (1).
Corollary 13. Given ε > 0, there exists real number K > 0 which is independent of ε and u such that u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I. Then for some v ∈ C 2 (I) satisfies the differential equation Proof. Setting φ (t)εE α (t) as εφ (t) in the inequality (8) and by applying same terminology as used in the Theorem 12, we can easily prove that the differential equation (1) has the Hyers-Ulam-Rassias stability.
Proof. Given ε > 0. Suppose that a twice continuously differentiable function u(t) satisfying the inequality for all t ∈ I. We have to prove that there exists real number K > 0 which is independent of ε and for all t ∈ I. Define a function p : (0, ∞) → R such that p(t) =: u (t) + l u (t) + m u(t) − r(t) for all t > 0. In view of (13), we have |p(t)| ≤ εE α (t). Taking Laplace transform to p(t), we have and thus In view of the (14), a function u 0 : (0, ∞) −→ R is a solution of (2) if and only if Assume that l and m are constants in F such that there exists µ and ν are in F with µ + ν = −l, µν = m and µ = ν. Then we have (s 2 + l s + m) = (s − µ) (s − ν), (15) implies that where q(t) = e µt − e νt µ − ν then we have v(0) = u(0) and v (0) = u (0). Taking Laplace transform to v(t), we obtain On the other hand, Using (17), we get L {v (t) + l v (t) + m v(t) − r(t)} = 0. Since L is one-to-one operator and linear, then we get v (t) + l v (t) + m v(t) = r(t). This means that v(t) is a solution of (2). It follows from (15) and (17) that The above equalities show that Taking modulus on both sides and using |p(t)| ≤ εE α (t), we get where the integrals t 0 e −ℜ(µ)x dx and t 0 e −ℜ(ν)x dx exists. Therefore, Then by the virtue of Definition 7, the linear differential equation (2) has the Mittag-Leffler-Hyers-Ulam stability. This completes the proof.
By using the methodology as applied in Theorem 14, we can establish the following corollary which shows the Hyers-Ulam stability of the non-homogeneous differential equation (2).
Corollary 15. Let ε > 0 and for each u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I. Then there exists real number K > 0 which is independent of ε and u such that r(t) for all t ∈ I.
Proof. Replacing εE α (t) as ε in the inequality (13) and by using same terminology of the Theorem 14, we can prove that the differential equation (2) has the Hyers-Ulam stability.
In analogous to Theorem 14, we also have the following result which shows the Mittag-Leffler-Hyers-Ulam-Rassias stability of the differential equation (2).
Corollary 17. For every ε > 0, there exists positive real number K which is independent of ε and u such that u(t) ∈ C 2 (I) satisfying the inequality u (t) + l u (t) + m u(t) ≤ εφ (t) for all t ∈ I. Then for some v ∈ C 2 (I) satisfies the differential equation such that |u(t) − v(t)| ≤ Kεφ (t), for all t ∈ I.
Proof. Setting φ (t)εE α (t) as εφ (t) in the inequality (18) and by applying same terminology as used in the Theorem 16, we can easily prove that the differential equation (2) has the Hyers-Ulam-Rassias stability.