AIJRSTEM19-

American International Journal of Research in Science, Technology, Engineering & Mathematics Available online at http://www.iasir.net ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 2328-3629 AIJRSTEM is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research) Hyers-Ulam Stability of nth Order System of Differential Equation R. Murali and A. Ponmana Selvan PG and Research Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur-635 601, Vellore Dist. Tamil Nadu, India. Abstract— In this paper, we are going to study the Hyers-Ulam stability, Generalised Hyers-Ulam stability, Hyers-Ulam-Rassias stability and Generalised Hyers-Ulam-Rassias stability of the system of nth order differential equation of the form 𝒙(𝒏) (𝒕) = 𝒇 (𝒕, 𝒙(𝒕), 𝒙′ (𝒕), 𝒙′′ (𝒕), … , 𝒙(𝒏−𝟏) (𝒕)) with initial conditions 𝒙(𝒂) = 𝒙𝟎 , 𝒙′ (𝒂) = 𝒙𝟏 , 𝒙′′ (𝒂) = 𝒙𝟐 , … . , 𝒙(𝒏−𝟏) (𝒕) = 𝒙𝒏−𝟏 in Banach spaces. Keywords—Hyers-Ulam stability; Generalised Hyers-Ulam stability; Hyers-Ulam-Rassias stability; Generalised Hyers-Ulam-Rassias stability; system of differential equations; Initial conditions I. INTRODUCTION In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions. Only the simplest differential equations are solvable by explicit formulas. However, some properties of solutions of a given differential equation may be determined without finding their exact form. Equilibria are not always stable. Since stable and unstable equilibrium play quite different roles in the dynamics of a system, it is useful to be able to classify equilibrium points based on their stability. Definition of Hyers-Ulam stability and Hyers-Ulam-Rassias stability have applicable significance since it means that if one is studying the Hyers-Ulam stable system then one does not have to reach the exact solution. (Which usually is quite difficult or time consuming). This is quite useful in many applications, for example Numerical Analysis, Optimization, Biology, and Economics etc., where finding the exact solution is quite difficult. The stability problem for various functional equations was originated from a famous talk of S.M. Ulam [20]. In 1940, Ulam was raised the question: Suppose one has a function y(t) which is close to solve an equation. Is there an exact solution x(t) of the equation which is close to y(t) ? (See [4, 11]). In 1941, D.H. Hyers [4] gave an affirmative answer to the equation of Ulam for additive Cauchy equation in Banach Spaces. The result of Hyers was generalized by T. Aoki [22] and Th. M. Rassias [16]. After that many Mathematicians have extended the Ulam’s problem in various directions [see for example 3, 12, 15]. A generalization of Ulam's problem was recently proposed by replacing functional equations with differential equations: The differential equation 𝜑 (𝑓, 𝑥, 𝑥 ′ , 𝑥 ′′ , … 𝑥 (𝑛) ) = 0has the Hyers-Ulam stability, if for a given 𝜀 > 0 and a function 𝑥 such that |𝜑 (𝑓, 𝑥, 𝑥 ′ , 𝑥 ′′ , … 𝑥 (𝑛) )| ≤ 𝜀 , there exists a solution 𝑥𝑎 of the differential equation such that‖𝑥(𝑡) − 𝑥𝑎 (𝑡)‖ ≤ 𝐾(𝜀), and lim 𝐾(𝜀) = 0 . 𝑛→∞ If the preceding statement is also true when we replace 𝜀 and 𝐾(𝜀) by 𝜑(𝑡) and 𝜙(𝑡), where 𝜙 , 𝜑 are appropriate functions not depending on 𝑥 and 𝑥𝑎 explicitly, then we say that the corresponding differential equation has the generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability. Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations [13, 14]. Thereafter, in 1998, C. Alsina and R. Ger [1] were the first authors who investigated the Hyers-Ulam stability of differential equations. They proved in [1] the following Theorem. Theorem 1.1. Assume that a differentiable function 𝑓 ∶ 𝐼 → ℝ is a solution of the differential inequality ‖𝑥 ′ (𝑡) − 𝑥(𝑡)‖ ≤ 𝜀 . Where 𝐼 is an open sub interval of ℝ . Then there exists a solution 𝑔 ∶ 𝐼 → ℝ of the differential equation 𝑥 ′ (𝑡) = 𝑥(𝑡) such that for any 𝑡 ∈ 𝐼 , we have, ‖𝑓(𝑡) − 𝑔(𝑡)‖ ≤ 3 𝜀 . This result of C. Alsina and R. Ger [1] has been generalized by Takahasi [12]. They proved in [12] that the Hyers - Ulam stability holds true for the Banach Space valued differential equation 𝑥 ′ (𝑡) = 𝜆 𝑥(𝑡) . After that, many Mathematicians have extended Ulam’s problem in various direction (See [2, 4, 5, 8, 10]). AIJRSTEM 19-212; © 2019, AIJRSTEM All Rights Reserved Page 71 Murali et al., American International Journal of Research in Science, Technology, Engineering & Mathematics,26(1), March-May 2019, pp. 71-75 Those previous results were extended to the Hyers-Ulam stability of linear differential equation of first order and higher order with constant coefficients in [6, 7, 8, 9, 10, 19, 23] and in [2], [11], [12], [13], [14], [17], [18], [21] and [26-29] respectively. Furthermore, Jung has proved the Hyers-Ulam stability of linear differential equations (See [5-9]). Rus investigated the Hyers-Ulam stability of linear differential equation and integral equations using the Gronwall’s lemma and the technique of weakly Picard’s operators (See [24], [25]). In 2014, Q.H. Alqifiary and J.K. Miljanovic [30] are examine the relation between practical stability and Hyers-Ulam-stability and Hyers-Ulam-Rassias stability as well. In addition, by practical stability we gave a sufficient condition in order that the first order nonlinear Systems of differential equations has local generalized Hyers-Ulam stability and local generalized Hyers-Ulam-Rassias stability. Recently, R. Murali and A. PonmanaSelvan [31, 32] are established the Hyers-Ulam stability, Generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability and Generalised Hyers-Ulam-Rassias stability of the system of second orderdifferential equation of the form 𝑥′′(𝑡) = 𝑓(𝑡, 𝑥(𝑡), 𝑥′(𝑡)) with initial conditions 𝑥(𝑎) = 𝑥0 𝑎𝑛𝑑 𝑥′(𝑎) = 𝑥1 and third order differential equation 𝑥′′′(𝑡) = 𝑓(𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡)) with initial conditions 𝑥(𝑎) = 𝑥0 , 𝑥 ′ (𝑎) = 𝑥1 𝑎𝑛𝑑 𝑥′′(𝑎) = 𝑥2 in Banach spaces. Encouraged by the above ideas, the purpose of this paper is to study the Hyers-Ulam stability, Generalised Hyers-Ulam stability, Hyers-Ulam-Rassias stability and Generalized Hyers-Ulam-Rassias stability of the system of nth order differential equation of the form 𝑥 (𝑛) (𝑡) = 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡)) (1.1) in Banach space with initial conditions 𝑥(𝑎) = 𝑥0 , 𝑥 ′ (𝑎) = 𝑥1 , 𝑥 ′′ (𝑎) = 𝑥2 , … . , 𝑥 (𝑛−1) (𝑡) = 𝑥𝑛−1 (1.2) 𝑛 for all 𝑡 ∈ 𝐼, 𝑥(𝑡) ∈ 𝐶 (𝐼, 𝔹) and where 𝐼 = [𝑎, 𝑏), 𝑎 ∈ ℝ, 𝑏 ∈ ℝ, −∞ < 𝑎 < 𝑏 < +∞. II. PRELIMINARIES Let (𝔹, ‖. ‖) be a Banach Space (Real or Complex), and 𝐼 = [𝑎, 𝑏) , 𝑎 ∈ ℝ , 𝑏 ∈ ℝ, 𝑎 < 𝑏 ≤ +∞ ,𝜀 be a positive real number, 𝑓: 𝐼 × 𝔹n → 𝔹 continuous operator and 𝜙: 𝐼 → ℝ+ be a continuous function. Let us consider the nth order system of differential equation𝑥 (𝑛) (𝑡) = 𝑓(𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡)) for all 𝑡 ∈ 𝐼, where 𝑓 is defined and continuous on 𝐼 × 𝔹n . Let 𝐺 be a closed and bounded set of 𝔹 containing the origin then there exists a real number 𝑀 > 0 such that 𝐺 = {𝑥: ‖𝑥‖ ≤ 𝑀} and let 𝐺0 be a subset of 𝐺. Definition 2.1 Let 𝑥 ∗ (𝑡, 𝑥0 , 𝑥1 , 𝑥2 , … , 𝑥𝑛−1 ) be the solution of the system (1.1) satisfying the intial conditios: 𝑥(𝑎) = 𝑥0 , 𝑥 ′ (𝑎) = 𝑥1 , 𝑥 ′′ (𝑎) = 𝑥2 , … . , 𝑥 (𝑛−1) (𝑡) = 𝑥𝑛−1 If for every 𝜀 > 0, 𝑥0 , 𝑥1 , 𝑥2 , … , 𝑥𝑛−1 ∈ 𝐺0 and each 𝑥 ∗ (𝑡, 𝑥0 , 𝑥1 , 𝑥2 , … , 𝑥𝑛−1 ) is in 𝐺 for all 𝑡 ∈ 𝐼, then the origin is said to be (𝐺0 , 𝐺, 𝜀)-practically stable. The solution which starts initially in 𝐺0 remain thereafter in 𝐺. Definition 2.2 We say that the system of nth order differential equation (1.1) has the Hyers-Ulam stability with initial condition (1.2), if there exists a constant 𝑆𝑓 > 0 such that for every 𝜀 > 0 and for each solution 𝑥(𝑡) ∈ 𝐶 𝑛 (𝐼, 𝔹) satisfying the inequality ‖𝑥 (𝑛) (𝑡) − 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡))‖ ≤ 𝜀 for all 𝑡 ∈ 𝐼. Then there exists some 𝑦 ∈ 𝐶 𝑛 (𝐼, 𝔹) satisfies the differential equation (1.1) with initial conditions (1.2) such that ‖𝑥(𝑡) − 𝑦(𝑡)‖ ≤ 𝑆𝑓 𝜀, for all 𝑡 ∈ 𝐼. We call such 𝑆 as the Hyers-Ulam stability constant for (1.1) with (1.2). Definition 2.3 We say that the differential equation (1.1) has the Hyers-Ulam-Rassias stability with initial condition (1.2) with respect to 𝜙 if there exists a constant 𝑆𝑓,𝜙 > 0 such that for every 𝜀 > 0 and for each solution 𝑥(𝑡) ∈ 𝐶 n (𝐼, 𝔹) satisfies the inequality ‖𝑥 (𝑛) (𝑡) − 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡))‖ ≤ 𝜀𝜙(𝑡), for all 𝑡 ∈ 𝐼. Then there exists some 𝑦 ∈ 𝐶 n (𝐼, 𝔹) satisfies the differential equation (1.1) with initial conditions (1.2) such that ‖𝑥(𝑡) − 𝑦(𝑡)‖ ≤ 𝑆𝑓,𝜙 𝜀𝜙(𝑡), for all 𝑡 ∈ 𝐼. We call such 𝑆 as the Hyers-Ulam-Rassias stability constant for (1.1) with initial condition (1.2). Definition 2.4 The differential equation (1.1) is said to have the Generalized Hyers-Ulam stability with initial condition (1.2), if there exists 𝜃𝑓 ∈ 𝐶(ℝ+ , ℝ+ ) with 𝜃𝑓 (0) = 0 such that 𝑥 ∈ 𝐶 n (𝐼, 𝔹) satisfying the inequality ‖𝑥 (𝑛) (𝑡) − 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡))‖ ≤ 𝜀, for all 𝑡 ∈ 𝐼. Then there exists some 𝑦 ∈ 𝐶 𝑛 (𝐼, 𝔹) satisfies the differential equation (1.1) with initial conditions AIJRSTEM 19-212; © 2019, AIJRSTEM All Rights Reserved Page 72 Murali et al., American International Journal of Research in Science, Technology, Engineering & Mathematics,26(1), March-May 2019, pp. 71-75 (1.2) such that ‖𝑥(𝑡) − 𝑦(𝑡)‖ ≤ 𝜃𝑓 (𝜀), for all 𝑡 ∈ 𝐼. Definition 2.5 The differential equation (1.1) is said to have the Generalized Hyers-Ulam-Rassias stability with initial condition (1.2) with respect to 𝜙 if there exists a constant 𝑆𝑓,𝜙 > 0 such that for every 𝜀 > 0 and for each solution 𝑥(𝑡) ∈ 𝐶 n (𝐼, 𝔹) satisfies the inequality ‖𝑥 (𝑛) (𝑡) − 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡))‖ ≤ 𝜙(𝑡), for all 𝑡 ∈ 𝐼. Then there exists some 𝑦 ∈ 𝐶 𝑛 (𝐼, 𝔹) satisfies the differential equation (1.1) with initial conditions (1.2) such that ‖𝑥(𝑡) − 𝑦(𝑡)‖ ≤ 𝑆𝑓,𝜙 𝜙(𝑡), for all 𝑡 ∈ 𝐼. We call such 𝑆 as the Generalized Hyers-Ulam-Rassias stability constant for (1.1) with initial condition (1.2). Remark 2.6 Consider the system of differential equation (1.1) with initial condition 𝑥(𝑎) = 𝑥0 ∈ 𝐺0 , 𝑥 ′ (𝑎) = 𝑥1 ∈ 𝐺0 , 𝑥 ′′ (𝑎) = 𝑥2 ∈ 𝐺0 , … . , 𝑥 (𝑛−1) (𝑡) = 𝑥𝑛−1 ∈ 𝐺0 (2.1) where 𝑓 is defined and continuous on 𝐼 × 𝔹 and equilibrium state is at the origin: 𝑓(𝑡, 0,0,0, … , 0) = 0, for all 𝑡 ∈ 𝐼 . The system (1.1) to be (𝐺0 , 𝐺, 𝜀)-practically stable it is sufficient that there exists a continuous non increasing on the system (1.1) solutions Lyapunov function 𝑉(𝑥, 𝑡) such that ℘ = {𝑥 ∈ 𝔹: 𝑉 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡)) ≤ 1} ⊆ 𝐺, 𝑡 ∈ 𝐼, (2.2) 𝐺0 ⊆ {𝑥 ∈ 𝔹: 𝑉 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡)) ≤ 1}. (2.3) Proof. We prove this remark by contradiction. Suppose that the conditions (2.2) and (2.3) are satisfied but there are 𝜇 ∈ 𝐼 and 𝑥0 , 𝑥1 , 𝑥2 , … , 𝑥𝑛−1 ∈ 𝐺0 such that the solution 𝑥(𝑡) = 𝑥(𝑎, 𝑥0 , 𝑥1 , 𝑥2 , … , 𝑥𝑛−1 ) of (1.1) leaves the set 𝐺 . From (2.2) follows the inequality 𝑉 (𝜇, 𝑥(𝜇), 𝑥 ′ (𝜇), 𝑥 ′′ (𝜇), … , 𝑥 (n−1) (𝜇)) > 1 which contradicts the condition (2.3). Hence the equilibrium of the system (1.1) is (𝐺0 , 𝐺, 𝜀)-practically stable. III. MAIN RESULTS In this section, we prove the Hyers-Ulam stability, Generalised Hyers-Ulam stability, Hyers-Ulam-Rassias stability and Generalised Hyers-Ulam-Rassias stability of the differential equation (1.1) with (1.2). Firstly, we prove the Hyers-Ulam stability of (1.1) with (1.2). Theorem 3.1 Assume that there exists a constant 𝑆𝑓 > 0 such that for every 𝜀 > 0 and for each solution 𝑥(𝑡) ∈ 𝐶 n (𝐼, 𝔹) satisfying the inequality ‖𝑥 (𝑛) (𝑡) − 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡))‖ ≤ 𝜀 with initial conditions (1.2) for all 𝑡 ∈ 𝐼. Then there exists some 𝑦 ∈ 𝐶 n (𝐼, 𝔹) satisfies (1.1) with initial condition 𝑦(𝑎) = 𝑦0 , 𝑦 ′ (𝑎) = 𝑦1 , 𝑦 ′′ (𝑎) = 𝑦2 , … , and 𝑦 (𝑛−1) (𝑎) = 𝑦𝑛−1 such that ‖𝑥(𝑡) − 𝑦(𝑡)‖ ≤ 𝑆𝑓 𝜀, ∀ 𝑡 ∈ 𝐼. Proof. Given that for every 𝜀 > 0, and for each solution 𝑥(𝑡) ∈ 𝐶 𝑛 (𝐼, 𝔹) satisfying ‖𝑥 (𝑛) (𝑡) − 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡))‖ ≤ 𝜀, (3.1) for all 𝑡 ∈ 𝐼. Now, we are going to prove that there exists a real number 𝑆𝑓 > 0 and for some 𝑦 in 𝐶 𝑛 (𝐼, 𝔹) satisfies the inequality (1.1) with𝑦(𝑎) = 𝑦0 , 𝑦 ′ (𝑎) = 𝑦1 , 𝑦 ′′ (𝑎) = 𝑦2 , … , and 𝑦 (𝑛−1) (𝑎) = 𝑦𝑛−1 such that ‖𝑥(𝑡) − 𝑦(𝑡)‖ ≤ 𝑆𝑓 𝜀, for all 𝑡 ∈ 𝐼. Let 𝐺 be a closed and bounded set, then there exists a real number 𝑀 > 0 such that 𝐺 = {𝑥: ‖𝑥‖ ≤ 𝑀}. Now, let 𝑥 ∗ = 𝑓(𝑎, 𝑥0 , 𝑥1 , 𝑥2 , … , 𝑥𝑛−1 ) satisfies the inequality (3.1) for arbitrary 𝜀, then 𝑥 ∗ be the solution of the differential equation (1.1) with (1.2). Since the equilibrium of (1.1) is (𝐺0 , 𝐺, 𝜀)-practically stable, then 𝑥 ∗ ∈ 𝐺. Hence ‖𝑥 ∗ ‖ ≤ 𝑀, since 𝑀 > 0, and 𝜀 > 0 then there exists 𝑆𝑓 > 0 such that 𝑀 = 𝑆𝑓 𝜀. Then we have ‖𝑥 ∗ ‖ ≤ 𝑆𝑓 𝜀, AIJRSTEM 19-212; © 2019, AIJRSTEM All Rights Reserved Page 73 Murali et al., American International Journal of Research in Science, Technology, Engineering & Mathematics,26(1), March-May 2019, pp. 71-75 for all 𝑡 ∈ 𝐼. Obviously, 𝑦(𝑡) ≡ 0 be the solution of (1.1) with (1.2) such that ‖𝑥 ∗ (𝑡) − 𝑦(𝑡)‖ ≤ 𝑆𝑓 𝜀, for all 𝑡 ∈ 𝐼. Hence by the virtue of Definition 2.2, the system of nth order differential equation has the HyersUlam stability. The following corollary shows the Generalized Hyers-Ulam stability of the system (1.1) with (1.2). Corollary 3.2 Assume that there exists 𝜃𝑓 > 0, for every 𝜀 > 0 and for each solution 𝑥(𝑡) ∈ 𝐶 n (𝐼, 𝔹) satisfies the inequality ‖𝑥 (𝑛) (𝑡) − 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡))‖ ≤ 𝜀, with initial condition 𝑥(𝑎) = 𝑥0 , 𝑥 ′ (𝑎) = 𝑥1 , 𝑥 ′′ (𝑎) = 𝑥2 , … . , 𝑥 (𝑛−1) (𝑡) = 𝑥𝑛−1 for all 𝑡 ∈ 𝐼. Then there exists some 𝑦 ∈ 𝐶 𝑛 (𝐼, 𝔹) satisfies the differential equation (1.1) with initial condition 𝑦(𝑎) = 𝑦0 , 𝑦 ′ (𝑎) = 𝑦1 , 𝑦 ′′ (𝑎) = 𝑦2 , … , and 𝑦 (𝑛−1) (𝑎) = 𝑦𝑛−1 such that‖𝑥(𝑡) − 𝑦(𝑡)‖ ≤ 𝜃𝑓 (𝜀), for all 𝑡 ∈ 𝐼. Proof. We can prove this corollary by the similar way as in the proof of Theorem 3.1. Next, we study the Hyers-Ulam-Rassias stability of the system (1.1) with initial condition (1.2). Theorem 3.3 Assume that there exists a constant 𝑆𝑓,𝜙 > 0 such that for every 𝜀 ∈ (0, 𝜀0 ] and for each solution 𝑥(𝑡) ∈ 𝐶 𝑛 (𝐼, 𝔹) satisfying the inequality ‖𝑥 (𝑛) (𝑡) − 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡))‖ ≤ 𝜀𝜙(𝑡), with initial condition (1.2) for all 𝑡 ∈ 𝐼. Then there exists some 𝑦 ∈ 𝐶 n (𝐼, 𝔹) satisfying the differential equation (1.1) with initial conditions 𝑦(𝑎) = 𝑦0 , 𝑦 ′ (𝑎) = 𝑦1 , 𝑦 ′′ (𝑎) = 𝑦2 , … , and 𝑦 (𝑛−1) (𝑎) = 𝑦𝑛−1 such that ‖𝑥(𝑡) − 𝑦(𝑡)‖ ≤ 𝑆𝑓,𝜙 𝜀𝜙(𝑡), ∀ 𝑡 ∈ 𝐼. Proof. Given that for every 𝜀 ∈ (0, 𝜀0 ], and for each solution 𝑥(𝑡) ∈ 𝐶 𝑛 (𝐼, 𝔹) satisfying ‖𝑥 (𝑛) (𝑡) − 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡))‖ ≤ 𝜀𝜙(𝑡), (3.2) for all 𝑡 ∈ 𝐼. Now, we prove that there exists real number 𝑆𝑓,𝜙 > 0 and for some 𝑦 ∈ 𝐶 𝑛 (𝐼, 𝔹) satisfying the inequality (1.1) with 𝑦(𝑎) = 𝑦0 , 𝑦 ′ (𝑎) = 𝑦1 , 𝑦 ′′ (𝑎) = 𝑦2 , … , and 𝑦 (𝑛−1) (𝑎) = 𝑦𝑛−1 such that ‖𝑥(𝑡) − 𝑦(𝑡)‖ ≤ 𝑆𝑓,𝜙 𝜀𝜙(𝑡), for all 𝑡 ∈ 𝐼. Let 𝐺 be a closed and bounded set, then there exists a real number 𝑀 > 0 such that 𝐺 = {𝑥: ‖𝑥‖ ≤ 𝑀}. If the equilibrium is (𝐺0 , 𝐺, 𝜀0 )-practically stable and there exists 𝜀1 > 0 such that 𝜀1 ≤ 𝜙(𝑡)𝜀 ≤ 𝜀0 for all 𝑡 ∈ 𝐼. Now, let 𝑥 ∗ = 𝑓(𝑎, 𝑥0 , 𝑥1 , 𝑥2 , … , 𝑥𝑛−1 ) satisfies the inequality (3.2) for arbitrary 𝜀1 , then 𝑥 ∗ be the solution of the differential equation (1.1) with (1.2). Since the equilibrium of (1.1) is (𝐺0 , 𝐺, 𝜀0 ) - practically stable, then 𝑥 ∗ ∈ 𝐺. Hence ‖𝑥 ∗ ‖ ≤ 𝑀, since 𝑀 > 0, and 𝜀1 > 0 then there exists 𝑆𝑓,𝜙 > 0 such that 𝑀 = 𝑆𝑓,𝜙 𝜀1 . Then we have ‖𝑥 ∗ ‖ ≤ 𝑆𝑓,𝜙 𝜀1 , for all 𝑡 ∈ 𝐼, hence ‖𝑥 ∗ ‖ ≤ 𝑆𝑓,𝜙 𝜀𝜙(𝑡). Obviously, 𝑦(𝑡) ≡ 0 be the solution of (1.1) with (1.2) such that ‖𝑥 ∗ (𝑡) − 𝑦(𝑡)‖ ≤ 𝑆𝑓,𝜙 𝜀𝜙(𝑡), for all 𝑡 ∈ 𝐼. Hence by the virtue of Definition 2.3, the system of nth order differential equation has the Hyers-Ulam stability. The following corollary shows the Generalized Hyers-Ulam-Rassias stability of the system (1.1) with (1.2). Corollary 3.4 Assume that there exists a constant 𝑆𝑓,𝜙 > 0 such that for every 𝜀 ∈ (0, 𝜀0 ] and for each solution 𝑥(𝑡) ∈ 𝐶 n (𝐼, 𝔹) satisfying the inequality ‖𝑥 (𝑛) (𝑡) − 𝑓 (𝑡, 𝑥(𝑡), 𝑥 ′ (𝑡), 𝑥 ′′ (𝑡), … , 𝑥 (𝑛−1) (𝑡))‖ ≤ 𝜙(𝑡), with (1.2) for all 𝑡 ∈ 𝐼. Then there exists some 𝑦 ∈ 𝐶 𝑛 (𝐼, 𝔹) satisfies (1.1) with initial condition 𝑦(𝑎) = 𝑦0 , 𝑦 ′ (𝑎) = 𝑦1 , 𝑦 ′′ (𝑎) = 𝑦2 , … , and 𝑦 (𝑛−1) (𝑎) = 𝑦𝑛−1 such that ‖𝑥(𝑡) − 𝑦(𝑡)‖ ≤ 𝑆𝑓,𝜙 𝜙(𝑡), ∀ 𝑡 ∈ 𝐼. Proof. We can prove this corollary by the similar way as in the proof of Theorem 3.3. AIJRSTEM 19-212; © 2019, AIJRSTEM All Rights Reserved Page 74 Murali et al., American International Journal of Research in Science, Technology, Engineering & Mathematics,26(1), March-May 2019, pp. 71-75 IV. CONCLUSION In this paper, we studied the Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Generalized Hyers-Ulam stability and Generalized Hyers-Ulam-Rassias stability of the system of nth order differential equation. That is, we obtained the sufficient criteria for the Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Generalized Hyers-Ulam stability and Generalized Hyers-Ulam-Rassias stability of the system of nth order differential equation. ACKNOWLEDGMENT The authors are very grateful to the editors for their valuable Comments. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] C. Alsina, R. Ger, “On Some inequalities and stability results related to the exponential function,” Journal of Inequalities Appl. 2 (1998) 373 - 380. P. Gavruta, S. M. Jung and Y. Li, Hyers - Ulam stability for the second order linear differential equations with boundary conditions, Elec. 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