International Journal of Analysis and Applications ISSN 2291-8639 Volume 1, Number 2 (2013), 71-78 http://www.etamaths.com ON HYERS-ULAM STABILITY FOR NONLINEAR DIFFERENTIAL EQUATIONS OF NTH ORDER MAHER NAZMI QARAWANI Abstract. This paper considers the stability of nonlinear differential equa- tions of nth order in the sense of Hyers and Ulam. It also considers the Hyers- Ulam stability for superlinear Emden-Fowler differential equation of nth order. Some illustrative examples are given. 1. INTRODUCTION In 1940, Ulam [1] posed the stability problem of of functional equations: Given a group G1and a metric group G2 with metric ρ(., .). Given ε > 0, does there exist a δ > 0 such that if f : G1 → G2 satisfies ρ(f(x),h(x)) < δ for all x,y ∈ G1,then a homomorphism h : G1 → G2 exists with ρ(f(x),h(x)) < ε,for all x,y ∈ G1? The problem for approximately additive mappings, on Banach spaces, was solved by Hyers [2]. The result obtained by Hyers was generalized by Rassias [3]. During the last two decades many mathematicians have extensively investigated the stability problems of functional equations (see [4-11]). Alsina and Ger [12] were the first mathematicians who investigated the Hyers- Ulam stability of the differential equation g′ = g.They proved that if a differentiable function y : I → R satisfies |y′ −y| ≤ ε for all t ∈ I,then there exists a differentiable function g : I → R satisfying g′(t) = g(t) for any t ∈ I such that |g −y| ≤ 3ε,for all t ∈ I. This result of alsina and Ger has been generalized by Takahasi et al [13] to the case of the complex Banach space valued differential equation y′ = λy. Furthermore, the results of Hyers-Ulam stability of differential equations of first order were also generalized by Miura et al. [14], Jung [15] and Wang et al. [16]. Li [17] established the stability of linear differential equation of second order in the sense of the Hyers and Ulam y′′ = λy. Li and Shen [18] proved the stability of nonhomogeneous linear differential equation of second order in the sense of the Hyers and Ulam y′′ + p(x)y′ + q(x)y + r(x) = 0, while Gavruta et al. [19] proved the Hyers-Ulam stability of the equation y′′ + β(x)y = 0 with boundary and initial conditions. The author in his study [20] estabilshed the Hyers-Ulam stability of the equations of the second order z′′ = F(x,z) with the initial conditions z(x0) = 0 = z ′(x0). In this paper we investigate the Hyers-Ulam stability of the following nonlinear differential equation of nth order 2010 Mathematics Subject Classification. 39A99, 39A30. Key words and phrases. Hyers-Ulam stability, superlinear, Emden-Fowler differential equation. c©2013 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 71 72 QARAWANI (1) y(n) = f(t,y,y′,y′′, ...,y(n−1)) with the initial conditions (2) y(t0) = y0 , y ′(t0) = y1, ... , y (n−1)(t0) = yn−1 where y ∈ C(n)(I),I = [t0, t1], (t, [y]) ≡ (t,y,y′,y′′, ...,y(n−1)) ∈ D, t ∈ I, −∞ < t0 < t1 < ∞, and f(t,y,y′,y′′, ...,y(n−1)) is defined on a closed bounded set D ⊂ Rn+1 that satisfies the condition (3) |f(t, [y]) −f(t, [z])| ≤ g(t) |y(t) −z(t)| (t1 − t0) n−1 where g(t) : I → (0,∞) is integrable function. Moreover we establish the Hyers-Ulam stability of the problem (1),(2) for f satisfying the Lipschitz condition (4) |f(t, [y]) −f(t, [z])| ≤ A0 n−1∑ i=0 ∣∣y(i)(t) −z(i)(t)∣∣ where A0 > 0. Definition 1 We will say that the equation (1) has the Hyers -Ulam stability if there exists a positive constant K > 0 with the following property: For every ε > 0, y ∈ C(n)(I), if (5) |y(n) −f(t, [y])| ≤ ε with the initial condition (2), then there exists a solution z(t) ∈ C(n)(I) of the equation (1), such that |y(t) − z(t)| ≤ Kε, where K is a constant that does not depend on ε nor on y(t). 2. MAIN RESULTS ON HYERS-ULAM STABILITY Theorem 1 If y ∈ C(n)(I) and f(t,y,y′,y′′, ...,y(n−1)) satisfies condition (3) on a closed bounded set D ⊂ Rn+1, then the initial value problem(1),(2) is stable in the sense of Hyers and Ulam. Proof. Let ε > 0 and y(t) be an approximate solution of the initial value prob- lem (1),(2).We will show that there exists a function z(t) ∈ C(n)(I) satisfying the equation (1) and the initial condition (2) such that |y(t) −z(t)| ≤ Kε From the inequality (5) we have (6) −ε ≤ y(n) −f(t, [y]) ≤ ε Integrating the last inequality n times, we obtain (7) ∣∣∣∣∣y(t) −n−1∑k=0 (t−t0) kyk k! − 1 (n−1)! t∫ t0 (t−s)n−1f(s, [y])ds ∣∣∣∣∣ ≤ (t− t0) nε n! ON HYERS-ULAM STABILITY 73 It is clear that z(t) = n−1∑ k=0 (t−t0)kyk k! + t∫ t0 f(s, [z]) (t−s)n−1 (n−1)! ds satisfies equation (1) and the initial condition (2). Consider the difference |y(t) −z(t)| ≤ ∣∣∣∣∣y(t) −n−1∑k=0 (t−t0) kyk k! − 1 (n−1)! t∫ t0 (t−s)n−1f(s, [y])ds ∣∣∣∣∣ + ∣∣∣∣∣∣ t∫ t0 f(s, [y]) −f(s, [z])) (t−s)n−1 (n− 1)! ds ∣∣∣∣∣∣ ≤ (t− t0)nε n! + 1 (n− 1)! t∫ t0 g(s) |y(s) −z(s)| (t1 − t0) n−1 (t1 − t0) n−1 ds(8) Applying Gronwall’s inequality, we obtain from inequalities (7) and (8) |y(t) −z(t)| ≤ (t1 − t0)nε n! exp   1 (n− 1)! t∫ t0 g(t)ds   Whence max t0≤t≤t1 |y(t) −z(t)| ≤ Kε Hence the initial value problem (1),(2) is stable in the sense of Hyers and Ulam. Theorem 2 If y ∈ C(n)(I) and f(t,y,y′,y′′, ...,y(n−1)) satisfies the Lipschitz condition (4) on a closed bounded set D ⊂ Rn+1, then the initial value prob- lem(1),(2) is stable in the sense of Hyers and Ulam. Proof. Given ε > 0, assume that y is an approximate solution of Eq. (1). We will show that there exists a function z(t) ∈ C(n)(I) satisfying equation (1) such that |y(t) −z(t)| ≤ Kε From the inequality (5) we have (9) −ε ≤ y(n) −f(t, [y]) ≤ ε By integrating the inequality (9) k times, we obtain (10) ∣∣∣∣∣y(n−k)(t) − k−1∑j=0 (t−t0) jyn−k+j j! − 1 (n−1)! t∫ t0 (t−s)n−1f(s, [y])ds ∣∣∣∣∣ ≤ (t− t0) kε k! where 1 ≤ k ≤ n. We can easily verify that the function z(t) z(n−k)(t) = k−1∑ j=0 (t−t0)jyn−k+j j! + t∫ t0 f(s, [z]) (t−s)k−1 (k−1)! ds must satisfy the initial value problem (1),(2) 74 QARAWANI Now let ∆(n−k) ≡ ∣∣y(n−k) − z(n−k)∣∣ . Then, using the inequalities (4),(10), we get the estimation ∆(n−k) ≤ ∣∣∣∣∣y(n−k)(t) − k−1∑j=0 (t−t0) jyn−k+j j! − 1 (n−1)! t∫ t0 (t−s)n−1f(s, [y])ds ∣∣∣∣∣ + 1 (n− 1)! t∫ t0 |f(s, [y]) −f(s, [z])|(t−s)n−1ds ≤ (t1 − t0)kε k! + A0n (n− 1)! t∫ t0 ∣∣∣y(n−k)(s) −z(n−k)(s)∣∣∣ (t−s)n−1ds(11) Thus, according to (4),(10) and (11), from Gronwall’s inequality it follows that ∣∣∣y(n−k)(t) −z(n−k)(t)∣∣∣ ≤ (t1 − t0)kε k! exp ( A0(t1 − t0)n (n− 1)! ) Consequently for k = n, we have max t0≤t≤t1 |y(t) −z(t)| ≤ (t1 − t0)nε n! exp ( A0(t1 − t0)n (n)! ) Hence the initial value problem (1),(2) is stable in the sense of Hyers and Ulam. Remark 1 Suppose that y ∈ C(n)(I) satisfies the inequality (9) with the zero initial condition y(t0) = 0, y ′(t0) = 0, ... , y (n−1)(t0) = 0. If f(t, [z]) satisfies Lipschitz condition (4) and f(t, 0, , ..., 0) ≡ 0, then one can similarly show that the zero solution z0 ≡ 0 of equation (1) is stable in the sense of Hyers and Ulam. 3. HYERS-ULAM STABILITY FOR SUPERLINEAR NTH ORDER DIFFERENTIAL EQUATION In this section we investigate the Hyers Ulam stability of solutions for superlinear nth order differential equation (12) y(n) = h(t) |y|α sgny , α > 1 with the initial condition (13) y(t0) = y0 , y ′(t0) = y1, ... , y (n−1)(t0) = yn−1 where y ∈ C(n)(I),I = [t0, t1], −∞ < t0 < t1 < ∞, and h(t) : I → R is continuous. Theorem 3 If y ∈ C(n)(I),and h(t) : I → R is continuous, then the initial value problem (12),(13) is stable in the sense of Hyers and Ulam. Proof. Given ε > 0,Suppose y(t) is an approximate solution of the initial value problem (12),(13).We show that there exists an exact solution z(t) ∈ C(n)(I) satisfying the equation (12) such that |y(t) −z(t)| ≤ Kε where k is a constant that does not explicitly depend on ε nor on y(t). From the inequality (5) we have (14) −ε ≤ y(n) −h(t) |y|α sgny ≤ ε By integrating the last inequality n times, we obtain ON HYERS-ULAM STABILITY 75 (15) ∣∣∣∣∣y(t) −n−1∑k=0 (t−t0) kyk k! − 1 (n−1)! t∫ t0 h(s) |y|α sgny.(t−s)n−1ds ∣∣∣∣∣ ≤ (t− t0) nε n! where 1 ≤ k ≤ n. We can easily verify that the function z(t) z(t) = n−1∑ k=0 (t−t0)kyk k! + 1 (n−1)! t∫ t0 h(s) |z|α sgnz.(t−s)n−1ds must satisfy the initial value problem (12),(13). Now since the derivative ∣∣∣∂(h(t)yα)∂y ∣∣∣ is bounded on S, then the function f(t,y) = h(t) |y|α sgny satisfies Lipschitz condition |f(t,y) −f(t,z)| ≤ L |y(t) −z(t)| , (t,y), (t,z) ∈ S where S = [t0, t1] × [−M,M] ⊂ R2, and M = max t0≤t≤t1 |y(t)| . Since h is continuous on I, then ∃ B0 > 0, |h(t)| ≤ B0, and from the inequality (15), we get the estimation |y(t) −z(t)| ≤ (t1 − t0)nε n! + B0L (n− 1)! t∫ t0 |y(s) −z(s)|(t−s)n−1ds From Gronwall’s inequality it follows that |y(t) −z(t)| ≤ (t1 − t0)nε n! exp ( B0L(t1 − t0)n n! ) Consequently, we have max t0≤t≤t1 |y(t) −z(t)| ≤ (t1 − t0)nε n! exp ( (t1 − t0)n (n)! ) Hence the initial value problem (12),(13) is stable in the sense of Hyers and Ulam. Remark 2 Suppose that y ∈ C(n)(I) satisfies the inequality (6) with the zero initial condition y(t0) = 0, y ′(t0) = 0, ... , y (n−1)(t0) = 0. If the function h : I → R is continuous, then one can similarly establish the Hyers-Ulam stability of zero solution z0 ≡ 0 of (12). Example1 Consider the problem y(5) = 8y2 sin t + et(16) y(k)(t0) = 0, k = 0, 4(17) and the inequality (18) −ε ≤ y(5) −y2 sin t + et ≤ ε where (t,y) ∈ [t0, t1] × [−M1,M1] , M1 = max t0≤t≤t1 |y(t)| . 76 QARAWANI Integrating the inequality (18) five times and using the initial condition (17), we get that ∣∣∣∣∣∣y(t) − 13 t∫ t0 (y3 sin t + et)(t−s)4ds ∣∣∣∣∣∣ ≤ (t− t0) 5ε 5! One can easily show that z(t) z(t) = 1 3 t∫ t0 (z3 sin t + et)(t−s)4ds has to satisfy the initial value problem (16),(17). Now Let us estimate the difference: |y(t) −z(t)| ≤ ∣∣∣∣∣∣y(t) − 13 t∫ t0 (t−s)4(y3 sin t + et)ds ∣∣∣∣∣∣ + 1 3 t∫ t0 (t−s)4 ∣∣y3 −z3∣∣ |sin t|ds ≤ (t1 − t0)5ε 5! + M21 t∫ t0 (t−s)4 |y −z|ds Therefore, we obtain max t0≤t≤t1 |y(t) −z(t)| ≤ (t1 − t0)5ε 5! exp ( M21 (t1 − t0)5 5! ) Hence the initial value problem (16),(17) is stable in the sense of Hyers and Ulam. 4. A SPECIAL CASE OF EQUATION (11) Consider the equation (19) y(n) = h(t)y with the initial conditions (20) y(t0) = y0 , y ′(t0) = y1, ... , y (n−1)(t0) = yn−1 where y ∈ C(n)(I),I = [t0, t1], −∞ < t0 < t1 < ∞, and h(t) : I → R is continuous. Theorem 4 If y ∈ C(n)(I),and h(t) : I → R is continuous, then the initial value problem(19),(20) is stable in the sense of Hyers and Ulam. Proof. assume that ε > 0 and that y is n times continuously differentiable real- valued function on I = [t0, t1]. We will show that there exists a function z(t) ∈ c2(I) satisfying equation (19) such that |y(t) −z(t)| ≤ Kε We have (21) −ε ≤ y(n) −h(t)y ≤ ε By integrating the inequality (21) n times, we obtain ON HYERS-ULAM STABILITY 77 ∣∣∣∣∣y(t) −n−1∑k=0 (t−t0) kyk k! − 1 (n−1)! t∫ t0 (t−s)n−1h(s)yds ∣∣∣∣∣ ≤ (t− t0) nε n! where 1 ≤ k ≤ n. It is easily to verify that the function z(t) z(t) = n−1∑ k=0 (t−t0)kyk k! + 1 (n−1)! t∫ t0 (t−s)n−1h(s)zds satisfies the initial value problem (19),(20). One can get the estimation (22) |y(t) −z(t)| ≤ (t1 − t0)nε n! + B0 (n− 1)! t∫ t0 |y(t) −z(t)|(t−s)n−1ds Using Gronwall’s inequality we have Hence max t0≤t≤t1 |y(t) −z(t)| ≤ (t1 − t0)nε n! exp ( B0(t1 − t0)n (n)! ) Therefore, the initial value problem (19),(20) is stable in the sense of Hyers and Ulam. Example 2 Consider the equation (23) y(4) − (1 + cos t) y = 0 (24) y(0) = 0, y′(0) = 1, y′′(0) = −1, y′′′(0) = 0 and the inequality ∣∣∣ y(4) − (1 + cos t) y ∣∣∣ ≤ ε where 0 ≤ t ≤ b, b ∈ R. Integrating the last inequality four times, we get∣∣∣∣∣∣y(t) − t + t 2 2 − 1 6 t∫ 0 (t−s)3 (1 + cos t) yds ∣∣∣∣∣∣ ≤ t 3ε 6 One can easily find that z(t) z(t) = t− t 2 2 + 1 6 t∫ 0 (t−s)3 (1 + cos t) zds satisfies the equation (23) and initial condition (24) Then, we obtain an estimation |y(t) −z(t)| ≤ b3ε 6 exp(b4/12) Hence Eq. (23) has the Hyers -Ulam stability. 78 QARAWANI References [1] Ulam S.M., (1964), Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, Science edition. 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[17] Li Y., (2010), Hyers-Ulam Stability of Linear Differential Equations,Thai Journal of Mathe- matics,Vol. 8 No 2, pp 215–219. [18] Li Y. and Shen Y., (2009), Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order, International Journal of Mathematics and Mathematical Sciences, Vol. 2009, Article ID 576852, pp. 7. [19] Gavruta P., Jung S. and Li Y. , (2011), Hyers-Ulam Stability For Second-Order Lin- ear Differential Equations With Boundary Conditions, EJDE, Vol.2011, No. 80, pp1-7, .http://ejde.math.txstate.edu/Volumes/2011/80/gavruta.pdf. [20] Qarawani M. N., (2012), Hyers-Ulam Stability of a Generalized Second- Order Non- linear Differential Equation, Applied Mathematics,Vol. 3, No. 12, pp. 1857-1861. doi: 10.4236/am.2012.312252. Department of Mathematics, AlQuds Open University, Palestine, West-Bank, Salfit P.O.Box 37