Generalized Hyers-Ulam Stability of the Second-Order Linear Differential Equations

We prove the generalized Hyers-Ulam stability of the 2nd-order linear di ﬀ erential equation of the form y (cid:2)(cid:2) (cid:3) p (cid:4) x (cid:5) y (cid:2) (cid:3) q (cid:4) x (cid:5) y (cid:6) f (cid:4) x (cid:5) , with condition that there exists a nonzero y 1 : I → X in C 2 (cid:4) I (cid:5) such that y (cid:2)(cid:2) 1 (cid:3) p (cid:4) x (cid:5) y (cid:2) 1 (cid:3) q (cid:4) x (cid:5) y 1 (cid:6) 0 and I is an open interval. As a consequence of our main theorem, we prove the generalized Hyers-Ulam stability of several important well-known di ﬀ erential equations.


Introduction
The stability problem of functional equations started with the question concerning stability of group homomorphisms proposed by Ulam 1 during a talk before a Mathematical Colloquium at the University of Wisconsin, Madison.In 1941, Hyers 2 gave a partial solution of Ulam's problem for the case of approximate additive mappings in the context of Banach spaces.In 1978, Rassias 3 generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences f x y − f x − f y ≤ x p y p , > 0, p ∈ 0, 1 .This phenomenon of stability that was introduced by Rassias 3 is called the Hyers-Ulam-Rassias stability or the generalized Hyers-Ulam stability .
Let X be a normed space over a scalar field K, and let I be an open interval.Assume that for any function f : I −→ X satisfying the differential inequality a n t y n t a n−1 t y n−1 t • • • a 1 t y t a 0 t y t h t ≤ 1.1 for all t ∈ I and for some ≥ 0, there exists a function f 0 : I −→ X satisfying a n t y n t a n−1 t y n−1 t • • • a 1 t y t a 0 t y t h t 0, for all t ∈ I; here K t is an expression for with lim −→0 K 0. Then, we say that the above differential equation has the Hyers-Ulam stability.
If the above statement is also true when we replace and K by ϕ t and φ t , where ϕ, φ : I −→ 0, ∞ are functions not depending on f and f 0 explicitly, then we say that the corresponding differential equation has the Hyers-Ulam-Rassias stability or the generalized Hyers-Ulam stability .
The Hyers-Ulam stability of differential equation y y was first investigated by Alsina and Ger 4 .This result has been generalized by Takahasi et al. 5 for the Banach space-valued differential equation y λy.In 6 , Miura et al. also proved the Hyers-Ulam-Rassias stability of linear differential of first order, y g t y t 0, where g t is a continuous function, while the author 7 proved the Hyers-Ulam-Rassias stability of linear differential of the form c t y t y t .Jung 8 proved the Hyers-Ulam-Rassias stability of linear differential of first order of the form c t y t g t y t h t 0. In this paper, we investigate the generalized Hyers-Ulam stability of differential equations of the form y p x y q x y f x . 1.3 We assume that X is a complex Banach space, I a, b is an arbitrary interval, and y 1 : I −→ X is a nonzero solution of corresponding homogeneous equation of 1.3 , where y 1 p x y 1 q x y 1 0. 1.4

Main Results
Taking some idea from 8 , we are going to investigate the stability of the 2nd-order linear differential equations.For the sake of convenience, all the integrals and derivations will be viewed as existing and R ω denotes the real part of complex number ω.Moreover, let I a, b be an open interval, where a, b ∈ R {±∞} are arbitrarily given with a < b.
Theorem 2.1.Let X be a complex Banach space.Assume that p, q : I −→ C and f : I −→ X are continuous functions and y 1 : I −→ X is a nonzero twice continuously differentiable function which satisfies the differential equation 1.4 .If a twice continuously differentiable function y : I −→ X satisfies for all x ∈ I, where k y a /y 1 a ∈ X and ϕ : I −→ 0, ∞ is a continuous function, then there exists a unique x 0 ∈ X such that

2.2
Proof.We assume that for all x ∈ I.It follows from 1.4 , 2.1 , and 2.3 that For simplicity, we use the following notation: for all s ∈ I.By making use of this notation and by 2.5 , we get is assumed to be integrable on I, we may select l 0 ∈ I, for any given > 0, such that l, x ≥ l 0 implies z x − z l < .That is, {z l } l∈I is a Cauchy net.By completeness of X, there exists an x 0 ∈ X such that z l converges to x 0 as l −→ b.It follows from 2.7 and the previous argument that, for any x ∈ I, is a solution of 1.3 .

Journal of Applied Mathematics
Now, we prove the uniqueness property of x 0 .Assume that x 1 , x 2 ∈ X satisfy inequality 2.2 in place of x 0 .Then, we have 2.10 thus, Hence, every 2nd-order linear differential equation has the generalized Hyers-Ulam stability with the condition that there exists a solution of corresponding homogeneous equation or there exists a general solution in the ordinary differential equations.for all x ∈ I. On the other hand, by ordinary differential equations, we know that y 1 x exp mx is a solution of corresponding homogeneous equation of 2.14 .It follows from Theorem 2.1, Remark 2.3, and 2.14 that there exists a solution y 0 : I −→ R of 2.14 such that for all x ∈ I and that exp mx dt ds.

2.17
for all x ∈ I.By the trial of y 0 x x, we see that it is a solution of corresponding homogeneous equation of 2.20 .Then it follows from Theorem 2.1, Remark 2.3, and 2.21 that there exists a solution y 0 : I −→ R of 2.20 such that x 4 3x 2 2.22 for all x ∈ I, where k y a /a and x 0 ∈ R is unique and We know from the ordinary differential equations that Laguerre, Chebyshev, and Gauss hypergeometric differential equations have the general solution.Then we can show that those have generalized Hyers-Ulam stability.

Example 2 . 4 .
Consider the second-order linear differential equation with constant coefficients y by cy f x .2.14 Let b 2 − 4c ≥ 0, m −b ± √ b 2 − 4c /2, and let f : I −→ R, ϕ : I −→ 0, ∞ be continuous functions.Assume that y : I −→ R is a twice continuously differential function satisfying the differential inequality y by cy − f x ≤ ϕ x 2.15

Example 2 . 5 .
Consider 2.14 .Let b 2 − 4c < 0, m −b ± √ b 2 − 4c /2 α ± iβ, and let f : I −→ R, ϕ : I −→ 0, ∞ be continuous functions.Let y : I −→ R be a twice continuously differential function satisfying the differential inequality of 2.15 for all x ∈ I.It follows from the ordinary differential equations that y 1 x exp αx cos βx .Then it follows from Theorem 2.1, Remark 2.3, and 2.15 that there exists a solution y 0 : I −→ R of 2.14 such that y 0 x exp αx cos βx • x 0 cos 2 βa x a exp 2α b a − s cos 2 βs ds exp αx cos βx • exp v α b • cos βv dv • exp v α b ds k 2.18 for all x ∈ I, where k y a / exp αa cos βa and x 0 ∈ R is unique and y x −y 0 x ≤ exp αx cos βx • x a exp − 2α b s cos 2 βs • b s cos 2 βt • exp α b t •ϕ t dt ds.be an open interval, where a, b ∈ 1, ∞ are arbitrarily given with a < b, f : I −→ R and ϕ : I −→ 0, ∞ are continuous functions.Assume that y : I −→ R is a twice continuously differential function satisfying the differential inequality This implies that x 1 x 2 and the proof is complete.
Remark 2.3.If we replace C by R in the proof of Theorem 2.1 and we assume that p, q are real-valued continuous functions, then we can see that Theorem 2.1 is true for a real Banach space X.
We know that Eulars differential equation of second order has the general solution in ordinary differential equations, then we can use Theorem 2.1 and Remark 2.3 for the Hyers-Ulam-Rassias stability in this case., a 1 are arbitrary constants.By Theorem 2.1 and Remark 2.3, Legender's differential equation has Hyers-Ulam-Rassias stability.∈ R, and a 0 , a 1 are arbitrary constants.Thus Hermites differential equation has generalized Hyers-Ulam stability.It is well known from the ordinary differential equations that ∈ R, is a solution of Bessel's differential equation x 2 y xy x 2 − p 2 y 0 2.31 that p ≥ 0. Then Bessel's differential equation has Hyers-Ulam-Rassias stability.