Hyers-ulam stability of exact second-order linear differential equations
© Ghaemi et al; licensee Springer. 2012
Received: 1 December 2011
Accepted: 23 March 2012
Published: 23 March 2012
In this article, we prove the Hyers-Ulam stability of exact second-order linear differential equations. As a consequence, we show the Hyers-Ulam stability of the following equations: second-order linear differential equation with constant coefficients, Euler differential equation, Hermite's differential equation, Cheybyshev's differential equation, and Legendre's differential equation. The result generalizes the main results of Jung and Min, and Li and Shen.
Mathematics Subject Classification (2010): 26D10; 34K20; 39B52; 39B82; 46B99.
The stability of functional equations was first introduced by Ulam . Hyers  gave a partial solution of Ulam's problem for the case of approximate additive mappings in the context of Banach spaces. Rassias  generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences ║f(x + y) - f(x) - f(y)║ ≤ ε(║x║ p +║y║ p ) (ε > 0, p ∈ [0, 1)).
and ║f(t) - f0(t)║ ≤ K(ε) for all t ∈ I. Here limε→ 0K(ε) = 0. Then we say that the above differential equation has the Hyers-Ulam stability.
If the above statement is also true when we replace ε and K(ε) by φ(t) and ϕ(t), where φ, ϕ : I → [0, ∞) are functions not depending on f and f0 explicitly, then we say that the corresponding differential equation has the Hyers-Ulam stability.
The Hyers-Ulam stability of the differential equation y' = y was first investigated by Alsina and Ger . This result has been generalized by Takahasi et al.  for the Banach space-valued differential equation y' = λy. Jung  proved the Hyers-Ulam stability of a linear differential equation of first-order.
for all t ∈ I, where are continuous functions and φ : I → [0, ∞) is a function. Assume that
(a) g(t) and are integrable on (a, c) for each c ∈ I;
(b) is integrable on I.
for all t ∈ I.
2. Main results
In this section, let I = (a, b) be an open interval with -∞ ≤ a < b ≤ ∞. In the following theorem, and . Taking some idea from , we investigate the Hyers-Ulam stability of exact second-order linear differential equations. For the sake of convenience, we assume that all the integrals and derivations exist.
for all x ∈ I.
for all x ∈ I.
is a solution of (1) with the condition (2). □
for all x ∈ I, where p(x) = (μ(x)p0(x))-1 [μ(x)p1(x) - (μ(x)p0(x))' ].
- 1.Li and Shen  proved the Hyers-Ulam stability of second-order linear differential equations with constant coefficients(10)
where the characteristic equation λ2 + cλ + b = 0 has two positive roots.
It is well-known that μ(x) = exp(mx), where , is a solution of (11) and consequently, it is an integrating factor of (10). Now the following corollaries are the generalization of [, Theorems 2.1 and 2.2].
for all x ∈ I.
as desired. □
for all x ∈ I, where μ(x) = exp(αx) cos βx and p(u) = [c - α + β tan βx].
- 2.Let α and β be real constants. The following differential equation
is called the Euler differential equation. It is exact when α - β = 2. By Theorem 2.1, it has the Hyers-Ulam stability.
- 3.Hermite's differential equation
- 4.Chebyshev's differential equation
- 5.Legendre's differential equation
is exact when n(n + 1) = 0 and it has the Hyers-Ulam stability.
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