Abstract
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.
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1. Introduction
The stability of functional equations was first introduced by Ulam [1]. Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings in the context of Banach spaces. Rassias [3] 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)).
Let X be a normed space over a scalar field and let I be an open interval. Assume that for any function f : I → X (y = f(x)) satisfying the differential inequality
for all t ∈ I and some ε ≥ 0, there exists a function f0 : I → X (y = f0(x)) satisfying
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 [4]. This result has been generalized by Takahasi et al. [5] for the Banach space-valued differential equation y' = λy. Jung [6] proved the Hyers-Ulam stability of a linear differential equation of first-order.
Theorem 1.1. ([6]) Let be a continuously differentiable function satisfying the differential inequality
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.
Then there exists a unique real number x such that
for all t ∈ I.
In this article, we prove the Hyers-Ulam stability of exact second-order linear differential equations (see [7]). A general second-order differential equation is of the form
and it is exact if
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 [6], 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.
Theorem 2.1. Let be continuous functions with p 0 (x) ≠ 0 for all x ∈ I, and let φ : I → [0, ∞) be a function. Assume that is a twice continuously differentiable function satisfying the differential inequality
for all x ∈ I and (2) is true. Then there exists a solution of (1) such that
for all x ∈ I.
Proof. It follows from (2) and (3) that
So we have
Integrating (4) from a to x for each x ∈ I, we get
Dividing both sides of the inequality (5) by │p0(x)│, we obtain
If we set
and in (6), then we have
Now we are in the situation of Theorem 1.1, that is, there exists a unique such that
for all x ∈ I.
It is easy to show that
is a solution of (1) with the condition (2). □
If (1) is multiplied by a function μ(x) such that the resulting equation is exact, that is,
and
then we say that μ(x) is an integrating factor of the Equation (1) (see [7]).
Corollary 2.2. Let be continuous functions with p0(x) ≠ 0 and μ(x) ≠ 0 for all x ∈ I, and let φ : I → [0, ∞) be a function. Assume that is a twice continuously differentiable function satisfying the differential inequality
for all x ∈ I and (8) is true. Then there exists a solution of (7) such that
for all x ∈ I, where p(x) = (μ(x)p0(x))-1 [μ(x)p1(x) - (μ(x)p0(x))' ].
Proof. It follows from Theorem 2.1 that there exists a unique such that
is a solution of (7) with the condition (8), where
and
with
as desired. □
-
1.
Li and Shen [8] 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.
Now, it follows from (7) and (8) that μ(x) is an integrating factor for (10) if it satisfies
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 [[8], Theorems 2.1 and 2.2].
Corollary 2.3. Consider the Equation (10). Let c2 - 4b ≥ 0, be a continuous function and let φ : I → [0, ∞) be a function. Assume that is a twice continuously differentiable function satisfying the differential inequality
for all x ∈ I. Then there exists a solution of (10) such that
for all x ∈ I.
Proof. μ(x) = exp(mx) is an integrating factor of (10) when c2 - 4b ≥ 0 and (the paragraph preceding of this corollary). By (12), we obtain
for all x ∈ I. Using Corollary 2.2 with φ1(x) = exp(mx)φ(x) instead of φ(x) and with (13) instead of (9), we conclude that there exists a unique such that
where k = -[exp(ma)y'(a) -m exp(ma)y(a) + c exp(ma)y(a)], for all x ∈ I, is a solution of (10) and
as desired. □
Corollary 2.4. Consider the Equation (10). Let c2 - 4b < 0, be a continuous function and let φ : I → [0, ∞) be a function. Assume that is a twice continuously differentiable function satisfying the differential inequality
for all x ∈ I. Then there exists a solution of (10) such that
for all x ∈ I, where μ(x) = exp(αx) cos βx and p(u) = [c - α + β tan βx].
Proof. It is easy to show that
for all x ∈ I. Now, similar to Corollary 2.3, there exists a unique such that
has the required properties, where k = [exp(αa) cos βay'(a) - (exp(αa) cos βa)'y(a) + c exp(αa) cos βay(a)]. □
-
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.
In general, μ(x) is an integrating factor of Euler differential equation if it satisfies
The Equation (14) can be written as
By the trial of μ(x) = xm, we show that
From (15) we obtain
Now we can use the above corollaries for the Hyers-Ulam stability of Euler differential equation. This result is comparable with [[9], Theorem 2] and the main results of [10].
-
3.
Hermite's differential equation
is exact when λ = -1 and it has the Hyers-Ulam stability.
-
4.
Chebyshev's differential equation
is exact when n = ±1. By Theorem 2.1, it has the Hyers-Ulam stability.
-
5.
Legendre's differential equation
is exact when n(n + 1) = 0 and it has the Hyers-Ulam stability.
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Ghaemi, M.B., Gordji, M.E., Alizadeh, B. et al. Hyers-ulam stability of exact second-order linear differential equations. Adv Differ Equ 2012, 36 (2012). https://doi.org/10.1186/1687-1847-2012-36
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DOI: https://doi.org/10.1186/1687-1847-2012-36