Invariance of Hyers-Ulam stability of linear differential equations and its applications

Abstract

We prove that the generalized Hyers-Ulam stability of linear differential equations of nth order (defined on I) is invariant under any monotone one-to-one correspondence $\tau: I \to J$ which is n times continuously differentiable. Moreover, using this result, we investigate the generalized Hyers-Ulam stability of the linear differential equation of second order and the Cauchy-Euler equation.

1 Introduction

Throughout this paper, let n be a positive integer and let I and J be non-degenerate intervals of R. We will consider the (linear) differential equation of nth order

$$ \mathcal{F} \bigl( y^{(n)}, y^{(n-1)}, \ldots, y', y, x \bigr) = 0 $$

(1)

defined on I, where $y : I \to\mathbf{R}$ is an n times continuously differentiable function.

For an arbitrary $\varepsilon> 0$, assume that an n times continuously differentiable function $y : I \to\mathbf{R}$ satisfies the differential inequality

$$ \bigl| \mathcal{F} \bigl( y^{(n)}, y^{(n-1)}, \ldots, y', y, x \bigr) \bigr| \leq\varepsilon $$

(2)

for all $x \in I$. If for each function $y : I \to\mathbf{R}$ satisfying the inequality (2), there exists a solution $y_{0} : I \to\mathbf{R}$ of the differential equation (1) such that

$$ \bigl| y(x) - y_{0}(x) \bigr| \leq K(\varepsilon) $$

(3)

for any $x \in I$, where $K(\varepsilon)$ depends on ε only and satisfies $\lim_{\varepsilon\to0} K(\varepsilon) = 0$, then we say that the differential equation (1) satisfies (or has) the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain I is not the whole space R). When the above statement also holds even though we replace ε and $K(\varepsilon)$ with some appropriate $\varphi(x)$ and $\Phi(x)$, respectively, then we say that the differential equation (1) satisfies the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability). For a more detailed definition of the Hyers-Ulam stability, refer to [1–4].

Obłoza seems to be the first author who investigated the Hyers-Ulam stability of linear differential equations (see [5, 6]): Given real-valued constants a and b, let $g, r : (a,b) \to\mathbf{R}$ be continuous functions with $\int_{a}^{b} | g(x) | \,dx < \infty$. Assume that $\varepsilon> 0$ is an arbitrary real number. Obłoza proved that if a differentiable function $y : (a,b) \to\mathbf{R}$ satisfies the inequality $| y'(x) + g(x) y(x) - r(x) | \leq\varepsilon$ for all $x \in(a,b)$ and if a function $y_{0} : (a,b) \to\mathbf{R}$ satisfies $y'_{0}(x) + g(x) y_{0}(x) = r(x)$ for all $x \in(a,b)$ and $y(\tau) = y_{0}(\tau)$ for some $\tau\in(a,b)$, then there exists a constant $\delta> 0$ such that $| y(x) - y_{0}(x) | \leq\delta$ for all $x \in(a,b)$.

Thereafter, Alsina and Ger [7] proved that if a differentiable function $y : (a,b) \to\mathbf{R}$ satisfies the differential inequality $| y'(x) - y(x) | \leq\varepsilon$, then there exists a function $y_{0} : (a,b) \to\mathbf{R}$ such that $y_{0}'(x) = y_{0}(x)$ and $| y(x) - y_{0}(x) | \leq3\varepsilon$ for all $x \in(a,b)$. This result of Alsina and Ger was generalized by Takahasi et al. [8]. Indeed, they proved the Hyers-Ulam stability of the Banach space valued differential equation $y'(x) = \lambda y(x)$ (see also [9–19]).

Assume that there exists a monotone one-to-one correspondence $\tau: I \to J$, which is n times continuously differentiable. Let $\sigma: J \to I$ be the inverse of τ. If we make a change of variable $t = \tau(x)$ and define an m times continuously differentiable function $z : J \to\mathbf{R}$ by $z(t) = y(\sigma(t))$, where m is an appropriate positive integer (possibly $m = n$), then we can substitute $x = \sigma(t)$, $y(x) = z(t)$, and

$$y^{(k)}(x) = \sum_{i=1}^{k} a_{k,i} z^{(i)}(t) \prod_{j=1}^{k} \tau^{(j)}(x)^{b_{k,j}} $$

in (1) for each $k \in\{ 1, 2, \ldots, n \}$, where $a_{k,i} \in\mathbf{N}_{0}$ and $b_{k,j} \in\{ 0, 1, \ldots, k \}$, to reduce the linear differential equation (1) to another equation of the form

$$ \mathcal{G} \bigl( z^{(m)}, z^{(m-1)}, \ldots, z', z, t \bigr) = 0 $$

(4)

defined on J. For this case, an n times continuously differentiable function $y : I \to\mathbf{R}$ is a solution of the differential equation (1) if and only if the function $z : J \to\mathbf{R}$ is a solution of the differential equation (4).

The main goal of this paper is to prove that the (generalized) Hyers-Ulam stability of the linear differential equations is invariant under any monotone one-to-one correspondence which is n times continuously differentiable. In other words, if the differential equation (1) has the (generalized) Hyers-Ulam stability, then the reduced differential equation (4) also has the (generalized) Hyers-Ulam stability, and vice versa.

Moreover, we investigate the generalized Hyers-Ulam stability of the linear differential equation of second order and the Cauchy-Euler equation.

2 Hyers-Ulam stability is invariant

In the following main theorem, we prove that the (generalized) Hyers-Ulam stability of the linear differential equation of nth order is invariant.

Theorem 2.1

Assume that the linear differential equation (1) defined on I can be reduced to another differential equation (4) defined on J via a monotone one-to-one correspondence $\tau: I \to J$ which is n times continuously differentiable. If the differential equation (1) has the (generalized) Hyers-Ulam stability, so does the reduced differential equation (4).

Proof

If the differential equation (1) has the Hyers-Ulam stability and if an n times continuously differentiable function $y : I \to\mathbf{R}$ satisfies the inequality (2) for all $x \in I$ and for some $\varepsilon> 0$, then there exists a solution $y_{0} : I \to\mathbf{R}$ of the differential equation (1) such that the inequality (3) holds for any $x \in I$, where $K(\varepsilon)$ depends on ε only and satisfies $\lim_{\varepsilon\to0} K(\varepsilon) = 0$.

Since the differential equation (4) can be reduced from (1) by using a monotone one-to-one correspondence $\tau: I \to J$ and there exists the inverse $\sigma: J \to I$ of τ, if we define a function $z : J \to\mathbf{R}$ by $z(t) = y(\sigma(t))$, then we can reduce the inequality (2) to a new inequality

$$ \bigl| \mathcal{G} \bigl( z^{(m)}, z^{(m-1)}, \ldots, z', z, t \bigr) \bigr| \leq\varepsilon $$

(5)

for all $t \in J$. Moreover, if we set $z_{0}(t) = y_{0}(\sigma(t))$, then the inequality (3) is reduced to

$$ \bigl| z(t) - z_{0}(t) \bigr| \leq K(\varepsilon) $$

(6)

for all $t \in J$.

Finally, it is obvious that $z_{0}$ is a solution of the differential equation (4) by considering the last part of the Introduction.

To prove this theorem for the case of generalized Hyers-Ulam stability, we consider the inequalities

$$\bigl| \mathcal{F} \bigl( y^{(n)}, y^{(n-1)}, \ldots, y', y, x \bigr) \bigr| \leq\varphi(x) $$

and

$$\bigl| y(x) - y_{0}(x) \bigr| \leq\Phi(x) $$

instead of (2) and (3), respectively, where $\varphi, \Phi: I \to[0,\infty)$ are continuous functions. Then the inequalities (5) and (6) are replaced by

$$\bigl| \mathcal{G} \bigl( z^{(m)}, z^{(m-1)}, \ldots, z', z, t \bigr) \bigr| \leq\psi(t) $$

and

$$\bigl| z(t) - z_{0}(t) \bigr| \leq\Psi(t), $$

respectively, where $\psi:= \varphi\circ\sigma$ and $\Psi:= \Phi\circ\sigma$.

The rest of the proof runs analogously to the first part of this proof. □

By exchanging the roles of the monotone one-to-one correspondence $\tau: I \to J$ and its inverse $\sigma: J \to I$, we can prove a corollary to Theorem 2.1.

Corollary 2.2

If the differential equation (4) has the (generalized) Hyers-Ulam stability, so does the original differential equation (1).

3 Stability of linear differential equation of second order

Throughout this section, we assume that I is a non-degenerate interval of R. We now consider the linear inhomogeneous differential equation of the second order

$$ y''(x) + f(x) y'(x) + g(x) y(x) = r(x), $$

(7)

where $f, g, r : I \to\mathbf{R}$ are given continuous functions. The Hyers-Ulam stability of the differential equation (7) has been proved under various additional conditions (see [20–24]). We will now investigate the generalized Hyers-Ulam stability of the linear differential equation (7) under weaker conditions in comparison with those of [20–24].

The proof of the following lemma can be found in [25], Section 2.16.

Lemma 3.1

Assume that the homogeneous differential equation corresponding to (7),

$$ y''(x) + f(x) y'(x) + g(x) y(x) = 0, $$

(8)

has a general solution $y_{h} : I \to\mathbf{R}$ of the form

$$y_{h}(x) = c_{1} y_{1}(x) + c_{2} y_{2}(x), $$

where $c_{1}$ and $c_{2}$ are arbitrary real-valued constants. Then the inhomogeneous linear differential equation (7) has a general solution $y : I \to\mathbf{R}$ of the form

$$y(x) = c_{1} y_{1}(x) + c_{2} y_{2}(x) - y_{1}(x) \int_{a_{1}}^{x} \frac{y_{2}(t) r(t)}{W(y_{1}, y_{2})(t)} \,dt + y_{2}(x) \int_{a_{2}}^{x} \frac{y_{1}(t) r(t)}{W(y_{1}, y_{2})(t)} \,dt, $$

where $a_{1}$ and $a_{2}$ are arbitrarily chosen points of I and

$$W(y_{1}, y_{2}) (x) := y_{1}(x) y_{2}'(x) - y_{1}'(x) y_{2}(x) $$

is the Wronskian of $y_{1}$ and $y_{2}$.

We now investigate the generalized Hyers-Ulam stability of the linear inhomogeneous differential equation of the second order (7) in the class of twice continuously differentiable functions.

Theorem 3.2

Let $f, g, r : I \to\mathbf{R}$ be continuous functions. Assume that the homogeneous differential equation (8) has a general solution $y_{h} : I \to\mathbf{R}$ of the form $y_{h}(x) = c_{1} y_{1}(x) + c_{2} y_{2}(x)$, where $c_{1}$ and $c_{2}$ are arbitrary real-valued constants. If a twice continuously differentiable function $y : I \to\mathbf{R}$ satisfies the inequality

$$ \bigl| y''(x) + f(x) y'(x) + g(x) y(x) - r(x) \bigr| \leq\varphi(x) $$

(9)

for all $x \in I$, where $\varphi: I \to[0,\infty)$ is given such that each of the following integrals exists, then there exists a solution $y_{0} : I \to\mathbf{R}$ of (7) such that

$$\bigl| y(x) - y_{0}(x) \bigr| \leq \bigl| y_{1}(x) \bigr| \biggl| \int_{a_{1}}^{x} \biggl| \frac{y_{2}(t)}{W(y_{1}, y_{2})(t)} \biggr| \varphi(t) \,dt \biggr| + \bigl| y_{2}(x) \bigr| \biggl| \int _{a_{2}}^{x} \biggl| \frac{y_{1}(t)}{W(y_{1}, y_{2})(t)} \biggr| \varphi(t) \,dt \biggr| $$

for all $x \in I$, where $a_{1}$, $a_{2}$ are arbitrarily chosen points of I.

Proof

If we define a continuous function $s : I \to\mathbf{R}$ by

$$ s(x) := y''(x) + f(x) y'(x) + g(x) y(x) $$

(10)

for all $x \in I$, then it follows from (9) that

$$ \bigl| s(x) - r(x) \bigr| \leq\varphi(x) $$

(11)

for all $x \in I$. In view of Lemma 3.1 and (10), there exist real-valued constants $\alpha_{1}$ and $\alpha_{2}$ such that

$$\begin{aligned} y(x) = & \alpha_{1} y_{1}(x) + \alpha_{2} y_{2}(x) \\ &{} - y_{1}(x) \int_{a_{1}}^{x} \frac{y_{2}(t) s(t)}{W(y_{1}, y_{2})(t)} \,dt + y_{2}(x) \int_{a_{2}}^{x} \frac{y_{1}(t) s(t)}{W(y_{1}, y_{2})(t)}\, dt, \end{aligned}$$

(12)

where $a_{1}, a_{2} \in I$ are arbitrarily chosen and $W(y_{1}, y_{2})(t) \neq0$ for all $t \in I$ because $y_{1}$ and $y_{2}$ are linearly independent.

We now define a function $y_{0} : I \to\mathbf{R}$ by

$$\begin{aligned} y_{0}(x) := & \alpha_{1} y_{1}(x) + \alpha_{2} y_{2}(x) \\ &{} - y_{1}(x) \int_{a_{1}}^{x} \frac{y_{2}(t) r(t)}{W(y_{1}, y_{2})(t)} \,dt + y_{2}(x) \int_{a_{2}}^{x} \frac{y_{1}(t) r(t)}{W(y_{1}, y_{2})(t)} \,dt \end{aligned}$$

(13)

for each $x \in I$. According to Lemma 3.1, it is obvious that $y_{0}$ is a solution of (7). Moreover, it follows from (11), (12), and (13) that

$$\begin{aligned} &\bigl| y(x) - y_{0}(x) \bigr| \\ &\quad = \biggl| y_{1}(x) \int_{a_{1}}^{x} \frac{y_{2}(t)}{W(y_{1}, y_{2})(t)} \bigl( r(t) - s(t) \bigr) \,dt + y_{2}(x) \int_{a_{2}}^{x} \frac{y_{1}(t)}{W(y_{1}, y_{2})(t)} \bigl( s(t) - r(t) \bigr) \,dt \biggr| \\ &\quad \leq \bigl| y_{1}(x) \bigr| \biggl| \int_{a_{1}}^{x} \biggl| \frac{y_{2}(t)}{W(y_{1}, y_{2})(t)} \biggr| \varphi(t) \,dt \biggr| + \bigl| y_{2}(x) \bigr| \biggl| \int _{a_{2}}^{x} \biggl| \frac{y_{1}(t)}{W(y_{1}, y_{2})(t)} \biggr| \varphi(t) \,dt \biggr| \end{aligned}$$

(14)

for any $x \in I$. □

If we set $c := a_{1} = a_{2}$ in Theorem 3.2 and use the equality (14), then we obtain the following corollary.

Corollary 3.3

Let $f, g, r : I \to\mathbf{R}$ be continuous functions. Assume that the homogeneous differential equation (8) has a general solution $y_{h} : I \to\mathbf{R}$ of the form $y_{h}(x) = c_{1} y_{1}(x) + c_{2} y_{2}(x)$, where $c_{1}$ and $c_{2}$ are arbitrary real-valued constants. If a twice continuously differentiable function $y : I \to\mathbf{R}$ satisfies the inequality (9) for all $x \in I$, where $\varphi: I \to[0,\infty)$ is given such that the following integral exists, then there exists a solution $y_{0} : I \to\mathbf{R}$ of (7) such that

$$\bigl| y(x) - y_{0}(x) \bigr| \leq \biggl| \int_{c}^{x} \biggl| \frac{y_{1}(x)y_{2}(t) - y_{1}(t)y_{2}(x)}{W(y_{1}, y_{2})(t)} \biggr| \varphi(t) \,dt \biggr| $$

for all $x \in I$, where c is an arbitrarily chosen point of I.

4 Hyers-Ulam stability of Cauchy-Euler equation

In this section, we consider the (inhomogeneous) Cauchy-Euler (differential) equation

$$ x^{2} y''(x) + \alpha x y'(x) + \beta y(x) = r(x), $$

(15)

where α and β are real-valued coefficients and $r : (0,\infty) \to\mathbf{R}$ is a differentiable function, and we will investigate the generalized Hyers-Ulam stability of this differential equation. Indeed, the generalized Hyers-Ulam stability of the Cauchy-Euler equation (15) has been proved under some additional conditions (see [26, 27]).

By using Theorem 2.1 and Corollary 3.3, we prove the generalized Hyers-Ulam stability of the Cauchy-Euler equation (15) for the case of $(\alpha- 1)^{2} - 4 \beta> 0$.

Theorem 4.1

If the real-valued constants α and β are given with $(\alpha- 1)^{2} - 4 \beta> 0$, then the Cauchy-Euler equation (15) has the generalized Hyers-Ulam stability. In particular, let c be a positive real-valued constant and let $m_{1}$, $m_{2}$ be the distinct roots of the indicial equation $m^{2} + (\alpha- 1) m + \beta= 0$, i.e.,

$$ m_{1} = \frac{-(\alpha-1) - \sqrt{(\alpha-1)^{2} - 4\beta}}{2},\qquad m_{2} = \frac{-(\alpha-1) + \sqrt{(\alpha-1)^{2} - 4\beta}}{2}. $$

(16)

If $r : (0,\infty) \to\mathbf{R}$ is a differentiable function and $y : (0,\infty) \to\mathbf{R}$ is a twice continuously differentiable function such that the inequality

$$ \bigl| x^{2} y''(x) + \alpha x y'(x) + \beta y(x) - r(x) \bigr| \leq\varphi(x) $$

(17)

holds for any $x \in(0,\infty)$, where $\varphi: (0,\infty) \to[0,\infty)$ is a given function such that the following integral exists, then there exists a solution $y_{0} : (0,\infty) \to\mathbf{R}$ of the inhomogeneous Cauchy-Euler equation (15) such that

$$\bigl| y(x) - y_{0}(x) \bigr| \leq\frac{1}{m_{2} - m_{1}} \biggl| \int _{c}^{x} \biggl| \biggl( \frac{x}{\zeta} \biggr)^{m_{1}} - \biggl( \frac{x}{\zeta} \biggr)^{m_{2}} \biggr| \frac{\varphi(\zeta)}{\zeta} \,d\zeta\biggr| $$

for all $x \in(0,\infty)$.

Proof

If we define a monotone one-to-one correspondence $\tau: (0,\infty) \to\mathbf{R}$ by

$$\tau(x) := \ln x = t, $$

then $x = e^{t}$ for each $t \in\mathbf{R}$. We now define a twice continuously differentiable function $z : \mathbf{R} \to\mathbf{R}$ by

$$z(t) := y(x) = y \bigl( e^{t} \bigr) $$

and we get

$$\begin{aligned}& x y'(x) = x \frac{d}{dx} y(x) = x \frac{d}{dt} y \bigl( e^{t} \bigr) \frac{dt}{dx} = z'(t), \\& x^{2} y''(x) = x^{2} \frac{d}{dx} y'(x) = x^{2} \frac{d}{dt} \bigl( e^{-t} z'(t) \bigr) \frac{dt}{dx} = z''(t) - z'(t). \end{aligned}$$

Using these relations, we can reduce the Cauchy-Euler equation (15) to the linear differential equation

$$ z''(t) + (\alpha-1) z'(t) + \beta z(t) = r \bigl( e^{t} \bigr). $$

(18)

Similarly, the inverse $\sigma: \mathbf{R} \to(0,\infty)$ of τ given by $\sigma(t) := e^{t} = x$ reduces the linear differential equation (18) to the Cauchy-Euler equation (15).

Furthermore, by Corollary 3.3, the linear differential equation (18) has the generalized Hyers-Ulam stability. Therefore, due to Theorem 2.1, the Cauchy-Euler equation (15) has the generalized Hyers-Ulam stability.

In fact, as we did for (18), we can apply the monotone one-to-one correspondence τ to reduce the inequality (17) to

$$ \bigl| z''(t) + (\alpha-1) z'(t) + \beta z(t) - r \bigl( e^{t} \bigr) \bigr| \leq\varphi \bigl( e^{t} \bigr) $$

(19)

for all $t \in\mathbf{R}$. According to (13) and Corollary 3.3 with $z(t)$, $\alpha-1$, β, $r(e^{t})$, $\varphi(e^{t})$, and $(\ln c)$ instead of $y(x)$, $f(x)$, $g(x)$, $r(x)$, $\varphi(x)$, and a, respectively, there exist real-valued constants $c_{1}$ and $c_{2}$ such that

$$\begin{aligned} z_{0}(t) = & c_{1} e^{m_{1} t} + c_{2} e^{m_{2} t} \\ &{} - \frac{e^{m_{1} t}}{m_{2} - m_{1}} \int_{\ln c}^{t} e^{-m_{1} \eta} r \bigl( e^{\eta} \bigr)\, d\eta+ \frac{e^{m_{2} t}}{m_{2} - m_{1}} \int_{\ln c}^{t} e^{-m_{2} \eta} r \bigl( e^{\eta} \bigr)\, d\eta \end{aligned}$$

and

$$\bigl| z(t) - z_{0}(t) \bigr| \leq\frac{1}{m_{2} - m_{1}} \biggl| \int _{\ln c}^{t} \bigl| e^{m_{1} (t-\eta)} - e^{m_{2} (t-\eta)} \bigr| \varphi \bigl( e^{\eta} \bigr) \,d\eta \biggr| $$

for any $t \in\mathbf{R}$.

If we set $t = \ln x$ and $z_{0}(t) = y_{0}(x)$ in the previous equality for $z_{0}(t)$ and if we substitute ζ for $e^{\eta}$ in the integrals, then we get

$$y_{0}(x) = c_{1} x^{m_{1}} + c_{2} x^{m_{2}} - \frac{x^{m_{1}}}{m_{2} - m_{1}} \int_{c}^{x} \frac{r(\zeta)}{\zeta^{m_{1}+1}} \,d\zeta+ \frac{x^{m_{2}}}{m_{2} - m_{1}} \int_{c}^{x} \frac{r(\zeta)}{\zeta^{m_{2}+1}} \,d\zeta, $$

which is a solution of the inhomogeneous Cauchy-Euler equation (15). Moreover, if we set $t = \ln x$, $z(t) = y(x)$, $z_{0}(t) = y_{0}(x)$, and if we substitute ζ for $e^{\eta}$ in the integral of the last inequality for $| z(t) - z_{0}(t) |$, then we obtain the inequality for $| y(x) - y_{0}(x) |$ given in the statement of this theorem. □

If we set $\varphi(x) = \varepsilon$ in Theorem 4.1, then we get the following corollary.

Corollary 4.2

Assume that the real-valued constants α, β are given with $(\alpha- 1)^{2} - 4 \beta> 0$ and ε is an arbitrarily given positive constant. Let c be a positive real-valued constant and let $m_{1}$, $m_{2}$ be given as (16). If $r : (0,\infty) \to\mathbf{R}$ is a differentiable function and $y : (0,\infty) \to\mathbf{R}$ is a twice continuously differentiable function such that the inequality

$$ \bigl| x^{2} y''(x) + \alpha x y'(x) + \beta y(x) - r(x) \bigr| \leq\varepsilon $$

(20)

holds for any $x \in(0,\infty)$, then there exists a solution $y_{0} : (0,\infty) \to\mathbf{R}$ of the inhomogeneous Cauchy-Euler equation (15) such that

$$\bigl| y(x) - y_{0}(x) \bigr| \leq\textstyle\begin{cases} \frac{\varepsilon}{m_{1} m_{2}} + \frac{\varepsilon}{m_{2} - m_{1}} ( \frac{1}{m_{2}} ( \frac{x}{c} )^{m_{2}} - \frac{1}{m_{1}} ( \frac{x}{c} )^{m_{1}} ) & (\textit{for } m_{1} \neq0 \neq m_{2}), \\ \frac{\varepsilon}{m_{2}^{2}} ( ( \frac{x}{c} )^{m_{2}} - 1 ) - \frac{\varepsilon}{m_{2}} \ln\frac{x}{c} & (\textit{for } m_{1} = 0), \\ \frac{\varepsilon}{m_{1}^{2}} ( ( \frac{x}{c} )^{m_{1}} - 1 ) - \frac{\varepsilon}{m_{1}} \ln\frac{x}{c}& (\textit{for } m_{2} = 0) \end{cases} $$

for all $x \in(0,\infty)$.

Proof

According to Theorem 4.1, there exists a solution $y_{0} : (0,\infty) \to\mathbf{R}$ of the inhomogeneous Cauchy-Euler equation (15) such that

$$\begin{aligned} \bigl| y(x) - y_{0}(x) \bigr| \leq& \frac{1}{m_{2} - m_{1}} \biggl| \int _{c}^{x} \biggl| \biggl( \frac{x}{\zeta} \biggr)^{m_{1}} - \biggl( \frac{x}{\zeta} \biggr)^{m_{2}} \biggr| \frac{\varepsilon}{\zeta} \,d\zeta \biggr| \\ = & \textstyle\begin{cases} \frac{\varepsilon}{m_{2} - m_{1}} \int_{c}^{x} ( \frac{x^{m_{2}}}{\zeta^{m_{2}+1}} - \frac{x^{m_{1}}}{\zeta^{m_{1}+1}} ) \,d\zeta & (\mbox{for } c \leq x), \\ \frac{\varepsilon}{m_{2} - m_{1}} \int_{x}^{c} ( \frac{x^{m_{1}}}{\zeta^{m_{1}+1}} - \frac{x^{m_{2}}}{\zeta^{m_{2}+1}} ) \,d\zeta & (\mbox{for } x < c) \end{cases}\displaystyle \\ = & \frac{\varepsilon}{m_{2} - m_{1}} \int_{c}^{x} \biggl( \frac{x^{m_{2}}}{\zeta^{m_{2}+1}} - \frac{x^{m_{1}}}{\zeta^{m_{1}+1}} \biggr) \,d\zeta \end{aligned}$$

for all $x \in(0,\infty)$. We can integrate the last inequality case by case and obtain the inequality for $| y(x) - y_{0}(x) |$. □

We now consider the case when $(\alpha-1)^{2} - 4\beta= 0$ and use Theorem 2.1 and Corollary 3.3 to prove the generalized Hyers-Ulam stability of the inhomogeneous Cauchy-Euler equation (15).

Theorem 4.3

If the real-valued constants α and β are given with $\alpha\neq1$ and $\beta= \frac{(\alpha- 1)^{2}}{4}$, then the Cauchy-Euler equation (15) has the generalized Hyers-Ulam stability. In particular, let c be a positive real-valued constant and let $\lambda= -\frac{\alpha- 1}{2}$. If $r : (0,\infty) \to\mathbf{R}$ is a differentiable function and $y : (0,\infty) \to\mathbf{R}$ is a twice continuously differentiable function such that the inequality

$$ \biggl| x^{2} y''(x) + \alpha x y'(x) + \frac{(\alpha- 1)^{2}}{4} y(x) - r(x) \biggr| \leq\varphi(x) $$

(21)

holds for each $x \in(0,\infty)$, where $\varphi: (0,\infty) \to[0,\infty)$ is a given function such that the following integral exists, then there exists a solution $y_{0} : (0,\infty) \to\mathbf{R}$ of the inhomogeneous Cauchy-Euler equation (15) with $\beta= \frac{(\alpha-1)^{2}}{4}$ such that

$$\bigl| y(x) - y_{0}(x) \bigr| \leq \biggl| \int_{c}^{x} \biggl| \ln\frac{x}{\zeta} \biggr| \biggl( \frac{x}{\zeta} \biggr)^{\lambda} \frac{\varphi(\zeta)}{\zeta} \,d\zeta \biggr| $$

for all $x \in(0,\infty)$.

Proof

Analogously to the proof of Theorem 4.1, we define a monotone one-to-one correspondence $\tau: (0,\infty) \to\mathbf{R}$ and a twice continuously differentiable function $z : \mathbf{R} \to\mathbf{R}$ by $\tau(x) = \ln x = t$ and $z(t) = y(x) = y(e^{t})$, respectively. In a similar way to the first part of the proof of Theorem 4.1, the Cauchy-Euler equation (15) has the generalized Hyers-Ulam stability.

In particular, we apply the monotone one-to-one correspondence τ to reduce the inequality (21) to

$$\biggl| z''(t) + (\alpha-1) z'(t) + \frac{(\alpha-1)^{2}}{4} z(t) - r \bigl( e^{t} \bigr) \biggr| \leq\varphi \bigl( e^{t} \bigr) $$

for any $t \in\mathbf{R}$. According to (13) and Corollary 3.3 with $z(t)$, $\alpha-1$, $\frac{(\alpha-1)^{2}}{4}$, $r(e^{t})$, $\varphi(e^{t})$, and $(\ln c)$ instead of $y(x)$, $f(x)$, $g(x)$, $r(x)$, $\varphi(x)$, and a, respectively, there exist real-valued constants $c_{1}$ and $c_{2}$ such that

$$z_{0}(t) = c_{1} e^{\lambda t} + c_{2} t e^{\lambda t} - e^{\lambda t} \int_{\ln c}^{t} \eta e^{-\lambda\eta} r \bigl( e^{\eta} \bigr) \,d\eta+ t e^{\lambda t} \int_{\ln c}^{t} e^{-\lambda\eta} r \bigl( e^{\eta} \bigr) \,d\eta $$

and

$$\bigl| z(t) - z_{0}(t) \bigr| \leq \biggl| \int_{\ln c}^{t} | t-\eta| e^{\lambda(t-\eta)} \varphi \bigl( e^{\eta} \bigr) \,d\eta \biggr| $$

for all $t \in\mathbf{R}$.

If we set $t = \ln x$ and $z_{0}(t) = y_{0}(x)$ in the previous equality for $z_{0}(t)$ and if we substitute ζ for $e^{\eta}$ in the integrals, then we get

$$y_{0}(x) = c_{1} x^{\lambda} + c_{2} x^{\lambda} \ln x - x^{\lambda} \int_{c}^{x} (\ln\zeta) \frac{r(\zeta)}{\zeta^{\lambda+1}} \,d\zeta+ x^{\lambda} (\ln x) \int _{c}^{x} \frac{r(\zeta)}{\zeta^{\lambda+1}} \,d\zeta, $$

which is obviously a solution of the inhomogeneous Cauchy-Euler equation (15) with $\beta= \frac{(\alpha-1)^{2}}{4}$. Furthermore, if we set $t = \ln x$, $z(t) = y(x)$, $z_{0}(t) = y_{0}(x)$, and if we substitute ζ for $e^{\eta}$ in the integral of the previous inequality for $| z(t) - z_{0}(t) |$, then we get the inequality for $| y(x) - y_{0}(x) |$ described in the statement of the present theorem. □

If we set $\varphi(x) = \varepsilon$ in Theorem 4.3, then we obtain the following corollary.

Corollary 4.4

Assume that the real-valued constants α and β are given with $\alpha\neq1$, $\beta= \frac{(\alpha- 1)^{2}}{4}$ and ε is an arbitrarily given positive constant. Let c be a positive real-valued constant and let $\lambda= -\frac{\alpha- 1}{2}$. If $r : (0,\infty) \to\mathbf{R}$ is a differentiable function and $y : (0,\infty) \to\mathbf{R}$ is a twice continuously differentiable function such that the inequality

$$\biggl| x^{2} y''(x) + \alpha x y'(x) + \frac{(\alpha- 1)^{2}}{4} y(x) - r(x) \biggr| \leq\varepsilon $$

holds for all $x \in(0,\infty)$, then there exists a solution $y_{0} : (0,\infty) \to\mathbf{R}$ of the inhomogeneous Cauchy-Euler equation (15) with $\beta= \frac{(\alpha-1)^{2}}{4}$ such that

$$\bigl| y(x) - y_{0}(x) \bigr| \leq\frac{\varepsilon}{\lambda^{2}} + \frac{\varepsilon}{\lambda} \biggl( \frac{x}{c} \biggr)^{\lambda} \biggl( \ln\frac{x}{c} - \frac{1}{\lambda} \biggr) $$

for all $x \in(0,\infty)$.

Proof

According to Theorem 4.3, there exists a solution $y_{0} : (0,\infty) \to\mathbf{R}$ of the inhomogeneous Cauchy-Euler equation (15) with $\beta= \frac{(\alpha-1)^{2}}{4}$ such that

$$\begin{aligned} \bigl| y(x) - y_{0}(x) \bigr| \leq& \biggl| \int_{c}^{x} \biggl| \ln\frac{x}{\zeta} \biggr| \biggl( \frac{x}{\zeta} \biggr)^{\lambda} \frac{\varepsilon}{\zeta} \,d\zeta \biggr| \\ = & \textstyle\begin{cases} \int_{c}^{x} ( \frac{x}{\zeta} )^{\lambda} ( \ln\frac{x}{\zeta} ) \frac{\varepsilon}{\zeta} \,d\zeta & (\mbox{for } c \leq x), \\ \int_{x}^{c} ( \frac{x}{\zeta} )^{\lambda} ( \ln\frac{\zeta}{x} ) \frac{\varepsilon}{\zeta} \,d\zeta & (\mbox{for } x < c) \end{cases}\displaystyle \\ = & \int_{c}^{x} \frac{\varepsilon x^{\lambda}}{\zeta^{\lambda+1}} \ln \frac{x}{\zeta} \,d\zeta \\ = & \frac{\varepsilon}{\lambda^{2}} + \frac{\varepsilon}{\lambda} \biggl( \frac{x}{c} \biggr)^{\lambda} \biggl( \ln\frac{x}{c} - \frac{1}{\lambda} \biggr) \end{aligned}$$

for all $x \in(0,\infty)$. □

We apply Theorem 2.1 and Corollary 3.3 to prove the generalized Hyers-Ulam stability of the Cauchy-Euler equation (15) for the case of $(\alpha-1)^{2} - 4\beta< 0$.

Theorem 4.5

If the real-valued constants α and β are given with $(\alpha- 1)^{2} - 4 \beta< 0$, then the Cauchy-Euler equation (15) has the generalized Hyers-Ulam stability. In particular, let $c > 0$ be a given real-valued constant and let

$$\lambda= -\frac{\alpha- 1}{2} \quad\textit{and}\quad \mu= \frac{1}{2} \sqrt{4\beta- (\alpha- 1)^{2}}. $$

If $r : (0,\infty) \to\mathbf{R}$ is a differentiable function and $y : (0,\infty) \to\mathbf{R}$ is a twice continuously differentiable function such that the inequality (17) holds for all $x \in(0,\infty)$, where $\varphi: (0,\infty) \to[0,\infty)$ is a given function such that the following integral exists, then there exists a solution $y_{0} : (0,\infty) \to\mathbf{R}$ of the inhomogeneous Cauchy-Euler equation (15) such that

$$\bigl| y(x) - y_{0}(x) \bigr| \leq\frac{1}{\mu} \biggl| \int _{c}^{x} \frac{x^{\lambda}}{\zeta^{\lambda+1}} \biggl| \sin \biggl( \mu\ln \frac{x}{\zeta} \biggr)\biggr| \varphi(\zeta) \,d\zeta \biggr| $$

for all $x \in(0,\infty)$.

Proof

In a similar way to the proofs of Theorems 4.1 and 4.3, we conclude that the Cauchy-Euler equation (15) has the generalized Hyers-Ulam stability.

Using the monotone one-to-one correspondence $\tau: (0,\infty) \to\mathbf{R}$ defined by $\tau(x) = \ln x$, we can reduce the inequality (17) to (19) and we apply (13) and Corollary 3.3 to verify the existence of real-valued constants $c_{1}$ and $c_{2}$ such that

$$\begin{aligned} z_{0}(t) = & c_{1} e^{\lambda t} \cos\mu t + c_{2} e^{\lambda t} \sin\mu t - \frac{e^{\lambda t} \cos\mu t}{\mu} \int _{\ln c}^{t} e^{-\lambda\eta} (\sin\mu\eta) r \bigl( e^{\eta} \bigr) \,d\eta \\ &{} + \frac{e^{\lambda t} \sin\mu t}{\mu} \int_{\ln c}^{t} e^{-\lambda\eta} (\cos\mu\eta) r \bigl( e^{\eta} \bigr) \,d\eta \end{aligned}$$

and

$$\bigl| z(t) - z_{0}(t) \bigr| \leq\frac{1}{\mu} \biggl| \int _{\ln c}^{t} e^{\lambda(t-\eta)} \bigl| \sin\mu(t-\eta)\bigr| \varphi \bigl( e^{\eta} \bigr) \,d\eta \biggr| $$

for all $t \in\mathbf{R}$, where $\mu> 0$.

If we set $t = \ln x$ and $z_{0}(t) = y_{0}(x)$ in the previous equality for $z_{0}(t)$ and if we substitute ζ for $e^{\eta}$ in the integrals, then we get

$$\begin{aligned} y_{0}(x) = & c_{1} x^{\lambda}\cos(\mu\ln x) + c_{2} x^{\lambda}\sin(\mu\ln x) \\ &{} - \frac{x^{\lambda}\cos(\mu\ln x)}{\mu} \int_{c}^{x} \frac{\sin(\mu\ln\zeta)}{\zeta^{\lambda+1}} r(\zeta) \,d\zeta \\ &{} + \frac{x^{\lambda}\sin(\mu\ln x)}{\mu} \int_{c}^{x} \frac{\cos(\mu\ln\zeta)}{\zeta^{\lambda+1}} r(\zeta)\, d\zeta, \end{aligned}$$

which is obviously a solution of the inhomogeneous Cauchy-Euler equation (15) with $(\alpha- 1)^{2} - 4 \beta< 0$. Finally, if we let $t = \ln x$, $z(t) = y(x)$, $z_{0}(t) = y_{0}(x)$, and if we substitute ζ for $e^{\eta}$ in the integral of the inequality for $| z(t) - z_{0}(t) |$, then we obtain the inequality for $| y(x) - y_{0}(x) |$ given in the present theorem. □

If we set $\varphi(x) = \varepsilon$ in Theorem 4.5, then we can easily prove the following corollary.

Corollary 4.6

Assume that the real-valued constants α and β are given with $(\alpha- 1)^{2} - 4 \beta< 0$ and ε is an arbitrarily given positive constant. Let $c > 0$ be a given real-valued constant and let

$$\lambda= -\frac{\alpha- 1}{2} \quad\textit{and}\quad \mu= \frac{1}{2} \sqrt{4\beta- (\alpha- 1)^{2}}. $$

If a differentiable function $r : (0,\infty) \to\mathbf{R}$ and a twice continuously differentiable function $y : (0,\infty) \to\mathbf{R}$ satisfy the inequality (20) for all $x \in(0,\infty)$, then there exists a solution $y_{0} : (0,\infty) \to\mathbf{R}$ of the inhomogeneous Cauchy-Euler equation (15) such that

$$\begin{aligned} &\bigl| y(x) - y_{0}(x) \bigr| \\ &\quad \leq \textstyle\begin{cases} \varepsilon \vert \frac{1}{\beta\mu} ( \frac{x}{c} )^{\lambda} ( \lambda \sin ( \mu\ln\frac{x}{c} ) - \mu \cos ( \mu\ln\frac{x}{c} ) ) + \frac{(-1)^{m_{x}}}{\beta} e^{\lambda\frac{\pi}{\mu} m_{x}} \vert \\ \quad{}+ \varepsilon\sum_{m=1}^{m_{x}} \frac{1}{| \beta|} e^{\lambda\frac{\pi}{\mu} (m-1)} | e^{\lambda\frac{\pi}{\mu}} + 1 | \quad(\textit{for } x \geq c), \\ \varepsilon \vert \frac{(-1)^{m_{c}}}{\beta\mu} ( \frac{x}{c} )^{\lambda} e^{\lambda\frac{\pi}{\mu} m_{c}} ( \lambda \sin ( \mu\ln\frac{x}{c} ) - \mu \cos ( \mu\ln\frac{x}{c} ) ) + \frac{1}{\beta} \vert \\ \quad{}+ \varepsilon\sum_{m=1}^{m_{c}} | \frac{1}{\beta\mu} ( \frac{x}{c} )^{\lambda} e^{\lambda\frac{\pi}{\mu} (m-1)} ( \lambda \sin ( \mu\ln\frac{x}{c} ) - \mu \cos ( \mu\ln\frac{x}{c} ) ) \\ \quad{} \times ( e^{\lambda\frac{\pi}{\mu}} + 1 ) |\hspace{77.5pt} (\textit{for } 0 < x < c), \end{cases}\displaystyle \end{aligned}$$

where $m_{x}$ and $m_{c}$ are defined in (23) and (25).

Proof

According to Theorem 4.5, there exists a solution $y_{0} : (0,\infty) \to\mathbf{R}$ of the inhomogeneous Cauchy-Euler equation (15) such that

$$\begin{aligned} \bigl| y(x) - y_{0}(x) \bigr| \leq& \frac{\varepsilon}{\mu} \biggl| \int _{c}^{x} \frac{x^{\lambda}}{\zeta^{\lambda+1}} \biggl| \sin \biggl( \mu\ln \frac{x}{\zeta} \biggr) \biggr| \,d\zeta \biggr| \\ = & \textstyle\begin{cases} \frac{\varepsilon}{\mu} \int_{c}^{x} \frac{x^{\lambda}}{\zeta^{\lambda+1}} | \sin ( \mu\ln\frac{x}{\zeta} ) |\, d\zeta & (\mbox{for } c \leq x), \\ \frac{\varepsilon}{\mu} \int_{x}^{c} \frac{x^{\lambda}}{\zeta^{\lambda+1}} | \sin ( \mu\ln\frac{x}{\zeta} ) | \,d\zeta & (\mbox{for } x < c) \end{cases}\displaystyle \end{aligned}$$

(22)

for all $x \in(0,\infty)$.

If $0 < c \leq x$ then we set

$$ \gamma_{x}(m) := x e^{-\frac{m \pi}{\mu}} \quad\mbox{and} \quad m_{x} := \biggl[ \frac{\mu}{\pi} \ln\frac{x}{c} \biggr], $$

(23)

where $[z]$ denotes the greatest integer not exceeding the given real number z. Then we have

$$ [ c, x ] = \bigl[ c, \gamma_{x}(m_{x}) \bigr] \cup \bigcup_{m=1}^{m_{x}} \bigl[ \gamma_{x}(m), \gamma_{x}(m-1) \bigr] $$

(24)

for each $x \geq c$. Hence, it follows from (22) and (24) that

$$\begin{aligned} \bigl| y(x) - y_{0}(x) \bigr| \leq& \frac{\varepsilon}{\mu} \biggl| \int _{c}^{\gamma_{x}(m_{x})} \frac{x^{\lambda}}{\zeta^{\lambda+1}} \sin \biggl( \mu\ln \frac{x}{\zeta} \biggr) \,d\zeta \biggr| \\ &{} + \frac{\varepsilon}{\mu} \sum_{m=1}^{m_{x}} \biggl| \int_{\gamma_{x}(m)}^{\gamma_{x}(m-1)} \frac{x^{\lambda}}{\zeta^{\lambda+1}} \sin \biggl( \mu \ln\frac{x}{\zeta} \biggr) \,d\zeta \biggr| \end{aligned}$$

for any $x \geq c$. Moreover, if we substitute $\eta= \ln\frac{x}{\zeta}$ in the above integrals, then we have

$$\begin{aligned} &\bigl| y(x) - y_{0}(x) \bigr| \\ &\quad \leq \frac{\varepsilon}{\mu} \biggl| -\int_{\ln\frac{x}{c}}^{\frac{\pi}{\mu} m_{x}} e^{\lambda\eta} \sin(\mu\eta) \,d\eta \biggr| + \frac{\varepsilon}{\mu} \sum _{m=1}^{m_{x}} \biggl| -\int_{\frac{\pi}{\mu} m}^{\frac{\pi}{\mu} (m-1)} e^{\lambda\eta} \sin(\mu\eta) \,d\eta \biggr| \\ &\quad = \varepsilon \biggl\vert \frac{1}{\beta\mu} \biggl( \frac{x}{c} \biggr)^{\lambda} \biggl( \lambda \sin \biggl( \mu\ln\frac{x}{c} \biggr) - \mu \cos \biggl( \mu\ln\frac{x}{c} \biggr) \biggr) + \frac{(-1)^{m_{x}}}{\beta} e^{\lambda\frac{\pi}{\mu} m_{x}} \biggr\vert \\ &\qquad{} + \varepsilon\sum_{m=1}^{m_{x}} \frac{1}{| \beta|} e^{\lambda\frac{\pi}{\mu} (m-1)} \bigl| e^{\lambda\frac{\pi}{\mu}} + 1 \bigr| \end{aligned}$$

for all $x \geq c$.

If $0 < x < c$ then we set

$$ \gamma_{c}(m) := c e^{-\frac{m \pi}{\mu}} \quad\mbox{and} \quad m_{c} := \biggl[ \frac{\mu}{\pi} \ln\frac{c}{x} \biggr]. $$

(25)

Then we obtain

$$ [ x, c ] = \bigl[ x, \gamma_{c}(m_{c}) \bigr] \cup \bigcup_{m=1}^{m_{c}} \bigl[ \gamma_{c}(m), \gamma_{c}(m-1) \bigr] $$

(26)

for any $0 < x < c$. Thus, it follows from (22) and (26) that

$$\begin{aligned} \bigl| y(x) - y_{0}(x) \bigr| \leq& \frac{\varepsilon}{\mu} \biggl| \int _{x}^{\gamma_{c}(m_{c})} \frac{x^{\lambda}}{\zeta^{\lambda+1}} \sin \biggl( \mu\ln \frac{x}{\zeta} \biggr) \,d\zeta \biggr| \\ &{} + \frac{\varepsilon}{\mu} \sum_{m=1}^{m_{c}} \biggl| \int_{\gamma_{c}(m)}^{\gamma_{c}(m-1)} \frac{x^{\lambda}}{\zeta^{\lambda+1}} \sin \biggl( \mu \ln\frac{x}{\zeta} \biggr) \,d\zeta \biggr| \end{aligned}$$

for each $0 < x < c$. Furthermore, if we substitute $\eta= \ln\frac{x}{\zeta}$ in the last integrals, then we have

$$\begin{aligned} &\bigl| y(x) - y_{0}(x) \bigr| \\ &\quad \leq \frac{\varepsilon}{\mu} \biggl| -\int_{0}^{\frac{\pi}{\mu} m_{c} + \ln\frac{x}{c}} e^{\lambda\eta} \sin(\mu\eta) \,d\eta \biggr| + \frac{\varepsilon}{\mu} \sum _{m=1}^{m_{c}} \biggl| -\int_{\frac{\pi}{\mu} m + \ln\frac{x}{c}} ^{\frac{\pi}{\mu} (m-1) + \ln\frac{x}{c}} e^{\lambda\eta} \sin(\mu\eta) \,d\eta \biggr| \\ &\quad = \varepsilon \biggl\vert \frac{(-1)^{m_{c}}}{\beta\mu} \biggl( \frac{x}{c} \biggr)^{\lambda} e^{\lambda\frac{\pi}{\mu} m_{c}} \biggl( \lambda \sin \biggl( \mu\ln \frac{x}{c} \biggr) - \mu \cos \biggl( \mu\ln\frac{x}{c} \biggr) \biggr) + \frac{1}{\beta} \biggr\vert \\ &\qquad{} + \varepsilon\sum_{m=1}^{m_{c}} \biggl\vert \frac{1}{\beta\mu} \biggl( \frac{x}{c} \biggr)^{\lambda} e^{\lambda\frac{\pi}{\mu} (m-1)} \biggl( \lambda \sin \biggl( \mu\ln\frac{x}{c} \biggr) - \mu \cos \biggl( \mu\ln\frac{x}{c} \biggr) \biggr) \bigl( e^{\lambda\frac{\pi}{\mu}} + 1 \bigr) \biggr\vert \end{aligned}$$

for all $0 < x < c$. □