- Research Article
- Open Access
© Soon-Mo Jung. 2010
- Received: 13 October 2009
- Accepted: 24 November 2009
- Published: 25 November 2009
- Differential Equation
- General Solution
- Power Series
- Error Function
- Nonnegative Integer
such that for any , where is an expression of only, then we say that the above differential equation satisfies the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain is not the whole space ). We may apply this terminology for other differential equations. For more detailed definition of the Hyers-Ulam stability, refer to [1–6].
Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [7, 8]). Here, we will introduce a result of Alsina and Ger (see ): If a differentiable function is a solution of the differential inequality , where is an open subinterval of , then there exists a solution of the differential equation such that for any .
Using the conventional power series method, the author in  investigated the general solution of the inhomogeneous Legendre differential equation of the form
under some specific conditions, where is a real number and the convergence radius of the power series is positive. Moreover, he applied this result to prove that every analytic function can be approximated in a neighborhood of by the Legendre function with an error bound expressed by (see [13–15]).
Let us consider the error function and the complementary error function defined by
respectively. We recursively define the integrals of the error function as follows:
whose general solution is given by
(see [16, § ]).
In Section 2 of this paper, using power series method, we will investigate the general solution of the inhomogeneous differential equation:
where the radius of convergence of the power series is , whose value is in general permitted to have infinity. Moreover, using the idea from [12–14], we will prove the Hyers-Ulam stability of the differential equation (1.6) in a class of special analytic functions (see the class in Section 3).
In the following theorem, we solve the inhomogeneous differential equation (1.8).
where is a solution of the homogeneous differential equation (1.6).
It is not difficult to see that
which proves that is a particular solution of the inhomogeneous equation (1.8).
We now apply the ratio test to the power series expression of . If is an odd integer not less than , then . Hence, the power series is a polynomial. And it follows from the first conditions of (2.1) and (2.6) that
Moreover, the convergence region of the power series for is the same as those of power series for and . In this paper, the convergence region will denote the maximum open set where the relevant power series converges. Hence, the power series expression for has the same convergence region as that of . This implies that is well defined on and so does for in (2.3) because converges for all under our hypotheses.
Since every solution to (1.8) can be expressed as a sum of a solution of the homogeneous equation and a particular solution of the inhomogeneous equation, every solution of (1.8) is certainly in the form of (2.3).
In the following theorem, we prove that if an analytic function satisfies some given conditions, then it can be approximated by a combination of integrals of the error function (see the last part of Section 1 or [16, § ]).
By Abel's formula (see [17, Theorem ]), we have
Finally, it follows from Theorem 2.1, (3.1), (3.2), (3.7), and (3.10) that there exists a solution function of the homogeneous differential equation (1.6) such that
If is finite, then the local Hyers-Ulam stability of the differential equation (1.6) immediately follows from Theorem 3.1.
According to Theorem 3.1, there exists a solution of the differential equation (1.6) satisfying
The conditions in (3.1) and (3.2) may seem too strong to construct some examples for the coefficients 's. In this section, however, we will show that the sequence given in Remark 2.2 satisfies these conditions: let and , for all and choose some constants and . The second inequality in (3.2) has been verified in Remark 2.2.
It is not difficult to show that
By using (4.2), we will now prove that
It then follows from (4.3) and (4.4) that
The author would like to express his cordial thanks to the referee for useful remarks which have improved the first version of this paper. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0071206).
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