- Research Article
- Open Access
Hyers-Ulam Stability of Differential Equation
© Soon-Mo Jung. 2010
- Received: 13 October 2009
- Accepted: 24 November 2009
- Published: 25 November 2009
We solve the inhomogeneous differential equation of the form , where is a nonnegative integer, and apply this result to the proof of a local Hyers-Ulam stability of the differential equation in a special class of analytic functions.
- Differential Equation
- General Solution
- Power Series
- Error Function
- Nonnegative Integer
Assume that and are a topological vector space and a normed space, respectively, and that is an open subset of . If for any function satisfying the differential inequality
for all and for some , there exists a solution of the differential equation
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:
for any . Suppose that we are given a nonnegative integer , and we introduce a differential equation
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 this paper, denotes the set of all nonnegative integers.
In the following theorem, we solve the inhomogeneous differential equation (1.8).
where is a solution of the homogeneous differential equation (1.6).
for all .
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
If is an even integer, then we have . Thus, for each even integer , the power series is a polynomial. By the second conditions in (2.1) and (2.6), we get
Therefore, the power series expression of converges for all .
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 this section, let be a nonnegative integer and let be a constant with . For a given , let us denote by the set of all functions with the properties (a) and (b):
(a) is represented by a power series whose radius of convergence is at least ;
(b)it holds true that for all , where for each .
It should be remarked that the power series in (b) has the same radius of convergence as that of given in (a).
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, § ]).
for any .
for each .
By Abel's formula (see [17, Theorem ]), we have
for each .
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
for all .
If is finite, then the local Hyers-Ulam stability of the differential equation (1.6) immediately follows from Theorem 3.1.
for any .
We now deal with an asymptotic behavior of functions in under the additional conditions (3.1) and (3.2).
for any .
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.
The first inequality in (3.2) is also true for all as we see in the following:
where denotes the largest integer not exceeding .
It is not difficult to show that
for all .
By using (4.2), we will now prove that
as : if for some , then
If for some , then
It then follows from (4.3) and (4.4) that
for all sufficiently large integers , which proves that the sequence satisfies the second inequality in (3.1).
Finally, we will show that the sequence satisfies the first inequality in (3.1). It follows from (4.3) that
for each .
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).
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle Scholar
- Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27(4):222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
- Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.View ArticleMATHGoogle Scholar
- Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
- Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978, 72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar
- Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, London, UK; 1960:xiii+150.Google Scholar
- Obłoza M: Hyers stability of the linear differential equation. Rocznik Naukowo-Dydaktyczny. Prace Matematyczne 1993, (13):259–270.MathSciNetMATHGoogle Scholar
- Obłoza M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Naukowo-Dydaktyczny. Prace Matematyczne 1997, (14):141–146.MathSciNetMATHGoogle Scholar
- Alsina C, Ger R: On some inequalities and stability results related to the exponential function. Journal of Inequalities and Applications 1998, 2(4):373–380. 10.1155/S102558349800023XMathSciNetMATHGoogle Scholar
- Takahasi S-E, Miura T, Miyajima S: On the Hyers-Ulam stability of the Banach space-valued differential equation . Bulletin of the Korean Mathematical Society 2002, 39(2):309–315. 10.4134/BKMS.2002.39.2.309MathSciNetView ArticleMATHGoogle Scholar
- Miura T, Jung S-M, Takahasi S-E: Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations . Journal of the Korean Mathematical Society 2004, 41(6):995–1005. 10.4134/JKMS.2004.41.6.995MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M: Legendre's differential equation and its Hyers-Ulam stability. Abstract and Applied Analysis 2007, 2007:-14.Google Scholar
- Jung S-M: Approximation of analytic functions by Airy functions. Integral Transforms and Special Functions 2008, 19(11–12):885–891.MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M: Approximation of analytic functions by Hermite functions. Bulletin des Sciences Mathematiques 2009, 133(7):756–764. 10.1016/j.bulsci.2007.11.001MathSciNetView ArticleMATHGoogle Scholar
- Kim B, Jung S-M: Bessel's differential equation and its Hyers-Ulam stability. Journal of Inequalities and Applications 2007, 2007:-8.Google Scholar
- Abramowitz M, Stegun IA: Handbook of Mathematical Functions. Dover, New York, NY, USA; 1972.MATHGoogle Scholar
- Wade WR: An Introduction to Analysis. 2nd edition. Prentice Hall, Upper Saddle River, NJ, USA; 2000.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.