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Approximate perfect differential equations of second order
Advances in Difference Equations volume 2012, Article number: 225 (2012)
In this paper we prove the Hyers-Ulam stability of the perfect linear differential equation , where , , and .
MSC:34K20, 26D10, 39B82, 34K06, 39B72.
The question concerning the stability of group homomorphisms was posed by Ulam . Hyers  solved the case of approximately additive mappings in Banach spaces and T.M. Rassias generalized the result of Hyers .
Definition 1.1 Let X be a normed space over a scalar field and let I be an open interval. Assume that , are continuous functions. We say that the differential equation
has the Hyers-Ulam stability if, for any function satisfying the differential inequality
for all and some , there exists a solution of (1.1) such that for all , where is a function depending only on ε.
Jung  solved the inhomogeneous differential equation of the form , where n is a positive integer, and he used this result to prove the Hyers-Ulam stability of the differential equation in a special class of analytic functions.
Li and Shen  proved that if the characteristic equation has two different positive roots, then the linear differential equation of second order with constant coefficients has the Hyers-Ulam stability where , and (see also [9, 10]). Abdollahpour and Najati  proved that the third-order differential equation has the Hyers-Ulam stability. Ghaemi et al.  proved the Hyers-Ulam stability of the exact second-order linear differential equation
Definition 1.2 We say that the differential equation
is perfect if it can be written as .
It is clear that the differential equation (1.2) is perfect if and only if . The aim of this paper is to investigate the Hyers-Ulam stability of the perfect differential equation (1.2), where , , , and . More precisely, we prove that the equation (1.2) has the Hyers-Ulam stability.
2 Hyers-Ulam stability of the perfect differential equation
In the following theorem, we prove the Hyers-Ulam stability of the differential equation (1.2).
Throughout this section, a and b are real numbers with .
Theorem 2.1 The perfect differential equation
has the Hyers-Ulam stability, where , , and for all .
Proof Let and with
Let for all . It is clear that
Also, we have
for all . Now we define
for all . It is clear that and
Hence, (2.1) implies that
Also, we have
for all . Since , there exist constants and such that . Thus
for all . Since and , there exist constants such that for all . Hence, (2.4) implies that
for all . It follows from (2.3) that
for all . □
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CP was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299) and DYS was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).
The authors declare that they have no competing interests.
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.