Approximate perfect differential equations of second order
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In this paper we prove the Hyers-Ulam stability of the perfect linear differential equation , where , , and .
MSC:34K20, 26D10, 39B82, 34K06, 39B72.
KeywordsHyers-Ulam stability differential equation
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 .
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.
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 .
has the Hyers-Ulam stability, where , , and for all .
for all . □
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).
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