Hyers-Ulam Stability of Third Order Euler’s Differential Equations

We investigate the Hyers-Ulam stability of third order Euler's differential equations of the form on any open interval , or , where , and are complex constants. 1. Introduction In 1940, Ulam gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems [1]. Among such problems is a problem concerning the stability of functional equations: give conditions in order for a linear function near an approximately linear function to exist. In the following year, Hyers [2] gave an answer to the problem of Ulam for additive functions defined on Banach spaces. Let and be two real Banach spaces and let . Then, for every function satisfying there exists a unique additive function with the property Furthermore, the result of Hyers has been generalized by Rassias [3, 4]. Since then, the stability problems of various functional equations have been investigated by many authors (see, e.g., [1, 5–8]). A generalization of Ulam’s problem was recently proposed by replacing functional equations with differential equations. The differential equation has Hyers-Ulam stability; if for given and a function such that then there exists a solution of the differential equation such that and . If the preceding statement is also true when we replace and by and , respectively, where and are appropriate functions not depending on and explicitly, then we say that the corresponding differential equation has the generalized Hyers-Ulam stability. Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see, e.g., [9, 10]). Thereafter, Alsina and Ger published their work [11], which handles the Hyers-Ulam stability of the linear differential equation . If a differentiable function is a solution of the inequality for any , then there exists a constant such that , for all . In [1], Rezaei et al. have discussed the Hyers-Ulam stability of linear differential equations of first and th order by applying Laplace transform which is comparable with the other methods available in the literature. It is Jung et al. who have investigated the Hyers-Ulam stability of linear differential equations of different classes including the stability of the delay differential equation , where is a constant (see, e.g., [5–7, 12–14]). Among the works, we are motivated by the results of [13], where he has studied the Hyers-Ulam stability of the following Euler’s differential equations: where , , and are complex constants. We may note that the Hyers-Ulam stability of (6) depends