We investigate the approximate solutions of the delay differential equation with an initial condition, where and are real constants. We show that they can be “approximated” by solutions of the equation that are constant on the interval and, therefore, have quite simple forms. Our results correspond to the notion of stability introduced by Ulam and Hyers. 1. Introduction While investigating real world phenomena we very often use equations. In general, it is well known that those equations are satisfied, however, with some error. Sometimes that error is neglected and it is believed that this will have only a minor influence on the final outcome. Since it is not always the case, it seems to be of interest to know when we can neglect the error, why, and to what extent. One of the tools for systematic treatment of the problem described above seems to be the notion of Hyers-Ulam stability and some ideas inspired by it. That notion has not actually been made very precise so far, and we still seek a better understanding of it (see, e.g., [1, 2]). But, roughly speaking, we might say that it is connected with the investigation of the following question: when is a function satisfying an equation with some “small” (in some sense) error “close” to a solution of that equation? The study of the stability problem for functional equations starts from the famous talk of Ulam and the partial solution of Hyers to Ulam's problem (see [3, 4]). Thereafter, Hyers and Ulam (see, e.g., [4–8]), but also several other authors (see, e.g., [9–12]), attempted to study the stability problem for various functional equations. In particular, we should mention here the well-known paper [13] of Th.M. Rassias, in which he actually rediscovered the result of Aoki [9] (cf. [14]), and which has significantly influenced research of numerous mathematicians (see [15–21] and the references therein). Assume that and are normed spaces and that is an open subset of . Let be a class of differentiable functions mapping into . If for any real and any function satisfying the differential inequality there exists a solution of the differential equation such that (where depends on only), then we say that the above differential equation is Hyers-Ulam stable in the class of function . We may use this terminology for other differential equations. For more detailed definitions of the Hyers-Ulam stability and some discussions and critiques of that subject, refer to [1, 2, 4, 13, 15, 16, 18–21]. Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see
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