Introduction This work is motivated as an attempt to deal with an estimation of the relative error for the eigenvalues of the Sturm-Liouville problem. We are interested in finding upper bounds for such error. The uniform spectral continuity of this operator was treated in [14] in very simple manner. It is worth pointing out that the knowledge of the bounds for these two errors allows us to consider the eigenvalue problem for Sturm-Liouville operator in normal form to be perfectly posed for each

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... eigenvalue regardless of its index. Similar problem in the finite dimensional space setting was considered in [8] and [22] . Yet another approach to the bounds for the absolute errors of eigenvalues by the Gerschgorin estimate under unitary similarity has been recently investigated in [24]. Notation and the case of the absolute error Suppose that V = {u € H 2 (a, b) : au(a) + a'u'{a) = 0, (3u{b) + P'u'(b) = 0}. The constants a, a', ¡3, /?' are assumed to be real with a 2 +a' 2 > 0, f3 2 +f3' 2 > 0, and the interval [a, 6] is finite. For ueV, let Lu = -u" + qu, Lu = -u" + qu, where the real-valued functions q,q are assumed to belong to C([a, 6]). Now