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We show that some Gieseker stable sheaves on a projective K3 surface $X$ arestable with respect to a stability condition of Bridgeland on the derivedcategory of $X$ if the stability condition is in explicit subsets of the spaceof stability conditions depending on the sheaves. Furthermore we shall give twoapplications of the result. As a part of these applications, we show that thefine moduli space of Gieseker stable torsion free sheaves on a K3 surface withPicard number one is the moduli space of $\mu$-stable locally free sheaves ifthe rank of the sheaves is not a square number.

Stability of Gieseker stable sheaves on K3 surfaces in the sense of Bridgeland and some applications