Lazarsfeld-Mukai bundles on K3 surfaces associated to a pencil computing Clifford index

Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an ample curve on $X$. In this paper we will study the semistability of the Lazarsfeld-Mukai bundle $E_{C,A}$ associated to a line bundle $A$ ion $C$ such that $\mathrm{\mid }A\mathrm{\mid }$ is a pencil on $C$ and computes the Clifford index of $C$. We give a necessary and sufficient condition for $E_{C,A}$ being semistable.