Let $N$ be a finite simple centralizer near-ring that is not an exceptional near-field. A semiendomorphism of $N$ is a map â from $N$ into $N$ such that $(a + b)â = aâ + bâ,(aba)â = aâbâaâ$, and $1â = 1$ for all $a,b \in N$. It is shown that every semiendomorphism of $N$ is an automorphism of $N$. A Jordan-endomorphism of $N$ is a map â from $N$ into $N$ such that $(a + b)â = aâ + bâ,(ab + ba)â = aâbâ + bâaâ$, and $1â = 1$ for all $a,b \in N$. It is shown that every Jordan-endomorphism of $N$ is an automorphism assuming $2 \in N$ is invertible. The above results imply that every semiendomorphism (Jordan-endomorphism) of a "special" class of semisimple near-rings is an automorphism. These results are in contrast to the ring situation where semiendomorphisms tend to be either an automorphism or an antiautomorphism.