  $F$-Yang-Mills connections are critical points of $F$-Yang Mills functional
on the space of connections of a principal fiber bundle, which is a
generalization of Yang-Mills connections, $p$-Yang-Mills connections and
exponential Yang-Mills connections and so on. Here, $F$ is a strictly
increasing $C^{2}$-function. In this paper, we extend Simons theorem for an
instability of Yang-Mills connections to $F$-Yang-Mills connections. We derive
a sufficient condition that any non-flat, $F$-Yang-Mills connection over convex
hypersurfaces in a Euclidean space is instable. In the sphere case, this
condition is expressed by an inequality with respect to its dimension and a
degree of the differential of the function $F$. The proofs of the results are
given by extending Kobayashi-Ohnita-Takeuchi's calculation to $F$-Yang-Mills
connections.
