  This paper investigates online composite optimization in dynamic
environments, where each objective or loss function contains a time-varying
nondifferentiable regularizer. To resolve it, an online proximal gradient
algorithm is studied for two distinct scenarios, including convex and strongly
convex objectives without the smooth condition. In both scenarios, unlike most
of works, an extended version of the conventional path variation is employed to
bound the considered performance metric, i.e., dynamic regret. In the convex
scenario, a bound $\mathcal{O}(\sqrt{T^{1-\beta}D_\beta(T)+T})$ is obtained
which is comparable to the best-known result, where $D_\beta(T)$ is the
extended path variation with $\beta\in[0,1)$ and $T$ being the total number of
rounds. In strongly convex case, a bound $\mathcal{O}(\log
T(1+T^{-\beta}D_\beta(T)))$ on the dynamic regret is established. In the end,
numerical examples are presented to support the theoretical findings.
