The coordinatewise uniformly Kadec–Klee property in some Banach spaces

We introduce a new property $UKK_c$ for a Banach space and show for an Orlicz sequence space the following: $UKK_c\Leftrightarrow H_c \Leftrightarrow\Phi\in\delta_2$. Moreover, we show that the Orlicz-direct-sum spaces $\Bigl(\sum\limits_{n=1}^\infty\oplus X_n\Bigr)_{l_{\Phi}}$ and $\Bigr(\sum\limits_{n=1}^\infty\oplus X_n\Bigr)_{l_{(\Phi)}}$ have the property $H_c$ if every $X_n$ $(n\in\mathbb{N})$ has the property $H_c$ and $\Phi\in\delta_2$.