Automatic continuity of almost multiplicative maps between Fr\ʹʹ{e}chet algebras

For Fr$\acute{\mathbf{\text{e}}}$chet algebras $(A‎, ‎(p_n))$‎ ‎and $(B‎, ‎(q_n))$‎, ‎a linear map $T:A\rightarrow B$ is‎ ‎\textit{almost multiplicative} with respect to $(p_n)$ and‎ ‎$(q_n)$‎, ‎if there exists $\varepsilon\geq 0$ such that $q_n(Tab‎ - ‎Ta Tb)\leq \varepsilon p_n(a) p_n(b),$ for all $n \in \mathbb{N}$‎, ‎$a‎, ‎b \in A$‎, ‎and it is called \textit{weakly almost‎ ‎multiplicative} with respect to $(p_n)$ and $(q_n)$‎, ‎if there‎ ‎exists $\varepsilon\geq 0$ such that for every $k \in \mathbb{N}$‎, ‎there exists $n(k) \in \mathbb{N}$‎, ‎satisfying the inequality‎ ‎$q_k(Tab‎ - ‎Ta Tb)\leq \varepsilon p_{n(k)}(a) p_{n(k)}(b),$ for‎ ‎all $a‎, ‎b \in A$‎. ‎We investigate the automatic continuity of‎ ‎(weakly) almost multiplicative maps between certain classes of‎ ‎Fr$\acute{\mathbf{\text{e}}}$chet algebras‎, ‎such as Banach‎ ‎algebras and Fr$\acute{\mathbf{\text{e}}}$chet $Q$-algebras‎. ‎We‎ ‎also obtain some results on the automatic continuity of dense‎ ‎range (weakly) almost multiplicative maps between‎ ‎Fr$\acute{\mathbf{\text{e}}}$chet algebras‎.‎