Almost n-Multiplicative Maps‎ between‎ ‎Frechet Algebras

For the fr'{e}chet algebras (a,(pk)) and (b,(qk)) and n∈n, n≥2, a linear map t:a→b is called textit{almost n-multiplicative}, with respect to (pk) and (qk), if there exists ε≥0 such that qk(ta1a2⋯an−ta1ta2⋯tan)≤εpk(a1)pk(a2)⋯pk(an),for each k∈n and a1,a2,…,an∈a. the linear map t is called textit{weakly almost n-multiplicative}, if there exists ε≥0 such that for every k∈n there exists n(k)∈n with qk(ta1a2⋯an−ta1ta2⋯tan)≤εpn(k)(a1)pn(k)(a2)⋯pn(k)(an), for each k∈n and a1,a2,…,an∈a.the linear map t is called n-multiplicative if ta1a2⋯an=ta1ta2⋯tan, for every a1,a2,…,an∈a. in this paper, we investigate automatic continuity of (weakly) almost n-multiplicative maps between certain classes of fr'{e}chet algebras, including banach algebras. we show that if (a,(pk)) is a fr'{e}chet algebra and t:a→c is a weakly almost n-multiplicative linear functional, then either t is n-multiplicative, or it is continuous. moreover, if (a,(pk)) and (b,(qk)) are fr'{e}chet algebras and t:a→b is a continuous linear map, then under certain conditions t is weakly almost n-multiplicative for each n≥2. in particular, every continuous linear functional on a is weakly almost n-multiplicative for each n≥2.