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In the current work, a new notion of $n$-weak amenability of Banach algebras using homomorphisms, namely $(\varphi ,\psi )$-$n$-weak amenability is introduced. Among many other things, some relations between $(\varphi ,\psi )$-$n$-weak amenability of a Banach algebra $𝒜$ and $M_m\left(𝒜\right)$, the Banach algebra of $m\times m$ matrices with entries from $𝒜$, are studied. Also, the relation of this new concept of amenability of a Banach algebra and its unitization is investigated. As an example, it is shown that the group algebra $L^1\left(G\right)$ is ($\varphi ,\psi$)-$n$-weakly amenable for any...

In this paper, we investigate the general solution and Hyers–Ulam–Rassias stability of a new mixed type of additive and quintic functional equation of the form [...] f(3x+y)−5f(2x+y)+f(2x−y)+10f(x+y)−5f(x−y)=10f(y)+4f(2x)−8f(x) $$f\left(3x+y\right)-5f\left(2x+y\right)+f\left(2x-y\right)+10f\left(x+y\right)-5f\left(x-y\right)=10f\left(y\right)+4f\left(2x\right)-8f\left(x\right)$$ in the set of real numbers.