On the E -Hyperstability of the Inhomogeneous σ -Jens

In this paper, we study the hyperstability problem for the well-known $\sigma$ -Jensen’s functional equation $f\left(xy\right)+f\left(x\sigma \left(y\right)\right)=2f\left(x\right)$ for all $x,y\in S$ , where $S$ is a semigroup and $\sigma$ is an involution of $S$ . We present sufficient conditions on $\mathcal{E}\subset {\left({ℝ}_{+}\right)}^{S^2}$ so that the inhomogeneous form of $\sigma$ -Jensen’s functional equation $f\left(xy\right)+f\left(x\sigma \left(y\right)\right)=2f\left(x\right)+\phi \left(x,y\right)$ for all $x,y\in S$ , where the inhomogeneity $\phi$ is given, can be $\mathcal{E}$ -hyperstable on $S$ .