On the hyperstability of a pexiderised $\unicode[stix]{x1d70e}$ -quadratic functional equation on semigroups

Motivated by the notion of Ulam stability, we investigate some inequalities connected with the functional equation $$\begin{eqnarray}f(xy)+f(x\unicode[STIX]{x1D70E}(y))=2f(x)+h(y),\quad x,y\in G,\end{eqnarray}$$ for functions $f$ and $h$ mapping a semigroup $(G,\cdot )$ into a commutative semigroup $(E,+)$ , where the map $\unicode[STIX]{x1D70E}:G\rightarrow G$ is an endomorphism of $G$ with $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D70E}(x))=x$ for all $x\in G$ . We derive from these results some characterisations of inner product spaces. We also obtain a description of solutions to the equation and hyperstability results for the $\unicode[STIX]{x1D70E}$ -quadratic and $\unicode[STIX]{x1D70E}$ -Drygas equations.