Generalized Andrásfai--Erdős--Sós theorems for odd cycles

In this note, we establish Andrásfai–Erdős–Sós-type stability theorems for two generalized Turán problems involving odd cycles, both of which are extensions of the Erdős Pentagon Problem. Our results strengthen previous results by Lidický–Murphy~\cite{LM21} and Beke–Janzer~\cite{BJ24}, while also simplifying parts of their proofs.