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 Stability results and existence theorems for nonlinear delay-fractional differential equations with $\Phi_p$-operator Cemil - Tunc Keywords:Delay-Fractional differential equations with singularity; Caputo’s fractional derivative; Hyers-Ulam stability; existence and uniqueness of solution Abstract: \abstract{}The study about delay-fractional differential equations (fractional DEs) have recently been got a great attention of scientists in many different subjects based on mathematically modeling. In the study of fractional DEs the first question raises is whether the problem will have a solution or not. Also, whether the problem is stable or not? In order to ensure the answer of these questions, we study the study existence and uniqueness of solutions (EUS) as well Hyers-Ulam stability for our proposed problem a nonlinear fractional DE with an operator $\Phi_p$ and a non zero delay $\tau>0$ in Banach’s space $\mathcal{Y}$ of order $n-1<\sigma,\,\beta\leq n$, $n\geq 3$, in the Caputo's sense. The assumed singular fractional DE with $\Phi_p$-operator is more general and complex than studied for stabilities by \textit{Khan et al. Eur Phys J Plus, (2018);133:26.}\\