Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations

In this paper, we are dealing with singular fractional differential equations (DEs) having delay and $ \mho_p $ ($ p $-Laplacian operator). In our problem, we Contemplate two fractional order differential operators that is Riemann–Liouville and Caputo's with fractional integral and fractional differential initial boundary conditions.The SFDE is given by

$ \begin{equation*} \left\{\begin{split} &\mathcal{D}^{\gamma}\big[\mho^*_p[\mathcal{D}^{\kappa}x(t)]\big]+\mathcal{Q}(t)\zeta_1(t, x(t-\varrho^*)) = 0, \\& \mathcal{I}_0^{1-\gamma}\big(\mho^*_p[\mathcal{D}^{\kappa}x(t)]\big)|_{t = 0} = 0 = \mathcal{I}_0^{2-\gamma}\big(\mho^*_p[\mathcal{D}^{\kappa}x(t)]\big)|_{t = 0}, \\& \mathcal{D}^{\delta^*}x(1) = 0, \, \, x(1) = x'(0), \, \, x^{(k)}(0) = 0\text{ for $k = 2, 3, \ldots, n-1$}, \end{split}\right. \end{equation*} $

$ \zeta_1 $ is a continuous function and singular at $ t $ and $ x(t) $ for some values of $ t\in [0, 1] $. The operator $ \mathcal{D}^{\gamma}, \, $ is Riemann–Liouville fractional derivative while $ \mathcal{D}^{\delta^*}, \mathcal{D}^{\kappa} $ stand for Caputo fractional derivatives and $ \delta^*, \, \gamma\in(1, 2] $, $ n-1<\kappa\leq n, $ where $ n\geq3 $. For the study of the EUS, fixed point approach is followed in this paper and an application is given to explain the findings.