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

• * Corresponding author: Cemil Tunc
• 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.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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