# American Institute of Mathematical Sciences

September  2020, 13(9): 2475-2487. doi: 10.3934/dcdss.2020139

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

 1 Department of Mathematics, Shaheed BB University, Sheringal, Dir Upper 18000, Khybar Pakhtunkhwa, Pakistan 2 Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080 Van, Turkey 3 Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia

* Corresponding author: Cemil Tunc

Received  January 2019 Revised  February 2019 Published  September 2020 Early access  November 2019

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.
Citation: Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139
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