New approach to solutions of a class of singular fractional q-differential problem via quantum calculus

In the present article, by using the fixed point technique and the Arzelà–Ascoli theorem on cones, we wish to investigate the existence of solutions for a non-linear problems regular and singular fractional q-differential equation $$ \bigl({}^{c}D_{q}^{\alpha }f\bigr) (t) = w \bigl(t, f(t), f'(t), \bigl({}^{c}D_{q}^{ \beta }f \bigr) (t) \bigr), $$(cDqαf)(t)=w(t,f(t),f′(t),(cDqβf)(t)), under the conditions $f(0) = c_{1} f(1)$f(0)=c1f(1), $f'(0)= c_{2} ({}^{c}D_{q} ^{\beta } f) (1)$f′(0)=c2(cDqβf)(1) and $f''(0) = f'''(0) = \cdots =f^{(n-1)}(0) = 0$f″(0)=f‴(0)=⋯=f(n−1)(0)=0, where $\alpha \in (n-1, n)$α∈(n−1,n) with $n\geq 3$n≥3, $\beta , q \in J=(0,1)$β,q∈J=(0,1), $c_{1} \in J$c1∈J, $c_{2} \in (0, \varGamma _{q} (2- \beta ))$c2∈(0,Γq(2−β)), the function w is $L^{\kappa }$Lκ-Carathéodory, $w(t, x_{1}, x_{2}, x_{3})$w(t,x1,x2,x3) and may be singular and ${}^{c}D_{q}^{\alpha }$Dqαc the fractional Caputo type q-derivative. Of course, here we applied the definitions of the fractional q-derivative of Riemann–Liouville and Caputo type by presenting some examples with tables and algorithms; we will illustrate our results, too.