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

Abstract

Abstract 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 (cDqαf)(t)=w(t,f(t),f′(t),(cDqβf)(t)), $$ \bigl({}^{c}D_{q}^{\alpha }f\bigr) (t) = w \bigl(t, f(t), f'(t), \bigl({}^{c}D_{q}^{ \beta }f \bigr) (t) \bigr), $$ under the conditions f(0)=c1f(1) $f(0) = c_{1} f(1)$, f′(0)=c2(cDqβf)(1) $f'(0)= c_{2} ({}^{c}D_{q} ^{\beta } f) (1)$ and f″(0)=f‴(0)=⋯=f(n−1)(0)=0 $f''(0) = f'''(0) = \cdots =f^{(n-1)}(0) = 0$, where α∈(n−1,n) $\alpha \in (n-1, n)$ with n≥3 $n\geq 3$, β,q∈J=(0,1) $\beta , q \in J=(0,1)$, c1∈J $c_{1} \in J$, c2∈(0,Γq(2−β)) $c_{2} \in (0, \varGamma _{q} (2- \beta ))$, the function w is Lκ $L^{\kappa }$-Carathéodory, w(t,x1,x2,x3) $w(t, x_{1}, x_{2}, x_{3})$ and may be singular and Dqαc ${}^{c}D_{q}^{\alpha }$ 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.

Keywords

• Singularity
• Caputo q-derivative
• Quantum calculus
• q-differential