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

[摘要] 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 $$ \bigl({}^{c}D_{q}^{\alpha }f\bigr) (t) = w \bigl(t, f(t), f'(t), \bigl({}^{c}D_{q}^{ \beta }f \bigr) (t) \bigr), $$ under the conditions $f(0) = c_{1} f(1)$, $f'(0)= c_{2} ({}^{c}D_{q} ^{\beta } f) (1)$ and $f''(0) = f'''(0) = \cdots =f^{(n-1)}(0) = 0$, where $\alpha \in (n-1, n)$ with $n\geq 3$, $\beta , q \in J=(0,1)$, $c_{1} \in J$, $c_{2} \in (0, \varGamma _{q} (2- \beta ))$, the function w is $L^{\kappa }$-Carathéodory, $w(t, x_{1}, x_{2}, x_{3})$ and may be singular and ${}^{c}D_{q}^{\alpha }$ 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.

[学科分类] 航空航天科学

[关键词] Singularity;Caputo q -derivative;Quantum calculus;q -differential