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چکیده
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In the present articles, by using fixed point technique and Arzel\`{a}-Ascoli theorem on cones, we wish to investigate the existence of solutions for a nonlinear problems regular and singular fractional $q$-differential equation $$({}^cD_q^\alpha f)(t) = w \big(t, f(t), f'(t), ({}^cD_q^\beta f)(t) \big),$$ under conditions $f(0) = c_1 f(1)$, $f'(0)= c_2 ({}^cD_q^{\beta} f) (1)$ and $f''(0) = f'''(0) = \dots =f^{(n-1)}(0) = 0$, here $\alpha \in (n-1, n)$ with $n\geq 3$, $\beta, q \in J=(0,1)$, $c_1 \in J$, $c_2 \in (0, \Gamma_q (2- \beta))$, function $w$ is a $L^\kappa$-Carath\'{e}odory, $w(t, x_1, x_2, x_3)$ may be singular and ${}^cD_q^\alpha$ the fractional Caputo type $q$-derivative. Of course, here we applied the definitions of fractional $q$-derivative of Riemann-Liouville and Caputo type. by presenting some examples with tables and algorithms, we will try to illustrate our results, too.
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