Abstract
In this manuscript, by using the Caputo and Riemann-Liouville type fractional q-derivatives, we consider two fractional q-integro-differential equations of the forms Dq alpha c[x](t)+w1(t,x(t),phi (x(t)))=0 and <disp-formula id="Equa">Dq alpha c[x](t)=w2(t,x(t),integral 0t</mml:msubsup>x(r)<mml:mspace width="0.2em"></mml:mspace>dr,cDq alpha</mml:msubsup>[x](t))<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13662_2020_2766_Article_Equa.gif" position="anchor"> </disp-formula> for t is an element of [0,l] under sum and integral boundary value conditions on a time scale Tt0={t:t=t0qn}{0} for n is an element of N where t0 is an element of R and q in (0,1). By employing the Banach contraction principle, sufficient conditions are established to ensure the existence of solutions for the addressed equations. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.