*Mathematical Methods in the Applied Sciences*Solutions of sum-type singular fractional q-integro-differential
equation with $m$-point boundary value using quantum calculus

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In this study, we investigate the sum-type singular nonlinear fractional
q-integro-differential $m$-point boundary value problem. The existence
of positive solutions is obtained by the properties of the Green
function, standard Caputo $q$-derivative, Riemann-Liouville fractional
$q$-integral and the means of a fixed point theorem on a real Banach
space $(\mathcal{X},
\|.\|)$ which has a
partially order by using a cone $P \subset
\mathcal{X}$. The proofs are based on solving the
operator equation $\mathcal{O}_1 x +
\mathcal{O}_2 x = x $ such that the operator
$\mathcal{O}_1$, $\mathcal{O}_2$
are $r$-convex, sub-homogeneous, respectively and define on cone
$P$. As applications, we provide an example illustrating the primary
effects.