Unique positive solution for nonlinear Caputo-type fractional $q$-difference equations with nonlocal and Stieltjes integral boundary conditions

This paper contain a new discussion for the type of generalized nonlinear Caputo fractional $q$-difference equations with $m$-point boundary value problem and Riemann-Stieltjes integral $\tilde{\alpha }[x]:={\int }_0^{1}x(t)d\mathrm{\Lambda }(t)\mathrm{.}$ By applying the fixed point theorem in cones, we investigate an existence of a unique positive solution depends on $\lambda >0.$ We present some useful properties related to the Green’s function for $m-$point boundary value problem.