By employing weight coefficient methods and parameterization techniques, this paper investigates certain novel local fractional half-discrete Hilbert-type inequalities with nonhomogeneous kernels on fractal sets. Furthermore, both the canonical equivalences and its degenerate forms are presented as applications. The main results of this paper can be seen as generalizations and extensions of classical half-discrete Hilbert-type inequalities to the realm of local fractional calculus.

This study explores a system of singular nonlinear higher-order katugampola fractional differential equations (KFDEs) with nonlocal coupled Riemann-Stieltjes integral boundary conditions. First, the addressed KFDEs are reformulated as the equivalent integral equations through explicit construction of some appropriate Green's functions. Second, through synergistic application of Schauder's and Guo-Krasnoselskii's fixed-point theorems, some explicit parameter interval-dependent existence criteria are established for at least one or two positive solutions to the considered KFDEs with the help of the properties of Green's functions. Finally, some concrete examples are provided to validate the effectiveness of the main theoretical results.