Study of (k,Θ)-Hilfer fractional differential and inclusion systems on the glucose graph

This article combines $$-Hilfer fractional calculus with glucose molecular graph, defines fractional differential and inclusion systems on each edge of a glucose molecular graph by the assumption that 0 or 1 marks the vertices, and studies the single-valued and multi-valued $$-Hilfer type fractional boundary value problems on the glucose molecular graph. On the one hand, the existence and uniqueness of solutions in the single-valued case are proved by using several fixed point theorems. On the other hand, in the multi-valued case, we consider that the right side of the inclusion has convex valued and non-convex value. By applying Leray-Schauder nonlinear alternative method of multi-valued maps as well as Covitz-Nadler fixed point theorem of multi-valued contractions, two existence results are obtained respectively. On this basis, we also get the topological structure of the solution set, which is a pioneering work for ($$)-Hilfer fractional differential inclusion on the glucose graph. Finally, several examples are provided to verify the reliability of our proposed results.