In this article, we investigate the existence results of the solutions for the new types of inclusion problems involving Caputo and Hadamard fractional derivatives. Utilizing some extended contractions that are more general than the previous ones which have been used in the litratures, we deduce some new and interesting results. In this way, we use $\alpha-\psi$-contractions and multi-functions. Applying a new concept of ${\alpha_{s}}^*$-admissibility for Hardy-Rogers-type $(\hbar, \alpha)-$contractions, we get our findings. Finally, some examples are given to delineate the efficiency of the theoretical results.
