مشخصات پژوهش

‎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 literatures‎, ‎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.