In this paper, based on a lemma giving a sufficient condition for a sequence with elements from a b-metric space to be Cauchy, we obtain Caristi-Kirk type and Boyd&Wong–Browder-Matkowski-Rus type fixed point results in the framework of b-metric spaces. In addition, we extend Theorems 1, 2 and 3 from [M. Bota,V. Ilea, E. Karapinar, O. Mles¸nit¸e, On α*-φ-contractive multi-valued operators in b-metric spaces and applications, Applied Mathematics & Information Sciences, 9 (2015), 2611-2620].