We obtain some fixed point theorems for mappings acting in b-metric spaces. The results extend those obtained in [R.P. Agarwal, E. Karapınar, A.F. Roldán-López-de-Hierro, On an extension of contractivity conditions via auxiliary functions, Fixed Point Theory Appl., 2015:104, 20015] using families of control functions, here also through conditions that involve α-admissibility of type S. We furnish an illustrative example to demonstrate the validity of the hypotheses and the degree of usefulness of our results. As an application, the existence of solution for functional equations arising in dynamic programming is discussed, followed by suitable examples.