In this paper, we establish new results on the existence, uniqueness, and convergence for a one-parameter family of mappings, where each member has a unique fixed point that is also the unique common fixed point of the entire family. These results apply to both contractive-type mappings and those that do not adhere to any contractive conditions. Our theorem includes, as special cases, the results of [B. D. Gel'man, Caristi's inequality and $\alpha$-contraction mappings, Funct. Anal. Appl., 53 (2019), 224--228], [R. P. Pant, V. Rakočević, D. Gopal, A. Pant, M. Ram, A general fixed point theorem, Filomat, 35 (2021), 4061--4072], as well as the well-known fixed-point theorems of Banach, Kannan, Chatterjea, and Ćirić.