2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167756In the context of smooth interval maps, we study an inducing scheme approach to prove existence and uniqueness of equilibrium states for potentials $ϕ$ with he `bounded range' condition $\sup ϕ- \inf ϕ< \htop$, first used by Hofbauer and Keller. We compare our results to Hofbauer and Keller's use of Perron-Frobenius operators. We demonstrate that this `bounded range' condition on the potential is important even if the potential is Hölder continuous. We also prove analyticity of the pressure in this context.Added Lemma 6 to deal with the disparity between leading eigenvalues and operator norms. Added extra references and corrected some typosDynamical Systems37D35, 37E05, 37D25text