This paper presents the stability theorem for the $T$-Picard iteration scheme and establishes the existence and uniqueness theorem for fixed points concerning $T$-mean nonexpansive mappings within $b$-metric-like spaces. The outcome of our fixed point theorem substantiated the existence and uniqueness of solutions to the Fredholm-Hammerstein integral equations defined on time scales. Additionally, we provided two numerical examples from distinct time scales to support our findings empirically.