In this paper, we consider a fixed point problem related to some contraction mappings and introduce new classes of Picard operators for such mappings in the framework of F-metric space, yielding some interesting and novel results. As application of the obtained results, we investigate the Hyers-Ulam stability of a fixed point problem, a Cauchy functional equation, and an integral equation. Also, we present the well-posedness of the fixed point problem and integral equation. Some illustrative examples are also provided to support the new findings.