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.