This paper is concerned with $p(\ge 2)$-cyclic self-mappings $T:\bigcup _{i\in \overline{p}}{A}_i\to \bigcup _{i\in \overline{p}}{A}_i$ in a metric space ($X$ $d$), with $A_i\subset X$, $T\left(A_i\right)\subseteq A_{i+1}$ for $i=\mathrm{1,2},\dots ,p$, under a general contractive condition which includes as particular cases several of the existing ones in the literature. The existence and uniqueness of fixed points and best proximity points is discussed as well as the convergence to them of the iterates generated by the self-mapping from given initial points.