An ordered $L$-space $(X,\rightarrow ,\leq )$ consists of a nonempty set $X$, a convergence structure $\rightarrow$ on the set of the sequences in $X$ and an ordering relation $\leq$ on $X$, the two being connected by some conditions of compatibility. In this paper, the authors establish several sufficient conditions in order that an operator $f:X\rightarrow X$ defined on an ordered $L$-space to be a Picard operator (that is, to have a unique fixed point $x^*$ and, for each $x\in X$, the sequence $(f^n(x))$ of successive approximations to converge to $x^*$). Finally, some applications to matrix equations are considered.