Let \((X_i,d_i)\), \(i\in\mathbb{N}^*\), be complete metric spaces, \((X_0,\to)\) be an \(L\)-space and \(X=\prod_{i\geq 0} X_i\). Consider the operators \(A_k:X_0\times X_1\times\dots\times X_k\to X_k\), \(k\in\mathbb R\), and \(A:X\to X\), \(A(x_0,x_1,\dots,x_n,\dots)=(A_0(x_0),A_1(x_0,x_1),\dots,A_n(x_0,x_1,\dots x_n), \dots)\). The authors show that \(A\) is a weakly Picard operator (Picard operator, respectively) whenever \(A_0\) is weakly Picard (Picard, respectively), \(A_k\), \(k\geq 1\) are contractions, and \(A\) is continuous.
Finally, the authors give applications to an infinite system of integral equations and to triangular operators on the family of the sequences with elements in a Banach space.