On uniform convergence of some classes of infinite products

We find necessary and sufficient conditions $\{ \alpha_k(x) \}$ must satisfy for the infinite product$$\prod\limits_{k=1}^{\infty} \bigl[ 1 + \alpha_k(x) u_k(x) \bigr]$$to converge uniformly under the condition that:1) the series $\sum\limits_{k=1}^{\infty} |\Delta u_k(x)|$ converges uniformly; 2) $\sum\limits_{k=1}^{\infty} |\Delta u_k(x)| = O(1)$.