In this article we introduce a new class of contraction maps, called $A$-contractions, which includes the contractions studied by R. Bianchini, M. S. Khan, S. Reich and T. Kannen. It is shown that the class of $A$-contractions is proper super class of Kannan’s and Khan’s contractions. Several results due to B. Ahmad, F. U. Rehman, Z. Chuanyi, N. Shioji et al. are extended to the $A$-contractions. We also show that a metric space is complete if and only if it has a fixed point property for $A$-contractions.