Summary: Many problems in integral and differential equations involve an equation in which there is almost a contraction mapping. Through some type of transformation we arrive at an operator of the form \(H(x) =x-f(x)\). The paper contains two main parts. First we offer several transformations which yield that operator. We then offer necessary and sufficient conditions to ensure that the operator is a large contraction. These operators yield unique fixed points. A partial answer to a question raised in D. R. Smart [Fixed point theorems. Cambridge: Cambridge University Press (1980; Zbl 0427.47036)] is given. The last section contains examples and applications.