Let

$F$ be a nonempty closed subset of

$BC(I,E)$ and

$A:F\to F$ be an operator controlled by the contraction conditions with a perturbed linear operator. The authors establish a theorem which ensures that

$A$ has a unique fixed point in

$F$. The authors also show that some known fixed point theorems concerned with integral operators can be derived from their theorem. In addition, the authors obtain a multivalued version of their theorem. As applications, the existence and uniqueness of solutions of impulsive periodic boundary value problems and functional differential inclusions are exhibited in the last section.