Let $T$ be a $\gamma$-contraction on a Banach space $Y$ and let $S$ be an almost $\gamma$-contraction, i.e. sum of an $\left(\epsilon ,\gamma \right)$-contraction with a continuous, bounded function which is less than $\epsilon$ in norm. According to the contraction principle, there is a unique element $u$ in $Y$ for which $u=Tu.$ If moreover there exists $v$ in $Y$ with $v=Sv$, then we will give estimates for $\parallel u-v\parallel .$ Finally, we establish some inequalities related to the Cauchy problem.

We consider a boundary value problem for a differential equation with deviating arguments and p-Laplacian: $-{\left({\varphi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+d/dtgradF\left(x\right)+g(t,x\left(t\right),x\left(\delta \left(t\right)\right)$, x’(t), x’(τ(t))) = 0, t ∈ [0,1]; $x\left(t\right)=\underline{\phi }\left(t\right),$ t ≤ 0; $x\left(t\right)=\overline{\phi }\left(t\right)$, t ≥ 1. An existence result is obtained with the help of the Leray-Schauder degree theory, with no restriction on the damping forces d/dt grad F(x).