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contraction principle; Cauchy problem
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( \varepsilon ,\gamma \right) $-contraction with a continuous, bounded function which is less than $\varepsilon $ 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 $\Vert u-v\Vert .$ Finally, we establish some inequalities related to the Cauchy problem.
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