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.

Let $H$ be a real Hilbert space, ${\Phi }_1:H\to ℝ$ a convex function of class ${𝒞}^1$ that we wish to minimize under the convex constraint $S$. A classical approach consists in following the trajectories of the generalized steepest descent system (cf. Brézis [5]) applied to the non-smooth function ${\Phi }_1+{\delta }_S$. Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function ${\Phi }_0:H\to ℝ$ whose critical points coincide with $S$ and a control...

Let H be a real Hilbert space, ${\Phi }_1:H\to$ a convex function of class ${𝒞}^1$ that we wish to minimize under the convex constraint S. A classical approach consists in following the trajectories of the generalized steepest descent system (cf.   Brézis [CITE]) applied to the non-smooth function ${\Phi }_1+{\delta }_S$. Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function ${\Phi }_0:H\to$ whose critical points coincide with S and...

We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem ${\partial }_{t}u+Xu=f,u_{{|}_{t=0}}=g,$ where $X$ is the vector fieldwith a boundedness condition on the divergence of each vector field $a_1,a_2$. This model was studied in the paper [LL] with a $W^{1,1}$ regularity assumption replacing our $BV$ hypothesis. This settles partly a question raised in the paper [Am]. We examine the details of the argument of...