The paper deals with the so-called convex feasibility problem in Banach space setting. Roughly speaking this problem is formulated as follows: Let

$C$ be a nonempty convex closed subset of a Banach space

$E$ and let

${C}_{1},{C}_{2},\cdots ,{C}_{r}$ be nonexpansive retracts of

$C$ such that

${\bigcap}_{i=1}^{r}{C}_{i}\ne \varnothing $. Assume that

$T$ is a mapping on

$C$ given by the formula

$T={\sum}_{i=1}^{r}{\alpha}_{i}{T}_{i}$, where

${\alpha}_{i}\in (0,1)$,

${\sum}_{i=1}^{r}{\alpha}_{i}=1$ and

${T}_{i}=(1-{\lambda}_{i})I+{\lambda}_{i}{P}_{i}$, where

${\lambda}_{i}\in (0,1)$ and

${P}_{i}$ is a nonexpansive retraction of

$C$ onto

${C}_{i}$ $(i=1,2,\cdots ,r)$. One can find assumptions concerning the space

$E$ which guarantee that the set

$F\left(T\right)$ of fixed points of the mapping

$T$ can be represented as

$F\left(T\right)={\bigcap}_{i=1}^{r}{C}_{i}$ and for every

$x\in C$ the sequence

$\left\{{T}^{n}x\right\}$ converges weakly to an element of

$F\left(T\right)$. The authors prove that if

$E$ is a uniformly convex Banach space with a Fréchet differentiable norm (or a reflexive and strictly convex Banach space satisfying the Opial condition) then the convex feasibility problem has a positive solution. Apart from that the problem of finding a common fixed point for a finite commuting family of nonexpansive mappings in a strictly convex and reflexive Banach space is also considered.