In this paper we are concerned with applications $f$, mapping into itself a closed convex subset $K$ of a normed space $X$, with the property: $$\| f(x)-f(y) \| \le a \{ \| x-f(x) \| + \| y-f(y) \| \} + b \| x-y \|$$ ($a$, $b$ not necessarily constants, $a \ge 0$, $b \ge 0$, $2a + b \le 1$). We prove that a fixed point of $f$ exists in $K$ under the following assumptions:
1)
is uniformly convex and in
there exists a point with bounded orbit;
2) $X$ is a Banach space, $K$ is weakly compact, $f$ is continuous and $\displaystyle \sup_{x,y \in K} b(x,y) < 1$. Thereafter, if $X$ is strictly convex, we prove that the set of all fixed points of $f$ in $K$ is closed and convex.