It is proved that: for every Banach space $X$ which has uniformly normal structure there exists a $k>1$ with the property: if $A$ is a nonempty bounded closed convex subset of $X$ and $T:A\to A$ is an asymptotically regular mapping such that $$\underset{n\to \infty }{lim\; inf}\left|\right||T^n\left|\right||<k,$$ where $\left|\right|\left|T\right|\left|\right|$ is the Lipschitz constant (norm) of $T$, then $T$ has a fixed point in $A$.

Let $X$ be a real Banach space. A multivalued operator $T$ from $K$ into $2^X$ is said to be pseudo-contractive if for every $x,y$ in $K$, $u\in T\left(x\right)$, $v\in T\left(y\right)$ and all $r>0$, $\parallel x-y\parallel \le \parallel (1+r)(x-y)-r(u-v)\parallel$. Denote by $G(z,w)$ the set $\{u\in K:\parallel u-w\parallel \le \parallel u-z\parallel \}$. Suppose every bounded closed and convex subset of $X$ has the fixed point property with respect to nonexpansive selfmappings. Now if $T$ is a Lipschitzian and pseudo-contractive mapping from $K$ into the family of closed and bounded subsets of $K$ so that the set $G(z,w)$ is bounded for some $z\in K$ and some $w\in T\left(z\right)$, then $T$ has a fixed point in $K$. ...

Let $C$ be a nonempty closed convex subset of a Banach space $E$ and $T:C\to C$ a $k$-Lipschitzian rotative mapping, i.eṡuch that $\parallel Tx-Ty\parallel \le k·\parallel x-y\parallel$ and $\parallel T^{n}x-x\parallel \le a·\parallel x-Tx\parallel$ for some real $k$, $a$ and an integer $n>a$. The paper concerns the existence of a fixed point of $T$ in $p$-uniformly convex Banach spaces, depending on $k$, $a$ and $n=2,3$.