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We prove the following theorem: Let p > 1 and let E be a real p-uniformly convex Banach space, and C a nonempty bounded closed convex subset of E. If $T=T_s:C\to C:s\in G=[0,\infty )$ is a Lipschitzian semigroup such that $g=limin{f}_{G\ni \alpha \to \infty }in{f}_{G\ni \delta \ge 0}1/\alpha {ʃ}_0^{\alpha }\parallel T_{\beta +\delta }{\parallel }^{p}d\beta <1+c$, where c > 0 is some constant, then there exists x ∈ C such that $T_{s}x=x$ for all s ∈ G.

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 $C$ 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 $n=2,3$.

Using modified Halpern iterations, by elementary method, we extend and improve results obtained by W.A. Kirk (Proc. Amer. Math. Soc. (1971), 294) and others, which have recently been presented in Chapter 11 of (2001).