### Structure of the fixed-point set of mappings with lipschitzian iterates

#### Abstract

We prove, by asymptotic center techniques and some inequalities

in Banach spaces, that if $E$ is $p$-uniformly convex Banach space,

$C$ is a nonempty bounded closed convex

subset of $E$, and $T\colon C\rightarrow C$ has lipschitzian iterates

(with some restrictions), then the set of fixed-points is not only

connected but even a retract of $C$. The results presented in this

paper improve and extend some results in

[J. Górnicki, < i> A remark on fixed point theorems for lipschitzian mappings< /i> , J. Math.

Anal. Appl. < b> 183< /b> (1994), 495–508],

[J. Górnicki, < i> The methods of Hilbert spaces and structure of the fixed-point set of

lipschitzian mapping< /i> , Fixed Point Theory and Applications, Hindawi Publ. Corporation,

2009, Article ID 586487].

in Banach spaces, that if $E$ is $p$-uniformly convex Banach space,

$C$ is a nonempty bounded closed convex

subset of $E$, and $T\colon C\rightarrow C$ has lipschitzian iterates

(with some restrictions), then the set of fixed-points is not only

connected but even a retract of $C$. The results presented in this

paper improve and extend some results in

[J. Górnicki, < i> A remark on fixed point theorems for lipschitzian mappings< /i> , J. Math.

Anal. Appl. < b> 183< /b> (1994), 495–508],

[J. Górnicki, < i> The methods of Hilbert spaces and structure of the fixed-point set of

lipschitzian mapping< /i> , Fixed Point Theory and Applications, Hindawi Publ. Corporation,

2009, Article ID 586487].

#### Keywords

Retraction; asymptotic center; fixed point; uniformly convex Banach space; strongly ergodic matrix

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