Relation between Fixed Point and Asymptotical Center of Nonexpansive Maps
Fixed Point Theory and Applications volume 2011, Article number: 175989 (2011)
We introduce the concept of asymptotic center of maps and consider relation between asymptotic center and fixed point of nonexpansive maps in a Banach space.
Many topics and techniques regarding asymptotic centers and asymptotic radius were studied by Edelstein , Bose and Laskar , Downing and Kirk , Goebel and Kirk , and Lan and Webb . Now, We recall that definitions of asymptotic center and asymptotic radius.
Let be a nonempty subset of a Banach space and a bounded sequence in . Consider the functional defined by
The infimum of over is said to be the asymptotic radius of with respect to and is denoted by . A point is said to be an asymptotic center of the sequence with respect to if
The set of all asymptotic centers of with respect to is denoted by .
We present new definitions of asymptotic center and asymptotic radius that is for a mapping and obtain new results.
Let be a bounded closed convex subset of . A sequence is said to be an asymptotic center for a mapping if, for each ,
Let be a nonempty subset of . We say that has the fixed-point property for continuous mappings of with asymptotic center if every continuous mapping admitting an asymptotic center has a fixed point.
Let be a nonempty subset of . We say that has Property if for every bounded sequence , the set is a nonempty and compact subset of .
Let be a normed space and a nonempty subset of . It is clear that
(i)if is a compact set, then in nonempty compact set and so has Property ;
(ii)if is a open set, since , therefore is empty and so fail to have Property .
2. Main Results
Our new results are presented in this section.
Let be a Banach space and let be a nonempty closed bounded and convex subset of . If satisfies Property , then every continuous mapping asymptotically admitting a center in has a fixed point.
Assume that is a continuous mapping and is a asymptotic center. Let has set of asymptotic center . Since has Property , is nonempty and compact and it is easy to see that it is also convex. In order to obtain the result, it will be enough to show that is -invariant since in this case we may apply Schauder's Fixed-Point Theorem [4, Theorem 18.10]. Indeed, let . Since is a asymptotic center for , we have
Let be a Banach space and let be a nonempty closed bounded and convex subset of . If has the fixed-point property for continuous mappings admitting an asymptotic center, then has Property .
Suppose that fails to have Property . There exists such that either or is noncompact. In the second case, by Klee's theorem in  there exists a continuous function without fixed points (). Since a closed convex subset of a normed space is a retract of the space, there exists a continuous mapping such that for all . Define by . Clearly is a continuous mapping. Moreover,
that is, is an asymptotic center for . Therefore, by Proposition 2.1, has a fixed point in , . Hence sets a contradiction.
Concerning the first case we proceed as follows.
Let . We take such that . For each positive integer , we consider the following nonempty sets:
Since , we have that
Fix an arbitrary and define, by induction, a sequence such that and the segment does not meet . Given , there exists a unique positive integer such that . In this case we define
It is a routine to check that is a continuous mapping from to . Furthermore, for every .
Let be a continuous retraction from into the closed convex subset . We can define by . It is clear that is a asymptotic center for and that is fixed-point free.
Proposition 2.1 (Theorem 2.2) is a generalizations of Theorem 3.1 (Theorem 3.3) in . It can be verified that definition of space is not necessary here.
As an easy consequence of both Proposition 2.1 and Theorem 2.2, we deduce the following result.
Let be a nonempty closed bounded and convex subset of a Banach space . The following conditions are equivalent.
(1) has the fixed-point property for continuous mappings admitting asymptotic center in .
(2) has Property .
Let be a nonempty closed convex bounded subset of a Banach space . By we denote the family of all nonempty compact convex subsets of . On we consider the well-known Hausdorff metric . Recall that a mapping is said to be nonexpansive whenever
Let be a Banach space and let be a nonempty closed convex and bounded subset of satisfying Property . If is a nonexpansive mapping, then has a fixed point.
Let be a nonexpansive mapping. The multivalued analog of Banach's Contraction Principle allows us to find a sequence in such that .
For each , the compactness of guarantees that there exists satisfying .
Now we are going to show that for every ,
Taking any , from the compactness of we can find such that
By compactness again we can assume that converges to a point . From above it follows that
Now we define the mapping by . Since the mapping is upper semicontinuous and for every is a compact convex set we can apply the Kakutani-Bohnenblust-Karlin Theorem in  to obtain a fixed point for and hence for .
Let be a metric space and a mapping. Then a sequence in is said to be an approximating fixed-point sequence of if .
Let be a bounded closed and convex subset of a Banach space , a nonexpansive mapping and . Then a mappings define by is always asymptotically regular, that is, for every , .
Let be a Banach space and a closed bounded convex subset of , and . If is a nonexpansive mapping, then the sequence is an asymptotic center for .
The above comments guarantee that is an approximated fixed-point sequence for . Let us see that the sequence an asymptotic center for . Given we have
Therefore is asymptotic center for .
Let be a normed space, a nonexpansive mapping with an approximating fixed point sequence and be a nonempty subset of such that is a nonempty star-shaped subset of . Then has an approximating fixed-point sequence in .
Suppose . Therefore
and so .
Now, let be the star center of . For every define by
For every , is a contraction, so there exists exactly one fixed point of . Now
Therefore is the approximating fixed-point sequence in of .
Let be a normed space, a nonexpansive mapping with an approximating fixed-point sequence and be a nonempty subset of such that . Suppose is a nonempty weakly compact star-shaped subset of . If is demiclosed, then has a fixed point in .
By the last theorem, has an approximating fixed-point sequence . Because is weakly compact, there exists a subsequence of such that . Since is demiclosed on and , it follows that . Therefore, .
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Haddadi, M., Mazaheri, H. & Ghasemi, M. Relation between Fixed Point and Asymptotical Center of Nonexpansive Maps. Fixed Point Theory Appl 2011, 175989 (2011). https://doi.org/10.1155/2011/175989