The Methods of Hilbert Spaces and Structure of the Fixed-Point Set of Lipschitzian Mapping
The purpose of this paper is to prove, by asymptotic center techniques and the methods of Hilbert spaces, the following theorem. Let H be a Hilbert space, let C be a nonempty bounded closed convex subset of H, and let M=[an,k]n,k≥1 be a strongly ergodic matrix. If T:C→C is a li...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2009-01-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://dx.doi.org/10.1155/2009/586487 |
| Summary: | The purpose of this paper is to prove, by asymptotic center techniques and the methods of Hilbert spaces, the following theorem. Let H be a Hilbert space, let C be a nonempty bounded closed convex subset of H, and let M=[an,k]n,k≥1 be a strongly ergodic matrix. If T:C→C is a lipschitzian mapping such that lim⁡inf⁡n→∞inf⁡m=0,1,...∑k=1∞an,k·‖Tk+m‖2<2, then the set of fixed points Fix T={x∈C:Tx=x} is a retract of C. This result extends and improves the corresponding results of [7, Corollary 9] and [8, Corollary 1]. |
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| ISSN: | 1687-1820 1687-1812 |