An Extension of Gregus Fixed Point Theorem
Let C be a closed convex subset of a complete metrizable topological vector space (X,d) and T:C→C a mapping that satisfies d(Tx,Ty)≤ad(x,y)+bd(x,Tx)+cd(y,Ty)+ed(y,Tx)+fd(x,Ty) for all x,y∈C, where 0<a<1, b≥0, c≥0, e≥0, f≥0, and a+b+c+e+f=1. Then T has a uni...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2007-03-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://dx.doi.org/10.1155/2007/78628 |
| Summary: | Let C be a closed convex subset of a complete metrizable topological vector space (X,d) and T:C→C a mapping that satisfies d(Tx,Ty)≤ad(x,y)+bd(x,Tx)+cd(y,Ty)+ed(y,Tx)+fd(x,Ty) for all x,y∈C, where 0<a<1, b≥0, c≥0, e≥0, f≥0, and a+b+c+e+f=1. Then T has a unique fixed point. The above theorem, which is a generalization and an extension of the results of several authors, is proved in this paper. In addition, we use the Mann iteration to approximate the fixed point of T. |
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| ISSN: | 1687-1820 1687-1812 |