Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces
Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E, and let T:K→K be a uniformly continuous pseudocontraction. If f:K→K is any contraction map on K and if every nonempty closed convex and bounded subset of K has...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2006-03-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://dx.doi.org/10.1155/FPTA/2006/69758 |
| Summary: | Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E, and let T:K→K be a uniformly continuous pseudocontraction. If f:K→K is any contraction map on K and if every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers {αn}, {μn}, that the iteration process z1∈K, zn+1=μn(αnTzn+(1−αn)zn)+(1−μn)f(zn), n∈ℕ, strongly converges to the fixed point of T, which is the unique solution of some variational inequality, provided that K is bounded. |
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| ISSN: | 1687-1820 1687-1812 |