Viscosity Approximation Methods for Nonexpansive Nonself-Mappings in Hilbert Spaces
Viscosity approximation methods for nonexpansive nonself-mappings are studied. Let C be a nonempty closed convex subset of Hilbert space H, P a metric projection of H onto C and let T be a nonexpansive nonself-mapping from C into H. For a contraction f on C and {tn}⊆(0,1), let xn be the unique fixe...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2007-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2007/48648 |
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| Summary: | Viscosity approximation methods for nonexpansive nonself-mappings are studied.
Let C be a nonempty closed convex subset of Hilbert space H, P a metric projection of H onto C and let T be a nonexpansive nonself-mapping from C into H. For a contraction f on C and {tn}⊆(0,1), let xn be the unique fixed point of the contraction x↦tnf(x)+(1−tn)(1/n)∑j=1n(PT)jx. Consider also the iterative processes
{yn} and {zn} generated by yn+1=αnf(yn)+(1−αn)(1/(n+1))∑j=0n(PT)jyn, n≥0, and zn+1=(1/(n+1))∑j=0nP(αnf(zn)+(1−αn)(TP)jzn),n≥0, where y0,z0∈C,{αn} is a real sequence in an interval [0,1]. Strong convergence of the sequences {xn},{yn}, and {zn} to a fixed point of T which solves some variational inequalities is obtained under certain appropriate conditions on the real sequences {αn} and {tn}. |
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| ISSN: | 0161-1712 1687-0425 |