A general composite iterative algorithm for nonexpansive mappings in Hilbert spaces
- Ceng, Lu-Chuan
- Guu, Sy-Ming
- Yao, Jen-Chih
DOIAbstract
AbstractLet H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient α∈(0,1), F:H→H is a k-Lipschitzian and η-strongly monotone operator with k>0,η>0, and A:H→H is a strongly positive bounded linear operator with coefficient γ̄∈(1,2). Let 0<μ<2η/k2,0<γ<μ(η−μk22)/α=τ/α. It is shown that the sequence {xn} generated by the following general composite iterative method: {yn=(I−αnμF)Txn+αnγf(xn),xn+1=(I−βnA)Txn+βnyn,∀n≥0, where {αn}⊂[0,1] and {βn}⊂(0,1], converges strongly to a fixed point x̄∈Fix(T), which solves the variational inequality 〈(I−A)x̄,x−x̄〉≤0,∀x∈Fix(T)
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