Approximation of Fixed Points of Nonexpansive Mappings and Solutions of Variational Inequalities
نویسندگان
چکیده
Let E be a real q-uniformly smooth Banach space with constant dq, q ≥ 2. Let T : E → E and G : E → E be a nonexpansive map and an η-strongly accretive map which is also κ-Lipschitzian, respectively. Let {λn} be a real sequence in 0, 1 that satisfies the following condition: C1: limλn 0 and ∑ λn ∞. For δ ∈ 0, qη/dqk 1/ q−1 and σ ∈ 0, 1 , define a sequence {xn} iteratively in E by x0 ∈ E, xn 1 Tn xn 1 − σ xn σ Txn − δλn 1G Txn , n ≥ 0. Then, {xn} converges strongly to the unique solution x∗ of the variational inequality problem VI G,K search for x∗ ∈ K such that 〈Gx∗, jq y − x∗ 〉 ≥ 0 for all y ∈ K , where K : Fix T {x ∈ E : Tx x}/ ∅. A convergence theorem related to finite family of nonexpansive maps is also proved.
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