Convergence of a Hybrid Iterative Scheme for Fixed Points of Nonexpansive Maps, Solutions of Equilibrium, and Variational Inequalities Problems

Bashir Ali · Journal of Mathematics · 2013

Let 𝐾 be a closed, convex, and nonempty subset of a real 𝑞-uniformly smooth Banach space 𝐸, which is also uniformly convex.For some 𝜅 > 0, let 𝑇 𝑖 : 𝐾 → 𝐸 𝑖 ∈ N and 𝐴 : 𝐾 → 𝐸 be family of nonexpansive maps and 𝜅-inverse strongly accretive map, respectively.Let 𝐺 : 𝐾 × 𝐾 → R be a bifunction satisfying some conditions.Let 𝑃 𝐾 be a nonexpansive projection of 𝐸 onto 𝐾.For some fixed real numbers 𝛿 ∈ (0, 1), 𝜆 ∈ (0, (𝑞𝜅/𝑑 𝑞 ) 1/(𝑞-1) ), and arbitrary but fixed vectors 𝑥 1 , 𝑢 ∈ 𝐸, let {𝑥 𝑛 } and {𝑦 𝑛 } be sequences generated by 𝐺(𝑦 𝑛 , 𝜂)+(1/𝑟)⟨𝜂-𝑦 𝑛 , 𝑗 𝑞 (𝑦 𝑛 -𝑥 𝑛 )⟩ ≥ 0, ∀𝜂 ∈ 𝐾, 𝑥 𝑛+1 = 𝛼 𝑛 𝑢+(1-𝛿)(1-𝛼 𝑛 )𝑥 𝑛 +𝛿 ∑ 𝑖≥1 𝜎 𝑖𝑛 𝑇 𝑖 𝑃 𝐾 (𝑦 𝑛 -𝜆𝐴𝑦 𝑛 ), 𝑛 ≥ 1, where 𝑟 ∈ (0, 1) is fixed, and {𝛼 𝑛 }, {𝜎 𝑖,𝑛 } ⊂ (0, 1) are sequences satisfying appropriate conditions.If 𝐹 := [∩ ∞ 𝑖=1 𝐹(𝑇 𝑖 )] ∩ VI(𝐾, 𝐴) ∩ EP(𝐺) ̸ = 0, under some mild conditions, we prove that the sequences {𝑥 𝑛 } and {𝑦 𝑛 } converge strongly to some element in 𝐹.

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