Convergence of a Hybrid Iterative Scheme for Fixed Points of Nonexpansive Maps, Solutions of Equilibrium, and Variational Inequalities Problems
نویسنده
چکیده
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 ontoK. 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 byG(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.
منابع مشابه
Iterative algorithms for families of variational inequalities fixed points and equilibrium problems
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