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

Abstract EN

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 Ti: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 PK be a nonexpansive projection of E onto K.

For some fixed real numbers δ∈(0,1), λ∈(0,(qκ/dq)1/(q-1)), and arbitrary but fixed vectors x1,u∈E, let {xn} and {yn} be sequences generated by G(yn,η)+(1/r)〈η-yn, jq(yn-xn)〉≥0, ∀η∈K, xn+1=αnu+(1-δ)(1-αn)xn+δ∑i≥1σinTiPK(yn-λAyn), n≥1, where r∈(0,1) is fixed, and {αn},{σi,n}⊂(0,1) are sequences satisfying appropriate conditions.

If F:=[∩i=1∞F(Ti)]∩VI(K,A)∩EP(G)≠∅, under some mild conditions, we prove that the sequences {xn} and {yn} converge strongly to some element in F.