Abstract
We consider the variational inequality problem for a family of operators of a nonempty closed convex subset of a 2-uniformly convex Banach space with a uniformly Gâteaux differentiable norm, into its dual space. We assume some properties for the operators and get strong convergence to a common solution to the variational inequality problem by the hybrid method proposed by Haugazeau. Using these results, we obtain several results for the variational inequality problem and the proximal point algorithm.
Highlights
Let N and R be the set of all positive integers and the set of all real numbers, respectively
Suppose that C is a nonempty closed convex subset of E and A is a monotone operator of C into E∗; that is, ⟨x − y, Ax − Ay⟩ ≥ 0 holds for all x, y ∈ C
The set of all solutions to the variational inequality problem for A is denoted by VI(C, A)
Summary
Let N and R be the set of all positive integers and the set of all real numbers, respectively. Qn = {z ∈ C : ⟨xn − z, x − xn⟩ ≥ 0} , xn+1 = PCn∩Qn x for each n ∈ N, where A is an α-inverse strongly monotone operator of C into H with VI(C, A) ≠ 0, PC is the metric projection of H onto a nonempty closed convex subset C of H, and {λn} ⊂ [0, 2α] They proved that {xn} converges strongly to PVI(C,A)x; see [17, 18]. Motivated by [19], we propose a new family of operators and prove strong convergence theorems of the sequence generated by these mappings Using these results, we get several additional results for the problem of variational inequalities and the proximal point algorithm
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