A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings
Tóm tắt
The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented. Determine an element x
* ∈ K and u
* ∈ T(x
*) such that 〈 u
*, x − x
*〉 ≥ 0 for all x ∈ K
where T: K → P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x
0, y
0 ∈ K, u
0 ∈ T(y
0) and v
0 ∈ T(x
0), we have 〈 u
k
+ x
k+1 − y
k
, x − x
k+1 〉 ≥ 0, ∀x∈ K, for u
k
∈ T(y
k
) and for k ≥ 0 where 〈 v
k
+ y
k
− x
k
, x − y
k
〉 ≥ 0, ∀ x ∈ K and for v
k
∈ T(x
k
).
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