An approximation of solutions of variational inequalities
- PDF / 590,161 Bytes
- 12 Pages / 468 x 680 pts Page_size
- 15 Downloads / 233 Views
We use a Mann-type iteration scheme and the metric projection operator (the nearestpoint projection operator) to approximate the solutions of variational inequalities in uniformly convex and uniformly smooth Banach spaces. 1. Introduction Let (B, · ) be a Banach space with the topological dual space B∗ , and let ϕ,x denote the duality pairing of B ∗ and B, where ϕ ∈ B∗ and x ∈ B. Let f : B → B ∗ be a mapping and let K be a nonempty, closed, and convex subset of B. The (general) variational inequality defined by the mapping f and the set K is VI( f ,K) : find x∗ ∈ K such that
f (x∗ ),x − x∗ ≥ 0 for every x ∈ K.
(1.1)
The nonlinear complementarily problem defined by f and K is by definition as follows:
NCP( f ,K) : find x∗ ∈ K such that f (x∗ ),x ≥ 0,
for every x ∈ K and f (x∗ ),x∗ = 0.
(1.2)
It is known (see [5, 6]) that when K is a closed convex cone, problems NCP( f ,K) and VI( f ,K) are equivalent. To study the existence of solutions of the NCP( f ,K) and VI( f ,K) problems, many authors have used the techniques of KKM mappings, and the Fan-KKM theorem from fixed point theory (see [1, 5, 6, 7, 8, 9, 10]). In case B is a Hilbert space, Isac and other authors have used the notion of “exceptional family of elements” (EFE) and the LeraySchauder alternative theorem (see [5, 6]). In [1, 2], Alber generalized the metric projection operator PK to a generalized projection operator πK : B ∗ → K from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces and Alber used this operator to study VI( f ,K) problems and to Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 377–388 DOI: 10.1155/FPTA.2005.377
378
Solutions of variational inequalities
approximate the solutions by an iteration sequence. In [7], the author used the generalized projection operator and a Mann-type iteration sequence to approximate the solutions of the VI( f ,K) problems. In case B is a uniformly convex and uniformly smooth Banach space, the continuity property of the metric projection operator PK has been studied by Goebel, Reich, Roach, and Xu (see [4, 12, 13]). In this paper, we use the operator PK and a Mann-type iteration scheme to approximate the solutions of NCP( f ,K) problems. 2. Preliminaries Let (X, · ) be a normed linear space and let K be a nonempty subset of X. For every x ∈ X, the distance between a point x and the set K is denoted by d(x,K) and is defined by the following minimum equation d(x,K) = inf x − y .
(2.1)
y ∈K
The metric projection operator (or the nearest-point projection operator) PK defined on X is a mapping from X to 2K :
PK (x) = z ∈ X : x − z = d(x,K), ∀x ∈ X .
(2.2)
If PK (x) = ∅, for every x ∈ X, then K is called proximal. If PK (x) is a singleton for every x ∈ X, then K is said to be a Chebyshev set. Theorem 2.1. Let (B, · ) be a reflexive Banach space. Then B is strictly convex if and only if every nonempty closed convex subset K ⊂ B is a Chebyshev set. Since uniformly convex and uniformly smooth Banach spa
Data Loading...
