On solving variational inequalities defined on fixed point sets of multivalued mappings in Banach spaces
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Journal of Fixed Point Theory and Applications
On solving variational inequalities defined on fixed point sets of multivalued mappings in Banach spaces Hong-Kun Xu and Luigi Muglia Abstract. We are concerned with the problem of solving variational inequalities which are defined on the set of fixed points of a multivalued nonexpansive mapping in a reflexive Banach space. Both implicit and explicit approaches are studied. Strong convergence of the implicit method is proved if the space satisfies Opial’s condition and has a duality map weakly continuous at zero, and the strong convergence of the explicit method is proved if the space has a weakly continuous duality map. An essential assumption on the multivalued nonexpansive mapping is that the mapping be single valued on its nonempty set of fixed points. Mathematics Subject Classification. 47J20, 47J25, 49J40. Keywords. Accretive operator, variational inequality, iterative method, multivalued nonexpansive mapping, opial condition, duality map.
1. Introduction In this paper, we are concerned with the problem of solving variational inequalities in Banach spaces. More precisely, let X be a Banach space, let C be a nonempty closed convex subset of X, and let A :⊂ D(A) ⊂ X → X be an operator, such that C ⊂ D(A). The variational inequality (VI), associated with A and C, is the problem of finding a point x∗ ∈ C with the property: Ax∗ , j(x − x∗ ) ≥ 0 for all x ∈ C,
(1.1)
where j(x − x∗ ) ∈ J(x − x∗ ) and J is the normalized duality map on X defined by: J(x) := {x∗ ∈ X ∗ : x, x∗ = x2 , x∗ = x},
x ∈ X.
(1.2)
It is easily understood that the existence and uniqueness of solutions of VI (1.1) require the operator A to satisfy certain conditions. For instance, if A is η-strongly accretive for some η > 0 (i.e., Ax − Ay, j(x − y) ≥ ηx − y2 0123456789().: V,-vol
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for all x, y ∈ C), then VI (1.1) has at most one solution. Indeed, if w and z are solutions of VI (1.1), then adding up the inequalities Aw, j(z − w) ≥ 0
Az, j(w − z) ≥ 0
and
yields −ηz − w ≥ −Az − Aw, j(z − w) ≥ 0; hence, z = w. Observe that in a Hilbert space H, we may rewrite VI (1.1) as a variational inequality that is defined on the fixed point set F ix(T ) of a nonexpansive mapping T : C → C (i.e., T x − T y ≤ x − y for x, y ∈ C): 2
Ax∗ , x − x∗ ≥ 0
x ∈ F ix(T ).
(1.3)
As a matter of fact, we may take T = PC to be the metric projection onto C. This shows an equivalence of VIs defined on an arbitrary closed convex subset C and VIs defined on the fixed point set of an arbitrary nonexpansive mapping on C. This is, however, no longer true in the setting of Banach spaces, due to the fact that there exist closed convex sets of a Banach space X which are not fixed point sets of nonexpansive mappings T : X → X. Such an example can be found in [1, p. 25]. However, if X is strictly convex, then every nonexpansive mapping T : C → C has a closed convex fixed point set F ix(T ). In this paper, we will study VI (1.3) in the case where the feasible set C is
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