Some Algorithms for Finding Fixed Points and Solutions of Variational Inequalities

and Applied Analysis 3 to a fixed point of the mapping T , which is also a solution of VI 1.5 defined on the set of fixed points of T . As direct consequences, we obtain the unique minimum-norm fixed point of T . Namely, we find the unique solution of the quadratic minimization problem: ‖x̃‖ min{‖x‖ : x ∈ Fix T }. 2. Preliminaries and Lemmas Throughout this paper, when {xn} is a sequence inH, xn → x resp., xn ⇀ x denotes strong resp., weak convergence of the sequence {xn} to x. Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that f : C → H is called a contractive mapping with constant α ∈ 0, 1 if there exists a constant α ∈ 0, 1 such that ‖f x − f y ‖ ≤ α‖x − y‖, for all x, y ∈ C. For every point x ∈ H, there exists a unique nearest point in C, denoted by PCx, such that ‖x − PCx‖ ≤ ∥ ∥x − y∥∥, ∀y ∈ C. 2.1 PC is called the metric projection of H to C. It is well known that PC is nonexpansive and that, for x ∈ H, z PCx ⇐⇒ 〈 x − z, y − z〉 ≤ 0, ∀y ∈ C. 2.2 In a Hilbert space H, we have ∥ ∥x − y∥∥2 ‖x‖ ∥∥y∥∥2 − 2〈x, y〉, ∀x, y ∈ H. 2.3 We need the following lemmas for the proof of our main results. Lemma 2.1. In a real Hilbert spaceH, the following inequality holds: ∥ ∥x y ∥ ∥ 2 ≤ ‖x‖ 2〈y, x y〉, ∀x, y ∈ H. 2.4 Lemma 2.2 see 32 . Let {sn} be a sequence of nonnegative real numbers satisfying sn 1 ≤ 1 − λn sn λnδn, ∀n ≥ 0, 2.5 where {λn} ⊂ 0, 1 and {δn} satisfy the following conditions: i limn→∞ λn 0, ii ∑∞ n 0 λn ∞, iii lim supn→∞δn ≤ 0 or ∑∞ n 0 λnδn < ∞. Then, limn→∞ sn 0. 4 Abstract and Applied Analysis Lemma 2.3 see 33 . Let {xn} and {zn} be bounded sequences in a Banach space E and {γn} a sequence in 0, 1 that satisfies the following condition: 0 < lim inf n→∞ γn ≤ lim sup n→∞ γn < 1. 2.6 Suppose that xn 1 γnxn 1 − γn zn for all n ≥ 0 and lim sup n→∞ ‖zn 1 − zn‖ − ‖xn 1 − xn‖ ≤ 0. 2.7 Then, limn→∞‖zn − xn‖ 0. Lemma 2.4 Demiclosedness principle 34 . Let C be a nonempty closed convex subset of a real Hilbert spaceH and S : C → C a nonexpansive mapping with Fix S / ∅. If {xn} is a sequence in C weakly converging to x and { I − S xn} converges strongly to y, then I − S x y; in particular, if y 0, then x ∈ Fix S . Lemma 2.5 see 14, 16 . Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that the mapping G : C → H is monotone and weakly continuous along segments, that is, G x ty → G x weakly as t → 0. Then, the variational inequality x̃ ∈ C, 〈Gx̃, p − x̃〉 ≥ 0, ∀p ∈ C, 2.8 is equivalent to the dual variational inequality x̃ ∈ C, 〈Gp, p − x̃〉 ≥ 0, ∀p ∈ C. 2.9 Lemma 2.6 see 5 . Let H be a real Hilbert space and C a closed convex subset of H. If T is a kstrictly pseudocontractive mapping on C, then the fixed point set Fix T is closed convex, so that the projection PFix T is well defined. Lemma 2.7 see 5 . Let H be a Hilbert space, C a closed convex subset of H, and T : C → H a kstrictly pseudocontractive mapping. Define a mapping S : C → H by Sx λx 1 − λ Tx for all x ∈ C. Then, as λ ∈ k, 1 , S is a nonexpansive mapping such that Fix S Fix T . The following lemma can be easily proven, and, therefore, we omit the proof. Lemma 2.8. Let H be a real Hilbert space. Let V : H → H be an l-Lipschitzian mapping with constant l ≥ 0 and F : H → H a ρ-Lipschitzian and η-strongly monotone mapping with constants ρ, η > 0. Then, for 0 ≤ γl < μη, 〈( μF − γV )x − (μF − γV )y, x − y〉 ≥ (μη − γl)∥∥x − y∥∥2, ∀x, y ∈ C. 2.10 That is, μF − γV is strongly monotone with constant μη − γl. The following lemma is an improvement of Lemma 2.9 in 4 see also 14 . Abstract and Applied Analysis 5 Lemma 2.9. Let H be a real Hilbert space H. Let F : H → H be a ρ-Lipschitzian and η-strongly monotone mapping with constants ρ, η > 0. Let 0 < μ < 2η/ρ2 and 0 < t < ξ ≤ 1. Then, R : ρI − tμF : H → H is a contractive mapping with constant ξ − tτ , where τ 1 − √ 1 − μ 2η − μρ2 < 1. Proof. First we show that I − μF is strictly contractive. In fact, by applying the ρ-Lipschitz continuity and η-strongly monotonicity of F and 2.3 , we obtain, for x, y ∈ H,and Applied Analysis 5 Lemma 2.9. Let H be a real Hilbert space H. Let F : H → H be a ρ-Lipschitzian and η-strongly monotone mapping with constants ρ, η > 0. Let 0 < μ < 2η/ρ2 and 0 < t < ξ ≤ 1. Then, R : ρI − tμF : H → H is a contractive mapping with constant ξ − tτ , where τ 1 − √ 1 − μ 2η − μρ2 < 1. Proof. First we show that I − μF is strictly contractive. In fact, by applying the ρ-Lipschitz continuity and η-strongly monotonicity of F and 2.3 , we obtain, for x, y ∈ H, ∥ ∥ ( I − μF)x − (I − μF)y∥∥2 ∥∥(x − y) − μ(Fx − Fy)∥∥2 ∥ ∥x − y∥∥2 − 2μ〈Fx − Fy, x − y〉 μ2∥∥Fx − Fy∥∥2 ≤ ∥∥x − y∥∥2 − 2μη∥∥x − y∥∥2 μ2ρ2∥∥x − y∥∥2 ( 1 − μ ( 2η − μρ2 ))∥ ∥x − y∥∥2, 2.11