Approximate Iteration Algorithm with Error Estimate for Fixed Point of Nonexpansive Mappings

and Applied Analysis 3 The purpose of this paper is to present a general viscosity iteration process {xn}which is defined by xn 1 I − αnA Txn βnγf xn ( αn − βn ) xn 1.9 and to study the convergence of {xn}, where T is a nonexpansive mapping andA is a strongly positive linear operator, if {αn}, {βn} satisfy appropriate conditions, then iteration sequence {xn} converges strongly to the unique solution x∗ ∈ F T of variational inequality 1.7 . Meanwhile, an approximate iteration algorithm xn 1 I − sA Txn tγf xn s − t xn 1.10 is presented which is used to calculate the fixed point of nonexpansive mapping and solution of variational inequality; the convergence rate estimate is also given. The results presented in this paper extend, generalize and improve the results of Xu 7 , Marino and Xu 2 , and some others. 2. Preliminaries This section collects some lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive. Lemma 2.1 see 3 . Assume that {an} is a sequence of nonnegative real numbers such that an 1 ≤ 1 − λn an δn, 2.1 where {λn} is a sequence in (0,1) and {δn} is a sequence in −∞, ∞ such that i ∑∞ n 1 λn ∞; ii lim supn→∞δn/rn ≤ 0, or ∑∞ n 1 |δn| < ∞. Then limn→∞an 0. Lemma 2.2 see 5 . Let H be a Hilbert space, K a closed convex subset of H, and T : K → K a nonexpansive mapping with nonempty fixed points set F T . If {xn} is a sequence in K weakly converging to x and if { I − T xn} converges strongly to y, then I − T x y. The following lemma is not hard to prove. Lemma 2.3. Let H be a Hilbert space, K a closed convex subset of H, f : H → H a contraction with coefficient 0 < h < 1, and A a strongly positive linear bounded operator with coefficient δ > 0. Then, for 0 < γ < δ/h , 〈 x − y, A − γfx − A − γfy ≥ δ − γh∥x − y∥2, x, y ∈ H. 2.2 That is, A − γf is strongly monotone with coefficient δ − γh. 4 Abstract and Applied Analysis Recall the metric nearest point projection PK from a real Hilbert space H to a closed convex subset K of H is defined as follows: given x ∈ H, PKx is the only point in K with the property ‖x − PKx‖ min y∈K ∥ ∥x − y∥. 2.3 PK is characterized as follows. Lemma 2.4. LetK be a closed convex subset of a real Hilbert spaceH. Given that x ∈ H and y ∈ K. Then y PKx if and only if there holds the inequality 〈 x − y, y − z ≥ 0, ∀z ∈ K. 2.4 Lemma 2.5. Assume thatA is a strongly positive linear-bounded operator on a Hilbert spaceH with coefficient δ > 0 and 0 < ρ ≤ ‖A‖−1. Then ‖I − ρA‖ ≤ 1 − ρδ . Proof. Recall that a standard result in functional analysis is that if V is linear bounded selfadjoint operator onH, then ‖V ‖ sup{|〈Vx, x〉| : x ∈ H, ‖x‖ 1}. 2.5 Now for x ∈ H with ‖x‖ 1, we see that 〈( I − ρAx, x 1 − ρ〈Ax, x〉 ≥ 1 − ρ‖A‖ ≥ 0 2.6 i.e., I − ρA is positive . It follows that ∥I − ρA∥ supI − ρAx, x : x ∈ H, ‖x‖ 1 sup { 1 − ρ〈Ax, x〉 : x ∈ H, ‖x‖ 1 ≤ 1 − ρδ. 2.7 The following lemma is also not hard to prove by induction. Lemma 2.6. Assume that {an} is a sequence of nonnegative real numbers such that an 1 ≤ 1 − λn an ( λn μn ) M, 2.8 whereM is a nonnegative constant and {λn}, {μn} are sequences in 0, ∞ such that i ∑∞ n 0 λn ∞; ii ∑∞ n 0 μn < ∞. Then {an} is bounded. Notation. We use → for strong convergence and⇀ for weak convergence. Abstract and Applied Analysis 5 3. A general Iteration Algorithm with Bounded Linear Operator LetH be a real Hilbert space,A be a bounded linear operator onH, and T be a nonexpansive mapping on H. Assume that the fixed point set F T {x ∈ H : Tx x} of T is nonempty. Since F T is closed convex, the nearest point projection fromH onto F T is well defined. Throughout the rest of this paper, we always assume that A is strongly positive, that is, there exists a constant δ > 0 such that 〈Ax, x〉 ≥ δ‖x‖, ∀x ∈ H. 3.1and Applied Analysis 5 3. A general Iteration Algorithm with Bounded Linear Operator LetH be a real Hilbert space,A be a bounded linear operator onH, and T be a nonexpansive mapping on H. Assume that the fixed point set F T {x ∈ H : Tx x} of T is nonempty. Since F T is closed convex, the nearest point projection fromH onto F T is well defined. Throughout the rest of this paper, we always assume that A is strongly positive, that is, there exists a constant δ > 0 such that 〈Ax, x〉 ≥ δ‖x‖, ∀x ∈ H. 3.1 Note: δ > 0 is throughout reserved to be the constant such that 3.1 holds. Recall also that a contraction on H is a self-mapping f of H such that ∥ ∥f x − fy∥ ≤ h∥x − y∥, ∀x, y ∈ H, 3.2 where h ∈ 0, 1 is a constant which is called contractive coefficient of f . For given contraction f with contractive coefficient 0 < h < 1, and t ∈ 0, 1 , s ∈ 0, 1 , s ≥ t such that 0 ≤ t ≤ s < ‖A‖−1 and 0 < γ < δ/h, consider a mapping St,s onH defined by St,sx I − sA Tx tγf x s − t x, x ∈ H. 3.3 Assume that s − t s −→ 0, 3.4 it is not hard to see that St,s is a contraction for sufficiently small s, indeed, by Lemma 2.5 we have ∥St,sx − St,sy ∥ ≤ tγ∥f x − fy∥ ∥ I − sA Tx − Ty∥ ∥ s − t x − y∥ ≤ tγh 1 − sδ s − t∥x − y∥ ( 1 t ( γh − 1 − s δ − 1 ∥x − y∥. 3.5 Hence, St,s has a unique fixed point, denoted by xt,s, which uniquely solves the fixed point equation: xt,s I − sA Txt,s tγf xt,s s − t xt,s. 3.6 Note that xt,s indeed depends on f as well, but we will suppress this dependence of xt,s on f for simplicity of notation throughout the rest of this paper. We will also always use γ to mean a number in 0, δ/h . 6 Abstract and Applied Analysis The next proposition summarizes the basic properties of xt,s, t ≤ s . Proposition 3.1. Let xt,s be defined via 3.6 . i {xt,s} is bounded for t ∈ 0, ‖A‖−1 , s ∈ 0, ‖A‖−1 . ii lims→ 0‖xt,s − Txt,s‖ 0. iii {xt,s} defines a continuous surface for t, s ∈ 0, ‖A‖−1 × 0, ‖A‖−1 , t ≤ s intoH. Proof. Observe, for s ∈ 0, ‖A‖−1 , that ‖I − sA‖ ≤ 1 − sδ by Lemma 2.5. To show i pick p ∈ F T . We then have ∥ ∥xt,s − p ∥ ∥ ∥ ∥ I − sA Txt,s − p ) t ( γf xt,s −Ap ) − sAp tAp∥ ≤ 1 − sδ ∥xt,s − p ∥ ∥ t ∥ ∥γf xt,s −Ap ∥ ∥ s − t ∥Ap∥ ≤ 1 − sδ ∥xt,s − p ∥ s [ γh ∥xt,s − p ∥ ∥γf ( p ) −Ap∥ s − t ∥Ap∥ ≤ 1 − sδ − γh∥xt,s − p ∥ s ∥γf ( p ) −Ap∥ s − t ∥Ap∥. 3.7 It follows that ∥xt,s − p ∥ ≤ ∥γf ( p ) −Ap∥ δ − γh s − t s ∥Ap ∥ δ − γh < ∞. 3.8 Hence {xt,s} is bounded. ii Since the boundedness of {xt,s} implies that of {f xt,s } and {ATxt,s}, and observe that ‖xt,s − Txt,s‖ ∥tf xt,s − sATxt,s s − t xt,s ∥, 3.9 we have lim s→ 0 ‖xt,s − Txt,s‖ 0. 3.10 To prove iii take t, t0 ∈ 0, ‖A‖−1 , s, s0 ∈ 0, ‖A‖−1 , s ≥ t, s0 ≥ t0 and calculate ‖xt,s − xt0,s0‖ ∥ t − t0 γf xt,s t0γ ( f xt,s − f xt0,s0 ) − s − s0 ATxt,s I − s0A Txt,s − Txt0,s0 s − t xt,s − xt0,s0 s − s0 t0 − t xt0,s0 ∥ ≤ |t − t0|γ ∥f xt,s ∥ t0γh‖xt,s − xt0,s0‖ |s − s0|‖ATxt,s‖ 1 − s0δ ‖xt,s − xt0,s0‖ s − t ‖xt,s − xt0,s0‖ |s − s0| |t − t0| ‖xt0,s0‖, 3.11 Abstract and Applied Analysis 7 which implies that ( s0δ − t0γh t − s ‖xt,s − xt0,s0‖ ≤ |t − t0|γ ∥ ∥f xt,s ∥ ∥ |s − s0|‖ATxt,s‖ |s − s0| |t − t0| ‖xt0,s0‖ −→ 0 3.12and Applied Analysis 7 which implies that ( s0δ − t0γh t − s ‖xt,s − xt0,s0‖ ≤ |t − t0|γ ∥ ∥f xt,s ∥ ∥ |s − s0|‖ATxt,s‖ |s − s0| |t − t0| ‖xt0,s0‖ −→ 0 3.12 as t → t0, s → s0. Note that lim t→ t0, s→ s0 ( s0δ − t0γh t − s ) s0 δ − 1 − t0 ( γh − 1 > 0, 3.13 it is obvious that lim t→ t0, s→ s0 ‖xt,s − xt0,s0‖ 0. 3.14 This completes the proof of Proposition 3.1. Our first main result below shows that xt,s converges strongly as s → 0 to a fixed point of T which solves some variational inequality. Theorem 3.2. One has that xt,s converges strongly as s → 0 t ≤ s to a fixed point x̃ of T which solves the variational inequality: 〈( A − γfx̃, x̃ − z ≤ 0, z ∈ F T . 3.15 Equivalently, One has PF T I −A γf x̃ x̃, where PF T · is the nearest point projection from H onto F T . Proof. We first shows the uniqueness of a solution of the variational inequality 3.15 , which is indeed a consequence of the strong monotonicity ofA−γf . Suppose x̃ ∈ F T and x̂ ∈ F T both are solutions to 3.15 , then 〈( A − γfx̃, x̃ − x̂ ≤ 0, 〈( A − γfx̂, x̂ − x̃ ≤ 0. 3.16 Adding up 3.16 gets 〈( A − γfx̃ − A − γfx̂, x̃ − x̂ ≤ 0. 3.17 The strong monotonicity of A − γf implies that x̃ x̂ and the uniqueness is proved. Below we use x̃ ∈ F T to denote the unique solution of 3.15 . To prove that xt,s converges strongly to x̃, we write, for a given z ∈ F T , xt,s − z t ( γf xt,s − s t Az ) I − sA Txt,s − z s − t xt,s 3.18 8 Abstract and Applied Analysis to derive that ‖xt,s − z‖ 〈 t ( γf xt,s − s t Az ) I − sA Txt,s − z s − t xt,s, xt,s − z 〉 t 〈 γf xt,s −Az, xt,s − z 〉 〈 I − sA Txt,s − z , xt,s − z〉 s − t 〈xt,s −Az, xt,s − z〉 ≤ 1 − sδ ‖xt,s − z‖ t 〈 γf xt,s −Az, xt,s − z 〉 s − t 〈xt,s −Az, xt,s − z〉. 3.19