General viscosity iterative method for a sequence of quasi-nonexpansive mappings

Research Article

General viscosity iterative method for a sequence of quasi-nonexpansive mappings

Cuijie Zhang^∗, Yinan Wang

College of Science, Civil Aviation University of China, Tianjin 300300, China.

Communicated by Y. H. Yao

Abstract

In this paper, we study a general viscosity iterative method due to Aoyama and Kohsaka for the fixed point problem of quasi-nonexpansive mappings in Hilbert space. First, we obtain a strong convergence theorem for a sequence of quasi-nonexpansive mappings. Then we give two applications about variational inequality problem to encourage our main theorem. Moreover, we give a numerical example to illustrate our main theorem. c2016 All rights reserved.

Keywords: Quasi-nonexpansive mapping, variational inequality, fixed point, viscosity iterative method.

2010 MSC: 47H10, 47J20.

1. Introduction

Throughout the present paper, let H be a real Hilbert space with inner product h·,·i and norm k · k.

Let C be a nonempty closed convex subset of H and T :C → C be a mapping. In this paper, we denote the fixed-point set of T by F ix(T). A mapping T is said to be quasi-nonexpansive, if F ix(T) 6= ∅ and kT x−pk≤kx−pk for allx∈C andp∈F ix(T). We know that ifT :C →C is quasi-nonexpansive, then F ix(T) is closed and convex (see [3] for more general results). A mappingT is said to be nonexpansive, if kT x−T yk≤kx−ykfor allx, y∈C. A mappingT is called demiclosed at 0, if any sequence{x_n}weakly converges tox, and if the sequence {T x_n} strongly converges to 0, thenT x= 0.

The viscosity iterative method was proposed by Moudafi [11] firstly. Choose an arbitrary initialx0∈H, the sequence{x_n}is constructed by:

x_n+1= ε_n 1 +εn

f(x_n) + 1 1 +εn

T x_n, ∀n≥0,

∗Corresponding author

Email addresses: [email protected](Cuijie Zhang),[email protected](Yinan Wang) Received 2016-07-19

whereT is a nonexpansive mapping andf is a contraction with a coefficientα∈[0,1) on H, the sequence {ε_n} is in (0,1), such that:

(i) limn→∞εn= 0;

(ii) P∞

n=0εn=∞;

(iii) limn→∞(_ε¹

n − _ε¹

n+1) = 0.

Then limn→∞xn=x^∗, where x^∗ ∈C(C =F ix(T)) is the unique solution of the variational inequality h(I−f)x^∗, x−x^∗i ≥0,∀x∈F ix(T). (1.1) Maing´e considered the viscosity iterative method for quasi-nonexpansive mappings in Hilbert space in [9]. His focus was on the following algorithm:

x_n+1=α_nf(x_n) + (1−α_n)T_ωx_n,

where{α_n} is a slow vanishing sequence, andω ∈(0,1], T_ω := (1−ω)I +ωT,T has two main conditions:

(i) T is quasi-nonexpansive;

(ii) I−T is demiclosed at 0.

He proved the sequence{x_n}converges strongly to the unique solution of the variational inequality (1.1).

Tian and Jin considered the following iterative process in [13]:

xn+1=αnγf(xn) + (I−αnA)Tωxn, ∀n≥0,

where the sequence{α_n}satisfies certain conditions,ω∈(0,¹₂),Tω = (1−ω)I+ωT, and T is also satisfied the same conditions in Maing´e [9] . Then they proved that {x_n} converges strongly to the unique solution of the variational inequality:

h(γf −A)x^∗, x−x^∗i ≤0, ∀x∈F ix(T).

Recently, Aoyama and Kohsaka considered the following general iterative method in [1]:

xn+1 =αnfn(xn) + (1−αn)Snxn,

where fn is a θ-contraction with respect to Ω = ∩^∞_n=1F ix(Sn) and {f_n} is stable on Ω, and {S_n} is a sequence of strongly quasi-nonexpansive mappings of C into C. That is to say, Sn is quasi-nonexpansive andS_nx_n−x_n→0 whenever {x_n}is a bounded sequence inC and kx_n−pk − kS_nx_n−pk→0 for some point p ∈Ω. Then they proved that if the sequence {α_n} satisfies appropriate conditions, {x_n} converges strongly to the unique fixed point of a contractionP_Ω◦f1.

Many various iterative algorithms have been studied and extended by many authors, especially about quasi-nonexpansive mappings (see [1, 4, 6–13, 15]).

Motivated by the above results, we extend the iterative method to quasi-nonexpansive mappings. We consider the following iterative process:

x_n+1 =α_nf_n(x_n) +

n

X

i=1

(αi−1−α_i)S_i^λnx_n, (1.2) where S_i^λn = (1−λ_n)I +λ_nS_i, and {S_i}^∞_i=1 is a sequence of quasi-nonexpansive mappings. Under the appropriate conditions, we establish the strong convergence of the sequence{x_n}generated by (1.2).

2. Preliminaries

We denote the strong convergence and the weak convergence of{x_n} tox∈H by x_n→x and x_n* x, respectively.

Letf :C→C be a mapping, Ω is a nonempty subset ofC, and θis a real number in [0,1). A mapping f is said to be aθ-contraction with respect to Ω, if

kf(x)−f(z)k≤θkx−zk, ∀x∈C, z∈Ω.

f is said to be aθ-contraction, iff is aθ-contraction with respect toC. The following lemmas are useful for our main result.

Lemma 2.1([1]). Let Ωbe a nonempty subset ofC andf :C→C a θ-contraction with respect toΩ, where 0≤θ <1. IfΩis closed and convex, then P_Ω◦f is aθ-contraction onΩ, whereP_Ω is the metric projection of H onto Ω.

Lemma 2.2 ([1]). Let f :C→C be aθ-contraction, where 0≤θ <1andT :C→C a quasi-nonexpansive mapping. Thenf ◦T is a θ-contraction with respect to F ix(T).

LetD be a nonempty subset ofC. A sequence{f_n}of mappings of C intoH is said to be stable onD, if{f_n(z) :n∈N} is a singleton for everyz ∈D. It is clear that if{f_n} is stable onD, then f_n(z) =f₁(z) for all n∈N andz∈D.

Lemma 2.3 ([9]). Let Tω := (1−ω)I +ωT, with T be a quasi-nonexpansive mapping onH, F ix(T) 6=φ, and ω∈(0,1], q∈F ix(T). Then the following statements are reached:

(i) F ix(T) =F ix(T_ω);

(ii) Tω is a quasi-nonexpansive mapping;

(iii) kT_ωx−q k²≤kx−qk²−ω(1−ω)kT x−xk² for allx∈H.

Lemma 2.4 ([5]). Assume {s_n} is a sequence of nonnegative real numbers such that s_n+1≤(1−β_n)s_n+β_nδ_n, n≥0,

s_n+1≤s_n−η_n+t_n, n≥0,

where {β_n} is a sequence in (0,1), ηn is a sequence of nonnegative real numbers, and{δ_n} and{t_n} are two sequences in R such that:

(i) P∞

n=0β_n=∞;

(ii) limn→∞t_n= 0;

(iii) limk→∞ηn_k = 0 implies lim sup_k→∞δn_k ≤0 for any subsequence {n_k} ⊂ {n}.

Thenlimn→∞s_n= 0.

Lemma 2.5 ([10]). Assume Ais a strongly positive linear bounded operator on Hilbert space H with coeffi- cientγ >¯ 0 and0< ρ≤kAk⁻¹. Then kI−ρAk≤1−ρ¯γ.

3. Main results

In this section, we prove the following strong convergence theorem.

Theorem 3.1. Let H be a real Hilbert space,C a nonempty closed convex subset ofH, {S_n}a sequence of quasi-nonexpansive mappings ofC intoC such that Ω =∩^∞_i=1F ix(Si) is nonempty, andI−Si is demiclosed at 0. Assume that {f_n} is a sequence of mappings of C into C such that each f_n is a θ-contraction with

respect to Ωand {f_n} is stable onΩ, where 0≤θ <1. Let {x_n} be a sequence defined by x₁ ∈C and x_n+1 =α_nf_n(x_n) +

n

X

i=1

(αi−1−α_i)S_i^λnx_n,

forn∈N, whereS_i^λn = (1−λ_n)I+λ_nS_i,λ_n∈(0,1]and{λ_n}satisfies0<lim infn→∞λ_n≤lim sup_n→∞λ_n<

1. Suppose that {α_n} is a sequence in (0,1]such that α₀ = 1, α_n →0, P∞

n=1α_n=∞ and {α_n} is strictly decreasing. Then {x_n} converges to ω∈Ω, where ω is the unique fixed point of a contraction PΩ◦f1.

First, we show some lemmas, then we prove Theorem 3.1. In the rest of this section, we set β_n=α_n(1 + (1−2θ)(1−α_n)),

and

γn=α²_nkfn(xn)−ωk²+2αn n

X

i=1

(αi−1−αi)hS_i^λnxn−ω, f1(ω)−ωi.

Lemma 3.2. {x_n}, {S_ix_n} and {f_n(x_n)} are bounded, and moreover, kx_n+1−ωk≤α_nkf_n(x_n)−ωk+

n

X

i=1

(αi−1−α_i)kS_i^λnx_n−ωk, (3.1) and

kx_n+1−ω k²≤(1−β_n)kx_n−ωk² +γ_n, hold for everyn∈N.

Proof. From Lemma 2.3, we know S_i^λn is quasi-nonexpansive and F ix(Si) =F ix(S_i^λn) for all i∈N. Since f_n is aθ-contraction with respect to Ω,S_i^λn is quasi-nonexpansive, ω∈Ω⊂F ix(S_i) =F ix(S_i^λn), and{f_n} is stable on Ω, it follows that

kxn+1−ωk=kαnfn(xn) +

n

X

i=1

(αi−1−αi)S_i^λnxn−ωk

≤α_n(kf_n(x_n)−f_n(ω)k+kf_n(ω)−ωk) +

n

X

i=1

(αi−1−α_i)kS_i^λnx_n−ω k

≤αnθkxn−ωk+αnkf1(ω)−ωk+(1−αn)kxn−ωk

= (1−α_n(1−θ))kx_n−ω k+α_n(1−θ)kf₁(ω)−ω k 1−θ

(3.2)

for everyn∈N. Thus, by the induction onn, for everyi∈N, we have

kSixn−ωk≤kxn−ω k≤max{kx1−ωk,kf₁(ω)−ωk 1−θ }.

Therefore, it turns out that {x_n}and {S_ix_n} are bounded, and moreover,{f_n(x_n)}is also bounded.

Equation (3.1) follows from (3.2).

By assumption, for everyi∈N, it follows that

hS^λ_iⁿx_n−ω, f_n(x_n)−ωi ≤kS_i^λnx_n−ωk · kf_n(x_n)−f_n(ω)k +hS_i^λnxn−ω, fn(ω)−ωi

≤θkx_n−ω k²+hS_i^λnx_n−ω, f₁(ω)−ωi,

(3.3)

and thus

kxn+1−ωk²=kαn(fn(xn)−ω) +

n

X

i=1

(αi−1−αi)(S_i^λnxn−ω)k²

=α²_nkf_n(x_n)−ωk² +k

n

X

i=1

(αi−1−α_i)(S_i^λnx_n−ω)k²

+ 2αnh

n

X

i=1

(αi−1−αi)(S_i^λnxn−ω), fn(xn)−ωi

≤α²_nkf_n(x_n)−ωk² +(1−α_n)² kx_n−ωk² + 2αn

n

X

i=1

(αi−1−αi)hS_i^λnxn−ω, fn(xn)−ωi

≤α²_nkf_n(x_n)−ωk² +(1−α_n)² kx_n−ωk² +2α_n(1−α_n)θkx_n−ωk² + 2αn

n

X

i=1

(αi−1−αi)hS_i^λnxn−ω, f1(ω)−ωi

= (1−β_n)kx_n−ω k²+γ_n for everyn∈N.

Lemma 3.3. The following hold:

• 0< β_n≤1 for every n∈N;

• 2α_n(1−α_n)/β_n→1/(1−θ) and 2α_n/β_n→1/(1−θ);

• α²_nkfn(xn)−ωk² /βn→0;

• P∞

n=1β_n=∞.

Proof. Since 0< αn≤1 and−1<1−2θ≤1, we know that

0< α²_n=αn(1 + (−1)(1−αn))< βn≤αn(1 + (1−αn)) =αn(2−αn)≤1.

Fromαn→0 we have 2αn(1−αn)/βn→1/(1−θ) and 2αn/βn→1/(1−θ). Since{f_n(xn)}is bounded and

α²_n βn

= α_n

1 + (1−2θ)(1−αn) →0, it follows thatα²_nkf_n(x_n)−ωk² /β_n→0.

Finally, we proveP∞

n=1βn=∞. Suppose that 1−2θ≥0. Then it follows thatβn≥αnfor everyn∈N.

Thus, P∞

n=1βn = ∞. Next, we suppose that 1−2θ < 0. Then βn >2(1−θ)αn for every n ∈N. Thus, P∞

n=1β_n≥2(1−θ)P∞

n=1α_n=∞. This completes the proof.

Proof of Theorem 3.1. By Lemma 2.1, it implies that P_Ω ◦f₁ is a θ-contraction on Ω and hence it has a unique fixed point on Ω.

From Lemma 3.2, we know that

kxn+1−ω k²≤(1−βn)kxn−ω k²+α²_nkfn(xn)−ω k² + 2α_n

n

X

i=1

(αi−1−α_i)hS_i^λnx_n−ω, f₁(ω)−ωi

= (1−β_n)kx_n−ω k²+α²_nkf_n(x_n)−ω k² + 2αn

n

X

i=1

(αi−1−αi)hλ_n(Sixn−xn), f1(ω)−ωi + 2α_n

n

X

i=1

(αi−1−α_i)hx_n−ω, f₁(ω)−ωi, which implies that

kxn+1−ωk² ≤(1−βn)kxn−ωk² +βn

hα²_nkfn(xn)−ωk² β_n

+2α_n βn

λ_n

n

X

i=1

(αi−1−α_i)kx_n−S_ix_nk · kf₁(ω)−ω k +2α_n

βn

(1−α_n)hx_n−ω, f₁(ω)−ωii .

(3.4)

On the other hand, we obtain from Lemma 2.3 (iii) that kx_n+1−ωk²=kα_n(f_n(x_n)−ω) +

n

X

i=1

(αi−1−α_i)(S_i^λnx_n−ω)k²

=α²_nkfn(xn)−ωk² +k

n

X

i=1

(αi−1−αi)(S_i^λnxn−ω)k² + 2α_nh

n

X

i=1

(αi−1−α_i)S_i^λnx_n−ω, f_n(x_n)−ωi

≤α²_nkfn(xn)−ωk² +(1−αn)² kxn−ωk²

−(1−α_n)λ_n(1−λ_n)

n

X

i=1

(αi−1−α_i)kS_ix_n−x_nk²

+ 2αn n

X

i=1

(αi−1−αi)hS_i^λnxn−ω, fn(xn)−ωi.

(3.5)

By using (3.3), we have

(1−α_n)²kx_n−ωk² +2α_n

n

X

i=1

(αi−1−α_i)hS_i^λnx_n−ω, f_n(x_n)−ωi

≤(1−α_n)² kx_n−ωk²+2α_n(1−α_n)θkx_n−ωk² + 2α_n

n

X

i=1

(αi−1−α_i)hS_i^λnx_n−ω, f₁(ω)−ωi)

≤(1−β_n)kx_n−ωk² +2α_n(1−α_n)kx_n−ωk · kf₁(ω)−ω k.

(3.6)

Since S_i^λn is quasi-nonexpansive, from (3.5) and (3.6), it follows that

kx_n+1−ωk² ≤kx_n−ωk² +α²_nkf_n(x_n)−ωk²+2α_n(1−α_n)kx_n−ωk · kf₁(ω)−ω k

−(1−αn)λn(1−λn)

n

X

i=1

(αi−1−αi)kSixn−xnk² . Suppose that M is a positive constant such that

M ≥sup{α_nkf_n(x_n)−ωk² +2(1−α_n)kx_n−ωk · kf₁(ω)−ω k, n∈N}.

So we have

kxn+1−ωk²≤kxn−ωk² +αnM −(1−αn)λn(1−λn)

n

X

i=1

(αi−1−αi)kSixn−xnk². (3.7) Set

sn=kxn−ωk², tn=αnM, δ_n= α_n² kf_n(x_n)−ωk²

β_n +2α_n β_n λ_n

n

X

i=1

(αi−1−α_i)kx_n−S_ix_nk · kf₁(ω)−ω k + 2α_n

β_n (1−α_n)hx_n−ω, f₁(ω)−ωi, ηn= (1−αn)λn(1−λn)

n

X

i=1

(αi−1−αi)kSixn−xnk². Then (3.4) and (3.7) can be rewritten as the following forms, respectively,

s_n+1 ≤(1−β_n)s_n+β_nδ_n, s_n+1≤s_n−η_n+t_n.

Finally, we observe that the condition limn→∞α_n = 0 and Lemma 3.3 imply limn→∞t_n = 0 and P∞

n=1β_n=∞, respectively. In order to complete the proof by using Lemma 2.4, it suffices to verify that

k→∞lim η_nk = 0, implies

lim sup

k→∞

δnk ≤0, for any subsequence{n_k} ⊂ {n}.

In fact, for every i∈N, ifη_nk →0 as k→ ∞, then (1−αn_k)λn_k(1−λn_k)

nk

X

i=1

(αi−1−αi)kSixn_k−xn_k k²→0.

And since 0 < lim infn→∞λn ≤lim sup_n→∞λn < 1, there exist λ > 0 and λ > 0, such that 0 < λ ≤ λ_n≤λ <1. Since limn→∞α_n= 0, there exist some positive integern₀ and α <1, such thatα_n< α, when n > n₀, then

(1−α)λ(1−λ)(αi−1−α_i)kS_ix_nk−x_nk k²≤(1−α)λ(1−λ)

n_k

X

i=1

(αi−1−α_i)kS_ix_nk −x_nk k²

≤(1−αnk)λnk(1−λnk)

nk

X

i=1

(αi−1−αi)kSixnk−xnk k²→0. Therefore, since {α_n} is strictly decreasing, it follows that

kS_ix_nk−x_nk k→0 and

nk

X

i=1

(αi−1−α_i)kS_ix_nk−x_nk k²→0 for everyi∈N.

By using the condition that I −S_i is demiclosed at 0, we obtain ω_w(x_nk) ⊂ F = ∩^∞_i=1F ix(S_i). From Lemma 3.3, it turns out that

lim sup

k→∞

2αn_k(1−αn_k)

β_nk hx_nk−ω, f1(ω)−ωi= 1

1−θlim sup

k→∞

hx_nk−ω, f1(ω)−ωi

= 1

1−θ sup

z∈ωw(x_nk)

hz−ω, f₁(ω)−ωi ≤0.

Since limn→∞α_n = 0, Pnk

i=1(αi−1−α_i) k S_ix_nk −x_nk k²→ 0 and {f_n(x_n)},{S_ix_n} are bounded, it is easy to see that lim sup_k→∞δnk ≤0. From Lemma 2.4, we conclude thatxn→ω.

Remark 3.4. WhenS_n=S, we can remove the following conditions: α₀ = 1 and{α_n}is strictly decreasing.

In fact, the above conditions guarantee the coefficientsαi−1−α_i greater than 0 for everyi∈N. The following corollary is the direct consequence of Theorem 3.1.

Corollary 3.5. Let H be a real Hilbert space, C a nonempty closed convex subset of H, S : C → C a quasi-nonexpansive mapping, such that F ix(S) 6= ∅ and I −S is demiclosed at 0. Assume that α_n → 0, P∞

n=1αn = ∞, and fn satisfies the same conditions of Theorem 3.1. Let {x_n} be a sequence defined by x₁ ∈C and

x_n+1 =α_nf_n(x_n) + (1−α_n)S^λnx_n (3.8) for n∈N, where S^λn = (1−λn)I+λnS, and{λ_n} also satisfies the same conditions of Theorem3.1. Then {x_n} converges toω ∈Ω, where ω is the unique fixed point of a contractionP_Ω◦f1.

Remark 3.6. If f_n=f and λ_n =λfor alln∈N, (3.8) becomes the viscosity approximation process which is introduced by Maing´e (see [9]).

4. Application to variational inequality problem

In this section, by applying Theorem 3.1 and Corollary 3.5, first we study the following variational inequality problem, which is to find a pointx^∗∈Ω, such that

hF(x^∗), x−x^∗i ≥0, ∀x∈Ω, (4.1)

where Ω is a nonempty closed convex subset of a real Hilbert space H, and F : H → H is a nonlinear operator.

The problem (4.1) is denoted by V I(Ω, F). It is well-known that V I(Ω, F) is equivalent to the fixed point problem (see, [7]). If the solution set ofV I(Ω, F) is denoted by Γ, we know that Γ =F ix(PΩ(I−λF)), whereλ >0 is an arbitrary constant,P_Ω is the metric projection onto Ω, andI is the identity operator on H.

Assume that,F isη-strongly monotone andL-Lipschitzian continuous, that is,F satisfies the conditions hF x−F y, x−yi ≥ηkx−y k², ∀x, y∈Ω,

kF x−F yk≤Lkx−yk, ∀x, y∈Ω.

By using Corollary 3.5, we obtain the following convergence theorem for solving the problemV I(Ω, F).

Theorem 4.1. Let F be η-strongly monotone and L-Lipschitzian continuous with η > 0, L > 0. Assume thatS is a quasi-nonexpansive operator withΩ =F ix(S)6=∅, and I−S is demiclosed at 0. And{α_n}is a sequence in(0,1] such thatα_n→0, P∞

n=1α_n=∞. Let{x_n} be a sequence defined by x₁ ∈H and

x_n+1= (I−µα_nF)S^λnx_n, (4.2)

where S^λn = (1−λ_n)I+λ_nS, λ_n∈(0,1], 0<lim infn→∞λ_n≤lim sup_n→∞λ_n<1, and0< µ < _L^2η2. Then {x_n} converges strongly to the unique solution of V I(Ω, F).

Proof. Setf_n= (I−µF)S^λn forn∈N and θ=p

1−2µη+µ²L². Note that

k(I−µF)x−(I−µF)y k²=kx−yk²−2µhx−y, F x−F yi+µ² kF x−F yk²

≤kx−yk²−2µηkx−yk²+µ²L² kx−yk²

= (1−µ(2η−µL²))kx−yk² .

From 0< µ < _L^2η2, we obtain thatI−µF is aθ-contraction. SinceSis quasi-nonexpansive, from Lemma 2.3,S^λn is quasi-nonexpansive. By Lemma 2.2,fnis aθ-contraction with respect toF ix(S), and it is stable on Ω. Moreover, it follows from (4.2) that

xn+1 =αnfn(xn) + (1−αn)S^λnxn

for n ∈ N. Thus from Corollary 3.5, we have that {x_n} converges strongly to ω = P_{F ix(S)} ◦ f1(ω) = P_{F ix(S)}(I−µF)ω, which is the unique solution ofV I(Ω, F).

Remark 4.2. The iteration (4.2) is called the hybrid steepest descent method, (see[2, 14] for more details).

Finally, we study the following variational inequality problem, which is to find a pointx^∗∈F ix(S), such that

h(γf −A)x^∗, x−x^∗i ≥0, ∀x∈F ix(S), (4.3) where f is a α-contraction and A is strongly positive, that is, there exists a constant ¯γ > 0 such that hAx, xi ≥γ¯kx k² for all x∈H. Assume that 0< γ <¯γ/α. The problem (4.3) is denoted by V IP, where x^∗ is the unique solution ofV IP, and we have x^∗ =P_{F ix(S)}(I−A+γf)x^∗.

Theorem 4.3. Assume thatS :H→H is a quasi-nonexpansive operator with Ω =F ix(S)6=∅, andI−S is demiclosed at 0. Let {x_n} be a sequence defined by x1∈H and

xn+1 =αnγtf(xn) + (I−αntA)S^λnxn, ∀n≥0, (4.4) where S^λn = (1−λ_n)I+λ_nS, and0< t < _kAk¹ , {λ_n} and{α_n} satisfy the same conditions of Theorem4.1.

Then{x_n} converges strongly to the unique solution of the V IP. Proof. Setf_n=tγf+ (I−tA)S^λn. By using Lemma 2.5, note that

kf_n(x)−f_n(p)k=k(tγf+ (I−tA)S^λn)x−(tγf + (I−tA)S^λn)pk

≤tγαkx−pk+(1−tγ)kx−pk

=(1−t(¯γ−γα))kx−pk.

From 0 < γ < γ/α, we obtain that¯ f_n is a θ-contraction with respect to F ix(S), and it is stable on F ix(S). Moreover, it follows from (4.4) that

xn+1 =αnfn(xn) + (1−αn)S^λnxn

forn∈N. Thus from Corollary 3.5, we have that{x_n}converges strongly to the unique solution ofV IP. Remark 4.4. Let ξ_n=α_nt, sinceα_n→0 andP∞

n=1α_n=∞, we have ξ_n→0 andP∞

n=1ξ_n=∞, then (4.4) become that

xn+1=ξnγf(xn) + (I−ξnA)S^λnxn, which is introduced by Tian and Jin (see [13]).

5. Numerical example

In this section, we give an example to support Theorem 3.1.

Example 5.1. In Theorem 3.1, we assume that H=R. Takef_n(x) = ^x_n,S_ix=xcos^x_i, wherex∈[−π, π].

Given the parameter λ_n= ³⁺²ⁿ_6n for everyn∈N.

By the definitions of Si, we have ∩ⁿ_i=1F ix(Si) = {0}. Si is a quasi-nonexpansive mapping since, if x∈[−π, π] andq = 0, then

kS_ix−qk=kS_ix−0k=|x| · |cosx

i |≤|x|=|x−q |.

From Theorem 3.1, we can conclude that the sequence{x_n} converges strongly to 0, asn→ ∞. We can rewrite (1.2) as follows

xn+1 = 1

nαnxn+

n

X

i=1

(αi−1−αi)(4n−3

6n xn+3 + 2n

6n xncosxn

i ). (5.1)

Next, we give the parameterαn has three different expressions in (5.1), that is to say, we setα⁽¹⁾n = _n+1¹ , α⁽²⁾n = _2n+1¹ ,α⁽³⁾n = ^√_n+1¹ . Then, through taking a distinct initial guess x1 = 3, by using software Matlab, we obtain the numerical experiment results in Table 1, wherenis the iterative number, and the expression of error we take ^|xn+1_|x^−xn|

n| .

Table 1: The values of{xn}.

n α⁽¹⁾_n α⁽²⁾_n α⁽³⁾_n

xn error xn error xn error

50 0.0313 1.97×10⁻² -0.0699 1.04×10⁻² 0.0001 1.38×10⁻¹ 100 0.0159 9.90×10⁻³ -0.0488 5.20×10⁻³ 0.0000 9.89×10⁻² 500 0.0032 2.00×10⁻³ -0.0210 1.10×10⁻³

1000 0.0016 9.99×10⁻⁴ -0.0146 5.24×10⁻⁴ 5000 0.0003 1.99×10⁻⁴ -0.0063 1.04×10⁻⁴ 10000 0.0002 9.99×10⁻⁵ -0.0044 5.22×10⁻⁵

From Table 1, we can easily see that with iterative number increases,{x_n}approaches to the unique fixed point 0 and the errors gradually approach to zero. And with the change ofαn, the convergent speed of the sequence {x_n}will be changed, when α_n=α⁽³⁾n , the speed of the sequence {x_n}is more faster than others, and when α_n =α⁽²⁾_n the convergent speed of the sequence {x_n} become slower. Through this example, we can conclude that our algorithm is feasible.

Acknowledgment

The first author is supported by the Fundamental Science Research Funds for the Central Universities (Program No. 3122014k010).

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