# New Hybrid Steepest Descent Algorithms for Equilibrium Problem and Infinitely Many Strict Pseudo-Contractions in Hilbert Spaces

Authors: Peichao Duan
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We propose an explicit iterative scheme for finding a common
element of the set of fixed points of infinitely many strict pseudo-contractive mappings and
the set of solutions of an equilibrium problem by the general iterative method, which solves
the variational inequality. In the setting of real Hilbert spaces, strong convergence theorems
are proved. The results presented in this paper improve and extend the corresponding
results reported by some authors recently. Furthermore, two numerical examples are given
to demonstrate the effectiveness of our iterative scheme.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈,〉 and induced norm ∥·∥. Let C be a nonempty closed convex subset of H.LetA:C→H be a nonlinear mapping; we consider the problem of finding x*∈C such that
(1)〈Ax*,x-x*〉≥0,∀x∈C.
It is known as the variational inequality problem (denoted by VI(A,C)).Generally,A is assumed to be Lipschitzian and strongly monotone. The relative definitions are listed as follows.(i)
Ais called k-Lipschitzian on C, if there exists a constant k>0 such that
(2)∥Ax-Ay∥≤k∥x-y∥,∀x,y∈C.(ii)
A is said to be η-strongly monotone on C, if there exists a constant η>0 such that
(3)〈Ax-Ay,x-y〉≥η∥x-y∥2,∀x,y∈C.(iii)
A mappingS of C is said to be a κ-strict pseudo-contraction if there exists a constant κ∈[0,1) such that
(4)∥Sx-Sy∥2≤∥x-y∥2+κ∥(I-S)x-(I-S)y∥2
for all x,y∈C; see [1].(iv)
A mappingS of C is said to be a nonexpansive mapping if it is strictly pseudo-contractive with constant κ=0.Obviously, the class of strict pseudo-contractions strictly includes the class of nonexpansive mappings. We denote the set of fixed points ofS by Fix(S) (i.e., Fix(S)={x∈C:Sx=x}).LetF be a bifunction from C×C to ℝ, where ℝ is the set of real numbers.The equilibrium problem forF:C×C→ℝ is to determine its equilibrium points, that is, the set
(5){x∈C:F(x,y)≥0,∀y∈C}.
The set of such solutions is denoted by EP(F).Many problems in applied sciences such as physics, optimization, and economics reduce into finding some element ofEP(F). Some methods have been proposed to solve the equilibrium problem (5); see, for instance, [2–6]. In particular, Combettes and Hirstoaga [7] proposed several methods for solving the equilibrium problem. On the other hand, Mann [8] and Shimoji and Takahashi [9] considered iterative schemes for finding a fixed point of a nonexpansive mapping. Further, Acedo and Xu [10] projected new iterative methods for finding a fixed point of strict pseudo-contractions.In 2006, Marino and Xu [5] proposed a general iterative method and proved that the algorithm converged strongly. Recently, Tian [11] revealed the inner contact of Yamada’s algorithm [12] and viscosity iterative algorithm and then introduced a new general iterative algorithm combining a k-Lipschitzian and η-strong monotone operator. On this basis, Wang [13] considered a general composite iterative method for infinitely many strict pseudo-contractions in 2010. However, the W-mapping used in Wang’s paper requires many composite operations. Very recently, He and Sun [14] proposed a new operator Ln to replace the W-mapping for infinite family nonexpansive mappings.The mappingLn is defined as follows:
(6)Ln=∑i=1n‍ωisnTi(n=1,2,…),
where {ωi}⊂(0,1) such that ∑i=1∞‍ωi=1, sn=∑i=1n‍ωi, and {Ti} are infinite nonexpansive mappings. Because it does not contain many composite computations, it is more simple and easy to realize.In this paper, we combine the operatorLn and the general iterative algorithm to propose a new explicit iterative scheme involving equilibrium problem (5) and an infinite family of strict pseudo-contractions. Under certain assumptions, we will prove that the sequence converges strongly. Further an example will be given to demonstrate the effectiveness of our iterative scheme and another will be given to compare numerical results and convergence rate of the algorithm in this paper and [15].


## 2. Preliminaries

In the sequel, we will make use of the following lemmas in a real Hilbert spaceH.Lemma 1.
LetH be a real Hilbert space. There hold the following identities:(i)
(7)∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉,∀x,y∈H,(ii)
(8)∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2,&MMMMMMMMM∀t∈[0,1],∀x,y∈H.Lemma 2 (see [13]).
LetA:H→H be a k-Lipschitzian and η-strongly monotone operator on a Hilbert space H with k>0, η>0, 0<μ<2η/k2, and 0<t<1. Then S=(I-tμA):H→H is a contraction with contractive coefficient 1-tτ and τ=μ(2η-(μk2/2)).Lemma 3 (see [1]).
LetS:C→C be a κ-strict pseudo-contraction. Define T:C→C by Tx=λx+(1-λ)Sx for each x∈C. Then, as λ∈[κ,1), T is a nonexpansive mapping such that Fix(T)=Fix(S).Lemma 4.
LetV:C→H be an l-Lipschitz mapping with coefficient l≥0 and A:C→H a k-Lipschitzian continuous operator and η-strongly monotone operator with k>0, η>0. Then, for 0<γ<μη/l,
(9)〈x-y,(μA-γV)x-(μA-γV)y〉≥(μη-γl)∥x-y∥2,x,y∈H.
That is, μA-γV is strongly monotone with coefficient μη-γl.Proof.
SinceA is l-Lipschitz and η-strongly monotone, it is easy to get
(10)〈x-y,(μA-γV)x-(μA-γV)y〉=μ〈x-y,Ax-Ay〉-γ〈x-y,Vx-Vy〉≥(μη-γl)∥x-y∥2,x,y∈C.Lemma 5 (see [16]).
Assume that{an} is a sequence of nonnegative real numbers such that
(11)an+1≤(1-γn)an+δn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that(i)
(12)∑n=1∞‍γn=∞;(ii)
(13)limsupn→∞δnγn≤0or∑n=1∞‍|δn|<∞.Then, limn→∞an=0.Let{Sn} be a sequence of κn-strict pseudo-contractions. Define Sn′=θnI+(1-θn)Sn, θn∈[κn,1). Then, by Lemma 3, Sn′ is nonexpansive. In order to find the common fixed point set of infinite mappings, W-mapping is often used; see [9, 13, 15, 17, 18] and references therein. The mapping Wn is defined by
(14)Un,n+1=I,Un,n=tnSn′Un,n+1+(1-tn)I,Un,n-1=tn-1Sn-1′Un,n+(1-tn-1)I,⋮Un,i=tiSi′Un,i+1+(1-ti)I,⋮Un,2=t2S2′Un,3+(1-t2)I,Wn=Un,1=t1S1′Un,2+(1-t1)I,
where t1,t2,… are real numbers such that 0≤tn<1. Such a mapping Wn is called a W-mapping generated by S1′,S2′,… and t1,t2,…. As we have seen, W-mapping contains many composite computation of Sn′, and it is complicated and needs a large number of complex operations. In [14], He and Sun proposed a new hybrid steepest descent method for solving fixed point problem defined on the common fixed point set of infinite nonexpansive mappings.Lemma 6 (see [14]).
LetH be a real Hilbert and Ti:H→H(i=1,2,…) all nonexpansive mappings with ⋂i=1∞Fix(Ti)≠∅. Let T=∑i=1∞‍ωiTi(i=1,2,…), where {ωi}⊂(0,1) such that ∑i=1∞‍ωi=1. Then T is a nonexpansive mapping with Fix(T)=⋂i=1∞Fix(Ti).Lemma 7 (see [14]).
LetH be a real Hilbert and Ti:H→H(i=1,2,…) all nonexpansive mappings with ⋂i=1∞Fix(Ti)≠∅. Let T=∑i=1∞‍ωiTi, where {ωi}⊂(0,1) such that ∑i=1∞‍ωi=1. Assume Ln=∑i=1n‍ωiTi/sn, where sn=∑i=1n‍ωi. Then Ln uniformly converges to T in each bounded subset in H.For solving the equilibrium problem, let us assume that the bifunctionF satisfies the following conditions:(A1)
F(x,x)=0 for all x∈C;(A2)
F is monotone; that is, F(x,y)+F(y,x)≤0 for any x,y∈C;(A3)
for eachx,y,z∈C,limsupt→0F(tz+(1-t)x,y)≤F(x,y);(A4)
F(x,·) is convex and lower semicontinuous for each x∈C.We recall some lemmas which will be needed in the rest of this paper.Lemma 8 (see [2]).
LetC be a nonempty closed convex subset of H, let F be bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0 and x∈H. Then there exists z∈C such that
(15)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.Lemma 9 (see [7]).
Forr>0, x∈H, define a mapping Tr:H→C as follows:
(16)Tr(x)={z∈C∣F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}
for all x∈H. Then, the following statements hold:(i)
Tr is single valued;(ii)
Tr is firmly nonexpansive; that is, for any x,y∈H,
(17)∥Trx-Try∥2≤〈Trx-Try,x-y〉;(iii)
Fix(Tr)=EP(F);(iv)
EP(F) is closed and convex.Lemma 10 (see [19]).
Let{xn} and {zn} be bounded sequences in a Banach space and {βn} a sequence of real numbers such that 0<liminfn→∞βn≤limsupn→∞βn<1 for all n=0,1,2,… Suppose that xn+1=(1-βn)zn+βnxn for all n=0,1,2,… and limsupn→∞∥zn+1-zn∥-∥xn+1-xn∥≤0. Then limn→∞∥zn-xn∥=0.Lemma 11 (see [6]).
LetC, H, F, and Trx be as in Lemma 9. Then the following holds:
(18)∥Tsx-Ttx∥2≤s-ts〈Tsx-Ttx,Tsx-x〉
for all s,t>0 and x∈H.Lemma 12 (see [13]).
LetH be a Hilbert space, C a nonempty closed convex subset of H, and T:C→C a nonexpansive mapping with Fix(T)≠∅. If {xn} is a sequence in C weakly converging to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y.We adopt the following notations:(1)
xn⇀x stands for the weak convergence of {xn} to x,(2)
xn→x stands for the strong convergence of {xn} to x.


## 3. Main Result

Recall that, given a nonempty closed convex subsetC of a real Hilbert space H, for any x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(19)∥x-PCx∥≤∥x-y∥
for all y∈C. Such a PC is called the metric (or the nearest point) projection of H onto C. As we all know, y=PCx if and only if there holds the following relation:
(20)〈x-y,y-z〉≥0∀z∈C.Throughout the rest of this paper, we always assume thatV is an l-Lipschitzian mapping of H into itself with coefficient l≥0 and A is a k-Lipschitzian continuous operator and η-strongly monotone on H with k>0, η>0. Assume that 0<μ<2η/k2 and 0<γ<μ(η-(μk2/2))/α=τ/l.Define a mappingUn=βnI+(1-βn)LnTrn. Since both Ln and Trn are nonexpansive, it is easy to get that Un is also nonexpansive. Consider the mapping Gn on H defined by
(21)Gnx=αnγV(x)+(I-αnμA)Unx,∀x∈H,n∈N,
where αn∈(0,1). By Lemmas 2 and 9, we have
(22)∥Gnx-Gny∥≤αnγ∥Vx-Vy∥+(1-αnτ)∥Unx-Uny∥≤αnγl∥x-y∥+(1-αnτ)∥x-y∥=(1-αn(τ-γl))∥x-y∥.
Since 0<1-αn(τ-γl)<1, it follows that Gn is a contraction. Therefore, by the Banach contraction principle, Gn has a unique fixed point xnV∈H such that
(23)xnV=αnγV(xnV)+(I-αnμA)UnxnV.For simplicity, we will writexn for xnV provided no confusion occurs. Next we prove the sequence {xn} converges strongly to a x*∈Ω=⋂i=1∞Fix(Si)∩EP(F) which solves the variational inequality
(24)〈(γV-μA)x*,p-x*〉≤0,∀p∈Ω.
By the property of the projection, we can get x*=PΩ(I-μA+γV)x* equivalently.Theorem 13.
LetC be a nonempty closed convex subset of a real Hilbert space H and F a bifunction from C×C to ℝ satisfying (A1)–(A4). Let Si:C→C be family κi-strict pseudo-contractions for some 0≤κi<1. Assume the set Ω=⋂i=1∞F(Si)∩EP(F)≠∅. Let V be an l-Lipschitzian mapping of H into itself with l≥0, and let A be a k-Lipschitzian continuous operator and η-strongly monotone on H with k>0, η>0,  0<μ<2η/k2, and 0<γ<μ(η-(μk2/2))/l=τ/l. For every n∈ℕ, let Ln be the mapping generated by Si′ and 0<ωi<1 with ∑i=1∞‍ωi=1 according to (6). Given x1∈H, let {xn} and {un} be sequences generated by the following algorithm:
(25)un=Trnxn,yn=βnxn+(1-βn)Lnun,xn+1=αnγVxn+(I-μαnA)yn.
If {αn},{βn}, and {rn} satisfy the following conditions:(i)
{αn}⊂(0,1),limn→∞αn=0,  and ∑n=1∞‍αn=∞;(ii)
0<liminfn→∞βn≤limsupn→∞βn<1;(iii)
{rn}⊂(0,∞),liminfn→∞rn>0, and limn→∞|rn+1-rn|=0, then, {xn} converges strongly to x*∈Ω, which solves the variational inequality (24).Proof.
The proof is divided into several steps.
Step  1. Show first that {xn} is bounded.
Taking anyp∈Ω, by Lemma 9, we have
(26)∥un-p∥=∥Trnxn-Trnp∥≤∥xn-p∥.
It follows from (25) that
(27)∥yn-p∥=∥βn(xn-p)+(1-βn)(Lnun-Lnp)∥≤βn∥xn-p∥+(1-βn)∥un-p∥≤∥xn-p∥.
Further we get
(28)∥xn+1-p∥=∥αn(γVxn-μAp)+(I-μαnA)yn-(I-μαnA)p∥≤αn(∥γVxn-γVp∥+∥γVp-μAp∥)+(1-αnτ)∥yn-p∥≤αnlγ∥xn-p∥+αn∥γVp-μAp∥+(1-αnτ)∥yn-p∥=(1-αn(τ-lγ))∥xn-p∥+αn(τ-lγ)∥γVp-μAp∥τ-lγ≤max{∥xn-p∥,∥γVp-μAp∥τ-lγ}.
By induction, we obtain ∥xn-p∥≤max{∥x1-p∥,∥γVp-μAp∥/(τ-lγ)},n≥1. Hence, {xn} is bounded, so are {un} and {yn}. It follows from the Lipschitz continuity of A and V that {Axn}, {Aun}, and {Vxn} are also bounded. From the nonexpansivity of Ln, it follows that {Lnxn} is also bounded.
Step  2. Show that
(29)∥xn+1-xn∥⟶0.
Suppose xn+1=βnxn+(1-βn)zn, then zn=(xn+1-βnxn)/(1-βn)=(αnγVxn+(I-μαnA)yn-βnxn)/(1-βn).
Hence, we have(30)zn+1-zn=αn+1γVxn+1+(I-μαn+1A)yn+1-βn+1xn+11-βn+1-αnγVxn+(I-μαnA)yn-βnxn1-βn=αn+1(γVxn+1-μAyn+1)1-βn+1+yn+1-βn+1xn+11-βn+1-αn(γVxn-μAyn)1-βn-yn-βnxn1-βn=αn+1(γVxn+1-μAyn+1)1-βn+1+βn+1xn+1+(1-βn+1)Ln+1un+1-βn+1xn+11-βn+1-αn(γVxn-μAyn)1-βn-βnxn+(1-βn)Lnun-βnxn1-βn≤αn+1(γVxn+1-μAyn+1)1-βn+1-αn(γVxn-μAyn)1-βn+Ln+1un+1-Lnun.
Observe that
(31)∥un+1-un∥=∥Trn+1xn+1-Trnxn∥≤∥Trn+1xn+1-Trn+1xn∥+∥Trn+1xn-Trnxn∥≤∥xn+1-xn∥+∥Trn+1xn-Trnxn∥.
By the definition of Ln, we have
(32)∥Ln+1un+1-Lnun∥=∥Ln+1un+1-Ln+1un∥+∥Ln+1un-Lnun∥≤∥un+1-un∥+∥∑i=1n+1‍ωisn+1Si′un-∑i=1n‍ωisnSi′un∥≤∥xn+1-xn∥+∥Trn+1xn-Trnxn∥+∥ωn+1sn+1Sn+1′un+∑i=1n‍(ωisn+1-ωisn)Si′un∥≤∥xn+1-xn∥+∥Trn+1xn-Trnxn∥+∥ωn+1sn+1Sn+1′un∥+∥∑i=1n‍-ωn+1ωisn+1snSi′un∥≤∥xn+1-xn∥+∥Trn+1xn-Trnxn∥+2M1ωn+1sn+1,
where M1=supi,n≥1{∥Si′un∥}.
It follows from (30) and (32) that
(33)∥zn+1-zn∥≤αn+11-βn+1(∥γVxn+1∥+∥μAyn+1∥)+αn1-βn(∥γVxn∥+∥μAyn∥)+∥Ln+1un+1-Lnun∥≤(αn+11-βn+1+αn1-βn)M2+∥xn+1-xn∥+∥Trn+1xn-Trnxn∥+2M1ωn+1sn+1,
where M2=supn{∥γVxn∥+∥μAyn∥}.
Hence we get∥zn+1-zn∥-∥xn+1-xn∥≤∥Trn+1xn-Trnxn∥+((αn+1/(1-βn+1))+(αn/(1-βn)))M2+M1(ωn+1/Sn+1). Since ∑n=1∞‍ωn=1 is convergent, it is easy to see that ∑n=1∞‍ωn/sn is also convergent. Thus we have ωn/sn→0(n→∞).
From conditions (i) and (iii) and Lemma11, we obtain
(34)limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0.
By Lemma 10, we have limn→∞∥zn-xn∥=0. Thus
(35)limn→∞∥xn+1-xn∥=limn→∞(1-βn)∥zn-xn∥=0.
By Lemma11 and (30) and (29), we obtain
(36)∥un+1-un∥⟶0.
Step  3. Show that
(37)∥xn-Lxn∥⟶0,
where L=∑i=1∞‍ωiSi′(i=1,2,…).
Observe that(38)∥xn-Lnxn∥≤∥xn-Lnun∥+∥Lnun-Lnxn∥≤∥xn-Lnun∥+∥un-xn∥,∥xn-Lnun∥≤∥xn-xn+1∥+∥xn+1-yn∥+∥yn-Lnun∥=∥xn-xn+1∥+αn∥γVxn-μAyn∥+βn∥xn-Lnun∥.
From condition (i) and (25), we can obtain
(39)(1-βn)∥xn-Lnun∥≤∥xn-xn+1∥+αn∥γVxn-μAyn∥⟶0.
It follows from condition (ii) that
(40)∥xn-Lnun∥⟶0.
By Lemma 9, we get
(41)∥un-p∥2=∥Trnxn-Trnp∥2≤〈Trnxn-Trnp,xn-p〉=12(∥un-p∥2+∥xn-p∥2+∥xn-un∥2).
This implies that
(42)∥un-p∥2≤∥xn-p∥2-∥xn-un∥2.
By nonexpansivity of Ln, we have
(43)∥yn-p∥2≤βn∥xn-p∥2+(1-βn)∥un-p∥2≤∥xn-p∥2-(1-βn)∥xn-un∥2.
It follows from (25) that
(44)∥xn+1-p∥2=∥αn(γVxn-p)+(I-μαnA)yn-(I-μαnA)p+αn(p-μAp)∥2≤αn∥γVxn-p∥2+(1-αnτ)∥yn-p∥2+αn∥p-μAp∥2≤αn∥γVxn-p∥2+(1-αnτ)(∥xn-p∥2-(1-βn)∥xn-un∥2)+αn∥p-μAp∥2≤αn∥γVxn-p∥2+∥xn-p∥2-(1-βn)∥xn-un∥2+αn∥p-μAp∥2.
This implies that
(45)(1-βn)∥xn-un∥2≤αn(∥γVxn-p∥2+∥p-μAp∥2)+∥xn-p∥2-∥xn+1-p∥2≤αn(∥γVxn-p∥2+∥p-μAp∥2)+(∥xn-p∥+∥xn+1-p∥)∥xn+1-xn∥.
From conditions (i) and (ii) and (29), we have
(46)∥xn-un∥⟶0.
Thus, we get
(47)∥xn-Lnxn∥⟶0.
On the other hand, we have(48)∥xn-Lxn∥≤∥xn-Lnxn∥+∥Lnxn-Lxn∥.
Combining (47) and Lemma 7, we obtain (37).
Step  4. Show that
(49)limsupn→∞〈(γV-μA)x*,xn-x*〉≤0,
where x*=PΩ(I-μA+γV)x* is a unique solution of the variational inequality (24). Indeed, take a subsequence {xnj} of {xn} such that
(50)limsupn→∞〈(γV-μA)x*,xn-x*〉=limj→∞〈(γV-μA)x*,xnj-x*〉.
Since{xnj} is bounded, there exists a subsequence {xnjk} of {xnj} which converges weakly to q. Without loss of generality, we can assume xnj⇀q. From (37), we obtain Lxnj⇀q.
By the same argument as in the proof of Theorem13, we have q∈Ω. Since x*=PΩ(I-μA+γV)x*, it follows that
(51)limsupn→∞〈(γV-μA)x*,xn-x*〉=limj→∞〈(γV-μA)x*,xnj-x*〉=〈(γV-μA)x*,q-x*〉≤0.
Step  5. Show that
(52)xn⟶x*.
Since(53)〈(γV-μA)x*,xn+1-x*〉=〈(γV-μA)x*,xn+1-xn〉+〈(γV-μA)x*,xn-x*〉≤∥(γV-μA)x*∥∥xn+1-xn∥+〈(γV-μA)x*,xn-x*〉.
It follows from (29) and (51) that
(54)limsupn→∞〈(γV-μA)x*,xn+1-x*〉≤0,∥xn+1-x*∥2=∥αnγVxn+(I-μαnA)yn-x*∥2=∥(I-μαnA)yn-(I-μαnA)x*+αn(γVxn-μAx*)∥2≤∥(I-μαnA)yn-(I-μαnA)x*∥2+2αn〈γVxn-μAx*,xn+1-x*〉≤(1-αnτ)2∥yn-x*∥2+2αn〈γVxn-γVx*,xn+1-x*〉+2αn〈(γV-μA)x*,xn+1-x*〉≤(1-αnτ)2∥xn-x*∥2+αnlγ(∥xn-x*∥2+∥xn+1-x*∥2)+2αn〈(γV-μA)x*,xn+1-x*〉.
This implies that
(55)∥xn+1-x*∥2≤(1-αnτ)2+αnlγ1-αnlγ∥xn-x*∥2+2αn1-αnlγ〈(γV-μA)x*,xn+1-x*〉≤(1-2αn(τ-lγ)1-αnlγ)∥xn-x*∥2+2αn1-αnlγ〈(γV-μA)x*,xn+1-x*〉+(αnτ)21-αnlγM3,
where M3=supn∥xn-x*∥2, n≥1. Put γn=2αn(τ-lγ)/(1-αnlγ), δn=(2αn/(1-αnlγ))〈(γV-μA)x*,xn+1-x*〉+((αnτ)2/(1-αnlγ))M3. It is easy to see that limsupn→∞δn/γn≤0. Hence, by Lemma 5, the sequence {xn} converges strongly to x*.Remark 14.
If we extend the equilibrium problem to be system of equilibrium problems, we still obtain the desired result by the similar proof of Theorem13.


## 4. Numerical Result

In this section, we consider the following two simple examples to demonstrate the effectiveness, realization, and convergence of the algorithm in Theorem13. Further, we compare convergence rates of the algorithm in this paper and [15].First, we give an example as follows.Example 15.
In Theorem13, let H=R, C=[1/4,+∞), F≡0, for all x,y∈C. Define S0:x↦x,S1:x↦x+(π/4)-arctanx, and let Sn=Snmod2, n=1,2,…. Take A=I with Lipschitz constant k=1 and strongly monotone constant η=1, Vx=2x, for all x∈H with Lipschitz coefficient l=2. Give the parameters αn=1/20n, βn=1/4 for every n≥1, and fix μ=1 and γ=1/8. Then {xn} is the sequence generated by
(56)yn=14xn+34Lnxn,xn+1=18120n2xn+(1-120n)yn.
As n→∞, we have {xn}→x*=1.Letωi=1/2i, i=1,2,…; then we have ∑i=1∞‍ωi=1. Take the initial guess x1=1/2, using software MATLAB R2012, we obtain the numerical experiment results in Table 1.Table 1
x
1
=
1
/
2.
n (iterative number)
x
(
1
) (initial guess)
Errors (n)
94
0.9969
3.1
×
1
0
-
3
150
0.9981
1.9
×
1
0
-
3
450
0.9994
6.3
×
1
0
-
4LetR2 be the two-dimensional Euclidean space with usual inner product 〈x(1),x(2)〉=x1(1)x1(2)+x2(1)x2(2)(forallx(1)=(x1(1),x2(1))⊤,x(2)=(x1(2),x2(2))⊤∈R2) and induced norm ∥x∥=x12+x22(forallx=(x1,x2)⊤∈R2).Next, we consider another simple example.Example 16.
In Theorem13, let H=R2, C=[0,1]×[0,1],  F≡0, for all x,y∈C. Give S1=I,S0:x=(x1,x2)⊤↦(sin((x1+x2)/2),cos((x1+x2)/2))⊤, and let Sn=Snmod2, n=1,2,…ωn=1/2n∈(0,1), n≥1. Take A=I with Lipschitz constant k=1 and strongly monotone constant η=1, f(x)=((1/4)x1, -(1/4)x2)⊤, for all x∈H with contraction coefficient ρ=1/4. Give the parameters αn=1/10n, βn=1/2 for every n≥1, and fix μ=1 and γ=1. Then {x(n)} is the sequence generated by
(57)y(n)=12x(n)+12Lnx(n),x(n+1)=110n(14x1(n),-14x2(n))+10n-110ny(n),
As n→∞, we have {x(n)}→x*≈(0.8310,0.5562)⊤.For analysis of the rate of convergence, we use the concept introduced by Rhoades [20] as follows.Definition 17.
LetE be a closed interval on the real line and f:E→E a continuous function. Suppose that {xn}n=1∞ and {yn}n=1∞ are two iterations which converge to the fixed point p of f. Then, {xn}n=1∞ is said to converge faster than {yn}n=1∞ if
(58)∥xn-p∥≤∥yn-p∥,∀n≥1.Now we turn to numerical simulation using the algorithm (57). Take the initial guess x(1)=(1,0)⊤ and x(1)=(1,1)⊤, respectively. All the numerical experiment results are given in Tables 2(a) and 3(a). Then we realize the algorithm in [15], and the W-mapping is used in the paper. Further we obtain the corresponding numerical results which can be found in Tables 2(b) and 3(b).Table 2
(a)x(1)=(1,0)⊤. (b) x(1)=(1,0)⊤.
(a)
n (iterative number)
x
(
1
) (initial guess)
Errors (n)
10
(0.8289, 0.5521)
4.6
×
1
0
-
3
50
(0.8308, 0.5559)
3.8
×
1
0
-
4
100
(0.8310, 0.5561)
1.9
×
1
0
-
4
(b)
n (iterative number)
x
(
1
) (initial guess)
Errors (n)
10
(0.8271, 0.5521)
5.6
×
1
0
-
3
50
(0.8308, 0.5558)
4.7
×
1
0
-
4
100
(0.8309, 0.5561)
2.2
×
1
0
-
4Table 3
(a)x(1)=(1,1)⊤. (b) x(1)=(1,1)⊤.
(a)
n(iterative number)
x
(
1
) (initial guess)
Errors (n)
10
(0.8341, 0.5534)
9.7
×
1
0
-
4
50
(0.8308, 0.5559)
3.8
×
1
0
-
4
100
(0.8310, 0.5561)
1.9
×
1
0
-
4
(b)
n (iterative number)
x
(
1
) (initial guess)
Errors (n)
10
(0.8359, 0.5531)
2.3
×
1
0
-
3
50
(0.8308, 0.5558)
4.7
×
1
0
-
4
100
(0.8309, 0.5561)
2.1
×
1
0
-
4It is easy to see that the approximation values obtained by the algorithm (25) in this paper are more close to the common fixed point x* at the same iterative number. And from the computer programming point of view, the algorithm is easier to implement in this paper.



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*Source: 139123-2013-08-29.xml*