# Hybrid Iterations for the Fixed Point Problem and Variational Inequalities

Authors: Li-Jun Zhu; Shin Min Kang; Zhangsong Yao; Young Chel Kwun
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

A hybrid iterative algorithm with Meir-Keeler contraction is presented for solving the fixed point problem of the pseudocontractive mappings and the variational inequalities. Strong convergence analysis is given aslimn→∞d(STxn,TSxn).


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## Body

## 1. Introduction

Throughout, we assume thatH is a real Hilbert space with the inner 〈·,·〉 and the norm ∥·∥ and C⊂H is a nonempty closed convex set.Definition 1.
A mappingT:C→C is said to bepseudocontractive if
(1)〈Tu-Tu†,u-u†〉≤∥u-u†∥2
for all u,u†∈C.We useFix⁡(T) to denote the set of fixed points of T.Remark 2.
It is easily seen that (1) is equivalent to the following:
(2)∥Tu-Tu†∥2≤∥u-u†∥2+∥(I-T)u-(I-T)u†∥2
for all u,u†∈C.Definition 3.
A mappingT:C→C is said to be L-Lipschitzian if
(3)∥Tu-Tu†∥≤L∥u-u†∥
for all u,u†∈C, where L>0 is a constant.IfL=1, T is said to benonexpansive.One of our purposes of this paper is to find the fixed points of the pseudocontractive mappings in Hilbert spaces. In the literature, there are a large number references associated with the fixed point algorithms for the pseudocontractive mappings. See, for instance, [1–9]. The first interesting algorithm for finding the fixed points of the Lipschitz pseudocontractive mappings in Hilbert spaces was presented by Ishikawa [4] in 1974.Ishikawa’s Algorithm. For any x0∈C, define the sequence {xn} iteratively by
(4)yn=(1-ϖn)xn+ϖnTxn,xn+1=(1-ϱn)xn+ϱnTyn
for all n∈N, where {ϖn}⊂[0,1] and {ϱn}⊂[0,1] satisfy the following conditions:(a)
lim⁡n→∞ϖn=0;(b)
∑n=1∞‍ϖnϱn=∞.Ishikawa proved that the sequence{xn} generated by (4) converges strongly to a fixed point of T provided C is a compact set.Recently, Zhou [9] suggested the following algorithm.Zhou’s Algorithm. For any x0∈C, define the sequence {xn} iteratively by
(5)yn=(1-ϱn)xn+ϱnTxn,zn=(1-ϖn)xn+ϖnTyn,Cn={z∈C:∥zn-z∥2≤∥xn-z∥2-ϖnϱn(1-2ϱn-ϱn2L2)∥xn-Txn∥2},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=projCn∩Qn(x0),n∈N,
where {ϖn} and {ϱn} are two real sequences in (0,1) satisfying the following conditions:(a)
ϖn≤ϱn for all n∈N;(b)
0<lim⁡inf⁡n→∞ϱn≤lim⁡sup⁡n→∞ϱn≤ϱ<1/(1+L2+1).Zhou proved that the sequence{xn} generated by (5) converges strongly to projFix⁡(T)(x0) without the compactness assumption.Definition 4.
A mappingA:C→H is said to beinverse strongly monotone if there exists ζ>0 such that
(6)〈u-v,Au-Av〉≥ζ∥Au-Av∥2
for all u,v∈C.The variational inequality problem is to findu∈C such that
(7)〈Au,v-u〉≥0,∀v∈C.
The set of solutions of the variational inequality problem is denoted by VI⁡(C,A). It is well known that variational inequality theory has emerged as an important tool in studying a wide class of obstacles, unilateral and equilibrium problems, which arise in several branches of pure and applied sciences in a unified and general framework. For related work, please refer to [10–18] and the references therein.Motivated and inspired by the related work on the fixed point problem and the variational inequality problem in the literature, the purpose of this paper is continuous to study algorithmic approach to the fixed point problem of the pseudocontractive mappings and the variational inequality problem in Hilbert spaces. We suggest a hybrid algorithm with Meir-Keeler contraction and consequently we prove the strong convergence of the presented algorithm.


## 2.  Preliminaries

Recall that the metric projectionprojC:H→C satisfies
(8)∥u-projC(u)∥=inf⁡{∥u-u†∥:u†∈C}.
The metric projection projC is a typical firmly nonexpansive mapping, that is,
(9)∥projC(u)-projC(u†)∥2≤〈projC(u)-projC(u†),u-u†〉
for all u,u†∈H.It is well known that, in a real Hilbert spaceH, the following equality holds:
(10)∥ξu+(1-ξ)u†∥2=ξ∥u∥2+(1-ξ)∥u†∥2-ξ(1-ξ)∥u-u†∥2
for all u,u†∈H and ξ∈[0,1].Lemma 5 (see [9]).
LetH be a real Hilbert space and let C be a closed convex subset of H. Let T:C→C be a continuous pseudocontractive mapping. Then,(i)
Fix⁡(T)⊂C is a closed convex set;(ii)
(I-T) is demiclosed at zero.Let{Cn}⊂H be a sequence of nonempty closed convex sets. We define the symbols s-LinCn and w-LsnCn as follows.(1)
x*∈s-LinCn⇔ there exists {xn}⊂Cn such that xn→x* strongly.(2)
x†∈w-LsnCn⇔ there exist a subsequence {Cni} of {Cn} and a sequence {xn} in Cni such that xn⇀x† weakly.IfC0 satisfies the following:
(11)C0=s-LinCn=w-LsnCn,
then we say that {Cn} converges to C0 in the sense of Mosco [19] and we write C0=M-lim⁡n→∞Cn. It is easy to show that if {Cn} is nonincreasing with respect to inclusion, then {Cn} converges to ⋂n=1∞‍Cn in the sense of Mosco.Tsukada [20] proved the following theorem for the metric projection.Lemma 6 (see [20]).
LetH be a Hilbert space. Let {Cn} be a sequence of nonempty closed convex subsets of H. If C0=M-lim⁡n→∞Cn exists and is nonempty, then, for each x∈H, {projCn(x)} converges strongly to projC0(x), where projCn and projC0 are the metric projections of H onto Cn and C0, respectively.Let(E,d) be a complete metric space. A mapping ψ:E→E is called aMeir-Keeler contraction [21] if, for any ϵ>0, there exists δ>0 such that
(12)d(u,u†)<ϵ+δ⟹d(ψ(u),ψ(u†))<ϵ
for all u,u†∈E. It is well known that the Meir-Keeler contraction is a generalization of the contraction.Lemma 7 (see [21]).
A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.Lemma 8 (see [22]).
Letψ be a Meir-Keeler contraction on a convex subset C of a Banach space E. Then, for any ϵ>0, there exists σ∈(0,1) such that
(13)∥u-u†∥≥ϵ⟹∥ψ(u)-ψ(u†)∥≤σ∥u-u†∥
for all u,u†∈C.Lemma 9 (see [22]).
LetC be a convex subset of a Banach space E. Let T be a nonexpansive mapping on C and let ψ be a Meir-Keeler contraction on C. Then the following holds.(i)
Tψ is a Meir-Keeler contraction on C.(ii)
For eachζ∈(0,1), (1-ζ)T+αψ is a Meir-Keeler contraction on C.


## 3.  Main Results

In this section, we firstly introduce a hybrid iterative algorithm for finding the common element of the fixed point problem and the variational inequality problem.Algorithm 10.
LetH be a real Hilbert space and C⊂H a nonempty closed convex set. Let ψ:C→C be a Meir-Keeler contractive mapping. Let A:C→H be a inverse strongly monotone mapping. Let T:C→C be a κ-Lipschitz pseudocontractive mapping with κ>1. For x0∈C0=C arbitrarily, define a sequence {xn} iteratively by
(14)un=projC[xn-αAxn],vn=(1-ϱn)un+ϱnTun,wn=(1-ϖn)un+ϖnTvn,Cn+1={μ∈Cn:∥wn-μ∥≤∥xn-μ∥},xn+1=projCn+1ψ(xn),n∈N,
where α∈(0,2λ) is a constant and {ϖn} and {ϱn} are two real number sequences in (0,1) satisfying 0<c1<ϖn≤ϱn<c2<1/(1+κ2+1).Next, we show the strong convergence of (14).Theorem 11.
Suppose thatΛ=VI⁡(C,A)∩Fix⁡(T)≠∅. Then the sequence {xn} defined by (14) converges strongly to x†=projΛψ(x†).Remark 12.
Note thatΛ is a closed convex subset of C. Thus projΛ is well defined. Since ψ is a Meir-Keeler contraction of C, it follows that projΛψ is a Meir-Keeler contraction of C by Lemma 9. According to Lemma 7, there exists a unique fixed point x†∈C such that x†=projΛψ(x†).Proof.
The outline of our proof is as follows.
Step
1.
Λ
⊂
C
n for all n∈N;
Step
2.
C
n is closed and convex for all n∈N;
Step
3.
lim
⁡
n
→
∞
∥
x
n
-
ν
∥
=
0 where ν=proj⋂n=1∞‍Cnψ(ν);
Step
4.
ν
∈
Fix
⁡
(
T
);
Step
5.
ν
∈
VI
⁡
(
C
,
A
);
Step
6.
ν
=
proj
Λ
ψ
(
ν
)
=
x
†.
Proof of Step
1. We prove this step by induction. (i) Λ⊂C0 is obvious. (ii) Suppose that Λ⊂Ck for some k∈N. Pick up x*∈Λ⊂Ck. Then, we have
(15)∥un-x*∥=∥projC[xn-αAxn]-projC[x*-αAx*]∥≤∥(xn-αAxn)-(x*-αAx*)∥≤∥xn-x*∥.
By (2), we have
(16)∥Tun-x*∥2≤∥un-x*∥2+∥Tun-un∥2,(17)∥Tvn-x*∥2=∥T((1-ϱn)un+ϱnTun)-x*∥2≤∥(1-ϱn)(un-x*)+ϱn(Tun-x*)∥2+∥(1-ϱn)un+ϱnTun-Tvn∥2.
From (10), we obtain
(18)∥(1-ϱn)un+ϱnTun-Tvn∥2=∥(1-ϱn)(un-Tvn)+ϱn(Tun-Tvn)∥2=(1-ϱn)∥un-Tvn∥2+ϱn∥Tun-Tvn∥2-ϱn(1-ϱn)∥un-Tun∥2.
Since T is κ-Lipschitzian and un-vn=ϱn(un-Tun), by (18), we get
(19)∥(1-ϱn)un+ϱnTun-Tvn∥2≤(1-ϱn)∥un-Tvn∥2+ϱn3κ2∥un-Tun∥2-ϱn(1-ϱn)∥un-Tun∥2=(1-ϱn)∥un-Tvn∥2+(ϱn3κ2+ϱn2-ϱn)∥un-Tun∥2.
By (10) and (16), we have
(20)∥(1-ϱn)(un-x*)+ϱn(Tun-x*)∥2=∥(1-ϱn)(un-x*)+ϱn(Tun-x*)∥2=(1-ϱn)∥un-x*∥2+ϱn∥Tun-x*∥2-ϱn(1-ϱn)∥un-Tun∥2≤(1-ϱn)∥un-x*∥2+ϱn(∥un-x*∥2+∥un-Tun∥2)-ϱn(1-ϱn)∥un-Tun∥2=∥un-x*∥2+ϱn2∥un-Tun∥2.
From (17), (19), and (20), we deduce
(21)∥Tvn-x*∥2≤∥un-x*∥2+(1-ϱn)∥un-Tvn∥2-ϱn(1-2ϱn-ϱn2κ2)∥un-Tun∥2.
Since ϱn<c2<1/(1+κ2+1), we have
(22)1-2ϱn-ϱn2κ2>0
for all n∈N. This together with (21) implies that
(23)∥Tvn-x*∥2≤∥un-x*∥2+(1-ϱn)∥un-Tvn∥2.
By (10), (15), and (23) and noting that ϖn≤ϱn, we have
(24)∥wn-x*∥2=∥(1-ϖn)un+ϖnTvn-x*∥2=(1-ϖn)∥un-x*∥2+ϖn∥Tvn-x*∥2-ϖn(1-ϖn)∥un-Tvn∥2≤∥un-x*∥2-ϖn(ϱn-ϖn)∥Tvn-x*∥2≤∥un-x*∥2≤∥xn-x*∥2
and hence x*∈Ck+1. This indicates that Λ⊂Cn for all n∈N.
Proof of Step
2. In fact, it is obvious from the assumption that C0=C is closed convex. Suppose that Ck is closed and convex for some k∈N. For any μ∈Ck, we know that ∥yk-μ∥≤∥xk-μ∥ is equivalent to
(25)∥yk-xk∥2+2〈yk-xk,xk-μ〉≤0.
So Ck+1 is closed and convex. By induction, we deduce that Cn is closed and convex for all n∈N.
Proof of Step
3. Firstly, from Step 2, we note that {xn} is well defined. Since ⋂n=1∞‍Cn is closed convex, we also have that proj⋂n=1∞‍Cn is well defined and so proj⋂n=1∞‍Cnψ is a Meir-Keeler contraction on C. By Lemma 7, there exists a unique fixed point ν∈⋂n=1∞‍Cn of proj⋂n=1∞‍Cnψ. Since Cn is a nonincreasing sequence of nonempty closed convex subsets of H with respect to inclusion, it follows that
(26)∅≠Λ⊂⋂n=1∞‍Cn=M-lim⁡n→∞Cn.
Setting sn∶=projCnψ(ν) and applying Lemma 6, we can conclude that
(27)lim⁡n→∞sn=proj⋂n=1∞‍Cnψ(ν)=ν.
Now, we show thatlim⁡n→∞∥xn-ν∥=0. Assume that M=lim⁡¯n→∞∥xn-ν∥>0. Then, for any ϵ with 0<ϵ<M, we can choose δ1>0 such that
(28)lim⁡n→∞¯∥xn-ν∥>ϵ+δ1.
Since ψ is a Meir-Keeler contraction, for the positive ϵ, there exists another δ2>0 such that
(29)∥x-y∥<ϵ+δ2⟹∥ψ(x)-ψ(y)∥<ϵ
for all x,y∈C.
In fact, we can choose a commonδ>0 such that (28) and (29) hold. If δ1>δ2, then
(30)lim⁡n→∞¯∥xn-ν∥>ϵ+δ1>ϵ+δ2.
If δ1≤δ2, then, from (29), it follows that
(31)∥x-y∥<ϵ+δ1⟹∥ψ(x)-ψ(y)∥<ϵ
for all x,y∈C. Thus, we have
(32)lim⁡n→∞¯∥xn-ν∥>ϵ+δ,(33)∥x-y∥<ϵ+δ⟹∥ψ(x)-ψ(y)∥<ϵ
for all x,y∈C. Since sn→ν, there exists n0∈N such that
(34)∥sn-ν∥<δ
for all n≥n0.
Now, we consider two possible cases.
Case
1. There exists n1≥n0 such that
(35)∥xn1-ν∥≤ϵ+δ.
By (33) and (34), we get
(36)∥xn1+1-ν∥≤∥xn1+1-sn1+1∥+∥sn1+1-ν∥=∥projCn1+1ψ(xn1)-projCn1+1ψ(ν)∥+∥sn1+1-ν∥≤∥ψ(xn1)-ψ(ν)∥+∥sn1+1-ν∥≤ϵ+δ.
By induction, we can obtain that
(37)∥xn1+m-ν∥≤ϵ+δ
for all m≥1, which implies that
(38)lim⁡n→∞¯∥xn-ν∥≤ϵ+δ,
which contradicts (32). Therefore, we conclude that ∥xn-ν∥→0 as n→∞.
Case
2 (∥xn-ν∥>ϵ+δfor alln≥n0). Now, we prove that Case 2 is impossible. Suppose that Case 2 is true. By Lemma 8, there exists σ∈(0,1) such that
(39)∥ψ(xn)-ψ(ν)∥≤σ∥xn-ν∥
for all n≥n0. Thus we have
(40)∥xn+1-sn+1∥=∥projCn+1ψ(xn)-projCn+1ψ(ν)∥≤∥ψ(xn)-ψ(ν)∥≤σ∥xn-ν∥
for all n≥n0. It follows that
(41)lim⁡n→∞¯∥xn+1-ν∥=lim⁡n→∞¯∥xn+1-sn+1∥≤σlim⁡n→∞¯∥xn-ν∥<lim⁡n→∞¯∥xn-ν∥,
which gives a contradiction. Hence we obtain
(42)lim⁡n→∞∥xn-ν∥=0.
Proof of Step
4. By Step 3, we deduce immediately that {xn} is bounded. Observe that
(43)∥xn+1-xn∥≤∥xn-ν∥+∥ν-sn+1∥+∥sn+1-xn+1∥=∥xn-ν∥+∥ν-sn+1∥+∥projCn+1ψ(xn)-projCn+1ψ(ν)∥≤∥xn-ν∥+∥ν-sn+1∥+∥ψ(xn)-ψ(ν)∥.
Therefore, we have
(44)lim⁡n→∞∥xn+1-xn∥=0.
Since xn+1∈Cn+1, we have
(45)∥wn-xn+1∥≤∥xn-xn+1∥.
This together with (44) implies that
(46)lim⁡n→∞∥wn-xn+1∥=lim⁡n→∞∥wn-xn∥=0.
From (15) and (24), we have
(47)∥wn-x*∥2≤∥un-x*∥2≤∥(xn-αAxn)-(x*-αAx*)∥2=∥xn-x*∥2+α2∥Axn-Ax*∥2-2〈Axn-Ax*,xn-x*〉≤∥xn-x*∥2+α(α-2λ)∥Axn-Ax*∥2.
Then we have
(48)(2λ-α)α∥Axn-Ax*∥2≤∥xn-x*∥2-∥wn-x*∥2≤∥xn-wn∥(∥xn-x*∥+∥wn-x*∥).
By (46) and (48), we obtain
(49)lim⁡n→∞∥Axn-Ax*∥=0.
Since projC is firmly nonexpansive, we have
(50)∥un-x*∥2=∥projC[xn-αAxn]-projC[x*-αAx*]∥2≤〈(xn-αAxn)-(x*-αAx*),un-x*〉=12(∥(xn-αAxn)-(x*-αAx*)∥2+∥un-x*∥2iiiiiiiiiiiii-∥(xn-αAxn)-(x*-αAx*)+x*-un∥2)≤12(∥xn-x*∥2+∥un-x*∥2iiiiiiiiiiiii-∥(xn-un)-α(Axn-Ax*)∥2)=12(∥xn-x*∥2+∥un-x*∥2-∥xn-un∥2iiiiiiiiiiiii+2α〈xn-un,Axn-Ax*〉-α2∥Axn-Ax*∥2).
It follows that
(51)∥un-x*∥2≤∥xn-x*∥2-∥xn-un∥2+2α〈xn-un,Axn-Ax*〉-α2∥Axn-Ax*∥2.
From (24) and (51), we get
(52)∥wn-x*∥2≤∥un-x*∥2≤∥xn-x*∥2-∥xn-un∥2+2α〈xn-un,Axn-Ax*〉-α2∥Axn-Ax*∥2≤∥xn-x*∥2-∥xn-un∥2+2α∥xn-un∥∥Axn-Ax*∥
and so
(53)∥xn-un∥2≤∥xn-x*∥2-∥wn-x*∥2+2α∥xn-un∥∥Axn-Ax*∥≤∥xn-wn∥(∥xn-x*∥+∥wn-x*∥)+2α∥xn-un∥∥Axn-Ax*∥.
This together with (46) and (49) implies that
(54)lim⁡n→∞∥xn-un∥=0.
Note that
(55)∥un-Tun∥≤∥un-wn∥+∥wn-Tun∥≤∥un-wn∥+(1-ϖn)∥un-Tun∥+ϖn∥Tvn-Tun∥≤∥un-wn∥+(1-ϖn)∥un-Tun∥+ϖnκ∥vn-un∥≤∥un-wn∥+(1-ϖn)∥un-Tun∥+ϖnϱnκ∥un-Tun∥.
It follows that
(56)∥un-Tun∥≤1ϖn(1-ϱnκ)∥un-wn∥≤1c1(1-c2κ)∥un-wn∥⟶0.
Since xn→ν, we have un→ν by (54). So, from (56) and Lemma 5, we deduce that ν∈Fix⁡(T).
Proof of Step
5. Define a mapping ϕ by
(57)ϕ(v)={Av+NCv,v∈C,∅,v∉C.
Then ϕ is maximal monotone (see [15]). Let (v,w)∈G(ϕ). Since w-Av∈NCv and un∈C, we have 〈v-un,w-Av〉≥0. On the other hand, from un=projC[xn-αAxn], we have
(58)〈v-un,un-(xn-αAxn)〉≥0,
that is,
(59)〈v-un,un-xnα+Axn〉≥0.
Therefore, we have
(60)〈v-un,w〉≥〈v-un,Av〉≥〈v-un,Av〉-〈v-un,un-xnα+Axn〉=〈v-un,Av-Axn-un-xnα〉=〈v-un,Av-Aun〉+〈v-un,Aun-Axn〉-〈v-un,un-xnα〉≥〈v-un,Aun-Axn〉-〈v-un,un-xnα〉.
Noting that ∥un-xn∥→0 and A is Lipschitz continuous, we obtain 〈v-ν,w〉≥0. Since ϕ is maximal monotone, we have ν∈ϕ-1(0) and hence ν∈VI⁡(C,A).
Proof of Step
6. Since xn+1=projCn+1ψ(xn), we have
(61)〈ψ(xn)-xn+1,xn+1-y〉≥0
for all y∈Cn+1. Since Λ⊂Cn+1, we get
(62)〈ψ(xn)-xn+1,xn+1-y〉≥0
for all y∈Λ. Noting that xn→ν∈Λ, we deduce
(63)〈ψ(ν)-ν,ν-y〉≥0
for all y∈Λ. Thus ν=projΛψ(ν)=x†. This completes the proof.



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*Source: 728363-2014-08-27.xml*