# Strong Convergence to Common Fixed Points of a Countable Family of Asymptotically Strictly Quasi-ϕ-Pseudocontractions

Authors: Wei-Qi Deng
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Based on an original idea, namely, a specific way of choosing the indexes of the involved mappings, we propose a new hybrid shrinking iteration scheme for approximating some common fixed points of a countable family of asymptotically strictly quasi-ϕ-pseudocontractions and obtain a strong convergence theorem in the framework of Banach space. Our result extends other authors, related results existing in the current literature. As application, an iterative solution to a system of equilibrium problems is provided.


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## Body

## 1. Introduction

Fixed point theory as an important branch of nonlinear analysis theory has been applied in many disciplines. From the viewpoint of applications in real world, it is not only to show the existence of fixed points of nonlinear mappings but also to construct iterative algorithms to approximate their fixed points, which explains the popularity of iteration methods for finding the fixed points of various nonlinear mappings. For instance, iteration approximation of common fixed points of asymptotically nonexpansive mappings, an important class of nonlinear mappings introduced by Goebel and Kirk [1], has been studied by many authors (see, e.g., [2–7]).In this paper, we discuss the approximation of common fixed points of a family of asymptotically strictly quasi-ϕ-pseudocontractions.Throughout this paper we assume thatE is a real Banach space with dual E*, C is a nonempty closed convex subset of E, and J:E→2E* is the normalized duality mapping defined by
(1)Jx={f∈E*:〈x,f〉=∥x∥2=∥f∥2},∀x∈E.
In the sequel, we use F(T) to denote the set of fixed points of a mapping T.Definition 1 (see [8]).
A mappingT:C→C is said to be an asymptotically strictly quasi-ϕ-pseudocontraction, if F(T)≠∅, and there exists a nonnegative real sequence {μn} with μn→0 (as n→∞) and a constant κ∈[0,1) such that
(2)ϕ(p,Tnx)≤(1+μn)ϕ(p,x)+κϕ(x,Tnx),∀n≥1,x∈C,p∈F(T),
where ϕ:E×E→ℝ+ denotes the Lyapunov functional defined by
(3)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E,
where 〈·,·〉 stands for the duality product. It is obvious from the definition of ϕ that
(4)ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jy〉,∀x,y,z∈E,(5)(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2,∀x,y∈E.Example 2.
LetC be a unit ball in a real Hilbert space l2, and let T:C→C be a mapping defined by
(6)T(x1,x2,…)=(0,x12,a2x2,a3x3,…),
where {ai} is a sequence in (0,1) satisfying ∏i=2∞‍ai=1/2. It is shown in [8] that
(7)ϕ(p,Tny)≤(1+μn)ϕ(p,y),∀n≥1,y∈C,p∈F(T),
where ϕ(x,y)=∥x-y∥2,μn=(2∏i=2n‍ai)2-1 for all n≥1. This shows that the mapping T defined above is an asymptotically strictly quasi-ϕ-pseudocontraction.Definition 3 (see [9]).
A mappingT:C→C is said to be asymptotically regular on C if, for any bounded subset K of C,
(8)limn→∞supx∈K{∥Tn+1x-Tnx∥}=0.In 2012, Zhang [9] used the following hybrid projection algorithm for a single asymptotically strictly quasi-ϕ-pseudocontraction to obtain strong convergence under a limit condition only in the framework of reflexive, strictly convex, and smooth Banach spaces such that both E and E* have the Kadec-Klee property
(9)x0∈Echosenarbitrarily;C1=C,x1=ΠC1x0,Cn+1={Mn1-κu∈Cn:ϕ(xn,Tnxn)≤21-κ〈xn-u,Jxn-JTnxn〉+μnMn1-κ},xn+1=ΠCn+1x0,∀n≥1,
where Mn=sup{ϕ(p,xn):p∈F(T)} and ΠCn+1 is the generalized projection (see (10)) of E onto Cn+1. Their results improved the corresponding results of Zhou and Gao [10] and Qin et al. [8].Inspired and motivated by the studies mentioned above, we introduce an up-to-date iteration scheme for approximating common fixed points of a countable family of asymptotically strictly quasi-ϕ-pseudocontractions and obtain a strong convergence theorem. The result extends the corresponding one for one map in [9].


## 2. Preliminaries

Following Alber [11], the generalized projectionΠC:E→C is defined by
(10)ΠC=arginfy∈Cϕ(y,x),∀x∈E.Lemma 4 (see [11]).
LetE be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of E. Then the following conclusions hold:(1)
ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y) for all x∈C and y∈E;(2)
ifx∈E and z∈C, then z=ΠCx⇔〈z-y,Jx-Jz〉≥0, for all y∈C;(3)
forx,y∈E, ϕ(x,y)=0 if and only if x=y.Remark 5.
The following basic properties for a Banach spaceE can be found in [12]:(i)
ifE is uniformly smooth, then J is uniformly continuous on each bounded subset of E;(ii)
ifE is reflexive and strictly convex, then J-1 is norm-weak-continuous;(iii)
ifE is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J:E→2E* is single valued, one-to-one, and onto;(iv)
a Banach spaceE is uniformly smooth if and only if E* is uniformly convex;(v)
each uniformly convex Banach spaceE has the Kadec-Klee property; that is, for any sequence {xn}⊂E, if xn⇀x∈E and ∥xn∥→∥x∥, then xn→x as n→∞.Lemma 6 (see [13]).
LetE be a strictly convex and smooth Banach space, let C be a nonempty and closed convex subset of E, and let T be a quasi-ϕ-nonexpansive mapping from C into E. Then F(T) is closed and convex.Lemma 7 (see [14]).
The unique solutions to the positive integer equation(11)n=i+(m-1)m2,m≥i,n=1,2,3,…
are
(12)i=n-(m-1)m2,m=-[12-2n+14],n=1,2,3,…,
where [x] denotes the maximal integer that is not larger than x.


## 3. Main Results

Recall that an operatorT in a Banach space is closed if xn→x and Txn→y as  n→∞; then Tx=y.Theorem 8.
LetE be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property, let C be a nonempty closed and convex subset of E, and {Ti}:C→C be a countable family of closed and asymptotically strictly quasi-ϕ-pseudocontractions with a nonnegative real sequence {μn(i)} satisfying μn(i)→0(as n→∞ for each i≥1) and a sequence {κi}⊂[0,1). Let {xn} be the sequence generated by
(13)x1∈E;C1=C,Cn+1={νn1-κinz∈Cn:ϕ(xn,Tinmnxn)≤21-κin〈xn-z,Jxn-JTinmnxn〉+μmn(in)νn1-κin},xn+1=ΠCn+1x1,∀n≥1,
where νn:=supp∈Fϕ(p,xn); ΠCn+1 is the generalized projection of E onto Cn+1; in and mn are the solutions to the positive integer equation n=i+((m-1)m/2)(m≥i,n=1,2,…); that is, for each n≥1, there exist unique in and mn such that
(14)i1=1,i2=1,i3=2,i4=1,i5=2,i6=3,i7=1,i8=2,…;m1=1,m2=2,m3=2,m4=3,m5=3,m6=3,m7=4,m8=4,….
If {Ti} are asymptotically regular on C and F:=∩i=1∞F(Ti) is nonempty and bounded, then {xn} converges strongly to ΠFx1.Proof.
We divide the proof into several steps.
(i) BothF and Cn (for all n≥1) are closed and convex subsets in C.
For eachi≥1, the closedness of F(Ti) follows from that of Ti, and hence F is closed. In addition, using the same argument as presented in the proof of Theorem 2.1 in [9], we have that each Fi is convex and so is F.
Next, we show thatCn is closed and convex for all n≥1. With C1(=C) being closed and convex, we may assume that Cn is closed and convex for some n≥2. In view of the definition of Cn+1 we have that
(15)Cn+1={z∈C:φ(z)≤a}∩Cn,
where φ(z)=(2/κin-1)〈xn-z,Jxn-JTinmnxn〉 and a=μmn(in)(νn/1-κin)-ϕ(xn,Tinmnxn). This shows that Cn+1 is closed and convex.
(ii)F is a subset of ∩n=1∞Cn.
It is obvious thatF⊂C1. Suppose that F⊂Cn for some n≥2. It follows from (4) that
(16)ϕ(p,Tinmnxn)=ϕ(p,xn)+ϕ(xn,Tinmnxn)+2〈p-xn,Jxn-JTinmnxn〉
for any p∈F⊂Cn. Then, from (16) we have
(17)ϕ(xn,Tinmnxn)≤μmn(in)1-κinϕ(p,xn)+21-κin〈xn-p,Jxn-JTinmnxn〉≤μmn(in)νn1-κinϕ(p,xn)+21-κin〈xn-p,Jxn-JTinmnxn〉.
This implies that p∈Cn+1, and so F⊂Cn+1.
(iii) Considerxn→x*∈C as n→∞.
In fact, sincexn=ΠCnx1, from Lemma 4 (2) we have 〈xn-y,Jx1-Jxn〉≥0, for all y∈Cn. Again since F⊂∩n=1∞Cn, we have 〈xn-p,Jx1-Jxn〉≥0, for all p∈F. It follows from Lemma 4 (1) that for each p∈F and for each n≥1,
(18)ϕ(xn,x1)=ϕ(ΠCnx1,x1)≤ϕ(p,x1)-ϕ(p,xn)≤ϕ(p,x1),
which implies that {ϕ(xn,x1)} is bounded and hence so is {xn}. Since for all n≥1, xn=ΠCnx1 and xn+1=ΠCn+1x1∈Cn+1⊂Cn, we have ϕ(xn,x1)≤ϕ(xn+1,x1). This implies that {ϕ(xn,x1)} is nondecreasing; hence the limit
(19)limn→∞ϕ(xn,x1)exists.
Since E is reflexive, there exists a subsequence {xni} of {xn} such that xni⇀x*∈C as i→∞. Since Cn is closed and convex and Cn+1⊂Cn, this implies that Cn is weakly closed and x*∈Cn for each n≥1. In view of xni=ΠCnix1, we have
(20)ϕ(xni,x1)≤ϕ(x*,x1),∀i≥1.
Since the norm ∥·∥ is weakly lower semicontinuous, we have
(21)liminfi→∞ϕ(xni,x1)=liminfi→∞(∥xni∥2-2〈xni,Jx1〉+∥x1∥2)≥∥x*∥2-2〈x*,Jx1〉+∥x1∥2=ϕ(x*,x1),
and so
(22)ϕ(x*,x1)≤liminfi→∞ϕ(xni,x1)≤limsupi→∞ϕ(xni,x1)≤ϕ(x*,x1).
This implies that limi→∞ϕ(xni,x1)=ϕ(x*,x1), and so ∥xni∥→∥x*∥ as i→∞. Since xni⇀x*, by virtue of Kadec-Klee property of E, we obtain that
(23)limi→∞xni=x*.
Since {ϕ(xn,x1)} is convergent, this, together with limi→∞ϕ(xni,x1)=ϕ(x*,x1), shows that limn→∞ϕ(xn,x1)=ϕ(x*,x1). If there exists some subsequence {xnj} of {xn} such that xnj→y as j→∞, then from Lemma 4 (1) we have that
(24)ϕ(x*,y)=limi,j→∞ϕ(xni,xnj)=limi,j→∞ϕ(xni,ΠCnjx1)≤limi,j→∞(ϕ(xni,x1)-ϕ(ΠCnjx1,x1))=limi,j→∞(ϕ(xni,x1)-ϕ(xnj,x1))=ϕ(x*,x1)-ϕ(x*,x1)=0.
That is, x*=y and so
(25)limn→∞xn=x*.
(iv)x* is a member of F.
Setℕi={k∈ℕ:k=i+((m-1)m/2),m≥i,m∈ℕ} for each i≥1. Note that Tikmk=Timk,κik=κi, and μmk(ik)=μmk(i) whenever k∈ℕi for each i≥1. For example, by Lemma 7 and the definition of ℕ1, we have ℕ1={1,2,4,7,11,16,…} and i1=i2=i4=i7=i11=i16=···=1. By virtue of xn+1=ΠCn+1x1∈Cn+1, we have
(26)ϕ(xk,Timkxk)≤21-κi〈xk-xk+1,Jxk-JTimkxk〉+μmk(i)νk1-κi
for all k∈ℕi. Note that {mk}k∈ℕi={i,i+1,i+2,…}; that is, mk↑∞ as ℕi∋k→∞. It then follows from (25) that
(27)limℕi∋k→∞ϕ(xk,Timkxk)=0,∀i≥1.
In view of (5), we have
(28)limℕi∋k→∞(∥xk∥-∥Timkxk∥)=0,∀i≥1.
This, in view of (25), implies that
(29)limℕi∋k→∞∥Timkxk∥=∥x*∥,∀i≥1,
which yields that
(30)limℕi∋k→∞∥JTimkxk∥=∥Jx*∥,∀i≥1.
This shows that {JTimkxk}k∈ℕi is bounded for each i≥1. Note that both E and E* are reflexive. We may assume that JTimkxk⇀f∈E* as ℕi∋k→∞. In view of the reflexivity of E, we know that there exists an element y∈E such that Jy=f. Then we have
(31)ϕ(xk,Timkxk)=∥xk∥2-2〈xk,JTimkxk〉+∥Timkxk∥2=∥xk∥2-2〈xk,JTimkxk〉+∥JTimkxk∥2.
Taking liminfℕi∋k→∞ on both sides of the above equation yields that
(32)0≥∥x*∥2-2〈x*,f〉+∥f∥2=∥x*∥2-2〈x*,Jy〉+∥Jy∥2=∥x*∥2-2〈x*,Jy〉+∥y∥2=ϕ(x*,y).
That is, x*=y, which in turn implies that f=Jx*. It follows that JTimkxk⇀Jx*∈E* as ℕi∋k→∞. In the light of the Kadec-Klee property of E*, we obtain from (30) that limℕi∋k→∞JTimkxk=Jx*. Since J-1:E*→E is norm-weak-continuous, Timkxk⇀x* as ℕi∋k→∞. This implies, from (29) and the Kadec-Klee property of E, that
(33)limℕi∋k→∞Timkxk=x*,∀i≥1.
Again note that {mk}k∈ℕi={i,i+1,i+2,…}; that is, mk+1-1=mk whenever k∈ℕi for each i≥1. We then have
(34)∥Timk+1xk-x*∥≤∥Timk+1xk-Timkxk∥+∥Timkxk-xk∥=∥Timk+1xk-Timk+1-1xk∥+∥Timkxk-xk∥.
It follows from (33) and the asymptotic regularity of each Ti that
(35)limℕi∋k→∞∥Timk+1xk-x*∥=0,∀i≥1.
That is,
(36)limℕi∋k→∞Ti(Timkxk)=x*,∀i≥1.
In view of the closedness of Ti, it follows from (33) that Tix*=x*, namely, for each i≥1, x*∈F(Ti) and hence x*∈F.
(v) Considerx*=ΠFx1, and so xn→ΠFx1 as n→∞.
Putu=ΠFx1. Since u∈F⊂Cn and xn=ΠCnx1, we have ϕ(xn,x1)≤ϕ(u,x1), for all n≥1. Then
(37)ϕ(x*,x1)=limn→∞ϕ(xn,x1)≤ϕ(u,x1),
which implies that x*=u since u=ΠFx1, and hence xn→x*=ΠFx1. This completes the proof.


## 4. Applications to a System of Equilibrium Problems

The kind of equilibrium problems is a unifying model for several problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games, and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems. Many authors have proposed some useful methods to solve the equilibrium problem (EP) and generalized equilibrium problem (GEP) (see, e.g., [15–19]).LetE be a smooth, strictly convex, and reflexive Banach space, C a nonempty closed and convex subset of E, and {fi}i=1∞:C×C→ℝ a countable family of bifunctions satisfying the conditions: for each i≥1,(A1)
fi(x,x)=0;(A2)
fi is monotone; that is, fi(x,y)+fi(y,x)≤0;(A3)
limsupt↓0fi(x+t(z-x),y)≤fi(x,y);(A4)
the mappingy↦fi(x,y) is convex and lower semicontinuous.A system of equilibrium problems for {fi}i=1∞ is to find an x*∈C such that
(38)fi(x*,y)≥0,∀y∈C,i≥1,
whose set of common solutions is denoted by EP:=∩i=1∞EP(fi), where EP(fi) denotes the set of solutions to the equilibrium problem for fi(i=1,2,…). It will be shown in Theorem 9 that such a system of problems can be reduced to approximation of some fixed points of a countable family of nonlinear mappings.Letr>0. Define a countable family of mappings {Ti}i=1∞:C→C as follows:
(39)Ti(x)={z∈C:fi(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},∀i≥1.
It is shown in [15] that (1)
{Ti}i=1∞ is a sequence of single-valued mappings;(2)
{Ti}i=1∞ is a sequence of closed quasi-ϕ-nonexpansive mappings; that is,
(40)ϕ(p,Tix)≤ϕ(p,x),∀x∈C,p∈F(Ti);(3)
F:=∩i=1∞F(Ti)=EP.Now, we have the following result.Theorem 9.
LetE be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property, C a nonempty closed and convex subset of E, and {Ti}i=1∞:C→C be a countable family of mappings defined by (39). Let {xn} be the sequence generated by(41)x1∈E;C1=C,Cn+1={z∈Cn:ϕ(xn,Tinxn)≤2〈xn-z,Jxn-JTinxn〉},xn+1=ΠCn+1x1,∀n≥1,
where ΠCn+1 is the generalized projection of E onto Cn+1; in is the solution to the positive integer equation n=i+((m-1)m/2)(m≥i,n=1,2,…). If F:=∩i=1∞F(Ti)≠∅, then {xn} strongly converges to ΠFx1 which is a common solution of the system of equilibrium problems for {fi}i=1∞.Proof.
Firstly, it follows from Lemma6 that F is closed and convex. Secondly, {Ti}i=1∞ are obviously asymptotically strictly quasi-ϕ-pseudocontractions with μ1(i)=0 and κi=0 for all i≥1, since they are quasi-ϕ-nonexpansive. Finally, (13) clearly reduces to (41) with mn=1 for all n≥1. Therefore conclusion can be obtained immediately from Theorem 8.



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