Inertial algorithm for approximating a common fixed point for a countable family of relatively nonexpansive maps

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R E S E A R C H

Open Access

Inertial algorithm for approximating

a common ﬁxed point for a countable

family of relatively nonexpansive maps

C.E. Chidume

*

_{, S.I. Ikechukwu and A. Adamu}

*_{Correspondence:}

[email protected] African University of Science and Technology, Abuja, Nigeria

Abstract

In this paper, we study an inertial algorithm for approximating a common ﬁxed point for a countable family of relatively nonexpansive maps in a uniformly convex and uniformly smooth real Banach space. We prove a strong convergence theorem. This theorem is an improvement of the result of Matsushita and Takahashi (J. Approx. Theory 134:257–266, 2005) and the result of Donget al.(Optim. Lett. 12:87–102, 2018). Finally, we give some applications of our theorem.

MSC: 47H09; 47H05; 47J25; 47J05

Keywords: Inertial algorithm; Relatively nonexpansive maps; Generalised projection; Strong convergence

1 Introduction

An inertial-type algorithm was ﬁrst proposed by Polyak [3] as an acceleration process in solving a smooth convex minimisation problem. An inertial-type algorithm is a two-step iterative method in which the next iterate is deﬁned by making use of the previous two iterates. It is well known that incorporating an inertial term in an algorithm speeds up or accelerates the rate of convergence of the sequence generated by the algorithm. Conse-quently, a lot of research interest is now devoted to the inertial-type algorithm (see e.g. [2, 4, 5] and the references contained in them).

LetEbe a real Banach space andE∗be its dual space. LetCbe a nonempty, closed and convex subset ofE, and letTbe a map fromCinto itself. The fixed point set ofTis defined as follows:F(T) :={x∈C:Tx=x}. The normalised duality mapJfromEto 2E∗_{is defined}

byJx={x∗∈E∗:x,x∗=x2=x∗2,∀x∈E}, where·,·denotes the duality pairing. The following properties of the normalised duality map will be needed in the sequel (see e.g. Ibaraki and Takahashi [6]):

1. IfEis a reﬂexive, strictly convex and smooth real Banach space, thenJis surjective, injective and single-valued. IfEis uniformly smooth, thenJis uniformly continuous on bounded sets.

2. In a real Hilbert spaceH, the duality mapJis the identity map onH.

The Lyapunov functionalψ:E×E→[0,∞) is deﬁned byψ(x,y) =x2– 2x,Jy+y2, ∀x,y∈E, whereJis the normalised duality map fromEtoE∗. It was ﬁrst introduced by

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Alber and has been extensively studied by many authors (see e.g. Alber [7], Chidumeet al. [8, 9], Kamimura and Takahashi [10], Reich [11, 12], Takahashi and Zembayashi [13], Zegeye [14] and a host of other authors). It is easy to see from the deﬁnition ofψthat, in a real Hilbert spaceH,ψ(x,y) =x–y2,∀x,y∈H. Furthermore, for anyx,y,z∈Eand α∈(0, 1), we have the following properties:

(N0) (x–y)2≤ψ(x,y)≤(x+y)2,

(N1) ψ(x,y) =ψ(x,z) +ψ(z,y) + 2z–x,Jy–Jz,

(N2) ψ(x,J–1(αJy+ (1 –α)Jz))≤αψ(x,y) + (1 –α)ψ(x,z),

(N3) ψ(x,y)≤ xJx–Jy+yx–y.

Let E be a smooth, strictly convex and reﬂexive real Banach space, and let C⊆E be nonempty, closed and convex. The mapC:E→Cdeﬁned byCx:=u0∈Csuch that

ψ(u0,x) =arg infy∈Cψ(y,x) is called thegeneralised projection, wherearg infy∈Cψ(y,x) is

the set of all the minimisers ofψ. We remark that in a real Hilbert spaceH, the gener-alised projectionCcoincides with the metric projectionPCfromHontoC.

A pointx∗∈Cis called anasymptotic ﬁxed pointofTif there exists a sequence{un_{} ⊆}_C

such thatunx∗andun–Tun →0, asn→ ∞(see e.g. Changet al. [15]). Here we shall denote the set of asymptotic ﬁxed points ofTbyF(T).

A real Banach spaceEis called anOpialspace (see e.g. Chidume [16]) if whenever{un_}

is a sequence inEsuch thatun_x_∈_E_{, then}

lim inf

n→∞ u

n_–_x_<_{lim inf} n→∞ u

n_–_y_, _∀_y_∈_E_,_y₌_x_.

Deﬁnition 1.1 A mapT:C→Cis said to berelatively nonexpansiveif:

(i) F(T) =F(T)=∅, and

(ii) ψ(p,Tx)≤ψ(p,x),∀x∈C,p∈F(T).

Remark1 Every real Hilbert space is an Opial space, and so if{un_}_{is a sequence in}_H_such

thatun_x∗_and_un_–_Tun_→_{0, it is well known that if}_T_{is nonexpansive, then}_Tx∗₌_x∗ andF(T) =F(T). Moreover, sinceψ(x,y) =x–y2,∀x,y∈H, it follows thatTis relatively nonexpansive.

LetB:={x∈E:x= 1}be the unit sphere ofE. A Banach spaceEis said to bestrictly convex if, for all x,y∈B, x=y⇒ x+₂y < 1. The space E is said to have the Kadec– Kleeproperty if whenever{un} is a sequence inE that converges weakly tou0∈Eand un_→_u0_{, as}_n_{→ ∞, then}_{_un_}_{converges strongly to}_u0_{. A space}_E_{is said to be}

uni-formly convexif, for each∈(0, 2], there existsδ> 0 such thatx–y ≥⇒ x+₂y< 1 –δ, ∀x,y∈B. It is well known that a uniformly convex real Banach space is reﬂexive, strictly convex and has the Kadec–Klee property [17, 18].

A functionδ: (0, 2]→[0, 1] called themodulus of convexityofEis deﬁned as follows: δ() :=inf{1 –x₂+y:x,y∈E,x=y= 1,x–y ≥}. It follows thatE is uniformly convex ifδ() > 0,∀∈(0, 2]. A real Banach spaceEis said to besmoothif the limit

lim

t→0

x+ty–x

t exists for allx,y∈B. (1.1)

It is said to beuniformly smoothif the limit in (1.1) above is attained uniformly, for all

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In 2003, Nakajo and Takahashi [19] studied the following CQiterative algorithm for approximating a ﬁxed point of a nonexpansive map in a real Hilbert space:u0_∈_C_and

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

vn₌_α

nun+ (1 –αn)Tun,

Cn={z∈C:vn–z ≤ un–z},

Qn={z∈C:un–z,un–u0 ≤0},

un+1₌_P

Cn∩Qnu

0_, _n_≥_1,

(1.2)

whereαn∈[0, 1],Cis a nonempty, closed and convex subset of a real Hilbert space,Tis

a nonexpansive map fromCinto itself andPCn∩Qn is the metric projection fromConto

Cn∩Qn. They proved that the sequence generated by algorithm (1.2) converges strongly

to a ﬁxed point ofT.

In 2005, Matsushita and Takahashi [1] studied the following iterative algorithm for ap-proximating a ﬁxed point of a relatively nonexpansive map in a uniformly convex and uniformly smooth real Banach space:u0_∈_C_and

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

vn₌_J–1₍_α

nJun+ (1 –αn)JTun),

Cn={z∈C:ψ(z,vn)≤ψ(z,un)},

Qn={z∈C:un–z,Jun–Ju0 ≤0},

un+1=Cn∩Qnu0, n≥1.

(1.3)

They proved that the sequence{un_}_{generated by algorithm (1.3) converges strongly to a}

ﬁxed point ofT.

Recently, Donget al. [2] studied the followinginertialCQ algorithm for nonexpansive maps in areal Hilbert space. They proved the following theorem:

Theorem 1.2(Donget al., [2, Theorem 4.1]) Let T:H→H be a nonexpansive map such that F(T)=∅.Let{αn} ⊂[α1,α2],α1∈(–∞, 0],α2∈[0,∞),{βn} ⊂[β, 1],β∈(0, 1].Set

u0_,_u1_∈_{H arbitrarily}_._{Deﬁne a sequence}_{_un_}_{by the following algorithm}_:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

wn₌_un₊_α

n(un–un–1),

vn= (1 –βn)wn+βnT(wn),

Cn={z∈H:vn–z ≤ wn–z},

Qn={z∈H:un–z,un–u0 ≤0},

un+1₌_P

Cn∩Qnu0, n≥0.

(1.4)

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2 Preliminaries

Lemma 2.1(Alber [7]) Let C be a nonempty closed and convex subset of a strictly convex and reﬂexive real Banach space E.If x∈E and u0∈C,then

u0=Cx ⇐⇒

u0–y,Ju0–Jx ≤0, ∀y∈C, and

ψ(y,Cx) +ψ(Cx,x)≤ψ(y,x), ∀y∈C,x∈E.

(2.1)

Lemma 2.2(Kamimura and Takahashi [10]) Let E be a smooth and uniformly convex real Banach space,and let{un_}_and_{_vn_}_{be two sequences of E}_._{If either}_{_un_}_or_{_vn_}_{is bounded}

andψ(un,vn)→0,as n→ ∞,thenun–vn →0,as n→ ∞.

Remark2 Using (N3), it is easy to see that the converse of Lemma 2.2 is also true whenever {un_}_and_{_vn_}_{are both bounded.}

Lemma 2.3(Matsushita and Takahashi [1]) Let E be a smooth and strictly convex real Banach space,and let C be a nonempty,closed and convex subset of E.Let T be a map from C into itself such that F(T)=∅andψ(y,Tx)≤ψ(y,x),∀(y,x)∈F(T)×C.Then F(T)

is closed and convex.

Lemma 2.4(Kohsaka and Takahashi [20, Theorem 3.3]) Let C be a nonempty,closed and convex subset of a uniformly convex and uniformly smooth real Banach space E,and let Ti:C→E,i= 1, 2, 3, . . . ,be a countable family of relatively nonexpansive maps such that

∞

i=1F(Ti)=∅.Suppose{αi} ⊂(0, 1)and{βi} ⊂(0, 1)are sequences such that

∞

i=1αi= 1

and T:C→E is deﬁned by

Tx=J–1

_∞

i=1 αi

βiJx+ (1 –βi)JTix

for each x∈C.

Then T is relatively nonexpansive and F(T) =∞_i₌₁F(Ti).

3 Main results

We ﬁrst prove the following lemma which will be central for the proof of our main theo-rem.

Lemma 3.1 Let E be a uniformly convex and uniformly smooth real Banach space.Let T:E−→E be a relatively nonexpansive map such that F(T)=∅.Let{un}be generated by the following algorithm:u0_,_u1_∈_{E and}

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C0=E,

wn₌_un₊_α

n(un–un–1),

vn₌_J–1_{[(1 –}_β₎_Jwn₊_β_JTwn_],

Cn+1={z∈Cn:ψ(z,vn)≤ψ(z,wn)},

un+1₌

Cn+1u0, n≥1,

(3.1)

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Proof We partition our proof into four steps.

Step 1. We show that{un}is well deﬁned andF(T)⊆Cn,∀n≥0.

Letz∈Cn+1, then ψz,vn_≤_ψ_z_,_wn

⇔ z2– 2z,Jvn +vn2≤ z2– 2z,Jwn +wn2

⇔ 2z,Jwn–Jvn ≤wn2–vn2. (3.2) Using (3.2) above, we observe thatCnis closed and convex,∀n≥0. We now show that

F(T)⊆Cn,∀n≥0. For n=0,F(T)⊆C0=E. AssumeF(T)⊆Cn, letp∈F(T). Then, by

(N2) and the fact thatT is relatively nonexpansive, we have that ψp,vn=ψ(p,J–1(1 –β)Jwn+βJTwn

≤(1 –β)ψp,wn+βψp,Twn

≤(1 –β)ψp,wn+βψp,wn=ψp,wn, (3.3)

which implies thatp∈Cn+1. So, by induction,F(T)⊆Cnfor alln≥0. Thus,{un}is well

deﬁned.

Step 2. We show that{un_}_,_{_vn_}_and_{_wn_}_{are bounded.}

We observe thatun₌

Cnu0andCn+1⊆Cnfor alln≥0. So, employing Lemma 2.1, we have thatψ(un_,_u0₎_≤_ψ₍_un+1_,_u0_{). Hence,}_{_ψ₍_un_,_u0_)}_{is nondecreasing. Furthermore, we} obtain thatψ(un,u0) =ψ(Cnu0,u0)≤ψ(p,u0) –ψ(p,un)≤ψ(p,u0), which implies that {ψ(un_,_u0₎_}_{is bounded; and hence by (}_N

0),{un}is bounded; and consequently,{ψ(un,u0)} is convergent. From Lemma 2.1, we have that

ψum,un=ψum,Cnu

0_≤_ψ_um_,_u0_–_ψ_un_,_u0_→_0, _as_n_,_m_{→ ∞}_. Hence,{un_}_{is Cauchy and this implies that}_un+1_–_un_→_{0, as}_n_{→ ∞}_.

Using the deﬁnition ofwn_{, we have that}_un_–_wn₌_α

n(un–1–un) ≤ un–1–un →0, as

n→ ∞. Since{wn_}_{is bounded, by Remark 2, we have that}_ψ₍_un_,_wn₎_→_{0, as}_n_{→ ∞. Since}

un+1_∈_C

n, it follows that 0≤ψ(un+1,vn)≤ψ(un+1,wn)→0, asn→ ∞, which implies that

un–vn →0, asn→ ∞; and consequently,{vn}is bounded.

Step 3. We show thatwn_–_Twn_→_{0, as}_n_{→ ∞}_.

Using Remark 2, we deduce thatψ(un_,_vn₎_→_{0, as}_n_{→ ∞}_{. By (}_N_{1) and the uniform}

continuity ofJon bounded sets, we have that

ψwn,vn=ψwn,un+ψun,vn+ 2un–wn,Jvn–Jun

≤ψwn,un+ψun,vn+ 2un–wnJun–Jvn→0, asn→ ∞. (3.4) Next, from the deﬁnition ofvn_{, we observe that}

_Jvn_–_Jwn₌_β_JTwn_–_Jwn_.

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Step 4. We show thatun_→ F(T)u0.

Since{wn_}_{is bounded, there exists}_{_wnk}, a subsequence of{_wn}, such that_wnk _x∗_, ask→ ∞. Using Step 3, we obtain thatwnk_–_Twnk →_{0, as}_k→ ∞_{. Since our map is} relatively nonexpansive, we have thatx∗∈F(T). Thus, it follows from Step 2 that there exists{unk}, a subsequence of {_un}, such that_unk _x∗_{, as}_k→ ∞. We now show that

x∗=F(T)u0. Setv=F(T)u0. Fromun₌

Cnu0andF(T)⊆Cn,∀n≥1, we haveψ(un,u0)≤ψ(v,u0). Employing the

weak lower semi-continuity of norm, we obtain

ψx∗,u0=x∗2– 2x∗,Ju0 +u02

≤lim infunk2_{– 2}_unk_,_Ju0 ₊_u02

=lim infψunk_,_u0≤_{lim sup}_ψ_unk_,_u0≤_ψ_v_,_u0_. _(3.5) However,

ψv,u0≤ψz,u0, for allz∈F(T) ⇒ ψv,u0≤ψx∗,u0≤ψv,u0

⇒ ψv,u0=ψx∗,u0. (3.6)

By the uniqueness ofF(T)u0,v=x∗. So, we deduce that x∗=F(T)u0. Next, we show that unk →_x∗_{, as}_k→ ∞. Using (3.5) and (3.6), we obtain that_ψ₍_unk_,_u0₎→_ψ₍_x∗_,_u0_), ask→ ∞. As a result of this, we obtain thatunk → _x∗, as_k→ ∞. By the Kadec– Klee property ofE, we conclude thatunk →_x∗_{, as}_k→ ∞. Consequently, since{_un}_is convergent, we obtain thatun_→_x∗_{, as}_n_{→ ∞}_{. Therefore,}_un_→

F(T)u0. This completes

the proof.

Using Lemma 3.1, we now prove our main theorem of this paper.

Theorem 3.2 Let E be a uniformly convex and uniformly smooth real Banach space.Let Ti:E→E,i= 1, 2, 3, . . . ,be a countable family of relatively nonexpansive maps such that

∞

i=1F(Ti)=∅.Suppose{αi} ⊂(0, 1)and{βi} ⊂(0, 1)are sequences such that

∞

i=1αi= 1

and T:E→E is deﬁned by Tx=J–1(∞_i₌₁αi(βiJx+ (1 –βi)JTix))for each x∈E.Let{un}be

generated by the following algorithm:u0_,_u1_∈_{E and} ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C0=E,

wn₌_un₊_α

n(un–un–1),

vn₌_J–1_{[(1 –}_β₎_Jwn₊_β_JTwn_],

un+1₌

Cn+1u0, n≥0,

(3.7)

whereαn∈(0, 1),β∈(0, 1),then{un}converges strongly to a point p=F(T)u0.

Proof From Lemma 2.4,Tis relatively nonexpansive andF(T) =∞_i₌₁F(Ti). The

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Corollary 3.3 Let E be uniformly convex and uniformly smooth real Banach spaces Lp(or

lpor Wpm( )), 1 <p<∞.Let Ti:E→E,i= 1, 2, 3, . . . ,be a countable family of relatively

nonexpansive maps such that∞_i₌₁F(Ti)=∅.Suppose that{αi} ⊂(0, 1)and{βi} ⊂(0, 1)

are sequences such that∞_i₌₁αi= 1and T:E→E is deﬁned by Tx=J–1(

∞

i=1αi(βiJx+

(1 –βi)JTix))for each x∈E.Let{un}be generated by the following algorithm:u0,u1∈E

and

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C0=E,

wn₌_un₊_α

n(un–un–1),

vn=J–1[(1 –β)Jwn+βJTwn],

un+1=Cn+1u

0_, _n_≥_1,

(3.8)

whereαn∈(0, 1),β∈(0, 1).Then{un}converges strongly toF(T)u0.

Corollary 3.4 Let H be a real Hilbert space.Let Ti:H→H,i= 1, 2, 3, . . . ,be a countable

family of nonexpansive maps such that∞_i₌₁F(Ti)=∅. Suppose{αi} ⊂(0, 1)and{βi} ⊂

(0, 1)are sequences such that∞_i₌₁αi= 1and T:H→H is deﬁned by Tx= (

∞

i=1αi(βix+

(1 –βi)Tix))for each x∈H.Let{un}be generated by the following algorithm:u0,u1∈H

and

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C0=H,

wn=un+αn(un–un–1),

vn_{= (1 –}_β₎_wn₊_β_Twn_,

Cn+1={z∈Cn:z–vn ≤ z–wn},

un+1₌

Cn+1u0, n≥1,

(3.9)

whereαn∈(0, 1),β∈(0, 1).Then{un}converges strongly toF(T)u0.

Proof Using Remark 1, we have thatTiis relatively nonexpansive for eachi≥1. Thus, the

conclusion follows from Theorem 3.2.

4 Conclusion

Theorem 3.2 is an improvement of the result of Matsushita and Takahashi [1] and the result of Donget al. [2] in the following sense:

• The algorithms studied in Matsushita and Takahashi [1] and Donget al. [2] require at

each step of the iteration processthe computation of two subsetsCnandQnofC; their

intersectionCn∩Qn, and the projection of the initial vector onto this intersection. In our iteration process, the subsetQnhas been dispensed with. Furthermore, the sequences{αn}and{βn}used in the algorithms of Matsushita and Takahashi [1] and

Donget al. [2], which are also to be computed at each step of the iteration process,

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• In Matsushita and Takahashi [1], the authors proved a strong convergence theorem for a relatively nonexpansive mapT:C→C. In our Theorem 3.2, a strong

convergence theorem is proved for acountable family of relatively nonexpansive maps

Ti:E→E,i∈N. Furthermore, unlike the algorithm of Matsushita and Takahashi, our algorithm has aninertial termwhich is known to improve the speed of convergence over algorithms without an inertial term (see e.g. [2–5, 21] and the references contained in them).

• In Donget al. [2, Theorem 4.1], the authors proved a strong convergence theorem in a real Hilbert space for one nonexpansive map. Our theorem is proved in the much more general uniformly convex and uniformly smooth real Banach spaces and for a countable family of relatively nonexpansive maps.

Acknowledgements

The authors thank the African Capacity Building Foundation (ACBF) for sponsoring this research work.

Funding

Research supported from ACBF Research Grant Funds to AUST.

Competing interests

The authors declare that they have no conﬂict of interest.

Authors’ contributions

CEC formulated the problem and suggested the method of proof of the Theorem to SII and AA. The computations using the method suggested by CEC were carried out by SII and AA. The analysis of the computations to arrive at the proof of the Theorem was done jointly by CEC, SII, and AA. All authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 9 November 2017 Accepted: 16 February 2018 References

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