R E S E A R C H
Open Access
Relaxed iterative algorithms for a system
of generalized mixed equilibrium problems
and a countable family of totally
quasi-Phi-asymptotically nonexpansive
multi-valued maps, with applications
CE Chidume
*, OM Romanus and UV Nnyaba
*Correspondence: [email protected] African University of Science and Technology, Abuja, Nigeria
Abstract
In this article, a Krasnoselskii-type and a Halpern-type algorithm for approximating a common fixed point of a countable family of totally quasi-
φ
-asymptoticallynonexpansive nonself multi-valued maps and a solution of a system of generalized mixed equilibrium problem are constructed. Strong convergence of the sequences generated by these algorithms is proved in uniformly smooth and strictly convex real Banach spaces with the Kadec-Klee property. Several applications of our theorems are also presented. Finally, our theorems are a significant improvement of several
important recent results.
MSC: 47H10; 47H04; 47J25; 47J20
Keywords: generalized mixed equilibrium problems; totally quasi-
φ
-asymptotically nonexpansive nonself multi-valued maps; strong convergence; equally continuous maps1 Introduction
In what follows, we assume thatX is a real Banach space with dual space X∗, K is a nonempty, closed, and convex subset ofX, and→andwill, respectively, denote strong and weak convergence.
(See,e.g., Wang and Zang [] for a similar definition for self maps.) LetG:K→Xbe any map. A pointu∈Kis called afixed pointofGif and only ifu∈Guand it is called an
asymptotic fixed pointofGif there exists a sequence{un}inKthat converges weakly tou andlimn→∞d(un,Gun) :=limn→∞infηn∈Gunun–ηn= (see Changet al.[]). We denote the set of fixed points and asymptotic fixed points ofGbyF(G) andFˆ(G), respectively.
A subsetKofXis said to be a retract ofX, if there exists a continuous mapP:X→K
such thatPu=u, for allu∈X. It is well known that every nonempty, closed, convex subset of a uniformly convex Banach spaceX is a retract ofX. A mapP:X→K is said to be a retraction ifP=P. A mapP:X→K is said to be a nonexpansive retraction, if it is nonexpansive and it is a retraction fromXtoK.
Define the Lyapunov functionalφ:X×X→Rby
φ(u,v) =u– u,Jv+v ∀u,v∈X. (.) From the definition ofφit is obvious that
u–v≤φ(u,v)≤u+v ∀u,v∈X. (.)
In what follows, we assume thatP:X→Kis a nonexpansive retraction.
Definition . A nonself multi-valued mapG:K→Xis said to berelatively
asymptoti-cally nonexpansiveifF(G)=∅,Fˆ(G) =F(G), and there exists a real sequence{βn} ⊂[,∞),
βn↓ such thatφ(p,ηn)≤βnφ(p,u)∀u∈K,p∈F(G),ηn∈G(PG)n–u,n≥ (see,e.g., Wang and Zang [] for a similar definition for self maps).
The following definitions appear in Bo and Yi [].
Definition . A nonself multi-valued mapG:K→Xis said to be
• quasi-φ-nonexpansiveifF(G)=∅andφ(p,ηn)≤φ(p,u)∀u∈K,p∈F(G),
ηn∈G(PG)n–u,n≥;
• quasi-φ-asymptotically nonexpansiveifF(G)=∅and there exists a real sequence
{βn} ⊂[,∞),βn↓such thatφ(p,ηn)≤βnφ(p,u)∀u∈K,p∈F(G),ηn∈G(PG)n–u,
n≥;
• totally quasi-φ-asymptotically nonexpansiveifF(G)=∅and there exist nonnegative real sequences{γn},{δn}withγn→,δn→(n→ ∞) and a strictly increasing and continuous functionρ:R+→R+withρ() = such that
φ(p,ηn)≤φ(p,u) +γnρ
φ(p,u)+δn
∀u∈K,p∈F(G),ηn∈G(PG)n–u,n≥. (.)
Remark From the definitions, it is easy to see that the class of relatively asymptot-ically nonexpansive multi-valued nonself maps and the class of quasi-φ-nonexpansive multi-valued nonself maps are proper subclasses of the class of quasi-φ-asymptotically nonexpansive multi-valued nonself maps and that the class of quasi-φ-asymptotically non-expansive multi-valued nonself maps is a proper subclass of the class of totally quasi-φ -asymptotically nonexpansive multi-valued nonself maps, but the converse may not be true.
Definition . A countable family of multi-valued nonself maps, Gi : K →X, i = , , , . . . , is said to be
• uniformly quasi-φ-asymptotically nonexpansiveif∞i=F(Gi)=∅and there exists a sequence{βn} ⊂[,∞),βn↓such that, for eachi≥,
φ(p,ηn)≤βnφ(p,u) ∀u∈K,p∈
∞
i=
F(Gi),ηn∈Gi(PGi)n–u,n≥
(see,e.g., Changet al.[]);
strictly increasing and continuous functionρ:R+→R+withρ() = such that, for
eachi≥,
φ(p,ηn)≤φ(p,u) +γnρ
φ(p,u)+δn
∀u∈K,p∈ ∞
i=
F(Gi),ηn∈Gi(PGi)n–u,n≥
(see,e.g., Yi []).
Remark From the definitions, it is easy to see that a countable family of uniformly
quasi-φ-asymptotically nonexpansive multi-valued nonself maps is a countable family of uni-formly totally quasi-φ-asymptotically nonexpansive multi-valued nonself maps.
Remark We also remark that a collection of countable families of uniformly totally quasi-φ-asymptotically nonexpansive multi-valued nonself maps is a subcollection of a collection of countable families of totally quasi-φ-asymptotically nonexpansive multi-valued nonself maps.
A motivation for the study of the class of totally quasi-φ-asymptotically nonexpansive self or nonself maps is the objective to unify various definitions of classes of maps, asso-ciated with the class of relatively nonexpansive self or nonself maps, which are extensions to arbitrary real Banach spaces of nonexpansive nonself maps, with nonempty fixed point sets in Hilbert spaces. Our objective is to prove general convergence theorems applicable to all these classes.
Definition .(See,e.g., Fenget al.[] for a similar definition for self maps) A multi-valued nonself mapG:K→Xis said to be
• equally continuousif forun,vn∈Kwe have
lim
n→∞un–vn= ⇒ nlim→∞ηnu–ηnv=
∀ηnu∈G(PG) n–un,η
nv∈G(PG) n–vn;
• uniformly continuousif forun,vn∈Kwe have
lim
n→∞un–vn= ⇒ nlim→∞ηnu–ηnv= ∀ηnu∈Gun,ηnv∈Gvn; • uniformlyL-Lipschitz continuousif there exists a constantL> such that
ηu–ηv ≤Lu–v ∀ηu∈G(PG)n–u,ηv∈G(PG)n–v,n≥.
Remark It is easy to see that the class of uniformlyL-Lipschitz multi-valued nonself maps is a proper subclass of the class of uniformly continuous multi-valued nonself maps and the class of uniformly continuous multi-valued nonself maps is a proper subclass of the class of equally continuous multi-valued nonself maps.
that
fu∗,v+ψ(v) –ψu∗+v–u∗,Au∗ ≥, ∀v∈K. (.)
The set of solutions of the generalized mixed equilibrium problem is denoted by GMEP(f,
A,ψ).
The class of generalized mixed equilibrium problems includes, as special cases, the class ofmixed equilibrium problems(A≡; see,e.g., Ceng and Yao [] and the references con-tained therein); the class ofgeneralized equilibrium problems(ζ ≡; see,e.g., Takahashi and Takahashi []); the class ofequilibrium problems(A≡,ζ≡; see,e.g., Fan [], Blum and Oettli [], and the references contained therein); the class ofvariational inequality problems(h≡,ζ≡; see,e.g., Stampacchia []); and the class ofconvex minimization problems(A≡,h≡).
The generalized mixed equilibrium problem has applications in physics, economics, fi-nance, transportation, network and structural analysis, ecology, image reconstruction, and elasticity. It includes, as special cases, fixed point problems, variational inequality prob-lems, complementarity probprob-lems, equilibrium probprob-lems, optimization probprob-lems, Nash equilibrium problems in noncooperative games, etc. (see,e.g., Blum and Otelli [], Dafer-mos and Nagurney [], Su [], Barbagallo [], Moudafi [], and the references con-tained therein). In other words, theGMEP(f,A,ψ) is a unifying model for several prob-lems arising in physics, engineering, science, optimization, finance, economics, etc. The projection method, which was first introduced by Haugazeau [], has been utilized to solve the mixed equilibrium problem, the generalized equilibrium problem, and equilib-rium problems in Banach spaces (see,e.g., Qinet al.[], Cholamjiaket al.[], Choet al.
[], Ceng and Yao [], and the references therein). The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact-ness assumptions imposed on maps or subsets of spaces.
Several strong and weak convergence theorems for asymptotically nonexpansive, rela-tively nonexpansive, quasi-φ-nonexpansive and quasi-φ-asymptotically nonexpansive self or nonself maps have been established by various authors in the setting of Banach spaces (see,e.g., Thianwan [], Nilsrakooet al.[], Wang [], Ma and Wang [], Chidumeet al.[, ], and the references contained therein).
In , Changet al.[] considered the class of uniformly quasi-φ-asymptotically non-expansive nonself mapsand studied, in a uniformly convex and uniformly smooth real Banach space, the following Halpern-type algorithm:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u∈X, chosen arbitrarily, K=K,
yn,i=J–(αnJu+ ( –αn)(βnJun+ ( –βn)JGi(PGi)n–un)), i≥,
Kn+={z∈Kn:supi≥φ(z,yn,i)≤αnφ(z,u) + ( –αn)φ(z,un) +θn},
un+=Kn+u, n≥,
(.)
where θn= (βn– )supu∈Fφ(u,un),F :=
∞
i=F(Gi), and{Gi}∞i= is a countable family of
(C) limn→∞αn= ;
(C) <lim infn→∞βn≤lim supn→∞βn< ; (C) Fis a bounded and convex subset ofK,
whereαn∈[, ] andβn∈(, ).
In the same year, Zhao and Chang [] proved that the sequence{un}, generated by algorithm (.), converges strongly toFuunder conditions (C), (C), and the following
condition:
(C∗) Fis a nonempty bounded subset ofK,
where{Gi}∞i=is a countable family of uniformlyL-Lipschitz continuous,closed and uni-formly totallyquasi-φ-asymptotically nonexpansivenonself maps.
Later in the same year, Yi [] established the results in the paper of Zhao and Chang [], when{Gi}∞i=is a countable family of uniformlyL-Lipschitz continuous anduniformly totallyquasi-φ-asymptotically nonexpansivenonself maps, under conditions (C), (C), and the following condition:
(C∗∗) Fis a nonempty subset ofK.
In , Bo and Yi [] proved that the sequence{un}, generated by the iterative algorithm
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u∈X, chosen arbitrarily, K=K,
yn=J–(αnJu+ ( –αn)(βnJun+ ( –βn)Jηn)), ηn∈G(PG)n–un,
Kn+={z∈Kn:φ(z,yn)≤αnφ(z,u) + ( –αn)φ(z,un) +θn},
un+=Kn+u, n≥,
converges strongly toFuunder conditions (C), (C), and (C∗∗), whereF:=F(G) and Gis a uniformlyL-Lipschitz continuous and totally quasi-φ-asymptotically nonexpansive
nonself multi-valuedmap, whileθn=γnsupu∈Fφ(u,un) +δn.
The results of Bo and Yi [], Yi [], Zhao and Chang [], and Changet al.[] are im-portant generalizations and improvements of imim-portant known results.
Motivated by these authors, it is our purpose in this paper to study the following Krasnoselskii-type and Halpern-type algorithms:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u∈X, chosen arbitrarily, K=K, u=Ku,
yn=J–(σJun+ ( –σ)Jη(minn)), (η
(in)
mn ∈Gin(PGin)mn–un),
zn=rinyn,
Kn+={v∈Kn:φ(v,zn)≤φ(v,un) +ωn},
un+=Kn+u, n≥,
(.) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u∈X, chosen arbitrarily, K=K, u=Ku, yn=J–(σnJu+ ( –σn)Jη(minn)), (η
(in)
mn ∈Gin(PGin)mn–un),
zn=rinyn,
Kn+={v∈Kn:φ(v,zn)≤σnφ(v,u) + ( –σn)φ(v,un) +ωn},
un+=Kn+u, n≥.
We also aim to prove, in a uniformly smooth and strictly convex real Banach spaceXwith Kadec-Klee property, that the sequences generated by these algorithms converge strongly to an element in W := (∞i=F(Gi))∩(
∞
i=GMEP(hi,Ai,ζi)), where{Gi}∞i= is a
count-able family ofequally continuous and totallyquasi-φ-asymptotically nonexpansive nonself
multi-valuedmaps;{Ai}∞i=,Ai:K→X∗is a sequence of continuous and monotone maps;
{hi}∞i=,hi:K×K→Ris a sequence of bifunctions satisfying appropriate conditions and
{ζi}∞i=,ζi:K→Ris a sequence of lower-semicontinuous and convex functions. Our the-orems are significant improvements and generalizations of numerous results for this class of nonlinear problems (in particular, the results of Bo and Yi [], Yi [], Zhao and Chang [], Changet al.[], Lv [], Wang and Zhang [], Dadashi and Postolache [], Yao and Postolache [], and the results of a host of other authors [see Remark below]).
2 Preliminaries
A mapJ:X→X∗ defined byJu:={u∗∈X∗:u,u∗=uu∗,u=u∗}is called a
normalized duality maponX, where·,·denotes the duality pairing between elements ofXandX∗.
We now present some lemmas that will be used in the sequel.
Lemma .(See Bo and Yi []) Let X be a smooth,strictly convex and reflexive Banach space and K be a nonempty,closed,convex subset of X.Let G:K→X be a total quasi-φ -asymptotically nonexpansive multi-valued mapping withδ= .Then F(G)is a closed and convex subset of K.
Lemma .(See Changet al.[]) Let X be a uniformly smooth and strictly convex real Banach space with Kadec-Klee property and let K be a nonempty closed convex subset of X.Let{un}and{yn}be two sequences in K such that un→u∗andφ(un,yn)→,whereφ
is the function defined by(.).Then yn→u∗.
Lemma .(See Alber []) Let K be a nonempty closed and convex subset of a reflexive strictly convex and smooth Banach space X.Then
φ(u,Ky) +φ(Ky,y)≤φ(u,v), ∀u∈K,v∈X. (.) LetK be a nonempty closed and convex subset of a Banach spaceX. For solving the generalized mixed equilibrium problem (.), we assume that a bifunctionh:K×K→R
satisfies the following conditions:
(B) h(u,u) = ,∀u∈X,
(B) his monotone, that is,h(u,v) +h(v,u)≤,∀u,v∈X, (B) for allu,y,z∈X,lim supt↓h(tz+ ( –t)u,v)≤h(u,v),
(B) for allu∈K,h(u,·) :K→Ris convex and lower-semicontinuous.
() There existsz∈Ksuch that
h(z,v) +ζ(v) –ζ(z) +v–z,Az+
rv–z,Jz–Ju ≥, ∀v∈K.
() If we define a mappingr:K→Kby
r(u) =
z∈K:h(z,v) +ζ(v) –ζ(z) +v–z,Az+
rv–z,Jz–Ju ≥,
∀v∈K
, u∈K,
then the mappingGrhas the following properties: (a) Gris single-valued;
(b) F(r) =GMEP(h,A,ζ) =Fˆ(r);
(c) GMEP(h,A,ζ)is a closed convex set ofK;
(d) φ(q,ru) +φ(ru,u)≤φ(q,u)∀q∈F(r),u∈X.
3 Main results
In what follows,inandmnare the unique solutions to the positive integer equationn=
i+(m–) m(m≥i,n= , , . . .). That is, for eachn≥, there exist uniqueinandmnsuch that
i= , i= , i= , i= ,
i= , i= , i= , i= , . . . ;
m= , m= , m= , m= ,
m= , m= , m= , m= , . . . .
See Deng []. We now prove the following strong convergence theorem using a Krasno-selskii-typealgorithm.
Theorem . Let X be a uniformly smooth and strictly convex real Banach space with Kadec-Klee property and let X∗ be its dual space.Let K be a nonempty closed and con-vex subset of X and hi :K ×K→R, i= , , , . . . , be a sequence of bifunctions
satis-fying (B)-(B). Let Ai:K →X∗, i= , , , . . . ,be a sequence of continuous monotone
maps and Gi:K→X,i= , , . . . ,be an infinite family of equally continuous and totally
quasi-φ-asymptotically nonexpansive nonself multi-valued maps with nonnegative real se-quences {γn(i)},{δn(i)}and a sequence of strictly increasing and continuous functions {ρi},
ρi:R+→R+,such thatγn(i)→,δn(i)→andρi() = .Letζi:K→R,i= , , , . . . ,be
a sequence of convex and lower-semicontinuous functions.Suppose for each i,δ(i)= and
:= (∞i=F(Gi))∩(∞i=GMEP(hi,Ai,ζi))is a nonempty subset of K.Then the sequence {un}generated by algorithm(.)converges strongly tou,whereσ∈(, ),rin∈[a,∞)
for some a> ,andωn=γm(inn)ρin[φ(p,un)] +δ
(in)
mn,p∈.
Proof The proof is presented in a number of steps.
Clearly,K=Kis closed and convex. AssumeKnis closed and convex for somen≥. It is easy to see thatKn+={v∈Kn: v,Jun–Jzn ≤ un–zn+ωn}. Consequently, it is closed and convex. Hence, Step is completed.
Step :⊂Knfor alln≥.
Again, we proceed by induction. Clearly,⊂K. Suppose⊂Knfor somen≥. Let
q∈. Then, by applying Lemma .(d), the definition ofφ, the convexity of · , and the totally quasi-φ-asymptotically nonexpansiveness ofGi, we have
φ(q,zn) =φ(q,rinyn)≤φ(q,yn) =φ
q,J–σJun+ ( –σ)Jη(minn)
=q– q,σJun+ ( –σ)Jη(minn) +σJun+ ( –σ)Jη
(in) mn
≤ q– σq,Jun– ( –σ)
q,Jη(in)
mn +σun
+ ( –σ)η(in) mn
=σ φ(q,un) + ( –σ)φq,η(in)
mn
≤σ φ(q,un) + ( –σ)φ(q,un) +γ(in)
mnρin
φ(q,un)+δ(in)
mn
≤φ(q,un) +γ(in)
mnρin
φ(q,un)+δ(in)
mn =φ(q,un) +ωn, which implies thatq∈Kn+. Therefore,⊂Knfor alln≥.
Step :{φ(un,u)}is convergent andlimn→∞ωn= .
Sinceun=KnuandKn+⊂Kn for alln≥, we have φ(un,u)≤φ(un+,u), which implies that{φ(un,u)}is nondecreasing. Furthermore, by Lemma ., we have
φ(un,u) =φ(Knu,u)≤φ(q,u) –φ(q,un)≤φ(q,u),
for alln≥ andq∈. Thus,{φ(un,u)} is bounded. Hence,{φ(un,u)}is convergent.
Furthermore, by inequality (.),{un}is bounded. Now, for eachi≥, defineKi:={k≥ :k=i+(m–) m,m≥i,m∈N}. Observe that if, for eachi≥,k∈Ki, thenγ(ik)
mk =γ
(i)
mk,
δ(ik) mk =δ
(i)
mk, andρik=ρi. Also,mk→ ∞ask→ ∞,k∈Ki. Therefore,limn→∞ωn= .
Step :un→u∗,zn→u∗, andyn→u∗asn→ ∞, for someu∗∈K.
By the boundedness of {un}and reflexivity of X, there exists a subsequence{unk} of
{un}such thatunku∗ask→ ∞. SinceKnkis weakly closed,u∗∈Knk⊂K. Thus,unk=
Knkuimplies thatφ(unk,u)≤φ(u∗,u)∀k≥. By the weak lower-semicontinuity of
· , we have
lim inf
k→∞ φ(unk,u) =lim infk→∞
unk
– u
nk,Ju+u
≥u∗– u∗,Ju +u=φ
u∗,u
,
which implies thatφ(u∗,u)≤lim infk→∞φ(unk,u)≤lim supk→∞φ(unk,u)≤φ(u∗,u). Hence, limk→∞φ(unk,u) = φ(u
∗,u
). Therefore, limk→∞unk=u
∗. By the
Kadec-Klee property of X, we have limk→∞unk = u
∗. Since {φ(un,u
)} is convergent and
limk→∞φ(unk,u) =φ(u
∗,u
), we havelimn→∞φ(un,u) =φ(u∗,u). Claim:un→u∗.
Suppose there exists a subsequence{unj}of{un}such thatunj→pasj→ ∞. By applying Lemma ., we have
φu∗,p= lim
j,k→∞φ(unk,unj) = lim
≤ lim j,k→∞
φ(unk,u) –φ(Knju,u)
= lim j,k→∞
φ(unk,u) –φ(uni,u)
=φu∗,u
–φu∗,u
= ,
which impliesu∗=p. Hence, the claim holds. Again, by Lemma . we have
φ(un+,un) =φ(un+,Knu)≤φ(un+,u) –φ(Knu,u) =φ(un+,u) –φ(un,u),
which implies thatφ(un+,un)→ asn→ ∞. Sinceun+∈Kn+, we haveφ(un+,zn)≤
φ(un+,un) +ωn. Consequently,φ(un+,zn)→ asn→ ∞. Therefore, by Lemma .,zn→
u∗. Fromφ(q,zn)≤φ(q,yn)≤φ(q,un) +ωn, we haveφ(q,yn)≤φ(q,un) +ωn. Combining this with the fact thatzn=rinynand Lemma .(d), we have
φ(zn,yn) =φ(rinyn,yn)≤φ(q,yn) –φ(q,rinyn)
≤φ(q,un) –φ(q,rinyn) +ωn=φ(q,un) –φ(q,zn) +ωn,
for anyq∈. This implies thatlimn→∞φ(zn,yn) = . Again, by Lemma . we haveyn→
u∗.
Step :u∗∈.
We first show thatu∗∈∞i=F(Gi). Note thatGik(PGik)mk
–=Gi(PGi)mk–wheneverk∈
Ki for eachi≥. From Step and by the uniform continuity ofJon bounded subsets ofX, we haveJyk–Ju∗ →,Juk–Ju∗ → ask→ ∞. Thus, for eachi≥,ηm(i)k ∈
Gik(PGik) mk–u
k, we have
Jyk–Ju∗=σJuk+ ( –σ)Jη(mi)k–Ju
∗
=( –σ)Jηm(i)
k–Ju
∗–σJu∗–Ju
k
≥( –σ)Jηm(i)
k–Ju
∗–σJu∗–Ju
k,
which implies thatlimk→∞Jηm(i)k–Ju∗= for eachi≥. By the norm-to-weak continuity ofJ–, we have, for eachi≥,η(i)
mku
∗ask→ ∞. Furthermore,
ηm(i)
k–u
∗=Jη(i)
mk–Ju
∗≤Jη(i)
mk–Ju
∗→.
Thus, for eachi≥,limk→∞ηm(i)k=u∗. Hence, by the Kadec-Klee property ofX, we have
lim k→∞η
(i)
mk=u
∗ ∀i≥. (.)
We now consider the sequence{w(mi)k}k∈Ki, generated by
wm(i)k+∈GiPηm(i)k⊂Gik(PGik)
By the continuity ofGiP, we have, for eachi≥,limk→∞w(mi)k+=w∗,w∗∈GiPu∗=Giu∗, sinceu∗∈K. Using the continuity of bothGiand (.), we obtain for eachi≥
w(mi)k+–η(mi)
k≤w
(i)
mk+–η
(i)
mk++η
(i)
mk+–uk++uk+–uk
+uk–ηm(i)k→, k→ ∞.
Therefore, for eachi≥,w(mi)k+→u
∗ask→ ∞. Hence, by the uniqueness of the limit, we
havew∗=u∗. Thus,u∗∈∞i=F(Gi).
We now show thatu∗∈∞i=GMEP(hi,Ai,ζi). Leti≥. Define a functioni:K×K→ Rby
i(u,v) =hi(u,v) +ζi(v) –ζi(u) +v–u,Aiu ∀u,v∈K.
Then, as shown by Zhang [], for eachi,i satisfies (B)-(B). Note that ifk∈Ki, for eachi≥, thenik =iandrik =ri. Now, the equationzk=rikyk implies that, for eachi≥,
i(zk,v) +
ri
v–zk,Jzk–Jyk ≥ ∀v∈K. (.)
By applying (B), we have, for eachi≥,
ri
v–zk,Jzk–Jyk ≥–i(zk,v)≥i(v,zk) ∀v∈K, (.)
which implies that
i(v,zk)≤
ri
v–zk,Jzk–Jyk ≤
ri
v–zkJzk–Jyk ≤
ri
v+MJzk–Jyk,
for allv∈K,i≥, and someM> . This implies thatlim infk→∞i(v,zk)≤ for allv∈K,
i≥. From (B), we obtain, fori≥,i(v,u∗)≤lim infk→∞i(v,zk)≤∀v∈K. Lett∈ (, ) andv∈K. Thenvt=tv+ ( –t)u∗∈K. Therefore, for eachi≥,i(vt,u∗)≤. From conditions (B) and (B) we have, for eachi≥,
=i(vt,vt)≤ti(vt,v) + ( –t)i
vt,u∗
≤ti(vt,v) ⇒ i(vt,v)≥ ∀v∈K.
By (B), we have, for eachi≥,i(u∗,v)≥lim supt↓i(vt,v)≥∀v∈K. Therefore,u∗∈
GMEP(hi,ζi,Ai) for eachi≥. Hence,u∗∈
∞
i=GMEP(hi,ζi,Ai).
Step :u∗=u.
Letv=u. Sinceu∗∈, we have
φ(v,u)≤φ
u∗,u
. (.)
Also, sinceun=Knuandv∈⊂Kn, we have
Sinceun→u∗, we have
φu∗,u
≤φ(v,u). (.)
From inequalities (.) and (.) we obtainφ(u∗,u) =φ(v,u). Thus,u∗=v=u. This
completes the proof.
We now prove the following strong convergence theorem using aHalpern-type algo-rithm.
Theorem . Let X be a uniformly smooth and strictly convex real Banach space with Kadec-Klee property and let X∗ be its dual space.Let K be a nonempty closed and con-vex subset of X and hi :K ×K→R, i= , , , . . . , be a sequence of bifunctions
satis-fying (B)-(B). Let Ai:K →X∗, i= , , , . . . ,be a sequence of continuous monotone
maps and Gi:K→X, i= , , . . . , be an infinite family of equally continuous and
to-tally quasi-φ-asymptotically nonexpansive nonself multi-valued maps with nonnegative real sequences{γn(i)},{δ(ni)}and a sequence of strictly increasing and continuous functions
{ρi},ρi:R+→R+such thatγn(i)→,δn(i)→,andρi() = .Letζi:K→R,i= , , , . . . ,
be a sequence of convex and lower-semicontinuous functions.Suppose,for each i,δ(i)=
and:= (∞i=F(Gi))∩(∞i=GMEP(hi,Ai,ζi))is a nonempty subset of K.Then the se-quence{un},generated by algorithm(.),converges strongly tou,where{σn} ⊂(, )
withlimn→∞σn= ,rin∈[a,∞)for some a> ,andωn=γ
(in)
mnρin[φ(p,un)] +δ
(in)
mn,p∈.
Proof As in the proof of Theorem ., the proof of this theorem is presented in six steps.
Step :Knis closed and convex for alln≥.
This follows easily by induction, just as in the proof of Theorem ..
Step :⊂Knfor alln≥. Clearly,⊂K. Suppose⊂Knfor somen≥. Letq∈. Then by applying Lemma .(d), the definition ofφ, the convexity of · , and the totally
quasi-φ-asymptotically nonexpansiveness ofGi, as in Step of the proof of Theorem ., we have
φ(q,zn) =φ(q,rinyn)
≤φ(q,yn)
=φq,J–σnJu+ ( –σn)Jη(minn)
=q– q,σnJu+ ( –σn)Jη(minn) +σnJu+ ( –σn)Jη
(in) mn
≤ q– σnq,Ju– ( –σn)
q,Jη(in)
mn +σnu
+ ( –σn)η(in)
mn
=σnφ(q,u) + ( –σn)φ
q,η(in)
mn
≤σnφ(q,u) + ( –σn)
φ(q,un) +γ(in)
mnρin
φ(q,un)+δ(in)
mn
≤σnφ(q,u) + ( –σn)φ(q,un) +γm(inn)ρin
φ(q,un)+δ(in)
mn
=σnφ(q,u) + ( –σn)φ(q,un) +ωn,
which implies thatq∈Kn+. Therefore, by induction,⊂Knfor alln≥.
Step :un→u∗,zn→u∗, andyn→u∗asn→ ∞, for someu∗∈K.
The verification of this step follows the same pattern as in the verification of Step in the proof of Theorem ..
Step :u∗∈.
We first show thatu∗∈∞i=F(Gi). Note thatGik(PGik) mk–=G
i(PGi)mk–wheneverk∈
Kifor eachi≥. From Step and the uniform continuity ofJon bounded subsets ofX, we haveJyk–Ju∗ → ask→ ∞. Thus, for eachi≥,η(mi)k ∈Gik(PGik)
mk–u
k, we have
Jyk–Ju∗=σnJu+ ( –σn)Jηm(i)k–Ju
∗
=( –σn)Jη(mi)
k–Ju
∗–σ
n
Ju∗–Ju
≥( –σn)Jη(mi)
k–Ju
∗–σ
nJu∗–Ju,
which implies thatlimk→∞Jηm(i)k–Ju∗= for eachi≥. The rest of the justification of this step follows the same pattern as in the justification of Step in the proof of Theo-rem ..
Step :u∗=u.
This is the same as Step of the proof of Theorem .. Hence, the proof is completed.
A prototype for the control parameter in Theorem . is the canonical choice,σn=n. 4 Applications
In this section, we present some applications of Theorem .. Similar applications of The-orem . also follow.
4.1 Countable family of totally quasi-φ-asymptotically nonexpansive nonself multi-valued maps and system of equilibrium problems
By settingA≡,ζ≡ in Theorem ., the sequence{un}, defined in Theorem ., con-verges strongly tou, where:= (
∞
i=F(Gi))∩(
∞
i=EP(hi)) andEP(h) is the set of solutions of the equilibrium problem forh.
4.2 Countable family of totally quasi-φ-asymptotically nonexpansive nonself multi-valued maps and system of convex optimization problems
By settingA≡,h≡ in Theorem ., the sequence{un}, defined in Theorem ., con-verges strongly tou, where:= (
∞
i=F(Gi))∩(
∞
i=CMP(ζi)) andCMP(ζ) is the set of solutions of the convex minimization problem forζ.
4.3 Countable family of totally quasi-φ-asymptotically nonexpansive nonself multi-valued maps and system of variational inequality problems
By settingh≡,ζ ≡ in Theorem ., the sequence{un}, defined in Theorem ., con-verges strongly tou, where:= (
∞
i=F(Gi))∩(
∞
i=VIP(K,Ai)) andVIP(K,A) is the
set of solutions of the variational inequality problem forAoverK.
4.4 Application in classical Banach spaces
Remark (See,e.g., Alber and Ryazantseva [], p.) The analytical representations of duality maps are known inlp,Lp(G), and Sobolev spacesWm(p G),p∈(,∞), andp–+ q–= .
4.5 Application in Hilbert spaces
The following theorem follows immediately from Theorem ..
Theorem . Let H be a real Hilbert space and K be a nonempty closed and convex subset of H.Let hi:K×K→R,i= , , , . . . ,be a sequence of bifunctions satisfying(B)-(B).Let
Ai:K→H be a sequence of continuous monotone maps and Gi:K→H,i= , , . . . ,be an
infinite family of equally continuous and totally quasi-asymptotically nonexpansive nonself multi-valued maps with nonnegative real sequences{γn(i)},{δn(i)}and a sequence of strictly
increasing and continuous functions{ρi},ρi:R+→R+,such thatγn(i)→,δ(ni)→,and
ρi() = .Letζi:K→R,i= , , , . . . ,be a sequence of convex and lower-semicontinuous
functions.Suppose,for each i,δ(i)= and:= (∞i=F(Gi))∩(
∞
i=GMEP(hi,Ai,ζi))is a
nonempty subset of K.Then the sequence{un},generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u∈H, chosen arbitrarily, K=K, u=Ku,
yn=σun+ ( –σ)η(minn), (η
(in)
mn ∈Gin(PGin)mn–un),
zn=Grinyn,
Kn+={v∈Kn:v–zn≤ v–un+ωn},
un+=PKn+u, n≥,
converges strongly to Pu,where PKis the metric projection of H onto K,σ∈(, ),rin∈ [a,∞)for some a> ,andωn=γm(inn)ρin(un–p) +δ
(in)
mn.
Remark
() Theorem . improves the results of Bo and Yi [] in the following ways:
• In Theorem ., a countable family of nonself multi-valued maps is considered, whereas in Bo and Yi [] a single nonself multi-valued map is considered. • The requirement thatGis uniformlyL-Lipschitz continuous in Bo and Yi [] is
weakened to: for eachi,Giis equally continuous in Theorem ..
• The algorithm in Theorem . involves only one control parameter{σn} ⊂(, ) satisfying condition(C), whereas the algorithm of Bo and Yi [] contains two control parameters{βn} ⊂(, )and{σn} ⊂[, ], satisfying conditions(C)and (C).
• The Banach spaces considered in Theorem . are uniformly smooth and strictly convex real Banach spaces with Kadec-Klee property, which include uniformly smooth and uniformly convex real Banach spaces studied in Bo and Yi [].
() Theorem . improves and generalizes the results in Zhao and Chang [] in a number of ways:
• The class of maps considered in Zhao and Chang [] is extended from the class ofuniformlyquasi-φ-asymptotically nonexpansivesingle-valuednonself maps to the slightly more general class of countable family of totally
• The requirement that, for eachi,Giis uniformlyLi-Lipschitz continuous in Zhao and Chang [] is weakened to the following statement: for eachi,Giis equally continuous in Theorem ..
• The results of Zhao and Chang [] are proved in uniformly smooth and uniformly convex real Banach spaces, while Theorem . is proved in the more general uniformly smooth and strictly convex real Banach spaces with the Kadec-Klee property.
• The control parameter in the algorithm considered in Theorem . is
{σn} ⊂(, )satisfying condition(C), whereas the algorithm of Zhao and Chang [] contains two control parameters{βn} ⊂(, )and{σn} ⊂[, ], satisfying conditions(C)and(C).
5 Conclusion
In this article, iterative schemes of the Krasnoselskii-type and the Halpern-type for ap-proximating a common point in the set of common fixed points of a countable family of totally quasi-φ-asymptotically nonexpansive nonself multi-valued maps and the set of so-lutions of a system of generalized mixed equilibrium problems are constructed. Strong convergence of the sequences generated by these algorithms is established in certain Ba-nach spaces. Among other applications, our theorems are applied to solve convex feasibil-ity problems, a system of convex minimization problems, a system of variational inequal-ity problems, and a system of generalized equilibrium problems in uniformly smooth and strictly convex real Banach spaces with the Kadec-Klee property. Finally, our theorems are important improvements of several important recent results on these classes of nonlinear problems.
Acknowledgements
The authors thank the referees for their time and comments. We also express our deepest gratitude to The African Capacity Building Foundation for funding this research work.
Funding
Research supported from ACBF Research Grant Funds to AUST.
Competing interests
The authors have no competing interests.
Authors’ contributions
All the authors carried out the work in this paper with the consultation of each other. 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.
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