R E S E A R C H
Open Access
A monotone projection algorithm for fixed
points of nonlinear operators
Changqun Wu
1*and Lijuan Sun
2*Correspondence:
houbuyong@yeah.net 1School of Business and
Administration, Henan University, Kaifeng, 475000, China Full list of author information is available at the end of the article
Abstract
In this paper, a monotone projection algorithm is investigated for equilibrium and fixed point problems. Strong convergence theorems for common solutions of the two problems are established in the framework of reflexive Banach spaces.
MSC: 47H09; 47J25; 90C33
Keywords: asymptotically quasi-
φ
-nonexpansive mapping; generalized asymptotically quasi-φ
-nonexpansive mapping; bifunction; equilibrium problem; fixed point1 Introduction and preliminaries
LetEbe a real Banach space with the dualE∗. Recall that the normalized duality mapping JfromEto E∗is defined by
Jx=f∗∈E∗:x,f∗=x=f∗,
where·,·denotes the generalized duality pairing. LetBE={x∈E:x= }be the unit ball ofE. Recall thatEis said to be smooth ifflimt→x+tyt–x exists for eachx,y∈BE. It is also said to be uniformly smooth iff the above limit is attained uniformly forx,y∈BE. Eis said to be strictly convex iffx+y< for allx,y∈Ewithx=y= andx=y. It is said to be uniformly convex ifflimn→∞xn–yn= for any two sequences{xn}and{yn} inEsuch thatxn=yn= andlimn→∞xn+yn= . It is well known thatEis uniformly smooth if and only ifE∗is uniformly convex. In what follows, we useand→to stand for weak and strong convergence, respectively. Recall thatEenjoys the Kadec-Klee property iff for any sequence{xn} ⊂E, andx∈Ewithxnx, andxn → x, thenxn–x → asn→ ∞. It is well known that ifEis a uniformly convex Banach space, thenEenjoys the Kadec-Klee property. LetEbe a smooth Banach space. Let us consider the functional defined by
φ(x,y) =x– x,Jy+y, ∀x,y∈E.
Recently, Alber [] introduced a generalized projection operatorCin a Banach spaceE which is an analogue of the metric projectionPCin Hilbert spaces. Recall that the general-ized projectionC:E→Cis a map that assigns to an arbitrary pointx∈Ethe minimum point of the functionalφ(x,y), that is,Cx=x, where¯ x¯is the solution to the minimiza-tion problemφ(x,¯ x) =miny∈Cφ(y,x). Existence and uniqueness of the operatorCfollows
from the properties of the functionalφ(x,y) and strict monotonicity of the mapping J. IfE is a reflexive, strictly convex and smooth Banach space, thenφ(x,y) = if and only ifx=y. In Hilbert spaces,C=PC. It is obvious from the definition of functionφ that (x–y)≤φ(x,y)≤(y+x),∀x,y∈E.
LetRbe the set of real numbers. LetFbe a bifunction fromC×CtoR, whereRdenotes the set of real numbers. Letϕ:C→Rbe a real-valued function andA:C→E∗ be a mapping. The so-called generalized mixed equilibrium problem is to findp∈Csuch that
F(p,y) +Ap,y–p+ϕ(y) –ϕ(p)≥, ∀y∈C. (.)
We useGMEP(F,A,ϕ) to denote the solution set of the equilibrium problem. That is,
GMEP(F,A,ϕ) :=p∈C:F(p,y) +Ap,y–p+ϕ(y) –ϕ(z)≥,∀y∈C.
Next, we give some special cases:
IfA= , then problem (.) is equivalent to findingp∈Csuch that
F(p,y) +ϕ(y) –ϕ(z)≥, ∀y∈C, (.)
which is called the mixed equilibrium problem.
IfF= , then problem (.) is equivalent to findingp∈Csuch that
Ap,y–p+ϕ(y) –ϕ(z)≥, ∀y∈C, (.)
which is called the mixed variational inequality of Browder type. Ifϕ= , then problem (.) is equivalent to findingp∈Csuch that
F(p,y) +Ap,y–p ≥, ∀y∈C, (.)
which is called the generalized equilibrium problem.
IfA= andϕ= , then problem (.) is equivalent to findingp∈Csuch that
F(p,y)≥, ∀y∈C, (.)
which is called the equilibrium problem.
For solving the above problem, let us assume that the bifunctionF:C×C→Rsatisfies the following conditions:
(A) F(x,x) = ,∀x∈C;
(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤,∀x,y∈C;
(A)
lim sup
t↓
Ftz+ ( –t)x,y≤F(x,y), ∀x,y,z∈C;
Iterative algorithms have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, trans-portation, network, elasticity and optimization; see [–] and the references therein. The computation of solutions of nonlinear operator equations (inequalities) is important in the study of many real world problems. Recently, the study of the convergence of various iter-ative algorithms for solving various nonlinear mathematical models forms the major part of numerical mathematics.
LetCbe a nonempty subset ofE, and letT :C→Cbe a mapping. In this paper, we useF(T) to stand for the fixed point set ofT. Recall thatT is said to be asymptotically regular onCiff for any bounded subsetK ofC,lim supn→∞{Tn+x–Tnx:x∈K}= . Recall thatT is said to be closed iff for any sequence{xn} ⊂Csuch thatlimn→∞xn=x andlimn→∞Txn=y, thenTx=y. Recall that a pointpinCis said to be an asymptotic fixed point of T iffC contains a sequence{xn} which converges weakly topsuch that
limn→∞xn–Txn= . The set of asymptotic fixed points ofTwill be denoted byF(T). T is said to be relatively nonexpansive iffF(T) =F(T)=∅and
φ(p,Tx)≤φ(p,x), ∀x∈C,∀p∈F(T).
T is said to be relatively asymptotically nonexpansive iffF(T) =F(T)=∅and
φp,Tnx≤( +μn)φ(p,x), ∀x∈C,∀p∈F(T),∀n≥, where{μn} ⊂[,∞) is a sequence such thatμn→ asn→ ∞.
Recall thatTis said to be quasi-φ-nonexpansive iffF(T)=∅and
φ(p,Tx)≤φ(p,x), ∀x∈C,∀p∈F(T).
Recall that T is said to be asymptotically quasi-φ-nonexpansive iff there exists a se-quence{μn} ⊂[,∞) withμn→ asn→ ∞such that
F(T)=∅, φp,Tnx≤( +μn)φ(p,x), ∀x∈C,∀p∈F(T),∀n≥.
Remark . The class of relatively asymptotically nonexpansive mappings, which is an extension of the class of relatively nonexpansive mappings, was first introduced in [].
Remark . The class of asymptotically quasi-φ-nonexpansive mappings, which is an extension of the class of quasi-φ-nonexpansive mappings, was considered in [–]. The class of quasi-φ-nonexpansive mappings and the class of asymptotically
quasi-φ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-φ -nonexpansive mappings and asymptotically quasi-φ-nonexpansive mappings do not re-quire the restrictionF(T) =F(T).
Recall thatTis said to be generalized asymptotically quasi-φ-nonexpansive iffF(T)=∅, and there exist two nonnegative sequences{μn} ⊂[,∞) withμn→ and{ξn} ⊂[,∞) withξn→ asn→ ∞such that
Remark . The class of generalized asymptotically quasi-φ-nonexpansive mappings [] is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings in the framework of Banach spaces which was studied by Agarwalet al.[].
In this paper, we consider a projection algorithm for a common solution of a family of generalized asymptotically quasi-φ-nonexpansive mappings and generalized mixed equi-librium problems. A strong convergence theorem is established in a Banach space. In order to prove our main results, we need the following lemmas.
Lemma .[] LetEbe a uniformly convex Banach space, and letr> . Then there exists a strictly increasing, continuous and convex functiong: [, r]→Rsuch thatg() = and
∞
i= (αixi)
≤
∞
i=
αixi
–αiαjg
xi–xj, ∀i,j∈ {, , . . . ,N}
for allx,x, . . . ,∈Br={x∈E:x ≤r}andα,α, . . . ,∈[, ] such that
∞
i=αi= .
Lemma .[] LetEbe a reflexive, strictly convex and smooth Banach space, letCbe a nonempty closed convex subset ofEandx∈E. Then
φ(y,Cx) +φ(Cx,x)≤φ(y,x), ∀y∈C.
Lemma .[] LetCbe a nonempty closed convex subset of a smooth Banach spaceE andx∈E. Thenx=Cxif and only if
x–y,Jx–Jx ≥, ∀y∈C.
Lemma .[] LetEbe a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and letCbe a nonempty closed and convex subset ofE. LetT:C→Cbe a generalized asymptotically quasi-φ-nonexpansive mapping. Then F(T) is closed and convex.
Lemma .[] LetCbe a closed convex subset of a smooth, strictly convex and reflexive Banach spaceE. LetA:C→E∗be a continuous and monotone mapping, letϕ:C→Rbe convex and lower semi-continuous, and letFbe a bifunction fromC×CtoRsatisfying (A)-(A). Letr> andx∈E. Then there existsz∈Csuch that
F(z,y) +Az,y–z+ϕ(y) –ϕ(z) +
ry–z,Jz–Jx ≥, ∀y∈C. Define a mappingTr:E→Cby
Trx=
z∈C:F(z,y) +Az,y–z+ϕ(y) –ϕ(z) +
ry–z,Jz–Jx ≥,∀y∈C
.
Then the following conclusions hold:
() Tris a single-valued firmly nonexpansive-type mapping,i.e., for allx,y∈E,
() F(Tr) =GMEP(F,A,ϕ)is closed and convex;
() Tris quasi-φ-nonexpansive;
() φ(q,Trx) +φ(Trx,x)≤φ(q,x),∀q∈F(Tr).
2 Main results
Theorem . Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property,and let C be a nonempty closed and convex subset of E. Letbe an index set and N be an integer.Let Aj:C→E∗ be a continuous and mono-tone mapping andϕj:C→Rbe a lower semi-continuous and convex function. Let Fj be a bifunction from C×C toRsatisfying(A)-(A)for every j∈.Let Tbe an iden-tity mapping,and let Ti:C→C be a generalized asymptotically quasi-φ-nonexpansive mapping for every≤i≤N.Assume that Ti is closed asymptotically regular on C and :=Ni=F(Ti)∩j∈GMEP(Fj,Aj,ϕj)is nonempty and bounded.Let{xn}be a sequence generated in the following manner:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E, chosen arbitrarily, C,j=C,
C=
j∈C,j, x=Cx, yn=J–(
N
i=αn,iJTinxn),
un,j∈C such that Fj(un,j,y) +Ajun,j,y–un,j+ϕj(y) –ϕj(un,j) +r
n,jy–un,j,Jun,j–Jyn ≥, ∀y∈C,
Cn+,j={z∈Cn:φ(z,un,j)≤φ(z,xn) +
N
i=μn,iMn+Nξn}, Cn+=
J∈Cn+,J, xn+=Cn+x,
where{αn,i}is a real number sequence in(, )for every i≤,{rn,j}is a real number sequence in [r,∞),where r is some positive real number,and Mn=sup{φ(z,xn) :z∈ }.Assume thatNi=αn,i= andlim infn→∞αn,αn,i> for every≤i≤N.Then the sequence{xn} converges strongly to x,where is the generalized projection from E onto .
Proof The proof is split into five steps.
Step . Show that the common solution set is convex and closed. This step is clear in view of Lemma . and Lemma ..
Step . Show that the setCnis convex and closed.
To show Step , it suffices to show, for any fixed but arbitraryi∈, thatCn,iis convex and closed. This can be proved by induction. It is clear thatC,j=Cis convex and closed. Assume thatCm,jis closed and convex for somem≥. We next prove thatCm+,jis convex and closed. It is clear thatCm+,jis closed. We only prove they are convex. Indeed,∀x,y∈ Cm+,j, we find thatx,y∈Cm,j, and
φ(x,um,j)≤φ(x,xm) + N
i=
and
φ(y,um,j)≤φ(y,xm) + N
i=
μn,iMn+Nξn.
Notice that the above two inequalities are equivalent to the following inequalities, respec-tively:
x,Jxm–Jum,j ≤ xm–um,j+ N
i=
μn,iMn+Nξn
and
y,Jxm–Jum,j ≤ xm–um,j+ N
i=
μn,iMn+Nξn.
These imply that
ax+ ( –a)y,Jxm–Jum,j
≤ xm–um,j+ N
i=
μn,iMn+Nξn, ∀a∈(, ).
SinceCm,jis convex, we see thatax+ ( –a)y∈Cm,j. Notice that the above inequality is equivalent to
φax+ ( –a)y,um,j
≤φax+ ( –a)y,xm+ N
i=
μn,iMn+Nξn.
This proves thatCm+,jis convex. This proves thatCnis closed and convex. This completes Step .
Step . Show that ⊂Cn.
It suffices to claim that ⊂Cn,jfor everyj∈. Note that ⊂C,j=C. Suppose that
⊂Cm,jfor somemand for everyj∈. Then, for∀z∈ ⊂Cm,j, we have
φ(z,um,j) =φ(z,Trm,jym)
≤φ(z,ym)
=φ
z,J–
αm,Jxm+ N
i=
αm,iJTimxm
=z–
z,αm,Jxm+ N
i=
αm,iJTimxm
+
αm,Jxm+ N
i=
αm,iJTimxm
≤ z– αm,z,Jxm– N
i=
αm,i
z,JTm i xm
+αm,xm+ N
i=
=αm,φ(z,xm) + N
i=
αm,iφ
z,Timxm
≤αm,φ(z,xm) + N
i=
αm,iφ(z,xm) + N
i=
αm,iμm,iφ(z,xm) + N
i=
αm,iξm
≤φ(z,xm) + N
i=
μm,iφ(z,xm) + N
i=
αm,iξm
≤φ(z,xm) + N
i=
μm,iMm+ N
i=
αm,iξm
≤φ(z,xm) + N
i=
μm,iMm+Nξm, (.)
which proves thatz∈Cm+,j. This completes Step . Step . Show thatxn→p, wherep∈ .
In view of Lemma ., we find thatφ(xn,x)≤φ(w,x) –φ(w,xn)≤φ(w,x) for∀w∈ ⊂ Cn. This shows that the sequenceφ(xn,x) is bounded. It follows that{xn}is also bounded. Since the framework of the space is reflexive, we may, without loss of generality, assume thatxnp, wherep∈Cn. Note thatφ(xn,x)≤φ(p,x). It follows that
φ(p,x)≤lim inf
n→∞ φ(xn,x)≤lim supn→∞ φ(xn,x)≤φ(p,x).
This gives thatlimn→∞φ(xn,x) =φ(p,x). Hence, we havelimn→∞xn=p. Since the spaceEenjoys the Kadec-Klee property, we find thatxn→pasn→ ∞.
Now, we are in a position to show thatp∈j∈GMEP(Fj,Aj,ϕj). By the construction ofCn, we have thatCn+⊂Cnandxn+=Cn+x∈Cn. It follows that
φ(xn+,xn) =φ(xn+,Cnx)
≤φ(xn+,x) –φ(Cnx,x) =φ(xn+,x) –φ(xn,x).
Lettingn→ ∞, we obtain thatφ(xn+,xn)→. In view ofxn+∈Cn+, we see that
φ(xn+,un,j)≤φ(xn+,xn) + N
i=
μn,iMn+Nξn.
sides of the equality above yields that
≥ p– p,u∗,j+u∗,j =p– p,Juj+Juj =p– p,Juj+uj =φp,uj.
That is,p=uj, which in turn implies thatJp=u∗,j. It follows thatJun
,jJp∈E∗. SinceE∗ enjoys the Kadec-Klee property, we obtain thatJun,j–Jp→ asn→ ∞. SinceJ–:E∗→E is demicontinuous, it follows thatun,jp. SinceE enjoys the Kadec-Klee property, we obtain thatun,j→pasn→ ∞. Note thatxn–un,j ≤ xn–p+p–un,j. This gives that
lim
n→∞xn–un,j= . (.)
SinceJis uniformly norm-to-norm continuous on any bounded sets, we have
lim
n→∞Jxn–Jun,j= . (.)
Notice that
φ(z,xn) –φ(z,un,j) =xn–un,j– z,Jxn–Jun,j
≤ xn–un,j
xn+un,j
+ zJxn–Jun,j. It follows from (.) and (.) that
lim
n→∞φ(z,xn) –φ(z,un,j) = . (.)
From (.), we find thatφ(z,yn)≤φ(z,xn) +Ni=μn,jMn+Nξn, wherez∈ . In view of un,j=Srn,jyn, we find from Lemma . that
φ(un,j,yn) =φ(Srn,jyn,yn)
≤φ(z,yn) –φ(z,Srn,jyn)
≤φ(z,xn) –φ(z,Srn,jyn) +
N
i=
μn,jMn+Nξn
=φ(z,xn) –φ(z,un,j) + N
i=
μn,jMn+Nξn.
From (.), we obtain that
lim
n→∞φ(un,j,yn) = .
This implies that un,j–yn → asn→ ∞. Sinceun,j→pasn→ ∞, we arrive at
assume thatJyny∗∈E∗. In view ofJ(E) =E∗, we see that there existsy∈Esuch that Jy=y∗. It follows that
φ(un,j,yn) =un,j– un,j,Jyn+Jyn.
Takinglim infn→∞on the both sides of the equality above yields that ≥φ(p,y). That is, p=y, which in turn implies thaty∗=Jp. It follows thatJynJp∈E∗. SinceE∗ enjoys the Kadec-Klee property, we obtain thatJyn–Jp→ asn→ ∞. Note thatJ–:E∗→Eis demicontinuous. It follows thatynp. SinceEenjoys the Kadec-Klee property, we obtain thatyn→pasn→ ∞. Sinceun,j–yn ≤ un,j–p+p–yn, we find thatlimn→∞un,i– yn= . SinceJis uniformly norm-to-norm continuous on any bounded sets, we have
limn→∞Jun,j–Jyn= . From the assumptionrn,i≥r, we see thatlimn→∞
Jun,j–Jyn
rn,j = .
Notice that
fj(un,j,y) + rn,j
y–un,j,Jun,j–Jyn ≥, ∀y∈C,
wherefj(un,j,y) =Fj(un,j,y) +Ajun,j,y–un,j+ϕj(y) –ϕj(un,j). From (A), we find that
y–un,j
Jun,j–Jyn rn,j ≥
rn,j
y–un,j,Jun,j–Jyn ≥fj(y,un,j), ∀y∈C.
Taking the limit asn→ ∞, we find thatfj(y,p)≤,∀y∈C. For <tj< andy∈C, define ytj =tjy+ ( –tj)p. It follows thatyt,j∈C, which yields thatfj(yt,j,p)≤. It follows from conditions (A) and (A) that =fj(yt,j,yt,j)≤tjfj(yt,j,y) + ( –tj)fj(yt,j,p)≤tjfj(yt,j,y). This yields thatfj(yt,j,y)≥. Lettingtj↓, we find from condition (A) thatfj(p,y)≥,∀y∈C. This implies thatp∈EP(fj) =GMEP(Fj,Aj,ϕj) for everyj∈.
Next, we statep∈Ni=F(Ti). SinceEis uniformly smooth, we know thatE∗is uniformly convex. It follows from Lemma . that
φ(z,un,j) =φ(z,Srn,jyn)
≤φ(z,yn)
=φ
z,J–
αn,Jxn+ N
i=
αn,iJTinxn
=z–
z,αn,Jxn+ N
i=
αn,iJTinxn
+
αn,Jxn+ N
i=
αn,iJTinxn
≤ z– αn,z,Jxn– N
i=
αn,i
z,JTinxn
+αn,xn+ N
i=
αn,iTinxn
–αn,( –αn,i)gJxn–JTinxn
=αn,φ(z,xm) + N
i=
αn,iφ
z,Tn ixn
–αn,( –αn,i)gJxn–JTinxn
≤αn,φ(z,xm) + N
i=
αn,iφ(z,xm) + N
i=
αn,iμn,iφ(z,xn) + N
i=
–αn,( –αn,i)gJxn–JTinxn
≤φ(z,xn) + N
i=
μn,iφ(z,xm) + N
i=
αn,iξn
–αn,( –αn,i)gJxn–JTinxn
≤φ(z,xn) + N
i=
μn,iMn+Nξn–αn,( –αn,i)gJxn–JTinxn.
This yields that
αn,( –αn,i)gJxn–JTinxn≤φ(z,xn) –φ(z,un,j) + N
i=
μn,iMn+Nξn.
In view oflim infn→∞αn,( –αn,i) > , we see from (.) thatlimn→∞g(Jxn–JTinxn) = It follows from the property ofgthat
lim
n→∞Jxn–JT n
ixn= . (.)
Sincexn→pasn→ ∞andJ:E→E∗is demicontinuous, we obtain thatJxnJp∈E∗. Note that |Jxn–Jp|=|xn–p| ≤ xn–p. This implies that Jxn → Jp as n→ ∞. SinceE∗enjoys the Kadec-Klee property, we see that
lim
n→∞Jxn–Jp= . (.)
On the other hand, we haveJTinxn–Jp ≤ JTinxn–Jxn+Jxn–Jp. Combining (.) with (.), one obtains thatlimn→∞JTinxn–Jp= . SinceJ–:E∗→Eis demicontinu-ous, one sees thatTinxnp. Notice that|Tinxn–p| ≤ JTinxn–Jp. This yields that
limn→∞Tinxn=p. Since the spaceEenjoys the Kadec-Klee property, we obtain that
limn→∞Tinxn–p= . Note thatTn+xn–p ≤ Tn+xn–Tnxn+Tnxn–p. SinceT is asymptotically regular, we find thatlimn→∞Tn+
i xn–p= . That is,TiTinxn–p→ asn→ ∞. It follows from the closedness ofTithatTip=pfor everyi∈ {, , . . . ,N}. This completes Step .
Step . Show thatp= x. Sincexn=Cnx, we see that
xn–z,Jx–Jxn ≥, ∀z∈Cn.
Since ⊂Cn, we find that
xn–w,Jx–Jxn ≥, ∀w∈ .
Lettingn→ ∞, we arrive at
p–w,Jx–Jp ≥, ∀w∈ .
From Lemma ., we can immediately obtain thatp= x. This completes the proof.
Remark . Theorem . mainly improves the corresponding results in Kim [], Yang et al.[], Hao [], Qinet al.[], Qinet al.[].
Remark . The framework of the space in Theorem . can be applicable toLp,p≥. IfN= and={}, then Theorem . is reduced to the following.
Corollary . Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property,and let C be a nonempty closed and convex subset of E.Let F be a bifunction from C×C toRsatisfying(A)-(A).Let Ti:C→C be a generalized asymptotically quasi-φ-nonexpansive mapping for every i∈ {, }.Assume that each Tiis closed asymptotically regular on C and F(T)∩F(T)∩EP(F)is nonempty and bounded. Let{xn}be a sequence generated in the following manner:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E, chosen arbitrarily, C=C,
x=Cx, yn=J–(α
n,Jxn+αn,JTnxn+αn,JTnxn),
un∈C such that F(un,y) +rny–un,Jun–Jyn ≥, ∀y∈C, Cn+={z∈Cn:φ(z,un)≤φ(z,xn) + (μn,+μn,)Mn+ ξn}, xn+=Cn+x,
where{αn,},{αn,},and{αn,}are real number sequences in(, ),{rn}is a real number sequence in[r,∞),where r is some positive real number,and Mn=sup{φ(z,xn) :z∈F(T)∩ F(T)∩EF(F)}.Assume that
i=αn,i= andlim infn→∞αn,αn,i> .Then the sequence
{xn} converges strongly toF(T)∩F(T)∩EP(F)x,whereF(T)∩F(T)EF(F) is the generalized projection from E onto F(T)∩F(T)∩EP(F).
Remark . Corollary . mainly improves the corresponding results in Qinet al.[]. To be more clear, the mapping is extended from quasi-φ-nonexpansive mappings to gen-eralized asymptotically quasi-φ-nonexpansive mappings and the framework of spaces is extended from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to this manuscript. Both authors read and approved the final manuscript.
Author details
1School of Business and Administration, Henan University, Kaifeng, 475000, China.2Kaifeng Vocational College of Culture
and Arts, Kaifeng, 475000, China.
Acknowledgements
The authors are grateful to the reviewers’ useful suggestions which improved the contents of the article.
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10.1186/1687-1812-2013-318
