R E S E A R C H
Open Access
Convergence theorems for equilibrium
problem and asymptotically
quasi-
φ
-nonexpansive mappings in the
intermediate sense
Jae Ug Jeong
**Correspondence:
[email protected] Department of Mathematics, Dongeui University, Busan, 614-714, South Korea
Abstract
In this paper, we introduce an iterative process which converges strongly to a common element of the set of fixed points of an asymptotically
quasi-
φ
-nonexpansive mapping in the intermediate sense and the solution set of generalized equilibrium problem in Banach spaces. Our theorems improve, generalize, and extend several results recently announced.MSC: 47H05; 47H09; 47H10
Keywords: fixed point; asymptotically quasi-
φ
-nonexpansive mapping; generalized f-projection operator; relatively nonexpansive mapping1 Introduction
LetEbe a real Banach space with the dual spaceE∗. LetCbe a nonempty closed convex subset ofE. LetT:C→Cbe a nonlinear mapping. We denote byF(T) the set of fixed points ofT.
A mappingTis said to be asymptotically nonexpansive if there exists a sequence{kn} ⊂ [,∞) withkn→ asn→ ∞such that
Tnx–Tny≤knx–y, ∀x,y∈C,n≥.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [] in . In uniformly convex Banach spaces, they proved that ifCis nonempty, bounded, closed, and convex, then every asymptotically nonexpansive self-mappingTon
Chas a fixed point. Further, the fixed point set ofT is closed and convex.
A mappingT is said to be asymptotically nonexpansive in the intermediate sense (see []) if it is continuous and the following inequality holds:
lim sup
n→∞
sup
x,y∈C
Tnx–Tny–x–y≤. (.)
IfF(T)=φ and (.) holds for allx∈K,y∈F(T), thenT is called asymptotically quasi-nonexpansive in the intermediate sense. It is well known that ifCis a nonempty closed
convex bounded subset of a uniformly convex Banach spaceE andT is a self-mapping ofCwhich is asymptotically nonexpansive in the intermediate sense, thenT has a fixed point (see []). It is worth mentioning that the class of mappings which are asymptoti-cally nonexpansive in the intermediate sense contains properly the class of asymptotiasymptoti-cally nonexpansive mappings.
Iterative approximation of a fixed point for asymptotically nonexpansive mappings in Hilbert or Banach spaces has been studied extensively by many authors (see [–] and the references therein).
LetEbe a smooth Banach space. The functionφ:E×E→Rdefined by
φ(x,y) =x– x,Jy+y, ∀x,y∈E,
is studied by Alber []. It follows from the definition ofφthat
x–y≤φ(x,y)≤x+y, ∀x,y∈E. (.)
Remark .
(i) IfEis a reflexive, strictly convex and smooth Banach space, then forx,y∈E, φ(x,y) = if and only ifx=y.
(ii) IfEis a real Hilbert space, thenφ(x,y) =x–y.
LetEbe reflexive, strictly convex and smooth Banach space. The generalized projection mapping, introduced by Alber [], is a mappingC:E→Cthat assigns to an arbitrary pointx∈Ethe minimum point of the functionalφ(y,x), that is,Cx=x, where isxis the solution to the minimization problem
φ(x,x) =inf
y∈Cφ(y,x).
A pointpinCis said to be an asymptotic fixed point ofTifCcontains a sequence{xn} which converges weakly topsuch thatlimn→∞xn–Txn= . The set of asymptotic fixed points ofT will be denoted byF˜(T). A mappingTis called relatively nonexpansive (see []) ifF˜(T) =F(T) andφ(p,Tx)≤φ(p,x) for allx∈Candp∈F(T).
Recently, Matsushita and Takahashi [] proved strong convergence theorems for ap-proximation of fixed points of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. More precisely, they proved the following theorem.
Theorem . Let E be a uniformly convex and uniformly smooth Banach space,let C be a nonempty closed convex subset of E,let T be a relatively nonexpansive mapping from C into itself and let{αn}be a sequence of real numbers such that≤αn≤andlim supn→∞αn< .
Suppose that{xn}is given by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x=x∈C,
yn=J–(αnJxn+ ( –αn)JTxn),
Hn={z∈C:φ(z,yn)≤φ(z,xn)},
Wn={z∈C:xn–z,Jx–Jxn ≥},
xn+=Hn∩Wnx, n= , , , . . . ,
where J is the normalized duality mapping on E.If F(T)is nonempty,then{xn}converges
strongly toF(T)x.
In [], Hao introduced the following iterative scheme for approximating a fixed point of asymptotically quasi-φ-nonexpansive mappings in the intermediate sense in a reflexive, strictly convex and smooth Banach spaceE:x∈E,C=C,x=Cx,
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
yn=J–(αnJxn+ ( –αn)JTnxn),
Cn+={z∈Cn:φ(z,yn)≤φ(z,xn) +ξn},
xn+=Cn+x, n= , , . . . ,
whereξn=max{,supp∈F(T),x∈C(φ(p,Tnx) –φ(p,x))}.
Motivated and inspired by the works mentioned above, in this paper, we introduce a new iterative scheme of the generalizedf-projection operator for finding a common ele-ment of the set of fixed points of asymptotically quasi-φ-nonexpansive mappings in the intermediate sense and the solution set of generalized equilibrium problem in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.
2 Preliminaries
LetEbe a real Banach space with the norm · and letE∗ be the dual space ofE. The normalized duality mappingJ:E→E∗is defined by
J(x) = f ∈E∗:x,f=x=f. By the Hahn-Banach theorem,J(x) is nonempty.
A Banach spaceEis called strictly convex ifx+y< for allx,y∈Uwithx=y, where
U={x∈E:x= }is the unit sphere ofE. A Banach spaceEis called smooth if the limit
lim
t→∞
x+ty–x t
exists for eachx,y∈U. It is also called uniformly smooth if the limit exists uniformly for allx,y∈U. In this paper, we denote the strong convergence and weak convergence of a sequence{xn}byxn→xandxnx, respectively.
Remark . The basic properties of a Banach spaceErelated to the normalized duality mappingJare as follows (see []):
() IfEis a smooth Banach space, thenJis single-valued and semicontinuous; () IfEis a uniformly smooth Banach space, thenJis uniformly norm-to-norm
continuous on each bounded subset ofE;
() IfEis a uniformly smooth Banach space, thenEis smooth and reflexive; () IfEis a reflexive and strictly convex Banach space, thenJ–is
norm-weak∗-continuous;
() Eis a uniformly smooth Banach space if and only ifE∗is uniformly convex.
Definition . A mappingT:C→Cis said to be
() quasi-φ-nonexpansive ifF(T)=φand φ(p,Tx)≤φ(p,x)
for allx∈Candp∈F(T);
() asymptotically quasi-φ-nonexpansive in the intermediate sense ifF(T)=φand lim sup
n→∞ p∈Fsup(T),x∈C
φp,Tnx–φ(p,x)≤
put
ξn=max
, sup
p∈F(T),x∈C
φp,Tnx–φ(p,x).
Remark . From the definition of asymptotically quasi-φ-nonexpansiveness in the in-termediate sense, it is obvious thatξn→ asn→ ∞and
φp,Tnx≤φ(p,x) +ξn, ∀p∈F(T),x∈C.
Recall thatTis said to be asymptotically regular onCif for any bounded subsetKofC,
lim sup
n→∞
Tn+x–Tnx:x∈K= .
Definition . A mappingT:C→Cis said to be closed if for any sequence{xn} ⊂C withxn→xandTxn→y,Tx=y.
Following Alber [], the generalized projectionC:E→Cis defined by
C(x) =
u∈C:φ(u,x) =min
y∈Cφ(y,x)
, ∀x∈E.
In , Wu and Huang [] introduced a generalizedf-projection operator in a Banach space, which extends the definition of the generalized projectionC. LetG:C×E∗→ R∪ {+∞}be a functional defined as follows:
G(y,w) =y– y,w+w+ ρf(y)
for all (y,w)∈C×E∗, whereρ is a positive number andf :C→R∪ {+∞}is proper, convex, and lower semicontinuous. From the definition ofG, it is easy to see the following properties:
(i) G(y,w)is convex and continuous with respect towwhenyis fixed;
(ii) G(y,w)is convex and lower semicontinuous with respect toywhenwis fixed.
Definition .([]) LetEbe a real smooth Banach space and letCbe a nonempty closed and convex subset ofE. We say thatfC:E→Cis a generalizedf-projection operator if
fCx=u∈C:G(u,Jx) =inf
y∈CG(y,Jx),∀x∈E
Lemma .([]) Let E be a Banach space and f:E→R∪ {+∞}be a lower semicontin-uous and convex function.Then there exist x∗∈E∗andα∈Rsuch that
f(x)≥x,x∗+α for all x∈E.
Lemma .([]) Let E be a reflexive smooth Banach space and let C be a nonempty closed convex subset of E.The following statements hold:
() fCxis a nonempty closed convex subset ofCfor allx∈E; () For allx∈F,x∈fCxif and only if
x–y,Jx–Jx+ρf(y) –ρf(x)≥
for ally∈C;
() IfEis strictly convex,thenfCis a single-valued mapping.
Letθ be a bifunction fromC×CtoR, whereRdenotes the set of real numbers. The equilibrium problem is to findx∈Csuch that
θ(x,y)≥ (.)
for ally∈C. The set of solutions of (.) is denoted byEP(θ).
For solving the equilibrium problem for a bifunctionθ:C×C→R, let us assume that
θ satisfies the following conditions:
(A) θ(x,x) = for allx∈C;
(A) θis monotone;i.e.,θ(x,y) +θ(y,x)≤for allx,y∈C; (A) for allx,y,z∈C,
lim
t↓θ
tz+ ( –t)x,y≤θ(x,y);
(A) for allx∈C,y→θ(x,y)is convex and lower semicontinuous.
Lemma .([]) Let C be a closed convex subset of a uniformly smooth,strictly convex,
and reflexive Banach space E and letθ be a bifunction from C×C to Rsatisfying the conditions(A)-(A).For all r> and x∈E,define a mapping Tθ
r :E→C as follows:
Tθ
rx=
z∈C:θ(z,y) +
ry–z,Jz–Jx ≥,∀y∈C
.
Then the following conclusions hold:
() Tθ
r is single-valued;
() Tθ
r is a firmly nonexpansive-type mapping,i.e.,for allx,y∈E,
Trθx–Trθy,JTrθx–JTrθy≤Trθx–Trθy,Jx–Jy;
() F(Tθ
() Tθ
r is quasi-φ-nonexpansive;
() φ(q,Tθ
rx) +φ(Trθx,x)≤φ(q,x),∀q∈F(Trθ).
Lemma .([]) Let E be a reflexive,strictly convex and smooth Banach space such that both E and E∗ have the Kadec-Klee property.Let C be a nonempty closed convex subset of E.Let T:C→C be a closed and asymptotically quasi-φ-nonexpansive mapping in the intermediate sense.Then F(T)is a closed convex subset of C.
Lemma .([]) Let E be a real reflexive smooth Banach space and let C be a nonempty closed and convex subset of E.Then,for any x∈E and x∈fCx,
φ(y,x) +G(x,Jx)≤G(y,Jx)
for all y∈C. 3 Main results
Theorem . Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.Let C be a nonempty closed convex subset of E.Letθ be a bifunc-tion from C×C toRsatisfying the conditions(A)-(A).Let T:C→C be a closed and asymptotically quasi-φ-nonexpansive mapping in the intermediate sense.Assume that T is asymptotically regular on C,F=F(T)∩EP(θ)is nonempty,and F(T)is bounded.Let f :E→R+be a convex and lower semicontinuous function with C⊂int(D(f))and f() = .
Let{αn}be a sequence in[, ]and{βn},{γn}be sequences in(, )satisfying the following
conditions:
(i) αn+βn+γn= ;
(ii) limn→∞αn= ;
(iii) <lim infn→∞βn≤lim supn→∞βn< .
Let{xn}be a sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E chosen arbitrarily,
C=C,
yn=J–(αnJx+βnJTnxn+γnJxn),
un∈C such thatθ(un,y) +rny–un,Jun–Jyn ≥, ∀y∈C,
Cn+={z∈Cn:G(z,Jun)≤αnG(z,Jx) + ( –αn)G(z,Jxn) +ξn},
xn+=fCn+x, ∀n≥,
(.)
whereξn=max{,supp∈F(T),x∈C(φ(p,Tnx) –φ(p,x))},{rn} is a real sequence in[a,∞)for
some a> andfCn
+is the generalized f -projection operator.Then{xn}converges strongly tofFx.
Proof It follows from Lemma . and Lemma . thatFis a closed convex subset ofC, so thatfFxis well defined for anyx∈C.
We split the proof into six steps.
Step . We first show thatCnis nonempty, closed, and convex for alln≥.
t)z∈Cn, wheret∈(, ). Notice that
G(z,Jun)≤αnG(z,Jx) + ( –αn)G(z,Jxn) +ξn, and
G(z,Jun)≤αnG(z,Jx) + ( –αn)G(z,Jxn) +ξn. The above inequalities are equivalent to
αnz,Jx+ ( –αn)z,Jxn– z,Jun
≤αnx+ ( –αn)xn–un+ξn (.)
and
αnz,Jx+ ( –αn)z,Jxn– z,Jun
≤αnx+ ( –αn)xn–un+ξn. (.)
Multiplyingtand –ton both sides of (.) and (.), respectively, we obtain
αnz,Jx+ ( –αn)z,Jxn– z,Jun
≤αnx+ ( –αn)xn–un+ξn.
Hence we have
G(z,Jun)≤αnG(z,Jx) + ( –αn)G(z,Jxn) +ξn.
This implies thatCn+is closed and convex for alln≥. This shows thatfCn+x is well
defined.
Step . We show thatF⊂Cnfor alln≥.
Forn= , we haveF⊂C=C. Now, assume thatF⊂Cnfor somen≥. Letq∈F. SinceT is asymptotically quasi-φ-nonexpansive with intermediate sense, we have from Remark . and Lemma . that
G(q,Jun) =Gq,JTrnθyn
=φq,Tθ
rnyn
+ ρf(q)
≤φ(q,yn) + ρf(q) =G(q,Jyn)
=Gq,αnJx+βnJTnxn+γnJxn
=q– αnq,Jx– βn
q,JTnxn
– γnq,Jxn +αnJx+βnJTnxn+γnJxn
+ ρf(q)
≤ q– αnq,Jx– βn
q,JTnxn
+αnJx+βnJTnxn
+γnJxn+ ρf(q) =αnG(q,Jx) +βnG
q,JTnxn
+γnG(q,Jxn)
=αnG(q,Jx) +βn φ
q,Tnxn
+ ρf(q)+γnG(q,Jxn)
≤αnG(q,Jx) +βn φ(q,xn) +ξn+ ρf(q)
+γnG(q,Jxn)
≤αnG(q,Jx) +βnG(q,Jxn) +γnG(q,Jxn) +ξn =αnG(q,Jx) + ( –αn)G(q,Jxn) +ξn,
which shows thatq∈Cn+. This implies thatF⊂Cn+and soF⊂Cnfor alln≥. Step . We prove that{xn}is bounded andlimn→∞G(xn,Jx) exists.
By Lemma ., we have the result that there existx∗∈E∗andα∈Rsuch that
f(x)≥x,x∗+α.
Sincexn∈Cn⊂E, it follows that
G(xn,Jx) =xn– xn,Jx+x+ ρf(xn)
≥ xn– xn,Jx+x+ ρ
xn,x∗+ ρα
=xn–
xn,Jx–ρx∗
+x+ ρα
≥ xn– xnJx–ρx∗+x+ ρα
=xn–Jx–ρx∗+x–Jx–ρx∗+ ρα.
For allq∈Fandxn=fCn x, we have
G(q,Jx)≥G(xn,Jx)
≥xn–Jx–ρx∗+x–Jx–ρx∗+ ρα.
This implies that the sequence {xn} is bounded and so is {G(xn,Jx)}. From (.) and Lemma ., we obtain
≤xn+–xn
≤
φ(xn+,xn)≤G(xn+,Jx) –G(xn,Jx). (.)
This shows that {G(xn,Jx)} is nondecreasing. It follows from the boundedness that
limn→∞G(xn,Jx) exists.
Step . Next, we prove thatxn→x,yn→x, andun→xasn→ ∞, wherexis some point inC.
By (.), we obtain
lim
n→∞φ(xn+,xn) = . (.)
andxn=fCnx, we obtain
G(x,Jx) =x– x,Jx+x+ ρf(x)
≤lim inf
n→∞ xn
– xn,Jx
+x+ ρf(xn)
=lim inf
n→∞ G(xn,Jx)
≤lim sup
n→∞
G(xn,Jx)
≤G(x,Jx),
which implies thatlimn→∞G(xn,Jx) =G(x,Jx). From Lemma ., we obtain
≤x–xn
≤φ(x,xn)
≤G(x,Jx) –G(xn,Jx).
Hence we havelimn→∞xn=x. In view of the Kadec-Klee property ofE, we find that
lim
n→∞xn=x. (.)
And we have
lim
n→∞xn–xn+= .
SinceJis uniformly norm-to-norm continuous, it follows that
lim
n→∞Jxn–Jxn+= .
Fromxn+=fCn+x∈Cn+⊂Cnand (.), we have
G(xn+,Jun)≤αnG(xn+,Jx) + ( –αn)G(xn+,Jxn) +ξn.
This is equivalent to the following:
φ(xn+,un)≤αnφ(xn+,x) + ( –αn)φ(xn+,xn) +ξn. (.)
Due to (.), (.), the assumption (ii), and Remark ., we have
lim
n→∞φ(xn+,un) = .
By (.), it follows that
asn→ ∞. SinceJis uniformly norm-to-norm continuous, we obtain
Jun → Jx (.)
asn→ ∞. This implies that{Jun}is bounded inE∗. SinceE∗ is reflexive, we assume thatJunu∈E∗asn→ ∞. In view ofJ(E) =E∗, there existsu∈Esuch thatJu=u. This implies thatJunJu. We have
φ(xn+,un) =xn+– xn+,Jun+un =xn+– xn+,Jun+Jun.
Takinglim infn→∞on both sides of the equality above, this yields
≥ x– x,u+u
=x– x,Ju+Ju
=x– x,Ju+u
=φ(x,u),
which shows thatx=uand soJunJx. It follows from (.) and the Kadec-Klee property ofE∗thatJun→Jxasn→ ∞. SinceJ–is norm-weak-continuous, we have
unx. (.)
From (.), (.), and the Kadec-Klee property ofE, we have
lim
n→∞un=x. (.)
On the other hand, we see from the weak lower semicontinuity of the norm that
φ(q,x) =q– q,Jx+x
≤lim inf
n→∞
q– q,Jun+un
=lim inf
n→∞ φ(q,un)
≤lim sup
n→∞ φ(q,un)
=lim sup
n→∞
q– q,Jun+un
≤φ(q,x),
which implies that
lim
By (.) and (.), we obtain limn→∞xn–un= . The uniform continuity of J on bounded sets gives
lim
n→∞Jxn–Jun= . (.)
Now, using the definition ofφ, we have, for allq∈F,
φ(q,xn) –φ(q,un) =xn–un– q,Jxn–Jun
≤ xn–un
xn+un
+ qJxn–Jun. From (.), we obtain
φ(q,xn) –φ(q,un)→
asn→ ∞. By (.), it follows that
lim
n→∞φ(q,xn) =φ(q,x). (.)
Hence, for anyq∈F⊂Cn, it follows from the convexity of · and Lemma . that
φ(q,un) =φq,Trnθyn
≤φ(q,yn)
=φq,J–αnJx+βnJTnxn+γnJxn
=q– q,αnJx+βnJTnxn+γnJxn
+αnJx+βnJTnxn+γnJxn
≤ q– αnq,Jx– βn
q,JTnxn
– γnq,Jxn +αnJx+βnJTnxn
+γnJxn =αnφ(q,x) +βnφ
q,Tnxn
+γnφ(q,xn)
≤αnφ(q,x) +βn
φ(q,xn) +ξn
+γnφ(q,xn)
≤αnφ(q,x) + ( –αn)φ(q,xn) +ξn. (.)
From (.), (.), (.), Remark ., and the assumption (ii), we obtain
lim
n→∞φ(q,yn) =φ(q,x).
From Lemma ., we see that for anyq∈Fandun=Trnθyn,
φ(un,yn) =φTrnθyn,yn
≤φ(q,yn) –φq,Trnθyn
Takingn→ ∞on both sides of the inequality above, we have
lim
n→∞φ(un,yn) = .
From (.), we have (un–yn)→ asn→ ∞. By (.), we have
yn → x (.)
asn→ ∞, and so
Jyn → Jx (.)
asn→ ∞. That is,{Jyn}is bounded inE∗. SinceE∗ is reflexive, we can assume that
Jyny∗∈E∗asn→ ∞. In view ofJ(E) =E∗, there existsy∈Esuch thatJy=y∗. It follows that
φ(un,yn) =un– un,Jyn+yn =un– un,Jyn+Jyn.
Takinglim infn→∞on both sides of the equality above, it follows that
≥ x– x,y∗+y∗
=x– x,Jy+Jy
=x– x,Jy+y
=φ(x,y).
From Remark .,x=y,i.e.,y∗=Jx. It follows thatJynJx∈E∗asn→ ∞. From (.) and the Kadec-Klee property ofE∗, we have
Jyn→Jx
asn→ ∞. SinceJ–is norm-weak∗-continuous,ynxasn→ ∞. From (.) and the Kadec-Klee property ofE, we have
lim
n→∞yn=x.
Step . We show thatx∈F. By Step , we get
lim
n→∞un–yn= .
The uniform continuity ofJon bounded sets gives
lim
From the assumptionrn≥aand (.), we see thatJunrn–Jyn→ asn→ ∞. But from (A) and (.), we note that
rn
y–un,Jun–Jyn ≥–θ(un,y)≥θ(y,un), ∀y∈C
and hence
y–un
Jun–Jyn
rn
≥θ(y,un), ∀y∈C,
which implied thatθ(y,x)≤ for ally∈C. Putyt=ty+ ( –t)xfor allt∈(, ] andy∈C. Then we getyt∈Candθ(yt,x)≤. Therefore, from (A) and (A), we obtain
=θ(yt,yt)≤tθ(yt,y) + ( –t)θ(yt,x)
≤tθ(yt,y).
Thus,θ(yt,y)≥ for ally∈C. Furthermore, ast→ ∞, we have from (A) thatθ(x,y)≥ for ally∈C. This implies thatx∈EP(θ).
Finally, we show thatx∈F(T). In view ofyn=J–(αnJx+βnJTnxn+γnJxn), we find that
Jun–Jyn=αn(Jun–Jx) +βn
Jun–JTnxn
+γn(Jun–Jxn).
Hence we have
βnJun–JTnxn≤ Jun–Jx+Jx–Jyn+αnJun–Jx +γnJun–Jxn.
From the assumptions (ii), (iii), and (.), we have
lim
n→∞Jun–JT nx
n= . (.)
Notice that
JTnxn–Jx≤JTnxn–Jun+Jun–Jx.
This implies from (.) that
lim
n→∞JT
nx
n–Jx= . (.)
The demicontinuity ofJ–:E∗→Eimplies thatTnx
nxasn→ ∞. We have
Tnxn–x=JTnxn–Jx≤JTnxn–Jx.
With the aid of (.), we see thatlimn→∞Tnxn=x. SinceEhas the Kadec-Klee prop-erty, we find that
lim
n→∞T nx
Since
Tn+x
n–x≤Tn+xn–Tnxn+Tnxn–x, we find from (.) and the asymptotic regularity ofTthat
lim
n→∞T n+x
n–x= ,
i.e.,TTnx
n–x→ asn→ ∞. It follows from the closedness ofTthatTx=x. So,x∈F(T) and hencex∈F=F(T)∩EP(θ).
Step . We show thatx=fFxand soxn→fFxasn→ ∞.
SinceF is a closed convex set, it follows from Lemma . thatfFx is single-valued, which is denoted byx˜. By the definition ofxn=fCnxandx˜∈F⊂Cn, we also have
G(xn,Jx)≤G(x˜,Jx)
for alln≥. By the definition ofG, we know that for anyx∈E,G(u,Jx) is convex and lower semicontinuous with respect touand so
G(x,Jx)≤lim inf
n→∞ G(xn,Jx)
≤lim sup
n→∞ G(xn,Jx)
≤G(x˜,Jx).
From the definition offFxandx∈F, we conclude that
x=x˜=fFx
andxn→x=fFxasn→ ∞. This completes the proof. Remark .
(i) Iff= , thenG(x,Jy) =φ(x,y)andfCn=Cn.
(ii) If we takef = ,θ= ,un=yn, andαn= for alln∈N, then the iterative scheme
(.) reduces to the following scheme: ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈Echosen arbitrarily,
C=C,
yn=J–(βnJTnxn+ ( –βn)Jxn),
Cn+={z∈Cn:φ(z,yn)≤φ(z,xn) +ξn},
xn+=Cn+x, ∀n≥,
whereξn=max{,supp∈F(T),x∈C(φ(p,Tnx) –φ(p,x))}, which is the algorithm
introduced by Hao [] and an improvement to (.).
Corollary . Let E be a uniformly smooth and strictly convex Banach space with the
Kadec-Klee property.Let C be a nonempty closed convex subset of E.Letθbe a bifunction from C×C toRsatisfying the conditions(A)-(A).Let T:C→C be a closed and quasi-φ-nonexpansive mapping.Assume thatF=F(T)∩EP(θ)is nonempty.Let f :E→R+be
a convex and lower semicontinuous function with C⊂int(D(f))and f() = .Let{αn}be a
sequence in[, ]and{βn},{γn}be sequences in(, )satisfying the following conditions:
(i) αn+βn+γn= ;
(ii) limn→∞αn= ;
(iii) <lim infn→∞βn≤lim supn→∞βn< .
Let{xn}be a sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E chosen arbitrarily,
C=C,
yn=J–(αnJx+βnJTxn+γnJxn)
un∈C such thatθ(un,y) +rny–un,Jun–Jyn ≥, ∀y∈C,
Cn+={z∈Cn:G(z,Jun)≤αnG(z,Jx) + ( –αn)G(z,Jxn)},
xn+=fCn+x, ∀n≥,
where {rn} is a real sequence in [a,∞)for some a> andfCn+ is the generalized f -projection operator.Then{xn}converges strongly tofFx.
Remark .
(i) By Remark ., Theorem . extends Theorem . of Hao [].
(ii) Theorem . generalizes Theorem . of Matsushita and Takahashi [] in the following respects:
• from the relatively nonexpansive mapping to the asymptotically quasi-φ-nonexpansive mapping in the intermediate sense;
• from a uniformly convex and uniformly smooth Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property; (iii) in view of the mappings and the frame work of the spaces, Theorem . generalizes
and improves Theorem . of Maet al.[], Theorem . of Qinet al.[], Theorem . of Qing and Lv [] and Theorem . of Saewan [].
We now provide a nontrivial family of mappings satisfying the conditions of Theo-rem ..
Example . LetE=Rwith the standard norm · =| · |andC= [, ]. LetT:C→C
be a mapping defined by
Tx=
⎧ ⎨ ⎩
We first show thatTis an asymptotically quasi-φ-nonexpansive mapping in the interme-diate sense withF(T) ={} =φ. In fact, forp= ∈F(T), we have
φp,Tnx= –Tnx
= n|x|
≤ | –x|=φ(p,x), ∀x∈
,
and
φp,Tnx= –Tnx
=
≤ | –x|=φ(p,x), ∀x∈
,
.
Therefore, we have
lim sup
n→∞
sup
p∈F(T),x∈C
φp,Tnx–φ(p,x)≤.
Next, we define a bifunctionθ:C×C→Rsatisfying the conditions (A)-(A) by
θ(x,y) =y–x.
Then the set of solutionsEP(θ) to the equilibrium problem forθ is obviously{}. Since F=F(T)∩EP(θ)=φandF(T) is bounded, it follows from Theorem . that the sequence defined by (.) converges strongly tofFx.
Competing interests
The author declares that he has no competing interests.
Author’s contributions
JUJ conceived of the study, its design, and its coordination. The author read and approved the final manuscript.
Acknowledgements
The author is grateful to the anonymous referees for useful suggestions, which improved the contents of the article.
Received: 28 April 2014 Accepted: 2 September 2014 Published:24 Sep 2014
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10.1186/1687-1812-2014-199
Cite this article as:Jeong:Convergence theorems for equilibrium problem and asymptotically
