R E S E A R C H
Open Access
The modified Ishikawa iterative algorithm
with errors for a countable family of Bregman
totally quasi-
D
-asymptotically nonexpansive
mappings in reflexive Banach spaces
Ren-Xing Ni
1and Jen-Chih Yao
2,3**Correspondence: [email protected] 2Center for General Education, China Medical University, Taichung, 40402, Taiwan
3Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article
Abstract
In this paper, a new modified Ishikawa iterative algorithm with errors by a shrinking projection method for generalized mixed equilibrium problems and a countable family of uniformly Bregman totally quasi-D-asymptotically nonexpansive mappings is introduced and investigated in the framework of a real Banach space. Strong convergence of the sequence generated by the proposed algorithm is derived under some suitable assumptions. These results are new and develop some recent results in this field.
MSC: 26B25; 46T99; 47H04; 47H05; 47H09; 47H10; 47J05; 47J20; 47J25; 52A41; 54C20
Keywords: asymptotically Bregman totally quasi-D-asymptotically nonexpansive mapping; generalized mixed equilibrium problem; Bregman distance; Bregman projection; fixed point; shrinking projection method; Banach space
1 Introduction and preliminaries
In this paper, without other specifications, letN∗ andRbe the sets of positive integers and real numbers, respectively,C be a nonempty, closed, and convex subset of a real Banach spaceE with the dual spaceE∗. The norm and the dual pair betweenE∗ andE
are denoted by · and·,·, respectively. Letg:E→R∪ {+∞}be a proper convex and lower semicontinuous function. Denote the domain ofg bydomg,i.e.,domg={x∈E:
g(x) < +∞}. The Fenchel conjugate ofg is the functiong∗:E∗→(–∞, +∞] defined by g∗(ζ) =supx∈E{ζ,x–g(x)}. Let T :E→C be a nonlinear mapping. For all x∈E and
x∗∈E∗, denote byF(T) ={x∈C:Tx=x}the set of fixed points ofT and byx,x∗the value ofx∗ atx. A mappingT is said to be nonexpansive ifTx–Ty ≤ x–yfor all
x,y∈E.
Let{xn}be a sequence inE, we denote the strong convergence of{xn}tox∈Ebyxn→x.
For anyx∈int(domg), the right-hand derivative ofgatxin the directiony∈Eis defined byg(x,y) :=limt→g(x+tyt)–g(x). The mappinggis called Gâteaux differentiable atxif, for
ally∈E,limt→g(x+tyt)–g(x) exists. In this case,g(x,y) coincides withg(x) and the value
attained uniformly fory= . We say thatgis uniformly Fréchet differentiable on a subset
CofEif the limit is attained uniformly forx∈Candy= .
The Legendre functiong:E→(–∞, +∞] is defined in []. From [], ifEis a reflexive Banach space, thengis the Legendre function if and only if it satisfies the conditions (L) and (L):
(L) The interior of the domain ofg,int(domg), is nonempty,gis Gâteaux differentiable onint(domg)anddom(g) =int(domg).
(L) The interior of the domain ofg∗,int(domg∗), is nonempty,g∗is Gâteaux differentiable onint(domg∗)anddomg∗=int(domg∗), where the function
g∗:E∗→(–∞, +∞]is the Fenchel conjugate ofg.
Examples of Legendre functions are given in [, ]. One important and interesting Leg-endre function is s · s( <s< +∞), in the Banach spaceEwhich is smooth and strictly
convex and, in particular, a Hilbert space.
By Bauschke et al.[], Theorem ., the conditions (L) and (L) also show that the functionsgandg∗ are strictly convex on the interior of their respective domains. From now on, we assume that the convex functiong:E→(–∞, +∞] is Legendre.
Definition . [, ] Let g :E→R be a Gâteaux differentiable and convex function. The functionD(·,·) :domg×int(domg)→[, +∞) defined byD(y,x) =g(y) –g(x) –y–
x,g(x)is called the Bregman distance with respect tog.
It follows from the strict convexity ofgthatD(x,y)≥ for allx,yinE. However,D(·,·) might not be symmetric andD(·,·) might not satisfy the triangular inequality.
Remark .[] The Bregman distance has the following properties:
() the three point identity, for anyx∈domgandy,z∈int(domg),
D(x,z) =D(x,y) +D(y,z) +g(y) –g(z),x–y;
() the four point identity, for anyy,w∈domgandx,z∈int(domg),
D(y,x) –D(y,z) –D(w,x) +D(w,z) =g(z) –g(x),y–w.
Definition .[] Letg:E→Rbe a Gâteaux differentiable and convex function. The Bregman projection ofx∈int(domg) onto the nonempty, closed and convex setC⊂domg
is the necessarily unique vectorProjgC(x)∈Csatisfying the following:
DProjgC(x),x=infD(y,x) :y∈C.
Definition .[] LetJ:E→E∗be the normalized duality mapping defined byJ(x) =
{x∗∈E∗:x,x∗=x=x∗},φ:E×E→R+ be the Lyapunov functional defined by
φ(x,y) =x– x,Jy+y,∀x,y∈E. The generalized projection
C(x) defined by
φC(x),x
=infφ(y,x) :y∈C.
Remark . () If E is a smooth Banach space and g(x) =x for all x∈E, then we
() IfE is a Hilbert space andg(x) =x for allx∈E, thenD(x,y) =x–yand the
Bregman projectionProjgC(x) is reduced to the metric projectionPC(x) ofxontoC. For
more details we refer the readers to [].
LetCbe a nonempty, closed, and convex subset ofEandTbe a mapping fromEtoC. A pointp∈Cis said to be an asymptotic fixed point ofT[] ifCcontains a sequence{xn}
which converges weakly topsuch thatlimn→∞xn–Txn= . A pointp∈Cis said to be a
strong asymptotic fixed point ofT [] ifCcontains a sequence which converges strongly topsuch thatlimn→∞xn–Txn= . We denote the sets of asymptotic fixed points and
strong asymptotic fixed points ofTbyF(T) andF(T), respectively.
Definition . () A mappingTfromEtoCis said to be Bregman relatively nonexpansive [, ], ifF(T) =F(T)=∅andD(p,Tx)≤D(p,x) for allx∈Eandp∈F(T).
()Tis said to be Bregman weak relatively nonexpansive [, , ], ifF(T) =F(T)=∅and
D(p,Tx)≤D(p,x) for allx∈Eandp∈F(T).
()T is said to be Bregman quasi-D-nonexpansive [, ], if F(T)= ∅andD(p,Tx)≤
D(p,x) for allx∈Eandp∈F(T).
()T is said to be Bregman firmly nonexpansive [], ifg(Tx) –g(Ty),Tx–Ty ≤
g(x) –g(y),Tx–Ty, ∀x,y∈E, or, equivalently, D(Tx,Ty) +D(Ty,Tx) +D(Tx,x) +
D(Ty,y)≤D(Tx,y) +D(Ty,x),∀x,y∈E.
() T is said to be Bregman strongly nonexpansive [], ifF(T)= ∅ andD(p,Tx)≤
D(p,x) for allx∈E andp∈F(T) and if whenever{xn} ⊂E is bounded, p∈F(T) and
limn→+∞[D(p,xn) –D(p,Txn)] = , it follows thatlimn→+∞D(Txn,xn) = .
()T is said to be relatively quasi-nonexpansive [], ifF(T) =F(T)=∅andφ(p,Tx)≤
φ(p,x) for allx∈Eandp∈F(T).
()Tis said to be weak relatively nonexpansive [–], ifF(T) =F(T)=∅andφ(p,Tx)≤
φ(p,x) for allx∈Eandp∈F(T).
()Tis said to be quasi-φ-nonexpansive [–], ifF(T)=∅andφ(p,Tx)≤φ(p,x) for allx∈Eandp∈F(T).
Definition . () A mappingT:E→Cis said to be Bregman totally quasi-D -asymptot-ically nonexpansive [], ifF(T)=∅and there exist nonnegative real sequences{vn},{un}
withvn,un→ (asn→+∞) and a strictly increasing continuous functionζ:R+→R+
withζ() = such that
Dp,Tnx≤D(p,x) +vn·ζ
D(p,x) +un, ∀n≥,x∈E,p∈F(T). (.)
() A mappingT:E→Cis said to be Bregman quasi-D-asymptotically nonexpansive [], ifF(T)=∅and there exists a sequence{kn} ⊂[, +∞) withlimn→+∞kn= such that
Dp,Tnx≤knD(p,x) for allx∈E,p∈F(T) andn≥. (.)
() A mappingT:E→Cis said to be Bregman quasi-D-asymptotically nonexpansive in the intermediate sense with sequence{vn}, ifF(T)=∅and there exists a sequence{vn}
in [, +∞) withlimn→+∞vn= such that
lim sup
n→+∞ x∈Esup,p∈F(T)
() A mappingT:E→Cis said to be totally quasi-φ-asymptotically nonexpansive [], ifF(T)=∅and there exist nonnegative real sequences{vn},{un}withvn,un→ (asn→
+∞) and a strictly increasing continuous functionζ:R+→R+withζ() = such that
φp,Tnx≤φ(p,x) +vn·ζ
φ(p,x) +un, ∀n≥,x∈E,p∈F(T). (.)
() A mapping T :E →C is said to be quasi-φ-asymptotically nonexpansive [], if F(T)=∅ and there exists a sequence {kn} ⊂[, +∞) withlimn→+∞kn= such that φ(p,Tnx)≤k
nφ(p,x) for allx∈E,p∈F(T) andn≥.
() A mappingT:E→Cis said to be quasi-φ-asymptotically nonexpansive in the inter-mediate sense with sequence{vn}, ifF(T)=∅and there exists a sequence{vn}in [, +∞)
withlimn→+∞vn= such that
lim sup
n→+∞ x∈Esup,p∈F(T)
φp,Tnx– ( +vn)φ(p,x) ≤. (.)
Remark . () Ifζ(t) =t,t≥, then (.) reduces to
Dp,Tnx≤( +vn)·D(p,x) +un, ∀n≥,x∈E,p∈F(T). (.)
In addition, if un≡ for alln≥, then Bregman totally quasi-D-asymptotically
non-expansive mappings coincide with Bregman quasi-D-asymptotically nonexpansive map-pings. Ifun≡ andvn≡ for alln≥, we obtain from (.) the class of mappings that
includes the class of Bregman quasi-nonexpansive mappings. If vn≡ and un=σn=
max{,supx∈E,p∈F(T)(D(p,Tnx) –D(p,x))}, for alln≥, then (.) reduces to (.) which
has been studied as mappings Bregman quasi-D-asymptotically nonexpansive in the in-termediate sense.
() From the definitions, it is obvious that ifF(T) =F(T)=∅, then a Bregman strongly nonexpansive mapping is a Bregman relatively nonexpansive mapping; a Bregman rela-tively nonexpansive mapping is a Bregman quasi-D-nonexpansive mapping. A Bregman quasi-D-nonexpansive mapping is a Bregman quasi-D-asymptotically nonexpansive map-ping, but the converse is not true.
If takingζ(t) =t,t≥,vn=kn– ,un= ,limn→+∞kn= , then (.) can be rewritten
as (.). This implies that each Bregman quasi-D-asymptotically nonexpansive mapping must be a Bregman total quasi-D-asymptotically nonexpansive mapping, but the converse is not true. In [], Changet al.gave an example of Bregman total quasi-D-asymptotically nonexpansive mapping. A Bregman relatively nonexpansive mapping is a Bregman weak relatively nonexpansive mapping, but the converse in not true in general. Indeed, for any mapping T :E→C, we have F(T)⊂F(T)⊂F(T). If T is Bregman relatively nonex-pansive, then F(T) =F(T) =F(T). In [], Naraghirad and Yao have given two examples of a Bregman weak relatively nonexpansive mapping which is not a Bregman relatively nonexpansive mapping, and a Bregman quasi-nonexpansive mapping which is neither a Bregman relatively nonexpansive mapping nor a Bregman weak relatively nonexpansive mapping.
A quasi-φ-nonexpansive mapping with a nonempty fixed point set F(T) is a quasi-φ -asymptotically nonexpansive mapping, but the converse may not be true. In the frame-work of Hilbert spaces, quasi-φ-(asymptotically) nonexpansive mappings is reduced to quasi-(asymptotically) nonexpansive mappings.
The idea of the definition of a total asymptotically nonexpansive mappings is to unify various definitions of classes of mappings associated with the class of asymptotically non-expansive mappings and to prove a general convergence theorems applicable to all these classes of nonlinear mappings.
Definition .[] LetEbe a Banach space. The functiong:E→Ris said to be a Bregman function if the following conditions are satisfied:
() gis continuous, strictly convex and Gâteaux differentiable; () the set{y∈E:D(x,y)≤r}is bounded for allx∈Eandr> .
The theory of fixed points with respect to Bregman distances have been studied in the last ten years and much intensively in the last six years. In [], Bauschke and Combettes introduced an iterative method to construct the Bregman projection of a point onto a countable intersection of closed and convex sets in reflexive Banach spaces. They proved strong convergence theorem of the sequence produced by their method; for more details, see [], Theorem .. To find a point of the intersection ofmclosed and convex subsets in a Banach space, in , Alber [] first studied the iterative method with Bregman pro-jections. In [], Alber investigated the generalized projections in a Banach space. For some recent articles on the existence of fixed points for Bregman nonexpansive type mappings, we refer the reader to [–, –].
It is well known that the following conclusions hold:
Lemma .[, ] Let E be a Banach space and g:E→R a Gâteaux differentiable func-tion which is locally uniformly convex on E.Let{yn}and{zn}be sequences in E such that either{yn}or{zn}is bounded.Thenlimn→+∞D(yn,zn) = ⇔limn→+∞yn–zn= .
Lemma . Let C be a nonempty closed convex subset of Banach space E and g:E→
(–∞, +∞]be a Legendre function which is total convex on bounded subsets of E.Let T:E→ C be a closed and Bregman totally quasi-D-asymptotically nonexpansive mapping with nonnegative real sequences{vn},{un}and a strictly increasing and continuous functionζ : R+→R+withζ() = .If vn,un→ (as n→+∞).Then F(T)is a closed convex subset of C.
Proof Let{xn}be a sequence inF(T) such thatxn→x∗ (asn→+∞). We haveTxn= xn→x∗(asn→+∞) and by the closeness ofT, we haveTx∗=x∗. This implies thatF(T)
is closed.
Letp,q∈F(T) andt∈(, ), and putw=tp+ ( –t)q. Next we prove thatw∈F(T). Indeed, in view of the definition ofD, we have
Dw,Tnw=g(w) –gTnw–gTnw,w–Tnw
=g(w) –gTnw–gTnw,tp+ ( –t)q–Tnw
Since
tDp,Tnw+ ( –t)Dq,Tnw
≤tD(p,w) +vnζ
D(p,w) +un
+ ( –t)D(q,w) +vnζ
D(q,w) +un
=tg(p) –g(w) –g(w),p–w+vnζ
D(p,w) +un
+ ( –t)g(q) –g(w) –g(w),q–w+vnζ
D(q,w) +un
=tg(p) + ( –t)g(q) –g(w) + ( –t)vnζ
D(q,w)
+un+tvnζ
D(p,w) . (.)
Substituting (.) into (.) and simplifying it, we have
≤Dw,Tnw≤tvnζ
D(p,w) + ( –t)vnζ
D(q,w) +un (asn→+∞).
Hence, we haveTnw→w. This implies thatT(Tnw) =Tn+w→w. SinceT is closed, we havew∈Tw,i.e.,w∈F(T). This completes the proof of Lemma ..
Definition .[] Letg:E→(–∞, +∞] be a convex and Gâteaux differentiable
func-tion.gis called
() totally convex atx∈int(domg)if its modulus of total convexity atx, that is, the functionvg:int(domg)×[, +∞)→[, +∞), defined by
vg(x,t) :=inf{D(y,x) :y∈domg,y–x=t}, is positive whenevert> ;
() totally convex if it is totally convex at every pointx∈int(domg);
() totally convex on bounded sets ifvg(B,t)is positive for any nonempty bounded
subsetBofEandt> , where the modulus of total convexity of the functiongon the setBis the functionvg:int(domg)×[, +∞)→[, +∞)defined by
vg(B,t) :=inf{vg(x,t) :x∈B∩domg}.
Definition .[, ] LetBbe the closed unit ball of a Banach spaceE. A functiong:
E→Ris said to be
() cofinite ifdomg∗=E∗;
() coercive iflimx→∞(g(x)/x) = +∞;
() sequentially consistent if for any two sequences{xn}and{yn}inEsuch that{xn}is
bounded,limn→+∞D(yn,xn) = ⇒limn→+∞yn–xn= ;
() locally bounded ifg(rB)is bounded for allr> ;
() locally uniformly smooth onEif the functionσr: [, +∞)→[, +∞), defined by
σr(t) = sup x∈rB,y∈E,y=,α∈(,)
αgx+ ( –α)ty+ ( –α)g(x–αty) –g(x)
×α( –α)–/,
() locally uniformly convex onE(or uniformly convex on bounded subsets ofE) if the gaugeρr: [, +∞)→[, +∞)of uniform convexity ofg, defined by
ρr(t) = inf x,y∈rB,x–y=t,α∈(,)
αg(x) + ( –α)g(y) –gαx+ ( –α)y
×α( –α) –/,
satisfiesρr(t) > ,∀r,t> .
Lemma .[] If g:E→(–∞, +∞]is Fréchet differentiable and totally convex,then g is cofinite.
Lemma .[] Let g:E→(–∞, +∞]be a convex function whose domain contains at least two points.Then the following statements hold:
() gis sequentially consistent if and only if it is totally convex on bounded sets. () Ifgis lower semicontinuous,thengis sequentially consistent if and only if it is
uniformly convex on bounded sets.
() Ifgis uniformly strictly convex on bounded sets,then it is sequentially consistent and the converse implication holds whengis lower semicontinuous,Fréchet differentiable on its domain and the Fréchet derivativegis uniformly continuous on bounded sets.
Lemma . [] Let g : E →R be uniformly Fréchet differentiable and bounded on bounded subsets of E.Theng is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E∗.
Lemma .([], Lemma .) Let g:E→R be a Gâteaux differentiable and totally convex function.If x∈E and the sequence{D(xn,x)}is bounded,then the sequence{xn}is also bounded.
Lemma .[] Let E be a Banach space,r> be a positive number and g:E→R be a continuous and convex function which is uniformly convex on bounded subsets of E.Then
g
m
n=
λnxn
≤
m
n=
λng(xn) –λiλjρr
xi–xj
for any given infinite subset{xn} ⊂Br() ={x∈E:x ≤r}and for any given sequence{λn} of positive numbers withmn=λn= ,for any i,j∈ {, , . . . ,m}with i<j,whereρr is the gauge of uniformly convexity of g.
Lemma .[] Let g:E→(–∞, +∞]be Gâteaux differentiable and totally convex on
int(domg).Let x∈int(domg)and C⊂int(domg)be a nonempty,closed,and convex set.If
ˆ
x∈C,then the following statements are equivalent:
() the vectorxˆis the Bregman projection ofxontoCwith respect tog; () the vectorxˆis the unique solution of the variational inequality:
g(x) –g(z),z–y ≥,∀y∈C;
Lemma .([], Theorem .) Let E be a reflexive Banach space and let g:E→R be a convex function which is bounded on bounded subsets of E.Then the following assertions are equivalent:
() gis strongly coercive and uniformly convex on bounded subsets ofE;
() domg∗=E∗,g∗is bounded on bounded subsets and uniformly smooth on bounded subsets ofE∗;
() domg∗=E∗,g∗is Fréchet differentiable andg∗is uniformly norm-to-norm continuous on bounded subsets ofE∗.
Lemma .([], Theorem .) Let E be a reflexive Banach space and let g:E→R be a continuous convex function which is strongly coercive.Then the following assertions are equivalent:
() gis bounded on bounded subsets and uniformly smooth on bounded subsets ofE; () g∗is Fréchet differentiable andg∗is uniformly norm-to-norm continuous on
bounded subsets ofE∗;
() domg∗=E∗,g∗is strongly coercive and uniformly convex on bounded subsets ofE∗.
For solving the equilibrium problem, let us assume that the bifunctionf :C×C→R
satisfies the following conditions:
(C) f(x,x) = ,∀x∈C;
(C) f is monotone,i.e.,f(x,y) +f(y,x)≤,∀x,y∈C;
(C) for eachy∈C, the functionx→f(x,y)is upper semicontinuous; (C) ∀x∈C,y→f(x,y)is convex and lower semicontinuous.
Lemma .[] Let E be a reflexive Banach space and g:E→R a convex,continuous and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subset of E.Let C be a nonempty,closed and convex subset of E and f :C×C→R a bifunction satisfying conditions(C)-(C)andEP(G)=∅,ϕ:C→R be a lower semicon-tinuous and convex functional,A:C→E∗be a continuous and monotone mapping.For r> and x∈E,define a mapping TG
r :E→C as follows:
TrGx=
z∈C:G(z,y) +
r
y–z,g(z) –g(x)≥,∀y∈C
, (.)
where G(x,y) =f(x,y) +ϕ(y) –ϕ(x) +Ax,y–x,∀x,y∈E.Then the following statements hold:
() dom(TG r) =E;
() TG
r is single-valued;
() TrGis a Bregman firmly nonexpansive mapping; () F(TG
r ) =GMEP(f,ϕ);
() GMEP(f,ϕ)is closed and convex ofC; () D(q,TG
r x) +D(TrGx,x)≤D(q,x),∀q∈F(TrG).
In , Saewanet al.[] studied the following generalized mixed equilibrium problem: findz∈Csuch that
wheref is a bifunction fromC×CtoR,ϕ:C→Ris a real-valued function andA:C→E∗
is a nonlinear mapping. Denote the set of solutions of the problem (.) byGMEP(f,ϕ),
i.e.,
GMEP(f,ϕ) =z∈C|f(z,y) +Az,y–z+ϕ(y) –ϕ(z)≥,∀y∈C.
Special cases: (I) IfA= , then the problem (.) is equivalent to findz∈Csuch that
f(z,y) +ϕ(y) –ϕ(z)≥, ∀y∈C, (.)
which is called the mixed equilibrium problem. Denote the set of solutions of (.) by
MEP(f,ϕ).
(II) Iff = , then the problem (.) is equivalent to findz∈Csuch that
Az,y–z+ϕ(y) –ϕ(z)≥, ∀y∈C, (.)
which is called the mixed variational inequality of Browder-type. Denote the set of solu-tions of (.) byVI(C,A,ϕ). In particular, we denoteVI(C,A, ) byVI(C,A).
(III) Ifϕ= , then the problem (.) is equivalent to findingz∈Csuch that
f(z,y) +Az,y–z ≥, ∀y∈C, (.)
which is called the generalized equilibrium problem. Denote the set of solutions of (.) byGEP(f).
(IV) IfA= ,ϕ= , then the problem (.) is equivalent to findingz∈Csuch that
f(z,y)≥, ∀y∈C, (.)
which is called the equilibrium problem. Denote the set of solutions of (.) byEP(f). It is well known that mixed equilibrium problems and their generalizations have been important tools for solving problems arising in the fields of linear or nonlinear pro-gramming, variational inequalities, complementary problems, optimization problems, and fixed point problems, and they have been widely applied to physics, structural analy-sis, management science, economics,etc.One of the most important and interesting top-ics in the theory of equilibria is to develop efficient and implementable algorithms for solving equilibrium problems and their generalizations (see,e.g., [–] and the refer-ences therein). Since the generalized mixed equilibrium problems have very close con-nections with both the fixed point problems and the variational inequalities problems, finding the common elements of these problems has drawn many researchers’ attention and has become one of the hot topics in the related fields in the past few years (see,e.g., [–, –] and the references therein). Some methods have been proposed to solve the generalized mixed equilibrium problem (see, for example, [–, , –]). Nu-merous problems in physics, optimization and economics help to find a solution of prob-lem (.).
recently been made so that strong convergence theorems are obtained; see, for example, [–, , –] and the references therein.
In [], Martinez-Yanes and Xu introduced the following iterative scheme for a single nonexpansive mappingTin a Hilbert spaceH:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily, yn=αnx+ ( –αn)Txn,
Cn={z∈C:z–yn≤ z–xn+αn(x+ xn–x,z)}, Qn={z∈C:xn–z,x–xn ≥},
xn+=PCn∩Qnx,
(.)
wherePCdenotes the metric projection ofHonto a closed and convex subsetCofH. They
proved that if{αn} ⊂(, ) andlimn→∞αn= , then the sequence{xn}converges strongly
toPF(T)x.
In [], Qin and Su extended the results of Martinez-Yanes and Xu [] from Hilbert spaces to Banach spaces and proved the following result: LetCbe a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space E and let T :C →C be a relatively nonexpansive mapping. Assume that {αn} ⊂(, ) and
limn→∞αn= . Define a sequence{xn}inCby the following algorithm: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily, yn=J–(αnJx+ ( –αn)JTxn), Cn={z∈C:φ(z,yn)≤φ(z,xn)}, Qn={z∈C:xn–z,Jx–Jxn ≥}, xn+=Cn∩Qnx,n≥.
(.)
IfF(T) is nonempty, then{xn}converges strongly toF(T)x.
In , Wangkeeree and Wangkeeree [] introduced the following hybrid projection algorithm for approximation of common fixed point of two families of relatively quasi-nonexpansive mappings, which is also a solution to a variational inequality problem in a Banach spaceE:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E chosen arbitrarily, C,i=C, C=
∞
i=C,i, xi=Cx,
wn,i=CJ–(Jxn–λn,iBxn),
zn,i=J–(βn(),iJxn+βn(),iJTixn+βn(),iJSiwn,i), yn,i=J–(αn,iJx+ ( –αn,i)Jzn,i),
Cn,i={z∈C:φ(z,yn,i)≤φ(z,xn) +αn,i(x+ Jxn–Jx,z)}, Cn+=
∞
i=Cn+,i, xn+=Cn+x.
(.)
They proved under appropriate conditions on the parameters that the sequence{xn}
gen-erated by (.) converges strongly to a common element of the set of common fixed points of the two families{Ti}and{Si}and the set of solutions to a variational inequality problem.
fea-sibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique has been applied in various ways in order to design and analyze not only iterative algorithms for solving feasibility and optimization problems, but also al-gorithms for solving variational inequalities, for approximating equilibria, for computing fixed points of nonlinear mappings, and so on (see,e.g., [, , –] and the references therein). In , Butnariu and Resmerita [] presented Bregman-type iterative algo-rithms and studied the convergence of the Bregman-type iterative method of solving some nonlinear operator equations.
In , by using the Bregman projection, Reich and Sabach [] presented the following proximal algorithms for finding common zeroes of maximal monotone operatorsAi:E→
E∗(i= , , . . . ,m) in a reflexive Banach spaceE. More precisely, they proved the following
strong convergence theorem.
Theorem RS Let E be a reflexive Banach space and let Ai:E→E ∗
(i= , , . . . ,m)be m maximal monotone operators such that Z:=mi=A–i (∗)=∅.Let g:E→R be a Legendre function that is bounded,uniformly Fréchet differentiable and totally convex on bounded subsets of E.Let{xn}be a sequence defined by the following iterative algorithm:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E chosen arbitrarily, yin=Resg
λin(xn+e
i n), Ci
n={z∈E:D(z,yni)≤D(z,xn+ein)}, Cn=
m
i=Cni,
Qn={z∈E:g(x) –g(xn),z–xn ≤}, xn+=ProjgCn∩Qn(x), ∀n≥.
(.)
If,for each i= , , . . . ,m,lim infn→+∞λin> and the sequences of errors{ei
n} ⊂E satisfy
limn→+∞ei
n= ,then each such sequence{xn}converges strongly toProjZg(x)as n→+∞. Further,under some suitable conditions,they obtained two strong convergence theorems of maximal monotone operators in a reflexive Banach space.Reich and Sabach[]also studied the convergence of two iterative algorithms for finitely many Bregman strongly non-expansive operators in a Banach space.
In [], Reich and Sabach proposed the following algorithms for finding common fixed points of finitely many Bregman firmly nonexpansive operatorsTi:E→E(i= , , . . . ,m)
in a reflexive Banach spaceE, ifF:=mi=F(Ti)=∅: ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E,
Qi=E, i= , , . . . ,m,
yi
n=Ti(xn+ein), Qi
n+={z∈Qni :g(xn+ein) –g(yni),z–yin ≤}, Qn=
m i=Qin,
xn+=ProjgQn+(x), ∀n≥.
(.)
Under some suitable conditions, they proved that the sequence{xn}generated by (.)
converges strongly toProjgF(x) and applied the result to the solution of convex feasibility
and equilibrium problems, where g:E→Rand{ei
n} ⊂E satisfyinglimn→+∞ein= for
Very recently, Chenet al.[] introduced the concept of weak Bregman relatively non-expansive mappings in a reflexive Banach space and gave an example to illustrate the ex-istence of a weak Bregman relatively nonexpansive mapping and the difference between a weak Bregman relatively nonexpansive mapping and a Bregman relatively nonexpansive mapping. They also proved the strong convergence of the sequences generated by the con-structed algorithms with errors for finding a fixed point of weak Bregman relatively non-expansive mappings and Bregman relatively nonnon-expansive mappings under some suitable conditions.
Motivated by the above mentioned results and the on-going research, in this paper, us-ing Bregman function and the shrinkus-ing projection method, we introduce new modified Ishikawa iterative algorithms with errors for finding a common element of solutions to the generalized mixed equilibrium problems (.) and fixed points to a countable fam-ily of Bregman totally quasi-D-asymptotically nonexpansive mappings in Banach spaces. We prove strong convergence theorems for the sequences generated by the proposed al-gorithm. Furthermore, these algorithms take into account possible computational errors. No assumptionF(T) =F(T) is imposed on the mappingTin reflexive Banach space set-ting. Our results improve and develop many known results in the current literature; see, for example, [, , , ].
2 Main results
We now state and prove the main result of this paper.
Theorem . Let E be a reflexive Banach space and g:E→R be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uni-formly smooth on bounded subsets of E.Let C be a nonempty,closed,and convex subset of E.
For each k= , , . . . ,m,let Ak:C→E∗be a continuous and monotone mapping,ϕk:C→R be a lower semicontinuous and convex functional,let fk:C×C→R be a bifunction sat-isfying(C)-(C)and Ti:E→int(domg),∀i∈N be an infinite family of closed and uni-formly Bregman totally quasi-D-asymptotically nonexpansive mappings with nonnegative real sequences{v(ni)},{un(i)}and a strictly increasing and continuous functionζ :R+→R+ withζ() = .Iflimn→+∞supi∈N∗{vn(i)}= andlimn→+∞supi∈N∗{u
(i)
n}= .Assume that Tiis uniformly asymptotically regular on E for all i≥,i.e.,limn→+∞supx∈KTn+
i x–Tinx= holds for any bounded subset K of E and F= [+i=∞F(Ti)]∩[
m
k=GMEP(fk,ϕk)]=∅.For all z,y∈C,Gk(z,y) =fk(z,y) +ϕk(y) –ϕk(z) +Akz,y–z,TrGkk,n(x) ={z∈C:Gk(z,y) +
rk,ny– z,g(z) –g(x) ≥,∀y∈C}.For an initial point x∈E,let Ci=C for each i≥and C=∞i=Ciand define the sequence{xn}by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yi
n=g∗[αng(xn) + ( –αn)g(zin)], zi
n=g∗[βng(xn) + ( –βn)g(Tin(xn+ein))], uin=TGm
rm,nT
Gm– rm–,n· · ·T
G r,nT
G r,ny
i n, Ci
n+={z∈Cn:D(z,uin)≤αnD(z,xn) + ( –αn)D(z,zin)≤D(z,xn) +ζni}, Cn+=
+∞ i= Cin+, xn+=ProjgCn+(x),
(.)
where the sequences{ζi
n},{ein},{rk,n},{αn},{βn}satisfy the following conditions:
() ζi
() for eachk= , , . . . ,m,{rk,n}n+=∞⊂(, +∞)satisfylim infn→+∞rk,n> ;
() {αn},{βn}are real sequences in[, ]such thatlim infn→∞( –αn)( –βn)βn> . Then the sequence{xn}converges strongly toProjgF(x).
Proof We define a bifunctionGk:C×C→Rby
Gk(x,y) =fk(x,y) +ϕk(y) –ϕk(x) +Akx,y–x, ∀x,y∈C.
Then we prove from Lemma . that the bifunctionGk satisfies conditions (C)-(C)
for eachk= , , . . . ,m. Therefore, the generalized mixed equilibrium problem (.) is equivalent to the following equilibrium problem: findx∈Csuch thatGk(x,y)≥,∀y∈C.
Hence,GMEP(fk,ϕk) =EP(Gk). By takingθnk=T Gk
rk,nT
Gk– rk–,n· · ·T
G r,nT
G
r,n,k= , , . . . ,m, and
θn=Ifor alln≥, we obtainun=θnmyn.
In view of Lemma . and Lemma ., we find that F is closed and convex, so that
ProjgF(x) is well defined for anyx∈E.
We divide the proof of Theorem . into six steps:
Step . We first show thatCnis closed and convex for eachn≥.
In fact, from the definition,C=
∞
i=Ci=Cfor alli≥ is closed and convex. Suppose
thatCi
n+is closed and convex for somen≥. For anyz∈Cin+, we know that
Dz,uin≤αnD(z,xn) + ( –αn)D
z,zin≤D(z,xn) +ζni
is equivalent to the following:
z–uin,αng(xn) + ( –αn)g
zin–guin≤αnD
uin,xn
+ ( –αn)D
uin,zin–guin
and
( –αn)
z–xn,g(xn) –g
zin≤–( –αn)D
xn,zin
+ζni, ∀i≥.
Since the left-hand sides of the last two inequalities are affine with respect tozas functions ofz,Ci
n+is closed and convex. HenceCn+=
+∞
i= Cin+is closed and convex for alln≥.
Step . Assume thatF⊂Cnfor alln≥. Then the sequence{xn}is bounded.
In fact, byxn+=ProjgCn+(x), it then follows from Lemma . that D(xn+,x) =D
ProjgC
n+(x),x
≤D(p,x) –D(p,xn+)≤D(p,x)
for eachp∈F⊂Cn,∀n≥. Hence, the sequence{D(xn+,x)}is bounded, by Lemma .,
{xn}is bounded and so are{Tixn},{yin},{zni}, and{uin}.
Step . Next, we show, by induction, thatF⊂Cnfor alln≥.
In fact, it is obvious thatF⊂C=C. Suppose thatF⊂Cnfor somen≥. Letp∈F, since Ti:E→C(∀i∈N) is an infinite family of closed and uniformly Bregman totally quasi-D-asymptotically nonexpansive mappings, by the definition ofD(·,·) and Remark ., for
eachi≥, we have
Dp,zin=Dp,g∗βng(xn) + ( –βn)g
Tinxn+ein
≤βnD(p,xn) + ( –βn)D
≤βnD(p,xn) + ( –βn)
Dp,xn+ein
+v(ni)·ζDp,xn+ein +u(ni)
=βnD(p,xn) + ( –βn)
D(p,xn) +D
xn,xn+ein
+xn–p,g
xn+ein
–g(xn)
+v(ni)·ζDp,xn+ein +u(ni)
=D(p,xn) + ( –βn)
Dxn,xn+ein
+xn–p,g
xn+ein
–g(xn)
+v(ni)·ζDp,xn+ein +u(ni)
≤D(p,xn) +ζni. (.)
Observe thatp∈Fimpliesp∈C. Thus, by (.), Lemma ., and the fact thatTGk
rk,n(k= , , . . . ,m) is a Bregman quasi-D-nonexpansive mapping, for eachp∈F, we have
Dp,uin=Dp,θnmyin
≤Dp,yin
=Dp,g∗αng(xn) + ( –αn)g
zi n
≤αnD(p,xn) + ( –αn)D
p,zin
≤αnD(p,xn) + ( –αn)
D(p,xn) +ζni
≤D(p,xn) +ζni. (.)
This shows thatp∈Cn+, which implies thatF⊂Cn+. HenceF⊂Cnfor alln≥.
Step . Now, we show that{xn}is Cauchy sequence.
In fact, combiningxn+=ProjgCn+(x)∈Cn+and Lemma ., we obtain ≤D(xn,xn+)≤ D(xn,x) –D(xn+,x) for alln≥. Thus, the sequence{D(xn,x)}is nondecreasing. It
fol-lows from the boundedness of{D(xn,x)}that the limit of{D(xn,x)}exists.
For any positive integerm, it then follows from Lemma . that
D(xn+m,xn+) =D
xn+m,ProjgCn+(x)
≤D(xn+m,x) –D
ProjgC
n+(x),x
=D(xn+m,x) –D(xn+,x), (.)
from which it follows from (.) that D(xn+m,xn+)→ as m,n→ ∞. We have from
Lemma . and the boundedness of{xn},
xn+m–xn+→, m,n→ ∞.
Hence, the sequence{xn}is Cauchy inC. SinceEis a Banach space andCis closed convex,
there existsp∈Csuch thatxn→pasn→ ∞. Now, sinceD(xn+m,xn+)→ asm,n→ ∞,
we have in particular that
lim
n→∞D(xn+,xn+) = (.)
and this further implies that
lim
n→∞xn+–xn+= (.)
Fromxn– (xn+ein)=ein → (asn→+∞,∀i≥), Lemma ., and the boundedness
of{g(xn+ein)}, we obtain
≤Dxn,xn+ein
=g(xn) –g
xn+ein
+ein,gxn+ein
≤g(xn) –g
xn+ein+ein·g
xn+ein→ asn→+∞,∀i≥. (.)
Sincegis uniformly smooth on bounded subsets ofE, by Lemma ., we find thatg(·)
is uniformly norm-to-norm continuous on any bounded sets andxn– (xn+ein)=ein →
(asn→+∞,∀i≥), and we obtain
lim
n→∞g(xn) –g
xn+ein= , ∀i≥. (.)
Thus, it follows from (.), (.),limn→+∞supi∈N∗{v
(i)
n}= , andlimn→+∞supi∈N∗{u
(i)
n}=
that
≤ζni
≤Dxn,xn+ein
+sup
p∈C
xn–p,g
xn+ein
–g(xn)
+v(ni)·sup
p∈C
ζDp,xn+ein +u(ni)
≤Dxn,xn+ein
+
sup
p∈C xn–p
·gxn+ein
–g(xn)
+v(ni)·sup
p∈C
ζDp,xn+ein +un(i)→ asn→+∞,∀i≥. (.)
Byxn+=ProjgCn+(x)∈Cn+⊂Cn+(⊂C) and by the definition ofCn+, it follows from
(.) and (.) that
≤Dxn+,uin+
≤D(xn+,xn+) +ζni+→, n→ ∞,∀i≥.
From Lemma ., we obtainlimn→∞xn+–uin+= . Therefore
xn+–uin+≤ xn+–xn++xn+–uin+→. (.)
It follows fromlimn→+∞xn–p= and (.) that
uin→p, n→ ∞,∀i≥. (.)
Step . Now we prove thatp∈[+i=∞F(Ti)]∩[
m
k=GMEP(fk,ϕk)].
(a) First we prove thatp∈+i=∞F(Ti).
Sincegis uniformly smooth on bounded subsets ofE, by Lemma ., we find thatg(·)
is uniformly norm-to-norm continuous on any bounded sets and from (.), we obtain
lim
n→∞g(xn) –g
It follows from the boundedness of the sequences{xn}andD(p,Tin(xn+ein))≤D(p,xn+ ein) +v(ni)·ζ[D(p,xn+ein)] +u
(i)
n for eachp∈Fandi≥ that the sequences{g(xn)}and
{g(Tn
i(xn+ein))} are bounded. Thus there existsr> such that{g(xn)} ⊂Br() and
{g(Tin(xn+ein))} ⊂Br(). For eachp∈F, we have from Lemma . and Lemma .
Dp,uin=Dp,θnmyin≤Dp,yin=Dp,g∗αng(xn) + ( –αn)g
zin
≤αnD(p,xn) + ( –αn)D
p,zin
≤αnD(p,xn) + ( –αn)·
βnD(p,xn) + ( –βn)D
p,Tinxn+ein
–βn( –βn)ρr∗g(xn) –g
Tinxn+ein
≤αnD(p,xn) + ( –αn)·
βnD(p,xn) + ( –βn)
Dp,xn+ein
+v(ni)
·ζDp,xn+ein
+u(ni) –βn( –βn)ρr∗g(xn) –g
Tinxn+ein
=αnD(p,xn) + ( –αn)·
βnD(p,xn) + ( –βn)
v(ni)·ζDp,xn+ein
+u(ni)+D(p,xn) +D
xn,xn+ein
+xn–p,g
xn+ein
–g(xn)
– ( –αn)βn( –βn)ρ∗rg(xn) –g
Tinxn+ein
≤αnD(p,xn) + ( –αn)·
D(p,xn) +ζni–βn( –βn)ρr∗g(xn)
–gTinxn+ein
≤αnD(p,xn) + ( –αn)D(p,xn) +ζni– ( –αn)βn( –βn)ρr∗g(xn)
–gTinxn+ein
=D(p,xn) +ζni– ( –αn)βn( –βn)ρr∗g(xn) –g
Tinxn+ein.
This implies that
≤( –αn)βn( –βn)ρr∗g(xn) –g
Tinxn+ein
≤D(p,xn) –D
p,uin+ζni. (.)
On the other hand, we have
D(p,xn) –D
p,uin=–Dxn,uin
+xn–p,g
uin–g(xn)
≤Dxn,uin
+xn–p ·g
uin–g(xn).
In view of (.) and (.), we obtain
D(p,xn) –D
p,uin→, n→ ∞. (.)
Combining (.) and (.),limn→+∞ζni= , and the assumptionlim infn→∞( –αn)βn( – βn) > , we have
ρr∗g(xn) –g
It follows from the property ofρr∗(·) that
lim
n→+∞g(xn) –g
Tinxn+ein= . (.)
Since xn→p as n→ ∞ and g(·) is uniformly norm-to-norm continuous on any
bounded sets, we obtain
g(xn) –g(p)→ asn→. (.)
Note that
gTinxn+ein
–g(p)≤g(p) –g(xn)+g(xn) –g
Tinxn+ein.
From (.) and (.), we see that
lim
n→+∞g
Tinxn+ein
–g(p)= . (.)
By Lemma ., note that g∗(·) is also uniformly norm-to-norm continuous on any
bounded sets. It follows from (.) that
lim
n→+∞T
n i
xn+ein
–p= . (.)
Noting thatTin+(xn+ein) –p ≤ Tin+(xn+ein) –Tin(xn+ein)+Tin(xn+ein) –p, the
uniformly asymptotic regularity ofTand (.), we havelimn→+∞Tin+(xn+ein) –p= .
That is,Ti(Tin(xn+ein))→pasn→ ∞, and it follows from the closeness ofTithatTip=p,
∀i≥,i.e. p∈+i=∞F(Ti).
(b) Now we prove thatp∈mk=GMEP(fk,ϕk) =
m
k=EP(Gk).
In fact, in view ofui
n=θnmyin, (.), and Lemma ., for eachq∈F(θnk), we have
≤Duin,yin=Dθnmyin,yin≤Dp,yin–Dp,θnmyin≤D(p,xn) –D
p,uin+ζni.
It follows from (.) andlimn→+∞ζni = thatD(ui
n,yin)→ asn→ ∞. Using
Lem-ma ., we see thatuin–yni → asn→ ∞. Furthermore,xn–yin ≤ xn–uin+uin– yin → asn→ ∞. Sincexn→p, n→ ∞andxn–yin →,n→ ∞, thenyin→p, n→ ∞. By the fact thatθk
n,k= , , . . . ,mis Bregman relatively nonexpansive and using
Lemma . again, we have
≤Dθnkyin,yin≤Dp,yni–Dp,θnkyin≤D(p,xn) –D
p,θnkyin+ζni. (.)
Observe that
Dp,uin=Dp,θnmyin=Dp,TGm
rm,nT
Gm– rm–,n· · ·T
G r,nT
G r,ny
i n
=Dp,TGm
rm,nT
Gm– rm–,n· · ·θ
k nyin
≤Dp,θnkyin. (.)
Using (.) and (.), we obtain ≤D(θnkyni,yin)≤D(p,xn) –D(p,uni) +ζni→,n→ ∞.
θk
nyin–yin+yin–p →,n→ ∞,k= , , . . . ,m. Similarly,limn→+∞θnk–yin–p= , k= , , . . . ,m. This further implies that
lim
n→+∞θ
k–
n yin–θnkyin= . (.)
Also, sinceg(·) is uniformly norm-to-norm continuous on any bounded sets and using (.), we obtainlimn→+∞g(θnkyin) –g(θnk–yin)= . From the assumption{rk,n}+n=∞⊂
(, +∞) satisfyinglim infn→+∞rk,n> for eachk= , , . . . ,m, we see that
lim
n→∞
g(θnkyin) –g(θnk–yin) rk,n
= . (.)
By Lemma ., we have, for each k= , , . . . ,m,Gk(θnkyni,y) + rk,ny–θ
k
nyin,g(θnkyin) –
g(θnk–yin) ≥,∀y∈C. Furthermore, replacingnbynjin the last inequality and using
condition (C), we obtain
y–θnk
jy
i nj·
g(θk njy
i
nj) –g(θ
k–
nj y
i nj) rk,nj
≥
rk,nj
y–θnk
jy
i nj,g
θnk
jy
i nj
–gθnk–
j y
i nj
≥–Gk
θnk
jy
i nj,y
≥Gk
y,θnk
jy
i nj
, ∀y∈C.
By taking the limit asj→+∞in the above inequality, for eachk= , , . . . ,m, we have from the condition (C), (.), andθk
njy
i
nj→pthatGk(y,p)≤,∀y∈C.
For <t≤ andy∈C, defineyt=ty+ ( –t)p. It follows fromy,p∈Cthatyt∈C, which
yieldsGk(yt,p)≤. It follows from the conditions (C) and (C) that
=Gk(yt,yt)≤tGk(yt,y) + ( –t)Gk(yt,p)≤tGk(yt,y),
that is,
Gk(yt,y)≥.
Lettingt→+, from the condition (C), we obtainG
k(p,y)≥,∀y∈C. This implies that p∈EP(Gk),k= , , . . . ,m,i.e.,p∈
m
k=GMEP(fk,ϕk) =
m
k=EP(Gk). Thus we havep∈F.
Step . Finally, we prove thatp=ProjgF(x).
From Lemma . and the definition ofxn+=ProjgCn+(x), we see thatxn+–z,g(x) –
g(xn+) ≥,∀z∈Cn+. SinceF⊂Cnfor eachn≥, we have
xn+–w,g(x) –g(xn+)
≥, ∀w∈F.
Letn→+∞in the last inequality, we see thatp–w,g(x) –g(p) ≥,∀w∈F. In view
of Lemma ., we can obtainp=ProjgF(x). This completes the proof of Theorem ..
Remark . () If we suppose thatTiis uniformlyLi-Lipschitz continuous onEfor each i∈N+, then the assumption thatT
iis closed and uniformly asymptotically regular onE
