R E S E A R C H
Open Access
New hybrid shrinking projection
algorithm for common fixed points of
a family of countable quasi-Bregman strictly
pseudocontractive mappings with
equilibrium and variational inequality and
optimization problems
Yongchun Xu
1and Yongfu Su
2**Correspondence:
2Department of Mathematics,
Tianjin Polytechnic University, Tianjin, 300387, China Full list of author information is available at the end of the article
Abstract
The purpose of this paper is to introduce and consider a new hybrid shrinking projection algorithm for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of a system of variational inequality problems, the set of solutions of a system of optimization problems, the common fixed point set of a uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings in reflexive Banach spaces. Strong convergence theorems have been proved under the appropriate conditions. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings are given in classical Banach spacesl2andL2; (3) the hybrid shrinking projection method presented in this paper modified some mistakes in the recent result of Ugwunnadi et al.(Fixed Point Theory Appl. 2014:231, 2014). These new results improve and extend the previously known ones in the literature.
MSC: 47H05; 47H09; 47H10
Keywords: Bregman distance; quasi-Bregman strictly pseudocontractive mapping; generalized projection; hybrid algorithm; equilibrium problem; variational inequality problem; optimization problem; fixed point
1 Introduction
LetCbe a nonempty subset of a real Banach space andTbe a mapping fromCinto itself. We denote byF(T) the set of fixed points ofT. Recall thatTis said to be asymptotically nonexpansive [] if there exists a sequence{kn} ⊂[, +∞) withlimn→∞kn= such that
Tnx–Tny≤knx–y, ∀x,y∈C,n≥.
It is well known thatTis said to be nonexpansive if
Tx–Ty ≤ x–y, ∀x,y∈C.
In the framework of Hilbert spaces, Takahashiet al.[] have introduced a new hybrid iterative scheme called a shrinking projection method for nonexpansive mappings. It is an advantage of projection methods that the strong convergence of iterative sequences is guaranteed without any compact assumption. Moreover, Schu [] has introduced a modi-fied Mann iteration to approximate fixed points of asymptotically nonexpansive mappings in uniformly convex Banach spaces. Motivated by [, ], Inchan [] has introduced a new hybrid iterative scheme by using the shrinking projection method with the modified Mann iteration for asymptotically nonexpansive mappings. The mappingTis said to be asymp-totically nonexpansive in the intermediate sense (cf.[]) if
lim sup
n→∞
sup
x,y∈C
Tnx–Tny–x–y≤. (.)
IfF(T) is nonempty and (.) holds for allx∈Candy∈F(T), thenTis said to be asymptot-ically quasi-nonexpansive in the intermediate sense. It is worth mentioning that the class of asymptotically nonexpansive mappings in the intermediate sense contains properly the class of asymptotically nonexpansive mappings since the mappings in the intermediate sense are not Lipschitz continuous in general.
Recently, many authors have studied further new hybrid iterative schemes in the frame-work of real Banach spaces; for instance, see [–]. Qin and Wang [] have introduced a new class of mappings which are asymptotically quasi-nonexpansive with respect to the Lyapunov functional (cf.[]) in the intermediate sense. By using the shrinking projection method, Hao [] has proved a strong convergence theorem for an asymptotically quasi-nonexpansive mapping with respect to the Lyapunov functional in the intermediate sense. In , Bregman [] discovered an elegant and effective technique for using of the so-called Bregman distance function (see Section ) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique is applied in various ways in order to design and analyze not only iterative algorithms for solving feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria, and for computing fixed points of nonlinear mappings.
Many authors have studied iterative methods for approximating fixed points of map-pings of nonexpansive type with respect to the Bregman distance; see [–]. In [], the author introduced a new class of nonlinear mappings which is an extension of asymp-totically quasi-nonexpansive mappings with respect to the Bregman distance in the in-termediate sense and proved the strong convergence theorems for asymptotically quasi-nonexpansive mappings with respect to Bregman distances in the intermediate sense by using the shrinking projection method.
mappings. Shahzad and Zegeye [] proved a strong convergence theorem for multival-ued Bregman relatively nonexpansive mappings, while Zegeye and Shahzad [] proved a strong convergence theorem for a finite family of Bregman weak relatively nonexpansive mappings.
Motivated and inspired by the above works, in Ugwunnadiet al.[] proved a new strong convergence theorem for a finite family of closed quasi-Bregman strictly pseudo-contractive mappings and a system of equilibrium problems in a real reflexive Banach space.
The purpose of this paper is to introduce and consider a new hybrid shrinking projection algorithm for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of a system of variational inequality problems, the set of so-lutions of a system of optimization problems, the common fixed point set of a uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings in reflex-ive Banach spaces. Strong convergence theorems have been proved under the appropriate conditions. The main innovative points in this paper are as follows: () the notion of uni-formly closed family of countable quasi-Bregman strictly pseudocontractive mappings is presented and the useful conclusions are given; () the relative examples of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings are given in classical Banach spaceslandL; () the hybrid shrinking projection method presented in this paper modified some mistakes in the recent result of Ugwunnadiet al.[]. These new results improve and extend the previously known ones in the literature.
2 Preliminaries
Throughout this paper, we assume thatEis a real reflexive Banach space with the dual space ofE∗and·,·is the pairing betweenEandE∗.
Letf :E→(–∞, +∞] be a function. The effective domain off is defined by
domf :=x∈E:f(x) < +∞.
Whendomf =∅, we say thatfis proper. We denote byint domf the interior of the effective domain off. We denote byranf the range off.
The functionf is said to be strongly coercive if
lim
x→∞ f(x)
x = +∞.
Given a proper and convex functionf :E→(–∞, +∞], the subdifferential off is a map-ping∂f :E→E∗defined by
∂f(x) =x∗∈E∗:f(y)≥f(x) +x∗,y–x,∀y∈E
for allx∈E.
The Fenchel conjugate function off is the convex functionf∗:E→(–∞, +∞) defined by
We know thatx∗∈∂f(x) if and only if
f(x) +f∗x∗=x∗,x
for allx∈E(see []).
Proposition . ([]) Let f :E → (–∞, +∞] be a proper, convex, and lower semi-continuous function.Then the following conditions are equivalent:
(i) ran∂f =E∗and∂f∗= (∂f)–is bounded on bounded subsets ofE∗;
(ii) f is strongly coercive.
Letf :E→(–∞, +∞] be a convex function andx∈int domf. For anyy∈E, we define the right-hand derivative off atxin the directionyby
f◦(x,y) =lim
t↓
f(x+ty) –f(x)
t . (.)
The functionf is said to be Gâteaux differentiable atxif the limit (.) exists for anyy. In this case, the gradient off atxis the function∇f(x) :E→E∗defined by∇f(x),y=f◦(x,y) for ally∈E. The functionfis said to be Gâteaux differentiable if it is Gâteaux differentiable at eachx∈int domf. If the limit (.) is attained uniformly iny= , then the function
f is said to be Fréchet differentiable atx. The functionf is said to be uniformly Fréchet differentiable on a subsetCofEif the limit (.) is attained uniformly forx∈Candy= . We know that iff is uniformly Fréchet differentiable on bounded subsets ofE, thenf is uniformly continuous on bounded subsets ofE(cf.[, ]). We will need the following results.
Proposition .([]) If a function f:E→R is convex,uniformly Fréchet differentiable,
and bounded on bounded subsets of E,then∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E∗.
Proposition .([]) Let f :E→R be a convex function which is bounded on bounded subsets of E.Then the following assertions are equivalent:
(i) f is strongly coercive and uniformly convex on bounded subsets ofE;
(ii) f∗is Fréchet differentiable and∇f∗is uniformly norm-to-norm continuous on bounded subsets ofdomf∗=E∗.
A functionf :E→(–∞, +∞] is said to be admissible if it is proper, convex, and lower semi-continuous on E and Gâteaux differentiable on int domf. Under these conditions we know thatf is continuous inint domf,∂f is single-valued and∂f =∇f; see [, ]. An admissible functionf :E→(–∞, +∞] is called Legendre (cf.[]) if it satisfies the following two conditions:
(L) the interior of the domain off,int domf, is nonempty,f is Gâteaux differentiable, anddom∇f =int domf;
Letf be a Legendre function onE. SinceEis reflexive, we always have∇f = (∇f∗)–. This fact, when combined with conditions (L) and (L), implies the following equalities:
ran∇f =domf∗=int domf∗ and ran∇f∗=domf =int domf.
Conditions (L) and (L) imply that the functionsf andf∗are strictly convex on the inte-rior of their respective domains. In [], authors gave an example of the Legendre function. Letf :E→(–∞, +∞] be a convex function onE which is Gâteaux differentiable on int domf. The bifunctionDf :domf ×int domf →[, +∞) given by
Df(x,y) =f(x) –f(y) –x–y,∇f(y)
is called the Bregman distance with respect tof (cf.[]). In general, the Bregman dis-tance is not a metric since it is not symmetric and does not satisfy the triangle inequality. However, it has the following important property, which is called the three point identity (cf.[]): for anyx∈domf andy,z∈int domf,
Df(x,y) +Df(y,z) –Df(x,z) =x–y,∇f(z) –∇f(y). (.) With a Legendre functionf :E→(–∞, +∞], we associate the bifunctionWf :domf∗×
domf →[, +∞) defined by
Wf(w,x) =f(x) –w,x+f∗(w).
Proposition . ([]) Let f :E→(–∞, +∞]be a Legendre function such that∇f∗ is bounded on bounded subsets ofint domf∗.Let x∈int domf.If the sequence{Df(x,xn)}is bounded,then the sequence{xn}is also bounded.
Proposition .([]) Let f :E→(–∞, +∞]be a Legendre function.Then the following statements hold:
(i) the functionWf(·,x)is convex for allx∈domf;
(ii) Wf(∇f(x),y) =Df(y,x)for allx∈int domf andy∈domf.
Letf :E→(–∞, +∞] be a convex function onE which is Gâteaux differentiable on int domf. The functionf is said to be totally convex at a pointx∈int domf if its modulus of total convexity atx,vf(x,·) : [, +∞)→[, +∞], defined by
vf(x,t) =infDf(y,x) :y∈domf,y–x=t,
is positive whenevert> . The functionf is said to be totally convex when it is totally convex at every point ofint domf. The functionf is said to be totally convex on bounded sets if, for any nonempty bounded setB⊂E, the modulus of total convexity off onB,
vf(B,t) is positive for anyt> , wherevf(B,·) : [, +∞)→[, +∞] is defined by
vf(B,t) =infvf(x,t) :x∈B∩int domf.
Proposition .([]) Let f :E→(–∞, +∞]be a convex function whose domain contains at least two points.If f is lower semi-continuous,then f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets.
Proposition .([]) Let f:E→R be a totally convex function.If x∈E and the sequence
{Df(xn,x)}is bounded,then the sequence{xn}is also bounded.
Let f :E →[, +∞) be a convex function on E which is Gâteaux differentiable on int domf. The functionf is said to be sequentially consistent (cf. []) if for any two sequences {xn} and{yn} in int domf and domf, respectively, such that the first one is bounded,
lim
n→∞Df(yn,xn) = ⇒ nlim→∞yn–xn= .
Proposition .([]) A function f:E→[, +∞)is totally convex on bounded subsets of E if and only if it is sequentially consistent.
LetCbe a nonempty, closed, and convex subset ofE. Letf :E→(–∞, +∞] be a con-vex function onEwhich is Gâteaux differentiable onint domf. The Bregman projection projfC(x) with respect tof (cf.[]) ofx∈int domf ontoCis the minimizer overCof the functionalDf(·,x) :→[, +∞], that is,
projfC(x) =argminDf(y,x) :y∈C
.
LetEbe a Banach space with dualE∗. We denote byJthe normalized duality mapping fromEto E∗defined by
Jx=f ∈E∗:x,f=x=f,
where·,·denotes the generalized duality pairing. It is well known that ifEis smooth, thenJis single-valued.
Proposition .([]) Let f:E→R be an admissible,strongly coercive,and strictly con-vex function.Let C be a nonempty,closed,and convex subset ofdomf.ThenprojfC(x)exists uniquely for all x∈int domf.
Letf(x) = x.
(i) IfEis a Hilbert space, then the Bregman projection is reduced to the metric projection ontoC.
(ii) IfEis a smooth Banach space, then the Bregman projection is reduced to the generalized projectionC(x)which is defined by
C(x) =argmin
φ(y,x) :y∈C,
whereφis the Lyapunov functional (cf.[]) defined by
Proposition .([]) Let f :E→(–∞, +∞]be a totally convex function.Let C be a nonempty,closed,and convex subset ofint domf and x∈int domf.If x∗∈C,then the fol-lowing statements are equivalent:
(i) The vectorx∗is the Bregman projection ofxontoC.
(ii) The vectorx∗is the unique solutionzof the variational inequality
z–y,∇f(x) –∇f(z)≥, ∀y∈C.
(iii) The vectorx∗is the unique solutionzof the inequality
Df(y,z) +Df(z,x)≤Df(y,x), ∀y∈C.
In recent years, the following notions have been presented by some authors.
A pointp∈Cis said to be asymptotic fixed point of a mapTif there exists a sequence
{xn}inCwhich converges weakly topsuch thatlimn→∞xn–Txn= . We denote byF(T)
the set of asymptotic fixed points ofT. A pointp∈Cis said to be strong asymptotic fixed point [] of a mappingT if there exists a sequence{xn}inCwhich converges strongly
to psuch thatlimn→∞xn–Txn= . We denote byF(T) the set of strong asymptotic
fixed points ofT. Letf :E→R, a mappingT:C→Cis said to be Bregman relatively nonexpansive [] ifF(T) =F(T) andDf(p,T(x))≤Df(p,x) for allx∈Candp∈F(T). The mappingT:C→Cis said to be Bregman weak relatively nonexpansive ifF(T) =F(T) andDf(p,T(x))≤Df(p,x) for allx∈C andp∈F(T). The mappingT:C→C is said to be quasi-Bregman relatively nonexpansive [] ifF(T)=∅andDf(p,T(x))≤Df(p,x)
for allx∈C andp∈F(T). In [] quasi-Bregman relatively nonexpansive is called left quasi-Bregman relatively nonexpansive. A mappingT:C→Cis said to be right quasi-Bregman relatively nonexpansive [] ifF(T)=∅andDf(T(x),p)≤Df(x,p) for allx∈C
andp∈F(T).
In [], authors presented the definition of quasi-Bregman strictly pseudocontractive mapping. In this paper, we extend this definition to the quasi-Bregman pseudocontractive mapping as follows.
Definition . Let C be a nonempty, closed, and convex subset of E and f : E →
(–∞, +∞] be an admissible function. Let T be a mapping from C into itself with a nonempty fixed point setF(T). The mappingT is said to be quasi-Bregmank -pseudo-contractive if there exists a constantk∈[, +∞) such that
Df(p,Tx)≤Df(p,x) +kDf(x,Tx), ∀p∈F(T),∀x∈C.
Ifk∈[, ), the mappingTis said to be quasi-Bregman strictly pseudocontractive. Ifk= , the mappingTis said to be quasi-Bregman pseudocontractive. The mappingTis said to be Bregman quasi-nonexpansive if
Df(p,Tx)≤Df(p,x), ∀p∈F(T),∀x∈C.
Definition . ([]) LetC be a nonempty, closed, and convex subset ofE. Let{Tn}
be a sequence of mappings fromCinto itself with a nonempty common fixed point set
F=∞n=F(Tn).{Tn}is said to be uniformly closed if for any convergent sequence{zn} ⊂C
such thatTnzn–zn → asn→ ∞, the limit of{zn}belongs toF.
The next lemmas have been proved in [], which is useful for the results of [], but in this paper we do not use Lemma . and Lemma ..
Lemma .([]) Let f :E→R be a Legendre function which is uniformly Fréchet dif-ferentiable and bounded on subsets of E,let C be a nonempty,closed,and convex subset of E,and let T:C→C be a quasi-Bregman strictly pseudocontractive mapping with respect to f.Then,for any x∈C,p∈F(T)and k∈[, ),the following holds:
Df(x,Tx)≤ –k
∇f(x) –∇f(Tx),x–p.
Lemma .([]) Let f :E→R be a Legendre function which is uniformly Fréchet differ-entiable on bounded subsets of E,let C be a nonempty,closed,and convex subset of E,and let T:C→C be a quasi-Bregman strictly pseudocontractive mapping with respect to f.
Then F(T)is closed and convex.
Lemma .([]) Let E be a real reflexive Banach space,f :E→(–∞, +∞]be a proper lower semi-continuous function,then f∗:E∗→(–∞, +∞]is a proper weak∗lower semi-continuous and convex function.Thus,for all z∈E,we have
Df
z,∇∗f N
i=
ti∇f(xi)
≤
N
i=
tiDf(z,xi).
LetEbe a real Banach space with the dualE∗andCbe a nonempty closed convex subset ofE. LetA:C→E∗ be a nonlinear mapping andF:C×C→Rbe a bifunction. Then consider the following generalized equilibrium problem of findingu∈Csuch that
ϕ(y) –ϕ(u) +F(u,y) +Au,y–u ≥, ∀y∈C. (.)
The set of solutions of (.) is denoted byEP,i.e.,
EP=u∈C:ϕ(y) –ϕ(u) +F(u,y) +Au,y–u ≥,∀y∈C.
WheneverA≡,ϕ(x)≡, problem (.) is equivalent to findingu∈Csuch that
F(u,y)≥, ∀y∈C, (.)
which is called the equilibrium problem. The set of its solutions is denoted byEP(F). WheneverF≡,ϕ(x)≡, problem (.) is equivalent to findingu∈Csuch that
which is called the variational inequality of Browder type. The set of its solutions is de-noted byVI(C,A).
WheneverF≡,A≡, problem (.) is equivalent to findingu∈Csuch that
ϕ(y)≥ϕ(u), ∀y∈C,
which is called the convex optimization problem. The set of its solutions is denoted by MIN(ϕ).
Problem (.) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see,e.g., [, ].
In order to solve the equilibrium problem for finding an elementx∈Csuch that
F(x,y)≥, ∀y∈C,
let us assume thatF:C×C→(–∞, +∞) satisfies the following conditions []:
(A) F(x,x) = for allx∈C,
(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤, for allx,y∈C, (A) for allx,y,z∈C,lim supt↓F(tz+ ( –t)x,y)≤F(x,y), (A) for allx∈C,F(x,·)is convex and lower semi-continuous.
Forr> , we define a mappingKr:E→Cas follows:
Tr(x) =
z∈C:F(z,y) +
r
y–z,∇f(z) –∇f(x)≥,∀y∈C
(.)
for allx∈E. The following two lemmas were proved in [].
Lemma . Let E be a reflexive Banach space and let f:E→R be a Legendre function.
Let C be a nonempty,closed,and convex subset of E and let F:C×C→R be a bifunc-tion satisfying(A)-(A).For r> ,let Tr:E→C be the mapping defined by(.).Then
domTr=E.
Lemma . Let E be a reflexive Banach space and let f :E→R be a convex,continuous,
and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E.Let C be a nonempty,closed,and convex subset of E and let F:
C×C→R be a bifunction satisfying(A)-(A).For r> ,let Tr:E→C be the mapping defined by(.).Then the following statements hold:
(i) Tris single-valued.
(ii) Tris a firmly nonexpansive-type mapping,i.e.,for allx,y∈E,
Trx–Try,∇f(Trx) –∇f(Try)≤Trx–Try,∇f(x) –∇f(y).
(iii) F(Tr) =F(Tr) =EP(F). (iv) EP(F)is closed and convex.
Lemma . Let E be a reflexive Banach space and let f :E→R be a convex,continuous,
and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E.Let C be a nonempty,closed,and convex subset of E and let F:
C×C→R be a bifunction satisfying(A)-(A).Let A:C→E∗be a monotone mapping,
i.e.,
Ax–Ay,x–y ≥, ∀x,y∈C.
Letϕ(x) :C→R be a convex lower semi-continuous functional.For r> ,let Kr:E→C be the mapping defined by
Kr(x) =
z∈C:G(z,y) +
r
y–z,∇f(z) –∇f(x)≥,∀y∈C
,
where
G(x,y) =ϕ(y) –ϕ(x) +F(x,y) +Ax,y–x.
Then the following statements hold:
(i) Kris single-valued.
(ii) Kris a firmly nonexpansive-type mapping,i.e.,for allx,y∈E,
Krx–Kry,∇f(Krx) –∇f(Kry)≤Krx–Kry,∇f(x) –∇f(y).
(iii) F(Kr) =F(Kr) =EP. (iv) EPis closed and convex.
(v) Df(p,Krx) +Df(Krx,x)≤Df(p,x),∀p∈EP(F),∀x∈E. Proof Let
G(x,y) =ϕ(y) –ϕ(x) +F(x,y) +Ax,y–x, ∀x,y∈C.
It is easy to show thatG(x,y) satisfies conditions (A)-(A). ReplacingF(x,y) byG(x,y) in
Lemma ., we can get the conclusions.
From [] we have the following conclusion.
Theorem . Let E be a p-uniformly convex Banach space with p≥.Then for all x,y∈ E,j(x)∈Jp(x),j(y)∈Jp(y),
j(x) –j(y),x–y≥ c p
cp–px–y
p,
where Jpis the generalized duality mapping from E into E∗and/c is the p-uniformly con-vexity constant of E.
3 Main results
We now prove the following theorem.
Theorem . Let C be a nonempty,closed,and convex subset of a real reflexive Banach space E and f :E→R be a strongly coercive Legendre function which is bounded,uniformly Fréchet differentiable, and totally convex on a bounded subset of E. Let{Fj}mj= be finite bifunctions from C×C to R satisfying(A)-(A)and let{Aj}mj=:C→E∗be finite monotone mappings,i.e.,
Ajx–Ajy,x–y ≥, ∀x,y∈C.
Let{ϕj(x)}mj=:C→R be finite convex lower semi-continuous functionals.Let{Tn}∞n= be
a uniformly closed family of countable quasi-Bregman strictly pseudocontractive map-pings from C into itself with uniformly k∈[, )such that F=mj=EPj∩(∞n=F(Tn))is nonempty.For given x∈C,let{Tn}∞n=be a sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x=x∈C=C,
yn=∇f∗(αn∇f(xn) + ( –αn)∇f(Tnxn)),
Gj(uj,n,y) +rn∇f(uj,n) –∇f(yn),y–un,j, ≥, ∀y∈C,j= , , , . . . ,m, Cn+={z∈Cn:Df(z,uj,n)≤Df(z,yn)≤Df(z,xn)
+–kk∇f(xn) –∇f(Tnxn),xn–z,j= , , , . . . ,m},
xn+=PfCn+x,
where
Gj(x,y) =ϕj(y) –ϕj(x) +Fj(x,y) +Ajx,y–x, EPj=u∈C:Gj(x,y)≥,∀y∈C
for j= , , , . . . ,m,and{αn},{βj,n}are sequences satisfyinglim supn→∞αn< ,{rn}is a sequence satisfyinglim infn→∞rn> .Then{xn}converges to q=PfFx.
Proof We divide the proof into six steps.
Step . We show thatCnis closed and convex for alln≥. Let
Dn=
z∈E:Df(z,yn)≤Df(z,xn) + k –k
∇f(xn) –∇f(Tnxn),xn–z,
Ej,n=
z∈E:Df(z,uj,n)≤Df(z,yn)
, j= , , , . . . ,m,
then
Cn+=C∩Cn∩Dn∩
m
j=
Ej,n
.
SinceC=Cis closed and convex, it is sufficient to prove that the setsDn,Ej,nare closed
Dnas follows:
Dn=
z∈E:Df(z,yn)≤Df(z,xn) + k
–k
∇f(xn) –∇f(Tnxn),xn–z
=
z∈E:Df(z,yn) –Df(z,xn)≤ k –k
∇f(xn) –∇f(Tnxn),xn–z
=
z∈E:f(xn) –f(yn) +z–xn,∇f(xn)–z–yn,∇f(yn)
≤ k
–k
∇f(xn) –∇f(Tnxn),xn–z
=
z∈E:z–xn,∇f(xn)–z–yn,∇f(yn)≤f(yn) –f(xn)
+ k
–k
∇f(xn) –∇f(Tnxn),xn–z
=
z∈E:
z,
–k∇f(xn) –∇f(yn) – k
–k∇f(Tnxn)
≤f(yn) –f(xn)
+
xn,
–k∇f(xn)
–xn,∇f(yn)–
xn, k
–k∇f(Tnxn)
.
From the above expression, we know thatDnis closed and convex for alln≥.
Next we show thatEj,nis closed and convex for alln≥,j= , , , . . . ,m. We rewriteEj,n
as follows:
Ej,n=
z∈E:Df(z,uj,n)≤Df(z,yn)
=z∈E:f(yn) –f(uj,n)≤
∇f(uj,n,z–uj,n)
–∇f(yn),z–yn
=z∈E:f(yn) –f(uj,n) +
∇f(uj,n),uj,n
–∇f(yn),yn≤∇f(uj,n) –∇f(yn),z
.
From the above expression, we know that Ej,n is closed and convex for alln≥, j=
, , , . . . ,m. ThereforeCnis closed and convex for alln≥.
Step . We show thatF⊂Cnfor alln≥. Note thatF⊂C=C. SupposeF⊂Cnfor
n≥, then for allp∈F ⊂Cn, sinceuj,n=Kr(j)(yn) for alln≥, j= , , , . . . ,m, from
Lemma ., we have
Df(p,uj,n) =Df
p,Kr(j)(yn)
≤Df(p,yn), j= , , , . . . ,m, (.)
where
Kr(j)(x) =
z∈C:Gj(z,y) +
r
y–z,∇f(z) –∇f(x)≥,∀y∈C
.
Since
Df(p,yn) =Dfp,∇f∗αn∇f(xn) + ( –αn)∇f(Tnxn)
=αnDf(p,xn) + ( –αn)Df(p,Tnxn)
≤αnDf(p,xn) + ( –αn)
≤Df(p,xn) +λDf(xn,Tnxn)
≤Df(p,xn) + k –k
∇f(xn) –∇f(Tnxn),xn–p, (.)
from (.) and (.) we know thatp∈Cn+, which impliesF⊂Cnfor alln≥. Step . We show that{xn}converges to a pointp∈C.
Sincexn=PfCnxandCn+⊂Cn, then we get
Df(xn,x)≤Df(xn+,x) for alln≥. (.)
Therefore{Df(xn,x)}is nondecreasing. On the other hand, by Proposition ., we have
Df(xn,x) =Df
PfCnx,x
≤Df(p,x) –Df(p,xn)≤Df(p,x)
for allp∈F⊂Cnand for alln≥. Therefore,Df(xn,x) is also bounded. This together with (.) implies that the limit of{Df(xn,x)}exists. Put
lim
n→∞Df(xn,x) =d. (.)
From Proposition ., we have, for any positive integerm, that
Df(xn+m,xn) =Df
xn+m,PfCnx
≤Df(xn+m,x) –Df
PfCnx,x
=Df(xn+m,x) –Df(xn,x)
for alln≥. This together with (.) implies that
lim
n→∞Df(xn+m,xn) =
holds uniformly for allm. Therefore, we get that
lim
n→∞xn+m–xn=
holds uniformly for allm. Then{xn}is a Cauchy sequence, therefore there exists a point p∈Csuch thatxn→p.
Step . We show that the limit of{xn}belongs to ∞
n=F(Tn).
Sincexn+∈Cn+, we have from the definition ofCn+that
Df(xn+,yn)≤Df(xn+,xn) + k
–k
∇f(xn) –∇f(Tnxn),xn–xn+
,
which implies thatlimn→∞Df(xn+,yn) = . Sincef is totally convex on bounded subsets
ofE,f is sequentially consistent (see []). It follows that
lim
From the uniform continuity of∇f, we have lim
n→∞∇f(xn) –∇f(yn)= .
Since
yn=∇f∗αn∇f(xn) + ( –αn)∇f(Tnxn)
,
we obtain that
lim
n→∞∇f(Tnxn) –∇f(xn)=nlim→∞
–αn
∇f(xn) –∇f(yn)= . (.)
Sincef is strongly coercive and uniformly convex on bounded subsets ofE,f∗is uniformly Fréchet differentiable on bounded sets. Moreover,f∗ is bounded on bounded sets, and from (.) we obtain
lim
n→∞Tnxn–xn= .
Since{Tn}is uniformly closed andxn→p, we havep∈∞n=F(Tn).
Step . We show that the limit of{xn}belongs toEPjfor allj= , , , . . . ,m.
We have proved that xn→pasn→ ∞. Now let us show thatp∈EPj for any j= , , , . . . ,m. Sincexn+∈Cn+, we have from the definition ofCn+that
Df(xn+,uj,n)≤Df(xn+,yn), j= , , , . . . ,m. Sincelimn→∞Df(xn+,yn) = , we have
lim
n→∞Df(xn+,uj,n) = , j= , , , . . . ,m.
Sincef is totally convex on bounded subsets ofE,f is sequentially consistent (see []). It follows that
lim
n→∞xn–uj,n= , j= , , , . . . ,m.
This together with (.) implies that
lim
n→∞yn–uj,n= , j= , , , . . . ,m.
Since∇f is uniformly norm-to-norm continuous on bounded subsets ofE, from (.) we havelimn→∞∇f(uj,n) –∇f(yn)= . Fromlim infn→∞rn> it follows that
lim
n→∞
∇f(uj,n) –∇f(yn)
rn = .
By the definition ofuj,n:=Kr(nj)yn, we have
G(uj,n,y) +
rn
y–uj,n,∇f(uj,n) –∇f(yn)
where
G(uj,n,y) =ϕ(y) –ϕ(uj,n) +F(uj,n,y) +Auj,n,y–uj,n.
We have from (A) that
rn
y–uj,n,∇f(uj,n) –∇f(yn)
≥–G(uj,n,y)≥G(y,uj,n), ∀y∈C.
Sincey→f(x,y) +Ax,y–xis convex and lower semi-continuous, lettingn→ ∞in the last inequality, from (A) we have
Gj(y,p)≤, ∀y∈C.
Fort, with <t< , andy∈C, letyt=ty+ ( –t)p. Sincey∈Candp∈C, thenyt∈Cand henceGj(yt,p)≤. So, from (A) we have
=Gj(yt,yt)≤tGj(yt,y) + ( –t)Gj(yt,p)≤tGj(yt,y).
Dividing byt, we have
Gj(yt,y)≥, ∀y∈C.
Lettingt→, from (A) we can get
Gj(p,y)≥, ∀y∈C,j= , , , . . . ,m.
So,p∈EPjfor allj= , , , . . . ,m.
Step . Finally, we prove thatp=PfFx, from Proposition . we have
Df
p,PfFx
+Df
PFfx,x
≤Df(p,x). (.)
On the other hand, sincexn=PfCnxandF⊂Cnfor alln, also from Proposition ., we have
DfPFfx,xn+
+Df(xn+,x)≤Df
PfFx,x
. (.)
By the definition ofDf(x,y), we know that
lim
n→∞Df(xn+,x) =Df(p,x). (.)
Combining (.), (.), and (.), we know thatDf(p,x) =Df(PfFx,x). Therefore, it fol-lows from the uniqueness ofPfFxthatp=PfFx. This completes the proof.
Remark . Theorem . includes the following three special cases.
in-equalities ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Au,y–u ≥,
Au,y–u ≥,
Au,y–u ≥, . . . ,
Amu,y–u ≥,
∀y∈C.
In this case, the iterative sequence{xn}is defined by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x=x∈C=C,
Ajuj,n,y–uj,n+rn∇f(uj,n) –∇f(xn),y–un,j, ≥, ∀y∈C,j= , , , . . . ,m, Cn+={z∈Cn:Df(z,uj,n)≤Df(z,yn)≤Df(z,xn)
+ k
–k∇f(xn) –∇f(Tnxn),xn–z,j= , , , . . . ,m}, xn+=PfCn+x.
() TakeTn≡I,ϕ(x)≡,A≡, whereIdenotes the identity operator, then the iterative sequence{xn}converges strongly to a solution of the system of equilibrium problems
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
F(u,y)≥,
F(u,y)≥,
F(u,y)≥, . . . ,
Fm(u,y)≥,
∀y∈C.
In this case, the iterative sequence{xn}is defined by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x=x∈C=C,
F(uj,n,y) +rn∇f(uj,n) –∇f(xn),y–un,j, ≥, ∀y∈C,j= , , , . . . ,m, Cn+={z∈Cn:Df(z,uj,n)≤Df(z,yn)≤Df(z,xn)
+–kk∇f(xn) –∇f(Tnxn),xn–z,j= , , , . . . ,m},
xn+=PfCn+x.
() TakeTn≡I,F(x,y)≡,A≡, whereIdenotes the identity operator, then the iter-ative sequence{xn}converges strongly to a solution of the system of convex optimization
problems ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ϕ(u) =miny∈Cϕ(y), ϕ(u) =miny∈Cϕ(y),
ϕ(u) =miny∈Cϕ(y), . . . ,
In this case, the iterative sequence{xn}is defined by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x=x∈C=C,
ϕ(y) –ϕ(uj,n) +rn∇f(uj,n) –∇f(xn),y–un,j, ≥, ∀y∈C,j= , , , . . . ,m, Cn+={z∈Cn:Df(z,uj,n)≤Df(z,yn)≤Df(z,xn)
+ k
–k∇f(xn) –∇f(Tnxn),xn–z,j= , , , . . . ,m}, xn+=PfCn+x.
4 Examples
LetEbe a Hilbert space andCbe a nonempty closed convex and balanced subset ofE. Let{xn}be a sequence inCsuch thatxn=r> ,{xn}converges weakly tox= , and
xn–xm ≥r> for alln= m. Define a countable family of mappings{Tn}:C→Cas
follows:
Tn(x) = ⎧ ⎨ ⎩
n+
n xn ifx=xn(∃n≥),
–x ifx=xn(∀n≥).
Conclusion . {Tn}has a unique common fixed point,that is,F=∞n=F(Tn) ={}for all n≥.
Proof The conclusion is obvious.
Conclusion . {Tn}is a uniformly closed family of countable quasi-Bregman(n+ ) -pseudocontractive mappings.
Proof Takef(x) =x, then
Df(x,y) =φ(x,y) =x–y
for allx,y∈Cand
Df(,Tnx) =Tnx=
⎧ ⎨ ⎩
(n+
n )
x
n ifx=xn,
x ifx=xn.
Therefore,
Df(,Tnxn)≤
n+
n
xn
=n
+ n+
n xn
=xn+n+
n xn
=xn+ (n+ ) xn
n
=xn+ (n+ )xn–Tnxn
=Df(,xn) + (n+ )Df(xn,Tnxn)
for allx∈C. On the other hand, for any strong convergent sequence{zn} ⊂Esuch that
zn→zandzn–Tnzn → asn→ ∞, it is easy to see that there exists a sufficiently large nature numberNsuch thatzn=xm for anyn,m>N. ThenTzn= –znforn>N, it follows fromzn–Tnzn → that zn→ and hencezn→z= . That is,z∈F.
Example . LetE=l, where
l=
ξ= (ξ,ξ,ξ, . . . ,ξn, . . .) : ∞
n=
|xn|<∞
,
ξ=
∞
n=
|ξn|
, ∀ξ∈l,
ξ,η=
∞
n=
ξnηn, ∀ξ= (ξ,ξ,ξ, . . . ,ξn, . . .),η= (η,η,η, . . . ,ηn, . . .)∈l.
Let{xn} ⊂Ebe a sequence defined by
x= (, , , , . . .),
x= (, , , , . . .),
x= (, , , , , . . .),
x= (, , , , , , . . .),
. . . ,
xn= (ξn,,ξn,,ξn,, . . . ,ξn,k, . . .),
. . . ,
where
ξn,k= ⎧ ⎨ ⎩
ifk= ,n+ , ifk= ,k=n+
for alln≥. It is well known thatxn=√,∀n≥ and{xn} converges weakly tox. Define a countable family of mappingsTn:E→Eas follows:
Tn(x) =
⎧ ⎨ ⎩
n+
n xn ifx=xn,
–x ifx=xn
Example . LetE=Lp[, ] ( <p< +∞) and
xn= –
n, n= , , , . . . .
Define a sequence of functions inLp[, ] by the following expression:
fn(x) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
xn+–xn ifxn≤x< xn++xn
, –
xn+–xn if xn++xn
≤x<xn+,
otherwise
for alln≥. Firstly, we can see, for anyx∈[, ], that
x
fn(t)dt→ =
x
f(t)dt, (.)
wheref(x)≡. It is well known that the above relation (.) is equivalent to{fn(x)} con-verges weakly tof(x) in a uniformly smooth Banach spaceLp[, ] ( <p< +∞). On the other hand, for anyn=m, we have
fn–fm=
fn(x) –fm(x)pdx
p
=
xn+
xn
fn(x) –fm(x)pdx+
xm+
xm
fn(x) –fm(x)pdx
p
=
xn+
xn
fn(x) p
dx+
xm+
xm
fm(x) p dx p =
xn+–xn
p
(xn+–xn) +
xm+–xm
p
(xm+–xm)
p
=
p
(xn+–xn)p–
+
p
(xm+–xm)p–
p
≥p+ p p> .
Let
un(x) =fn(x) + , ∀n≥.
It is obvious thatunconverges weakly tou(x)≡ and
un–um=fn–fm ≥
p+ pp> , ∀n≥. (.)
Define a mappingT:E→Eas follows:
Tn(x) =
⎧ ⎨ ⎩
n+
n un ifx=un(∃n≥),
Since (.) holds, by using Conclusions . and ., we know that{Tn}is a uniformly closed family of countable quasi-Bregman (n+ )-pseudocontractive mappings.
5 The mistakes in the result of Ugwunnadi et al. [24]
In [], from page , line – to page , line , there exists a mistake ratiocination as follows.
Mistake ratiocination Since xn+∈Cn+,it follows from(.), (.)that (∗) f(xn+) –f(wn) –
∇f(wn),xn+–wn
=Df(xn+,wn)
≤Df(xn+,yn)
≤Df(xn+,xn) + k
–k
∇f(xn) –∇f(Tnxn),xn–xn+
,
which implies from(.), (.), (.),and(.)that
lim
n→∞Df(xn+,yn) = .
However, (.), (.) are the following:
(.) Df(w,wn) =Df
w,∇f∗ m
j=
βj,n∇f(uj,n)
≤
m
j=
βj,nDf(w,uj,n)
≤
m
j=
βj,nDf(w,yn)
=Df(w,yn)
for anyw∈F,
(.) Df(w,yn) =Dfw,∇f∗αn∇f(xn) + ( –αn)∇f(Tnxn)
≤αnDf(w,xn) + ( –αn)Df(w,Tnxn)
≤αnDf(w,xn) + ( –αn)
Df(w,xn) +kDf(xn,Tnxn)
≤Df(w,xn) +kDf(xn,Tnxn)
≤Df(w,xn) + k –k
∇f(xn) –∇f(Tnxn),xn–w
for anyw∈F.
In fact, the authors attempt taking w=xn+ in (.) and (.) to get the (∗). This is an obvious mistake since (.) and (.) are right for onlyw∈F, butxn+ does not be-long toF. Therefore, the definition of an iterative sequence{xn}must be modified so that limn→∞Df(xn+,xn) = implieslimn→∞Df(xn+,yn) = .
Mistake ratiocination Also,since yn→p as n→ ∞,we have from Lemma.,for each j= , , , . . . ,m,
≤Df(p,uj,n) =Df
p,Resfg jyn
≤Df(p,yn)→ as n→ ∞.
In fact, we are proving thatp∈EPjfor anyj= , , , . . . ,m, therefore, if we do not know whetherp∈EPj, then the above inequalities are not right since ifp∈EPj, the above
in-equalities are right. In this paper, we have overcome these shortcomings by modifying the iterative scheme.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper was proposed by the corresponding author YS, and YS prepared the manuscript initially for basic structure. YX performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Hebei North University, Zhangjiakou, 075000, China.2Department of Mathematics, Tianjin
Polytechnic University, Tianjin, 300387, China.
Acknowledgements
This project is supported by the National Natural Science Foundation of China under grant (11071279).
Received: 9 April 2015 Accepted: 9 June 2015 References
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