R E S E A R C H
Open Access
Hybrid iterative algorithm for finite
families of countable Bregman
quasi-Lipschitz mappings with applications
in Banach spaces
Minjiang Chen
1, Jianzhi Bi
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 method for finding a common element of the setEPof solutions of a generalized equilibrium problem, the common fixed point setFof finite uniformly closed families of countable Bregman quasi-Lipschitz mappings in reflexive Banach spaces. It is proved that under appropriate conditions, the sequence generated by the hybrid shrinking projection method converges strongly to some point inEP∩F. Relative examples are given. Strong convergence theorems are proved. The application for Bregman asymptotically quasi-nonexpansive mappings is also given. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable Bregman quasi-Lipschitz mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable Bregman quasi-Lipschitz mappings are given in classical Banach spaces l2andL2; (3) the application for Bregman asymptotically quasi-nonexpansive
mappings is also given; (4) because the main theorems do not need the boundedness of the domain of mappings, so a corresponding technique for the proof is given. This new results improve and extend the previously known ones in the literature.
MSC: 47H05; 47H09; 47H10
Keywords: Bregman distance; Bregman quasi-Lipschitz mapping; generalized projection; hybrid algorithm; Bregman asymptotically quasi-nonexpansive mappings; finite families; equilibrium problem
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≥.
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 authors has 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 inter-mediate sense and has 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.
classical Banach spaceslandL; () the application for Bregman asymptotically
quasi-nonexpansive mappings is also given; () because the main theorems do not need the boundedness of the domain of mappings, so a corresponding technique for the proof is given. This 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·,·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
f∗x∗=supx∗,x–f(x),x∈E.
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 semicontin-uous 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 . (.)
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 semicontinuous onEand Gâteaux differentiable onint domf. Under these conditions we know thatf is continuous inint domf,∂f is single-valued and∂f =∇f; see [, ]. An ad-missible 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;
(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 [], author 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
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) =D
f(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) =inf
Df(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) =inf
vf(x,t) :x∈B∩int domf
.
We remark in passing thatf is totally convex on bounded sets if and only iff is uniformly convex on bounded sets; see [, ].
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 function f 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
φ(y,x) =y– y,Jx+x
for ally,x∈E.
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 this paper, we present the following definition.
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 set F(T). The mapping T is said to be Bregman quasi-Lipschitz if there exists a constantL≥ such that
The mappingT is said to be Bregman quasi-nonexpansive if
Df(p,Tx)≤Df(p,x), ∀p∈F(T),∀x∈C.
Bregman quasi-Lipschitz mappings are a more generalized class than the class of Bregman mappings. On the other hand, this class also contains the relatively quasi-Lipschitz mappings and quasi-quasi-Lipschitz mappings. Therefore, Bregman quasi-quasi-Lipschitz mappings are very important in the nonlinear analysis and fixed point theory and appli-cations.
Definition . LetCbe a nonempty, closed, and convex subset ofE. Let{Tn} be
se-quence of mappings from C into 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.
In Section , we will give two examples of a uniformly closed family of countable Breg-man quasi-Lipschitz mappings.
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:
F(u,y) +Au,y–u ≥, ∀y∈C. (.)
The set of solutions of (.) is denoted byEP,i.e.,
EP=u∈C:F(u,y) +Au,y–u ≥,∀y∈C.
WheneverE=Ha Hilbert space, problem (.) was introduced and studied by Takahashi and Takahashi [].
WheneverA≡, 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≡, problem (.) is equivalent to findingu∈Csuch that
Au,y–u ≥, ∀y∈C,
which is called the variational inequality of Browder type. The set of its solutions is de-noted byVI(C,A).
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, let us assume thatF:C×C→(–∞, +∞) satisfies the following conditions []:
(A) F(x,x) = for allx∈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) =Fˆ(Tr) =EP(F).
(iv) EP(F)is closed and convex.
(v) Df(p,Trx) +Df(Trx,x)≤Df(p,x),∀p∈EP(F),∀x∈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).Let A:C→E∗be a monotone mapping,
i.e.,
Ax–Ay,x–y ≥, ∀x,y∈C.
For r> ,let Kr:E→C be the mapping defined by
Kr(x) =
z∈C:F(z,y) +Az,y–z+
r
y–z,∇f(z) –∇f(x)≥,∀y∈C
.
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)
.
(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) =F(x,y) +Ax,y–x, ∀x,y∈C.
It is easy to show that,G(x,y) satisfies conditions (A)-(A). ReplacingF(x,y) byG(x,y) in
Lemma ., we can get the conclusions.
3 Main results
Theorem . Let f :E→(–∞, +∞]be a Legendre function which is totally convex on bounded subsets of E. Suppose that ∇f∗ is bounded on bounded subsets of int domf∗.
Let C be a nonempty,closed,and convex subset ofint domf.Let{Tn}:C→C be a
uni-formly closed family of countable Bregman quasi-Lipschitz mappings with the condition
limn→∞Ln= ,where
Df(p,Tnx)≤LnDf(p,x), ∀p∈F,∀x∈C. (.)
Let F be a common fixed point set of{Tn}.Then F is closed and convex.
Proof Firstly, we prove thatFis closed. Let{pn} ⊂F,pn→pasn→ ∞, thenTnpn–pn=
→ asn→ ∞. Since{Tn}is uniformly closed, we know thatp∈F. HenceFis closed.
Next we prove thatF is convex. Letp,p∈F,p=tp+ ( –t)p, wheret∈(, ). We
prove thatp∈F. By the three point identity (.), we know that
Df(x,y) =Df(x,z) +Df(z,y) +
x–z,∇f(z) –∇f(y).
This implies
Df(pi,Tnp) =Df(pi,p) +Df(p,Tnp) +
pi–p,∇f(p) –∇f(Tnp)
(.)
fori= , . Combining (.) and (.) yields
Df(p,Tnp)≤(Ln– )Df(pi,p) –
pi–p,∇f(p) –∇f(Tnp)
(.)
fori= , . Multiplyingtand ( –t) on both sides of (.) withi= andi= , respectively, yields
lim
n→∞Df(p,Tnp)≤nlim→∞
ξn–
tp+ ( –t)p–p,∇f(p) –∇f(Tnp)
= ,
where
ξn= (Ln– )
tDf(p,p) + ( –t)Df(p,p)
.
This implies that {Df(p,Tnp)}is bounded. By Propositions . and ., we see that the
sequence {Tnpn} is bounded andp–Tnp → asn→ ∞. Sincep→p, and{Tn}is
Next we will prove the main strong convergence theorem for the finite families of count-able Bregman quasi-Lipschitz mappings by using a new hybrid projection scheme. In this scheme, we will use some detailed technology.
Theorem . Let f :E→(–∞, +∞]be a Legendre function which is bounded,strongly coercive,uniformly Fréchet differentiable and totally convex on bounded subsets on E.Let C be a nonempty,closed,and convex subset ofint domf.Let{Tn(i)}∞n=:C→C be N uni-formly closed families of countable Bregman quasi-Lipschitz mappings with the condition
limn→∞L(ni)= for i= , , , . . . ,N.Let F= ∞n=
N i=F(T
(i)
n )and F∩EP be nonempty.Let
{xn}be a sequence of C generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈int domf, arbitrarily,
yi,n=∇f∗(αn∇f(xn) + ( –αn)∇f(Tn(i)xn)), i= , , , . . . ,N,
F(ui,n,y) +Aui,n,y–ui,n+rn∇f(ui,n) –∇f(yi,n),y–ui,n ≥, ∀y∈C,
Ci,n+={z∈Cn:Df(z,ui,n)≤Df(z,yi,n)≤Df(z,xn) +ξn}, n≥,
Ci,=C, Cn+= Ni=Ci,n+, xn=PCfnx,
where
ξn= (Ln– ) sup p∈F∩EP∩B(PfF∩EPx,)
Df(p,x),
B(x, ) =y∈E:Df(y,x)≤
,
Ln=max
L()n,L()n ,L()n , . . . ,L(N)
n
and{αn}is a sequence satisfyinglim supn→∞αn< .Then{xn}converges to q=PFf∩EPx. Proof We divide the proof into six steps.
Step . We show thatCnis closed and convex for alln≥. It is obvious thatCi,=Cis
closed and convex. Suppose thatCi,kis closed and convex for somek≥. We see for each
i= , , , . . . ,Nthat
Ci,k+=
z∈C:Df(z,ui,k)≤Df(z,yi,k)≤Df(z,xk) +ξk
∩Ci,k
and
Df(z,ui,k)≤Df(z,yi,k)≤Df(z,xk) +ξk
is equivalent to
⎧ ⎨ ⎩
∇f(xk) –∇f(yi,k),z ≤ f∗(∇f(xk)) –f∗(∇f(yi,k))+ξk,
∇f(yi,k) –∇f(ui,k),z ≤ f∗(∇f(yi,k)) –f∗(∇f(ui,k)).
(.)
Therefore
Ci,k+=
It is easy to see that ifz,z satisfy (.), the elementz=tz+ ( –t)zsatisfies also (.)
for allt∈(, ), so that the set
z∈C:zsatisfies (.)
is convex and closed, and henceCi,k+is closed and convex for alln≥. ThereforeCn+=
N
i=Ci,n+is closed and convex.
Step . We show thatF∩EP∩B(PfF∩EPx, )⊂Cnfor alln≥. It is obvious thatF∩
EP∩B(PfF∩EPx, )⊂Ci,=Cfor all ≤i≤N. Suppose thatF∩EP∩B(PfF∩EPx, )⊂Cn
for somen≥. Letp∈F∩EP∩B(PFf∩EPx, ). By Proposition ., we have Df(p,yi,n) =Df
p,∇f∗αn∇f(xn) + ( –αn)∇f
Tn(i)xn
=Wfp,∇f∗αn∇f(xn) + ( –αn)∇f
Tn(i)xn
≤αnWf
p,∇f(xn)
+ ( –αn)Wf
p,∇fTn(i)xn
=αnDf(p,xn) + ( –αn)Df
p,Tn(i)xn
≤αnDf(p,xn) + ( –αn)Df(p,xn) +ξn
≤Df(p,xn) +ξn. (.)
On the other hand, by Lemma ., we havep=Kr(p) and
Df(p,Kryi,n) +Df(Knyi,n,yi,n)≤Df(p,yi,n),
that is,
Df(p,ui,n) +Df(Knyi,n,yi,n)≤Df(p,yi,n). (.)
Combining (.) and (.) we know thatp∈Ci,n+ for all ≤i≤N, which implies that F∩EP∩B(PFf∩EPx, )⊂Ci,n+. ThereforeF∩EP∩B(PFf∩EPx, )⊂Cn+. By induction we
know thatF∩EP∩B(PFf∩EPx, )⊂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
PCfnx,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 toF.
Sincexn+∈Cn+, we have for all ≤i≤Nthat Df(xn+,ui,n)≤Df(xn+,yi,n)≤Df(xn+,xn) +ξn→
asn→ ∞. By Proposition ., we obtain that
lim
n→∞xn+–yi,n= ,nlim→∞xn+–ui,n= . (.)
From
yi,n=∇f∗
αn∇f(xn) + ( –αn)∇f
Tn(i)xn
,
we get
∇f(yi,n) =αn∇f(xn) + ( –αn)∇f
Tn(i)xn
,
which implies that
∇f(yi,n) –∇f(xn) = ( –αn)
∇fTn(i)xn
–∇f(xn)
.
By Proposition ., we have
lim
n→∞∇f(yi,n) –∇f(xn)= , so that
lim
n→∞∇f
Tn(i)xn
–∇f(xn)= .
By Propositions . and .,∇f∗ is uniformly continuous on bounded subsets ofEand thus
lim
n→∞T
(i)
nxn–xn= .
Since {Tn(i)} is an asymptotically countable family of Bregman weak relatively
nonex-pansive mappings and xn→p, so thatp∈ ∞n=F(T (i)
n) for each ≤i≤N. Therefore
Step . We show that the limit of{xn}belongs toEP.
We have proved that xn→p as n→ ∞. Now let us show that p∈EP. Since ∇f
is uniformly norm-to-norm continuous on bounded subsets of E, from (.) we have limn→∞∇f(ui,n) –∇f(yi,n)= . Fromlim infn→∞rn> it follows that
lim
n→∞
∇f(ui,n) –∇f(yi,n)
rn
= .
By the definition ofun:=Krnyn, we have
G(ui,n,y) +
rn
y–ui,n,∇f(ui,n) –∇f(yi,n)
≥, ∀y∈C,
where
G(ui,n,y) =F(ui,n,y) +Aui,n,y–ui,n.
We have from (A) that
rn
y–ui,n,∇f(ui,n) –∇f(yi,n)
≥–G(ui,n,y)≥G(y,ui,n), ∀y∈C.
Sincey→f(x,y) +Ax,y–xis convex and lower semi-continuous, lettingn→ ∞in the last inequality, from (A) we have
G(y,p)≤, ∀y∈C.
Fort, with <t< , andy∈C, letyt=ty+ ( –t)p. Sincey∈Candp∈C, thenyt∈Cand
henceG(yt,p)≤. So, from (A) we have
=G(yt,yt)≤tG(yt,y) + ( –t)G(yt,p)≤tG(yt,y).
Dividing byt, we have
G(yt,y)≥, ∀y∈C.
Lettingt→, from (A) we can get
G(p,y)≥, ∀y∈C.
So,p∈EP.
Step . Finally, we prove thatp=PfF∩EPx, from Proposition ., we have Df
p,PfF∩EPx
+Df
PFf∩EPx,x
≤Df(p,x). (.)
On the other hand, sincexn=PfCnxandF·EP⊂Cnfor alln, also from Proposition .,
we have
Df
PFf∩EPx,xn+
+Df(xn+,x)≤Df
PfF∩EPx,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(PFf∩EPx,x). Therefore,
it follows from the uniqueness ofPfF∩EPxthatp=PFf∩EPx. This completes the proof. Definition . LetCbe a nonempty, closed, and convex subset ofE. LetTbe a mapping from Cinto itself with a nonempty fixed point set F(T). The mapping T is said to be Lyapunov quasi-Lipschitz if there exists a constantL≥ such that
φ(p,Tx)≤Lφ(p,x), ∀p∈F(T),∀x∈C.
The mappingT is said to be Lyapunov quasi-nonexpansive if
φ(p,Tx)≤φ(p,x), ∀p∈F(T),∀x∈C.
If we choosef(x) =xfor allx∈E, then Theorem . reduces to the following
corol-lary.
Corollary . Let E be a smooth Banach space and C be a closed convex subset of E.Let {Tn(i)}∞n=:C→C be N uniformly closed families of countable Lyapunov quasi-Lipschitz mappings with the conditionlimn→∞L(ni)= for i= , , , . . . ,N.Let F= ∞n=
N i=F(T
(i)
n )
and F∩EP be nonempty.Let{xn}be a sequence of C generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈int domf, arbitrarily,
yi,n=J–(αnJxn+ ( –αn)JTn(i)xn), i= , , , . . . ,N,
F(ui,n,y) +Aui,n,y–ui,n+rnJ(ui,n) –J(yi,n),y–ui,n ≥, ∀y∈C,
Ci,n+={z∈Cn:φ(z,ui,n)≤φ(z,yi,n)≤φ(z,xn) +ξn}, n≥,
Ci,=C, Cn+= Ni=Ci,n+, xn=PCfnx,
where
ξn= (Ln– ) sup p∈F∩EP∩B(PfF∩EPx,)
φ(p,x),
B(x, ) =y∈E:φ(y,x)≤,
Ln=max
L()n,L()n ,L()n , . . . ,L(nN)
and{αn}is a sequence satisfyinglim supn→∞αn< .Then{xn}converges to q=PFf∩EPx. 4 Example
andxn–xm ≥r> for alln=m. Define a countable family of mappings{Tn}:C→C
as 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 Bregman quasi-Lipschitz
mappings with the conditionlimn→∞Ln=limn→∞nn+= .
Proof Takef(x) =x, then
Df(x,y) =φ(x,y) =x– x,Jy+y
for allx,y∈Cand
Df(,Tnx) =Tnx=
⎧ ⎨ ⎩
n+
n xn
ifx=x
n,
x ifx=x
n.
Therefore
Df(,Tnx)≤
n+
n x
=n+ n Df(,x)
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= (, , , , , , . . .),
. . . ,
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
for alln≥. By using Conclusions . and .,{Tn}is a uniformly closed family of
count-able Bregman quasi-Lipschitz mappings with the conditionlimn→∞Ln=limn→∞n+n = .
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) p
dx
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
=
xn+–xn
p
(xn+–xn) +
xm+–xm
p
(xm+–xm)
p
=
p
(xn+–xn)p–
+
p
(xm+–xm)p–
p
≥p+ pp> .
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≥),
–x ifx=un(∀n≥).
Since (.) holds, by using Conclusions . and ., we know that{Tn}is a uniformly closed
family of countable Bregman quasi-Lipschitz mappings with the conditionlimn→∞Ln=
limn→∞n+n = .
5 Application
The mapping T is said to be Bregman asymptotically quasi-nonexpansive (cf. []) if
F(T)=∅and there exists a sequence{kn} ⊂[, +∞) withlimn→∞kn= such that
Df
p,Tnx≤knDf(p,x), ∀p∈F(T),∀x∈C.
Every Bregman quasi-nonexpansive mapping is Bregman asymptotically quasi-nonexpan-sive withkn≡. LetSn=Tnfor alln≥, the above inequality becomes
Df(p,Snx)≤knDf(p,x), ∀p∈F(T),∀x∈C.
It is obvious that ∞n=F(Sn) = n∞=F(Tn) =F(T).
Lemma . Assume that T is uniformly Lipschitz,that is,there exists a constant L≥
such that
Tnx–Tny≤Lx–y, ∀x,y∈C
for all n≥.Then{Sn}={Tn}is uniformly closed.
Proof Assumezn–Snzn →,zn→pasn→ ∞, we havezn–Tnzn →, therefore
asn→ ∞. On the one hand,Tnp→p, on the other hand,Tn+p→Tp, these imply that p=Tp. Hencep∈ ∞n=F(Sn). This completes the proof.
Next we give an application of Theorem . to find the fixed point of Bregman asymp-totically quasi-nonexpansive mappings.
Theorem . Let f :E→(–∞, +∞]be a Legendre function which is bounded,strongly coercive,uniformly Fréchet differentiable,and totally convex on bounded subsets on E.Let C be a nonempty,closed,and convex subset ofint domf.Let{Ti}Ni=:C→C be an N uni-formly Lipschitz Bregman asymptotically quasi-nonexpansive mapping with a nonempty common fixed point set F= Ni=F(Ti)and F∩EP be nonempty.Let{xn}be a sequence of
C generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈int domf, arbitrarily,
yi,n=∇f∗(αn∇f(xn) + ( –αn)∇f(Tinxn)), i= , , , . . . ,N,
F(ui,n,y) +Aui,n,y–ui,n+rn∇f(ui,n) –∇f(yi,n),y–ui,n ≥, ∀y∈C,
Ci,n+={z∈Cn:Df(z,ui,n)≤Df(z,yi,n)≤Df(z,xn) +ξn}, n≥,
Ci,=C, Cn+= Ni=Ci,n+, xn=PCfnx,
where
ξn= (kn– ) sup p∈F∩EP∩B(PFf∩EPx,)
Df(p,x),
B(x, ) =y∈E:Df(y,x)≤
and{αn}is a sequence satisfyinglim supn→∞αn< .Then{xn}converges to q=PFf∩EPx. Proof LetSn=Tnfor alln≥, by using Lemma . and Theorem . we can obtain the
conclusion.
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 one family of countable Bregman quasi-Lipschitz mappings. MC performed all the steps of the proofs in this research for the finite families of countable Bregman quasi-Lipschitz mappings. JB performed the application to the Bregman asymptotically quasi-nonexpansive mappings. All authors read and approved the final manuscript.
Author details
1Department of Mathematics and Sciences, Shijiazhuang University of Economics, Shijiazhuang, 050031, 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: 27 March 2015 Accepted: 9 June 2015
References
2. Takahashi, W, Takeuchi, Y, Kubota, R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl.341, 276-286 (2008)
3. Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.43, 153-159 (1991)
4. Inchan, I: Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces. Int. J. Math. Anal.2, 1135-1145 (2008)
5. Bruck, RE, Kuczumow, T, Reich, S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math.65, 169-179 (1993)
6. Hecai, Y, Aichao, L: Projection algorithms for treating asymptotically quasi-φ-nonexpansive mappings in the intermediate sense. J. Inequal. Appl. (2013). doi:10.1186/1029-242X-2013-265
7. Qing, Y: Some results on asymptotically quasi-φ-nonexpansive mappings in the intermediate sense. J. Fixed Point Theory2012, Article ID 1 (2012)
8. Qin, X, Huang, S, Wang, T: On the convergence of hybrid projection algorithms for asymptotically quasi-φ-nonexpansive mappings. Comput. Math. Appl.61, 851-859 (2011)
9. Qin, X, Wang, L: On asymptotically quasi-φ-nonexpansive mappings in the intermediate sense. Abstr. Appl. Anal. 2012, Article ID 636217 (2012). doi:10.1155/2012/636217
10. Alber, YI: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes Pure Appl. Math., vol. 178, pp. 15-50. Dekker, New York (1996)
11. Hao, Y: Some results on a modified Mann iterative scheme in a reflexive Banach space. Fixed Point Theory Appl.2013, Article ID 227 (2013). doi:10.1186/1687-1812-2013-227
12. Bregman, LM: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys.7, 200-217 (1967)
13. Martin-Marquez, V, Reich, S, Sabach, S: Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete Contin. Dyn. Syst.6, 1043-1063 (2013)
14. Reich, S, Sabach, S: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal.73, 122-135 (2010)
15. Reich, S, Sabach, S: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optim. Appl., vol. 49, pp. 301-316. Springer, New York (2011)
16. Suantai, S, Cho, YJ, Cholamjiak, P: Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces. Comput. Math. Appl.64, 489-499 (2012)
17. Naraghirad, E, Yao, J-C: Bregman weak relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2013, Article ID 141 (2013)
18. Bauschke, HH, Borwein, JM, Combettes, PL: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math.3, 615-647 (2001)
19. Barbu, V, Precupanu, T: Convexity and Optimization in Banach Spaces. Springer, Dordrecht (2012) 20. Reich, S, Sabach, S: A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces.
J. Nonlinear Convex Anal.10, 471-485 (2009)
21. Zalinescu, C: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)
22. Tomizawa, Y: A strong convergence theorem for Bregman asymptotically quasi-nonexpansive mappings in the intermediate sense. Fixed Point Theory Appl.2014, Article ID 154 (2014)
23. Censor, Y, Lent, A: An iterative row-action method for interval convex programming. J. Optim. Theory Appl.34, 321-353 (1981)
24. Chen, G, Teboulle, M: Convergence analysis of a proximal-like minimization algorithm using Bregman functions. SIAM J. Optim.3, 538-543 (1993)
25. Butnariu, D, Iusem, AN, Zalinescu, C: On uniform convexity, total convexity and convergence of the proximal points and outer Bregman projection algorithms in Banach spaces. J. Convex Anal.10, 35-61 (2003)
26. Butnariu, D, Resmerita, E: Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal.2006, Article ID 84919 (2006)
27. Reich, S, Sabach, S: Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim.31, 22-44 (2010)
28. Butnariu, D, Iusem, AN: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic, Dordrecht (2000)
29. Alber, Y, Butnariu, D: Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces. J. Optim. Theory Appl.92, 33-61 (1997)
30. Takahashi, S, Takahashi, W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal.69, 1025-1033 (2008)
31. Martinet, B: Regularisation d’inequations variationnelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér.4, 154-159 (1970)
32. Kohsaka, F, Takahashi, W: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstr. Appl. Anal.2004(3), 239-249 (2004)
