fixedpointtheoryandapplications.springeropen.com... · 2017. 8. 27. · NiandYaoFixedPointTheoryandApplications20152015:35 DOI10.1186/s13663-015-0283-8 RESEARCH OpenAccess ThemodiﬁedIshikawaiterativealgorithm

Ni and Yao Fixed Point Theory and Applications (2015) 2015:35 DOI 10.1186/s13663-015-0283-8

R E S E A R C H Open Access

The modified Ishikawa iterative algorithmwith errors for a countable family of Bregmantotally quasi-D-asymptotically nonexpansivemappings in reflexive Banach spacesRen-Xing Ni1 and Jen-Chih Yao2,3*

*Correspondence:[email protected] for General Education,China Medical University, Taichung,40402, Taiwan3Department of Mathematics, KingAbdulaziz University, P.O. Box 80203,Jeddah, 21589, Saudi ArabiaFull list of author information isavailable at the end of the article

AbstractIn this paper, a new modified Ishikawa iterative algorithm with errors by a shrinkingprojection method for generalized mixed equilibrium problems and a countablefamily of uniformly Bregman totally quasi-D-asymptotically nonexpansive mappingsis introduced and investigated in the framework of a real Banach space. Strongconvergence of the sequence generated by the proposed algorithm is derived undersome suitable assumptions. These results are new and develop some recent results inthis field.MSC: 26B25; 46T99; 47H04; 47H05; 47H09; 47H10; 47J05; 47J20; 47J25; 52A41; 54C20

Keywords: asymptotically Bregman totally quasi-D-asymptotically nonexpansivemapping; generalized mixed equilibrium problem; Bregman distance; Bregmanprojection; fixed point; shrinking projection method; Banach space

1 Introduction and preliminariesIn this paper, without other specifications, let N∗ and R be the sets of positive integersand real numbers, respectively, C be a nonempty, closed, and convex subset of a realBanach space E with the dual space E∗. The norm and the dual pair between E∗ and Eare denoted by ‖ · ‖ and 〈·, ·〉, respectively. Let g : E → R ∪ {+∞} be a proper convex andlower semicontinuous function. Denote the domain of g by dom g , i.e., dom g = {x ∈ E :g(x) < +∞}. The Fenchel conjugate of g is the function g∗ : E∗ → (–∞, +∞] defined byg∗(ζ ) = supx∈E{〈ζ , x〉 – g(x)}. Let T : E → C be a nonlinear mapping. For all x ∈ E andx∗ ∈ E∗, denote by F(T) = {x ∈ C : Tx = x} the set of fixed points of T and by 〈x, x∗〉 thevalue of x∗ at x. A mapping T is said to be nonexpansive if ‖Tx – Ty‖ ≤ ‖x – y‖ for allx, y ∈ E.

Let {xn} be a sequence in E, we denote the strong convergence of {xn} to x ∈ E by xn → x.For any x ∈ int(dom g), the right-hand derivative of g at x in the direction y ∈ E is definedby g ′(x, y) := limt→

g(x+ty)–g(x)t . The mapping g is called Gâteaux differentiable at x if, for

all y ∈ E, limt→g(x+ty)–g(x)

t exists. In this case, g ′(x, y) coincides with �g(x) and the valueof the gradient of g at x. The mapping g is called Gâteaux differentiable if it is Gâteauxdifferentiable for any x ∈ int(dom g). g is called Fréchet differentiable at x if this limit is

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Ni and Yao Fixed Point Theory and Applications (2015) 2015:35 Page 2 of 24

attained uniformly for ‖y‖ = . We say that g is uniformly Fréchet differentiable on a subsetC of E if the limit is attained uniformly for x ∈ C and ‖y‖ = .

The Legendre function g : E → (–∞, +∞] is defined in []. From [], if E is a reflexiveBanach space, then g is the Legendre function if and only if it satisfies the conditions (L)and (L):

(L) The interior of the domain of g , int(dom g), is nonempty, g is Gâteaux differentiableon int(dom g) and dom(g) = int(dom g).

(L) The interior of the domain of g∗, int(dom g∗), is nonempty, g∗ is Gâteauxdifferentiable on int(dom g∗) and dom g∗ = int(dom g∗), where the functiong∗ : E∗ → (–∞, +∞] is the Fenchel conjugate of g .

Examples of Legendre functions are given in [, ]. One important and interesting Leg-endre function is

s ‖ · ‖s ( < s < +∞), in the Banach space E which is smooth and strictlyconvex and, in particular, a Hilbert space.

By Bauschke et al. [], Theorem ., the conditions (L) and (L) also show that thefunctions g and g∗ are strictly convex on the interior of their respective domains. Fromnow on, we assume that the convex function g : E → (–∞, +∞] is Legendre.

Definition . [, ] Let g : E → R be a Gâteaux differentiable and convex function.The function D(·, ·) : dom g × int(dom g) → [, +∞) defined by D(y, x) = g(y) – g(x) – 〈y –x,�g(x)〉 is called the Bregman distance with respect to g .

It follows from the strict convexity of g that D(x, y) ≥ for all x, y in E. However, D(·, ·)might not be symmetric and D(·, ·) might not satisfy the triangular inequality.

Remark . [] The Bregman distance has the following properties:() the three point identity, for any x ∈ dom g and y, z ∈ int(dom g),

D(x, z) = D(x, y) + D(y, z) + 〈�g(y) – �g(z), x – y〉;() the four point identity, for any y, w ∈ dom g and x, z ∈ int(dom g),

D(y, x) – D(y, z) – D(w, x) + D(w, z) = 〈�g(z) – �g(x), y – w〉.

Definition . [] Let g : E → R be a Gâteaux differentiable and convex function. TheBregman projection of x ∈ int(dom g) onto the nonempty, closed and convex set C ⊂ dom gis the necessarily unique vector Projg

C(x) ∈ C satisfying the following:

D(Projg

C(x), x)

= inf{

D(y, x) : y ∈ C}

.

Definition . [] Let J : E → E∗ be the normalized duality mapping defined by J(x) ={x∗ ∈ E∗ : 〈x, x∗〉 = ‖x‖ = ‖x∗‖}, φ : E × E → R+ be the Lyapunov functional defined byφ(x, y) = ‖x‖ – 〈x, Jy〉 + ‖y‖, ∀x, y ∈ E. The generalized projection �C(x) defined by

φ(�C(x), x

)= inf

{φ(y, x) : y ∈ C

}.

Remark . () If E is a smooth Banach space and g(x) = ‖x‖ for all x ∈ E, then wehave �g(x) = Jx for all x in E. Hence, D(·, ·) reduces to the usual map φ(·, ·) as D(x, y) =‖x‖ – 〈x, Jy〉 + ‖y‖ = φ(x, y), ∀x, y ∈ E. The Bregman projection Projg

C(x) reduces to thegeneralized projection �C(x) []. It is obvious from the definition of φ that (‖x‖ – ‖y‖) ≤φ(x, y) ≤ (‖x‖ + ‖y‖).


() If E is a Hilbert space and g(x) = ‖x‖ for all x ∈ E, then D(x, y) = ‖x – y‖ and theBregman projection Projg

C(x) is reduced to the metric projection PC(x) of x onto C. Formore details we refer the readers to [].

Let C be a nonempty, closed, and convex subset of E and T be a mapping from E to C.A point p ∈ C is said to be an asymptotic fixed point of T [] if C contains a sequence {xn}which converges weakly to p such that limn→∞ ‖xn – Txn‖ = . A point p ∈ C is said to be astrong asymptotic fixed point of T [] if C contains a sequence which converges stronglyto p such that limn→∞ ‖xn – Txn‖ = . We denote the sets of asymptotic fixed points andstrong asymptotic fixed points of T by F̂(T) and F̃(T), respectively.

Definition . () A mapping T from E to C is said to be Bregman relatively nonexpansive[, ], if F̂(T) = F(T) �= ∅ and D(p, Tx) ≤ D(p, x) for all x ∈ E and p ∈ F(T).

() T is said to be Bregman weak relatively nonexpansive [, , ], if F̃(T) = F(T) �= ∅ andD(p, Tx) ≤ D(p, x) for all x ∈ E and p ∈ F(T).

() T is said to be Bregman quasi-D-nonexpansive [, ], if F(T) �= ∅ and D(p, Tx) ≤D(p, x) for all x ∈ E and p ∈ F(T).

() T is said to be Bregman firmly nonexpansive [], if 〈�g(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 [], if F̂(T) �= ∅ and D(p, Tx) ≤D(p, x) for all x ∈ E and p ∈ F̂(T) and if whenever {xn} ⊂ E is bounded, p ∈ F̂(T) andlimn→+∞[D(p, xn) – D(p, Txn)] = , it follows that limn→+∞ D(Txn, xn) = .

() T is said to be relatively quasi-nonexpansive [], if F̂(T) = F(T) �= ∅ and φ(p, Tx) ≤φ(p, x) for all x ∈ E and p ∈ F(T).

() T is said to be weak relatively nonexpansive [–], if F̃(T) = F(T) �= ∅ and φ(p, Tx) ≤φ(p, x) for all x ∈ E and p ∈ F(T).

() T is said to be quasi-φ-nonexpansive [–], if F(T) �= ∅ and φ(p, Tx) ≤ φ(p, x) forall x ∈ E and p ∈ F(T).

Definition . () A mapping T : E → C is said to be Bregman totally quasi-D-asymptot-ically nonexpansive [], if F(T) �= ∅ and there exist nonnegative real sequences {vn}, {un}with vn, un → (as n → +∞) and a strictly increasing continuous function ζ : R+ → R+

with ζ () = such that

D(p, Tnx

) ≤ D(p, x) + vn · ζ [D(p, x)

]+ un, ∀n ≥ , x ∈ E, p ∈ F(T). (.)

() A mapping T : E → C is said to be Bregman quasi-D-asymptotically nonexpansive[], if F(T) �= ∅ and there exists a sequence {kn} ⊂ [, +∞) with limn→+∞ kn = such that

D(p, Tnx

) ≤ knD(p, x) for all x ∈ E, p ∈ F(T) and n ≥ . (.)

() A mapping T : E → C is said to be Bregman quasi-D-asymptotically nonexpansivein the intermediate sense with sequence {vn}, if F(T) �= ∅ and there exists a sequence {vn}in [, +∞) with limn→+∞ vn = such that

lim supn→+∞

supx∈E,p∈F(T)

[D

(p, Tnx

)– ( + vn)D(p, x)

] ≤ . (.)


() A mapping T : E → C is said to be totally quasi-φ-asymptotically nonexpansive [],if F(T) �= ∅ and there exist nonnegative real sequences {vn}, {un} with vn, un → (as n →+∞) and a strictly increasing continuous function ζ : R+ → R+ with ζ () = such that

φ(p, 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} ⊂ [, +∞) with limn→+∞ kn = such thatφ(p, Tnx) ≤ knφ(p, x) for all x ∈ E, p ∈ F(T) and n ≥ .

() A mapping T : E → C is said to be quasi-φ-asymptotically nonexpansive in the inter-mediate sense with sequence {vn}, if F(T) �= ∅ and there exists a sequence {vn} in [, +∞)with limn→+∞ vn = such that

lim supn→+∞

supx∈E,p∈F(T)

[φ(p, Tnx

)– ( + vn)φ(p, x)

] ≤ . (.)

Remark . () If ζ (t) = t, t ≥ , then (.) reduces to

D(p, Tnx

) ≤ ( + vn) · D(p, x) + un, ∀n ≥ , x ∈ E, p ∈ F(T). (.)

In addition, if un ≡ for all n ≥ , then Bregman totally quasi-D-asymptotically non-expansive mappings coincide with Bregman quasi-D-asymptotically nonexpansive map-pings. If un ≡ and vn ≡ for all n ≥ , we obtain from (.) the class of mappings thatincludes 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 all n ≥ , then (.) reduces to (.) whichhas been studied as mappings Bregman quasi-D-asymptotically nonexpansive in the in-termediate sense.

() From the definitions, it is obvious that if F̂(T) = F(T) �= ∅, then a Bregman stronglynonexpansive mapping is a Bregman relatively nonexpansive mapping; a Bregman rela-tively nonexpansive mapping is a Bregman quasi-D-nonexpansive mapping. A Bregmanquasi-D-nonexpansive mapping is a Bregman quasi-D-asymptotically nonexpansive map-ping, but the converse is not true.

If taking ζ (t) = t, t ≥ , vn = kn – , un = , limn→+∞ kn = , then (.) can be rewrittenas (.). This implies that each Bregman quasi-D-asymptotically nonexpansive mappingmust be a Bregman total quasi-D-asymptotically nonexpansive mapping, but the converseis not true. In [], Chang et al. gave an example of Bregman total quasi-D-asymptoticallynonexpansive mapping. A Bregman relatively nonexpansive mapping is a Bregman weakrelatively nonexpansive mapping, but the converse in not true in general. Indeed, for anymapping T : E → C, we have F(T) ⊂ F̃(T) ⊂ F̂(T). If T is Bregman relatively nonex-pansive, then F(T) = F̃(T) = F̂(T). In [], Naraghirad and Yao have given two examplesof a Bregman weak relatively nonexpansive mapping which is not a Bregman relativelynonexpansive mapping, and a Bregman quasi-nonexpansive mapping which is neither aBregman relatively nonexpansive mapping nor a Bregman weak relatively nonexpansivemapping.

() The class of quasi-φ-(asymptotically) nonexpansive mappings is more general thanthat of relatively nonexpansive mappings which requires the restriction F̂(T) = F(T).


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 toquasi-(asymptotically) nonexpansive mappings.

The idea of the definition of a total asymptotically nonexpansive mappings is to unifyvarious definitions of classes of mappings associated with the class of asymptotically non-expansive mappings and to prove a general convergence theorems applicable to all theseclasses of nonlinear mappings.

Definition . [] Let E be a Banach space. The function g : E → R is said to be a Bregmanfunction if the following conditions are satisfied:

() g is continuous, strictly convex and Gâteaux differentiable;() the set {y ∈ E : D(x, y) ≤ r} is bounded for all x ∈ E and r > .

The theory of fixed points with respect to Bregman distances have been studied in thelast ten years and much intensively in the last six years. In [], Bauschke and Combettesintroduced an iterative method to construct the Bregman projection of a point onto acountable intersection of closed and convex sets in reflexive Banach spaces. They provedstrong convergence theorem of the sequence produced by their method; for more details,see [], Theorem .. To find a point of the intersection of m closed and convex subsets ina Banach space, in , Alber [] first studied the iterative method with Bregman pro-jections. In [], Alber investigated the generalized projections in a Banach space. For somerecent 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 thateither {yn} or {zn} is bounded. Then limn→+∞ 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 withnonnegative real sequences {vn}, {un} and a strictly increasing and continuous function ζ :R+ → R+ with ζ () = . If vn, un → (as n → +∞). Then F(T) is a closed convex subset of C.

Proof Let {xn} be a sequence in F(T) such that xn → x∗ (as n → +∞). We have Txn =xn → x∗ (as n → +∞) and by the closeness of T , we have Tx∗ = x∗. This implies that F(T)is closed.

Let p, q ∈ F(T) and t ∈ (, ), and put w = tp + ( – t)q. Next we prove that w ∈ F(T).Indeed, in view of the definition of D, we have

D(w, Tnw

)= g(w) – g

(Tnw

)–

⟨�g

(Tnw

), w – Tnw

⟩

= g(w) – g(Tnw

)–

⟨�g

(Tnw

), tp + ( – t)q – Tnw

⟩

= g(w) + tD(p, Tnw

)+ ( – t)D

(q, Tnw

)– tg(p) – ( – t)g(q). (.)


Since

tD(p, Tnw

)+ ( – t)D

(q, Tnw

)

≤ t{

D(p, w) + vnζ[D(p, w)

]+ un

}

+ ( – t){

D(q, w) + vnζ[D(q, w)

]+ un

}

= t{

g(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

≤ D(w, Tnw

) ≤ tvnζ[D(p, w)

]+ ( – t)vnζ

[D(q, w)

]+ un (as n → +∞).

Hence, we have Tnw → w. This implies that T(Tnw) = Tn+w → w. Since T is closed, wehave w ∈ Tw, i.e., w ∈ F(T). This completes the proof of Lemma .. �

Definition . [] Let g : E → (–∞, +∞] be a convex and Gâteaux differentiable func-tion. g is called

() totally convex at x ∈ int(dom g) if its modulus of total convexity at x, that is, thefunction vg : int(dom g) × [, +∞) → [, +∞), defined byvg(x, t) := inf{D(y, x) : y ∈ dom g,‖y – x‖ = t}, is positive whenever t > ;

() totally convex if it is totally convex at every point x ∈ int(dom g);() totally convex on bounded sets if vg(B, t) is positive for any nonempty bounded

subset B of E and t > , where the modulus of total convexity of the function g onthe set B is the function vg : int(dom g) × [, +∞) → [, +∞) defined byvg(B, t) := inf{vg(x, t) : x ∈ B ∩ dom g}.

Definition . [, ] Let B be the closed unit ball of a Banach space E. A function g :E → R is said to be

() cofinite if dom g∗ = E∗;() coercive if lim‖x‖→∞(g(x)/‖x‖) = +∞;() sequentially consistent if for any two sequences {xn} and {yn} in E such that {xn} is

bounded, limn→+∞ D(yn, xn) = ⇒ limn→+∞ ‖yn – xn‖ = ;() locally bounded if g(rB) is bounded for all r > ;() locally uniformly smooth on E if the function σr : [, +∞) → [, +∞), defined by

σr(t) = supx∈rB,y∈E,‖y‖=,α∈(,)

[αg

(x + ( – α)ty

)+ ( – α)g(x – αty) – g(x)

]

× [α( – α)

]–/,

satisfies limt→σr(t)

t = , ∀r > ;


() locally uniformly convex on E (or uniformly convex on bounded subsets of E) if thegauge ρr : [, +∞) → [, +∞) of uniform convexity of g , defined by

ρr(t) = infx,y∈rB,‖x–y‖=t,α∈(,)

[αg(x) + ( – α)g(y) – g

(αx + ( – α)y

)]

× [α( – α)

]–/,

satisfies ρr(t) > , ∀r, t > .

Lemma . [] If g : E → (–∞, +∞] is Fréchet differentiable and totally convex, then g iscofinite.

Lemma . [] Let g : E → (–∞, +∞] be a convex function whose domain contains atleast two points. Then the following statements hold:

() g is sequentially consistent if and only if it is totally convex on bounded sets.() If g is lower semicontinuous, then g is sequentially consistent if and only if it is

uniformly convex on bounded sets.() If g is uniformly strictly convex on bounded sets, then it is sequentially consistent and

the converse implication holds when g is lower semicontinuous, Fréchet differentiableon its domain and the Fréchet derivative �g is uniformly continuous on bounded sets.

Lemma . [] Let g : E → R be uniformly Fréchet differentiable and bounded onbounded subsets of E. Then �g is uniformly continuous on bounded subsets of E from thestrong topology of E to the strong topology of E∗.

Lemma . ([], Lemma .) Let g : E → R be a Gâteaux differentiable and totally convexfunction. If x ∈ E and the sequence {D(xn, x)} is bounded, then the sequence {xn} is alsobounded.

Lemma . [] Let E be a Banach space, r > be a positive number and g : E → R be acontinuous 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 with

∑mn= λn = , for any i, j ∈ {, , . . . , m} with i < j, where ρr is the

gauge of uniformly convexity of g .

Lemma . [] Let g : E → (–∞, +∞] be Gâteaux differentiable and totally convex onint(dom g). Let x ∈ int(dom g) and C ⊂ int(dom g) be a nonempty, closed, and convex set. Ifx̂ ∈ C, then the following statements are equivalent:

() the vector x̂ is the Bregman projection of x onto C with respect to g ;() the vector x̂ is the unique solution of the variational inequality:

〈�g(x) – �g(z), z – y〉 ≥ , ∀y ∈ C;() the vector x̂ is the unique solution of the inequality: D(y, z) + D(z, x) ≤ D(y, x), ∀y ∈ C.


Lemma . ([], Theorem .) Let E be a reflexive Banach space and let g : E → R be aconvex function which is bounded on bounded subsets of E. Then the following assertionsare equivalent:

() g is strongly coercive and uniformly convex on bounded subsets of E;() dom g∗ = E∗, g∗ is bounded on bounded subsets and uniformly smooth on bounded

subsets of E∗;() dom g∗ = E∗, g∗ is Fréchet differentiable and �g∗ is uniformly norm-to-norm

continuous on bounded subsets of E∗.

Lemma . ([], Theorem .) Let E be a reflexive Banach space and let g : E → R bea continuous convex function which is strongly coercive. Then the following assertions areequivalent:

() g is bounded on bounded subsets and uniformly smooth on bounded subsets of E;() g∗ is Fréchet differentiable and �g∗ is uniformly norm-to-norm continuous on

bounded subsets of E∗;() dom g∗ = E∗, g∗ is strongly coercive and uniformly convex on bounded subsets of E∗.

For solving the equilibrium problem, let us assume that the bifunction f : C × C → Rsatisfies 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 each y ∈ C, the function x �→ 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 andstrongly coercive function which is bounded on bounded subsets and uniformly convex onbounded subset of E. Let C be a nonempty, closed and convex subset of E and f : C ×C → Ra bifunction satisfying conditions (C)-(C) and EP(G) �= ∅, ϕ : C → R be a lower semicon-tinuous and convex functional, A : C → E∗ be a continuous and monotone mapping. Forr > and x ∈ E, define a mapping TG

r : E → C as follows:

TGr x =

{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 statementshold:

() dom(TGr ) = E;

() TGr is single-valued;

() TGr is a Bregman firmly nonexpansive mapping;

() F(TGr ) = GMEP(f ,ϕ);

() GMEP(f ,ϕ) is closed and convex of C;() D(q, TG

r x) + D(TGr x, x) ≤ D(q, x), ∀q ∈ F(TG

r ).

In , Saewan et al. [] studied the following generalized mixed equilibrium problem:find z ∈ C such that

f (z, y) + 〈Az, y – z〉 + ϕ(y) – ϕ(z) ≥ , ∀y ∈ C, (.)


where f is a bifunction from C ×C to R, ϕ : C → R is a real-valued function and A : C → E∗

is a nonlinear mapping. Denote the set of solutions of the problem (.) by GMEP(f ,ϕ),i.e.,

GMEP(f ,ϕ) ={

z ∈ C|f (z, y) + 〈Az, y – z〉 + ϕ(y) – ϕ(z) ≥ ,∀y ∈ C}

.

Special cases: (I) If A = , then the problem (.) is equivalent to find z ∈ C such that

f (z, y) + ϕ(y) – ϕ(z) ≥ , ∀y ∈ C, (.)

which is called the mixed equilibrium problem. Denote the set of solutions of (.) byMEP(f ,ϕ).

(II) If f = , then the problem (.) is equivalent to find z ∈ C such 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 (.) by VI(C, A,ϕ). In particular, we denote VI(C, A, ) by VI(C, A).

(III) If ϕ = , then the problem (.) is equivalent to finding z ∈ C such that

f (z, y) + 〈Az, y – z〉 ≥ , ∀y ∈ C, (.)

which is called the generalized equilibrium problem. Denote the set of solutions of (.)by GEP(f ).

(IV) If A = , ϕ = , then the problem (.) is equivalent to finding z ∈ C such that

f (z, y) ≥ , ∀y ∈ C, (.)

which is called the equilibrium problem. Denote the set of solutions of (.) by EP(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 forsolving 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’ attentionand 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 solvethe generalized mixed equilibrium problem (see, for example, [–, , –]). Nu-merous problems in physics, optimization and economics help to find a solution of prob-lem (.).

It is well known that, in an infinite-dimensional Hilbert space, only weak convergencetheorems for the segmenting Mann iteration were established even for nonexpansivemappings. Attempts to modify the segmenting Mann iteration for nonexpansive map-pings and asymptotically nonexpansive mappings by hybrid projection algorithms have


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 singlenonexpansive mapping T in a Hilbert space H :

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

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∩Qn x,

(.)

where PC denotes the metric projection of H onto a closed and convex subset C of H . Theyproved that if {αn} ⊂ (, ) and limn→∞ αn = , then the sequence {xn} converges stronglyto PF(T)x.

In [], Qin and Su extended the results of Martinez-Yanes and Xu [] from Hilbertspaces to Banach spaces and proved the following result: Let C be a nonempty, closed,and convex subset of a uniformly smooth and uniformly convex Banach space E andlet T : C → C be a relatively nonexpansive mapping. Assume that {αn} ⊂ (, ) andlimn→∞ αn = . Define a sequence {xn} in C by 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∩Qn x, n ≥ .

(.)

If F(T) is nonempty, then {xn} converges strongly to �F(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 aBanach space E:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

x ∈ E chosen arbitrarily,C,i = C, C =

⋂∞i= C,i,

xi = �C x,wn,i = �CJ–(Jxn – λn,iBxn),zn,i = J–(β ()

n,i Jxn + β()n,i JTixn + β

()n,i JSiwn,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 pointsof the two families {Ti} and {Si} and the set of solutions to a variational inequality problem.

In , Bregman [] discovered an elegant and effective technique for using the so-called Bregman distance function D(·, ·) in the process of designing and analyzing fea-


sibility and optimization algorithms. This opened a growing area of research in whichBregman’s technique has been applied in various ways in order to design and analyze notonly iterative algorithms for solving feasibility and optimization problems, but also al-gorithms for solving variational inequalities, for approximating equilibria, for computingfixed points of nonlinear mappings, and so on (see, e.g., [, , –] and the referencestherein). In , Butnariu and Resmerita [] presented Bregman-type iterative algo-rithms and studied the convergence of the Bregman-type iterative method of solving somenonlinear operator equations.

In , by using the Bregman projection, Reich and Sabach [] presented the followingproximal algorithms for finding common zeroes of maximal monotone operators Ai : E →E∗ (i = , , . . . , m) in a reflexive Banach space E. More precisely, they proved the followingstrong convergence theorem.

Theorem RS Let E be a reflexive Banach space and let Ai : E → E∗ (i = , , . . . , m) be mmaximal monotone operators such that Z :=

⋂mi= A–

i (∗) �= ∅. Let g : E → R be a Legendrefunction that is bounded, uniformly Fréchet differentiable and totally convex on boundedsubsets of E. Let {xn} be a sequence defined by the following iterative algorithm:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

x ∈ E chosen arbitrarily,yi

n = Resgλi

n(xn + ei

n),Ci

n = {z ∈ E : D(z, yin) ≤ D(z, xn + ei

n)},Cn =

⋂mi= Ci

n,Qn = {z ∈ E : 〈�g(x) – �g(xn), z – xn〉 ≤ },xn+ = Projg

Cn∩Qn (x), ∀n ≥ .

(.)

If, for each i = , , . . . , m, lim infn→+∞ λin > and the sequences of errors {ei

n} ⊂ E satisfylimn→+∞ ei

n = , then each such sequence {xn} converges strongly to ProjgZ(x) as n → +∞.

Further, under some suitable conditions, they obtained two strong convergence theoremsof maximal monotone operators in a reflexive Banach space. Reich and Sabach [] alsostudied 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 fixedpoints of finitely many Bregman firmly nonexpansive operators Ti : E → E (i = , , . . . , m)in a reflexive Banach space E, if F :=

⋂mi= F(Ti) �= ∅:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

x ∈ E,Qi

= E, i = , , . . . , m,yi

n = Ti(xn + ein),

Qin+ = {z ∈ Qi

n : 〈�g(xn + ein) – �g(yi

n), z – yin〉 ≤ },

Qn =⋂m

i= Qin,

xn+ = ProjgQn+

(x), ∀n ≥ .

(.)

Under some suitable conditions, they proved that the sequence {xn} generated by (.)converges strongly to Projg

F (x) and applied the result to the solution of convex feasibilityand equilibrium problems, where g : E → R and {ei

n} ⊂ E satisfying limn→+∞ ein = for

each n ≥ and i = , , . . . , m.


Very recently, Chen et 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 aweak Bregman relatively nonexpansive mapping and a Bregman relatively nonexpansivemapping. 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 nonexpansive mappings under some suitableconditions.

Motivated by the above mentioned results and the on-going research, in this paper, us-ing Bregman function and the shrinking projection method, we introduce new modifiedIshikawa iterative algorithms with errors for finding a common element of solutions tothe 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 assumption F̂(T) = F(T) is imposed on the mapping T in reflexive Banach space set-ting. Our results improve and develop many known results in the current literature; see,for example, [, , , ].

2 Main resultsWe 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 coerciveBregman 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 → Rbe a lower semicontinuous and convex functional, let fk : C × C → R be a bifunction sat-isfying (C)-(C) and Ti : E → int(dom g), ∀i ∈ N be an infinite family of closed and uni-formly Bregman totally quasi-D-asymptotically nonexpansive mappings with nonnegativereal sequences {v(i)

n }, {u(i)n } and a strictly increasing and continuous function ζ : R+ → R+

with ζ () = . If limn→+∞ supi∈N∗{v(i)n } = and limn→+∞ supi∈N∗{u(i)

n } = . Assume that Ti isuniformly asymptotically regular on E for all i ≥ , i.e., limn→+∞ supx∈K ‖Tn+

i x – Tni x‖ =

holds for any bounded subset K of E and F = [⋂+∞

i= F(Ti)]∩ [⋂m

k= GMEP(fk ,ϕk)] �= ∅. For allz, y ∈ C, Gk(z, y) = fk(z, y) + ϕk(y) – ϕk(z) + 〈Akz, y – z〉, TGk

rk,n (x) = {z ∈ C : Gk(z, y) + rk,n

〈y –z,�g(z) – �g(x)〉 ≥ ,∀y ∈ C}. For an initial point x ∈ E, let Ci

= C for each i ≥ andC =

⋂∞i= Ci

and define the sequence {xn} by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

yin = �g∗[αn�g(xn) + ( – αn)�g(zi

n)],zi

n = �g∗[βn�g(xn) + ( – βn)�g(Tni (xn + ei

n))],ui

n = TGmrm,n TGm–

rm–,n · · ·TGr,n TG

r,n yin,

Cin+ = {z ∈ Cn : D(z, ui

n) ≤ αnD(z, xn) + ( – αn)D(z, zin) ≤ D(z, xn) + ζ i

n},Cn+ =

⋂+∞i= Ci

n+,xn+ = Projg

Cn+(x),

(.)

where the sequences {ζ in}, {ei

n}, {rk,n}, {αn}, {βn} satisfy the following conditions:() ζ i

n = D(xn, xn + ein) + supp∈C〈xn – p,�g(xn + ei

n) –�g(xn)〉+ v(i)n · supp∈C ζ [D(p, xn + ei

n)] +u(i)

n , ein ∈ E satisfying limn→+∞ supi∈N∗{‖ei

n‖} = for each n ≥ and i ≥ ;


() for each k = , , . . . , m, {rk,n}+∞n= ⊂ (, +∞) satisfy lim infn→+∞ rk,n > ;

() {αn}, {βn} are real sequences in [, ] such that lim infn→∞( – αn)( – βn)βn > .Then the sequence {xn} converges strongly to Projg

F (x).

Proof We define a bifunction Gk : C × C → R by

Gk(x, y) = fk(x, y) + ϕk(y) – ϕk(x) + 〈Akx, y – x〉, ∀x, y ∈ C.

Then we prove from Lemma . that the bifunction Gk satisfies conditions (C)-(C)for each k = , , . . . , m. Therefore, the generalized mixed equilibrium problem (.) isequivalent to the following equilibrium problem: find x ∈ C such that Gk(x, y) ≥ , ∀y ∈ C.Hence, GMEP(fk ,ϕk) = EP(Gk). By taking θ k

n = TGkrk,n TGk–

rk–,n · · ·TGr,n TG

r,n , k = , , . . . , m, andθ

n = I for all n ≥ , we obtain un = θmn yn.

In view of Lemma . and Lemma ., we find that F is closed and convex, so thatProjg

F (x) is well defined for any x ∈ E.We divide the proof of Theorem . into six steps:Step . We first show that Cn is closed and convex for each n ≥ .In fact, from the definition, C =

⋂∞i= Ci

= C for all i ≥ is closed and convex. Supposethat Ci

n+ is closed and convex for some n ≥ . For any z ∈ Cin+, we know that

D(z, ui

n) ≤ αnD(z, xn) + ( – αn)D

(z, zi

n) ≤ D(z, xn) + ζ i

n

is equivalent to the following:

⟨z – ui

n,αn�g(xn) + ( – αn)�g(zi

n)

–�g(ui

n)⟩ ≤ αnD

(ui

n, xn)

+ ( – αn)D(ui

n, zin)

– g(ui

n)

and

( – αn)⟨z – xn,�g(xn) – �g

(zi

n)⟩ ≤ –( – αn)D

(xn, zi

n)

+ ζ in, ∀i ≥ .

Since the left-hand sides of the last two inequalities are affine with respect to z as functionsof z, Ci

n+ is closed and convex. Hence Cn+ =⋂+∞

i= Cin+ is closed and convex for all n ≥ .

Step . Assume that F ⊂ Cn for all n ≥ . Then the sequence {xn} is bounded.In fact, by xn+ = Projg

Cn+(x), it then follows from Lemma . that

D(xn+, x) = D(Projg

Cn+(x), x

) ≤ D(p, x) – D(p, xn+) ≤ D(p, x)

for each p ∈ F ⊂ Cn, ∀n ≥ . Hence, the sequence {D(xn+, x)} is bounded, by Lemma .,{xn} is bounded and so are {Tixn}, {yi

n}, {zin}, and {ui

n}.Step . Next, we show, by induction, that F ⊂ Cn for all n ≥ .In fact, it is obvious that F ⊂ C = C. Suppose that F ⊂ Cn for some n ≥ . Let p ∈ 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 of D(·, ·) and Remark ., foreach i ≥ , we have

D(p, zi

n)

= D(p,�g∗[βn�g(xn) + ( – βn)�g

(Tn

i(xn + ei

n))])

≤ βnD(p, xn) + ( – βn)D(p, Tn

i(xn + ei

n))


≤ βnD(p, xn) + ( – βn){

D(p, xn + ei

n)

+ v(i)n · ζ [

D(p, xn + ei

n)]

+ u(i)n

}

= βnD(p, xn) + ( – βn){

D(p, xn) + D(xn, xn + ei

n)

+⟨xn – p,�g

(xn + ei

n)

– �g(xn)⟩+ v(i)

n · ζ [D

(p, xn + ei

n)]

+ u(i)n

}

= D(p, xn) + ( – βn){

D(xn, xn + ei

n)

+⟨xn – p,�g

(xn + ei

n)

– �g(xn)⟩

+ v(i)n · ζ [

D(p, xn + ei

n)]

+ u(i)n

}

≤ D(p, xn) + ζ in. (.)

Observe that p ∈ F implies p ∈ C. Thus, by (.), Lemma ., and the fact that TGkrk,n (k =

, , . . . , m) is a Bregman quasi-D-nonexpansive mapping, for each p ∈ F , we have

D(p, ui

n)

= D(p, θm

n yin)

≤ D(p, yi

n)

= D(p,�g∗[αn�g(xn) + ( – αn)�g

(zi

n)])

≤ αnD(p, xn) + ( – αn)D(p, zi

n)

≤ αnD(p, xn) + ( – αn)[D(p, xn) + ζ i

n]

≤ D(p, xn) + ζ in. (.)

This shows that p ∈ Cn+, which implies that F ⊂ Cn+. Hence F ⊂ Cn for all n ≥ .Step . Now, we show that {xn} is Cauchy sequence.In fact, combining xn+ = Projg

Cn+(x) ∈ Cn+ and Lemma ., we obtain ≤ D(xn, xn+) ≤

D(xn, x) – D(xn+, x) for all n ≥ . 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 integer m, it then follows from Lemma . that

D(xn+m, xn+) = D(xn+m, Projg

Cn+(x)

) ≤ D(xn+m, x) – D(Projg

Cn+(x), x

)

= D(xn+m, x) – D(xn+, x), (.)

from which it follows from (.) that D(xn+m, xn+) → as m, n → ∞. We have fromLemma . and the boundedness of {xn},

xn+m – xn+ → , m, n → ∞.

Hence, the sequence {xn} is Cauchy in C. Since E is a Banach space and C is closed convex,there exists p ∈ C such that xn → p as n → ∞. Now, since D(xn+m, xn+) → as m, n → ∞,we have in particular that

limn→∞ D(xn+, xn+) = (.)

and this further implies that

limn→∞‖xn+ – xn+‖ = (.)

from Lemma ..


From ‖xn – (xn + ein)‖ = ‖ei

n‖ → (as n → +∞, ∀i ≥ ), Lemma ., and the boundednessof {‖�g(xn + ei

n)‖}, we obtain

≤ D(xn, xn + ei

n)

= g(xn) – g(xn + ei

n)

+⟨ei

n,�g(xn + ei

n)⟩

≤ ∣∣g(xn) – g(xn + ei

n)∣∣ +

∥∥ein∥∥ · ∥∥�g

(xn + ei

n)∥∥ → as n → +∞,∀i ≥ . (.)

Since g is uniformly smooth on bounded subsets of E, by Lemma ., we find that �g(·)is uniformly norm-to-norm continuous on any bounded sets and ‖xn –(xn +ei

n)‖ = ‖ein‖ →

(as n → +∞, ∀i ≥ ), and we obtain

limn→∞

∥∥�g(xn) – �g

(xn + ei

n)∥∥ = , ∀i ≥ . (.)

Thus, it follows from (.), (.), limn→+∞ supi∈N∗{v(i)n } = , and limn→+∞ supi∈N∗{u(i)

n } = that

≤ ∣∣ζ in∣∣

≤ D(xn, xn + ei

n)

+∣∣∣sup

p∈C

⟨xn – p,�g

(xn + ei

n)

– �g(xn)⟩∣∣∣

+∣∣∣v(i)

n · supp∈C

ζ[D

(p, xn + ei

n)]∣∣∣ +

∣∣u(i)n

∣∣

≤ D(xn, xn + ei

n)

+[supp∈C

‖xn – p‖]· ∥∥�g

(xn + ei

n)

– �g(xn)∥∥

+∣∣v(i)

n∣∣ · sup

p∈Cζ[D

(p, xn + ei

n)]

+∣∣u(i)

n∣∣ → as n → +∞,∀i ≥ . (.)

By xn+ = ProjgCn+

(x) ∈ Cn+ ⊂ Cn+ (⊂ C) and by the definition of Cn+, it follows from(.) and (.) that

≤ D(xn+, ui

n+) ≤ D(xn+, xn+) + ζ i

n+ → , n → ∞,∀i ≥ .

From Lemma ., we obtain limn→∞ ‖xn+ – uin+‖ = . Therefore

∥∥xn+ – uin+

∥∥ ≤ ‖xn+ – xn+‖ +∥∥xn+ – ui

n+∥∥ → . (.)

It follows from limn→+∞ ‖xn – p‖ = and (.) that

uin → p, n → ∞,∀i ≥ . (.)

Step . Now we prove that p ∈ [⋂+∞

i= F(Ti)] ∩ [⋂m

k= GMEP(fk ,ϕk)].(a) First we prove that p ∈ ⋂+∞

i= F(Ti).Since g is uniformly smooth on bounded subsets of E, by Lemma ., we find that �g(·)

is uniformly norm-to-norm continuous on any bounded sets and from (.), we obtain

limn→∞

∥∥�g(xn) – �g(ui

n)∥∥ = , ∀i ≥ . (.)


It follows from the boundedness of the sequences {xn} and D(p, Tni (xn + ei

n)) ≤ D(p, xn +ei

n) + v(i)n · ζ [D(p, xn + ei

n)] + u(i)n for each p ∈ F and i ≥ that the sequences {�g(xn)} and

{�g(Tni (xn + ei

n))} are bounded. Thus there exists r > such that {�g(xn)} ⊂ Br() and{�g(Tn

i (xn + ein))} ⊂ Br(). For each p ∈ F , we have from Lemma . and Lemma .

D(p, ui

n)

= D(p, θm

n yin) ≤ D

(p, yi

n)

= D(p,�g∗[αn�g(xn) + ( – αn)�g

(zi

n)])

≤ αnD(p, xn) + ( – αn)D(p, zi

n)

≤ αnD(p, xn) + ( – αn) · [βnD(p, xn) + ( – βn)D(p, Tn

i(xn + ei

n))

– βn( – βn)ρ∗r(∥∥�g(xn) – �g

(Tn

i(xn + ei

n))∥∥)]

≤ αnD(p, xn) + ( – αn) · {βnD(p, xn) + ( – βn)[D

(p, xn + ei

n)

+ v(i)n

· ζ (D

(p, xn + ei

n))

+ u(i)n

]– βn( – βn)ρ∗

r(∥∥�g(xn) – �g

(Tn

i(xn + ei

n))∥∥)}

= αnD(p, xn) + ( – αn) · {βnD(p, xn) + ( – βn){

v(i)n · ζ [

D(p, xn + ei

n)]}

+ u(i)n + D(p, xn) + D

(xn, xn + ei

n)

+⟨xn – p,�g

(xn + ei

n)

– �g(xn)⟩}

– ( – αn)βn( – βn)ρ∗r(∥∥�g(xn) – �g

(Tn

i(xn + ei

n))∥∥)

≤ αnD(p, xn) + ( – αn) · (D(p, xn) + ζ in – βn( – βn)ρ∗

r(∥∥�g(xn)

– �g(Tn

i(xn + ei

n))∥∥))

≤ αnD(p, xn) + ( – αn)D(p, xn) + ζ in – ( – αn)βn( – βn)ρ∗

r(∥∥�g(xn)

– �g(Tn

i(xn + ei

n))∥∥)

= D(p, xn) + ζ in – ( – αn)βn( – βn)ρ∗

r(∥∥�g(xn) – �g

(Tn

i(xn + ei

n))∥∥)

.

This implies that

≤ ( – αn)βn( – βn)ρ∗r(∥∥�g(xn) – �g

(Tn

i(xn + ei

n))∥∥)

≤ D(p, xn) – D(p, ui

n)

+ ζ in. (.)

On the other hand, we have

∣∣D(p, xn) – D

(p, ui

n)∣∣ =

∣∣–D

(xn, ui

n)

+⟨xn – p,�g

(ui

n)

– �g(xn)⟩∣∣

≤ D(xn, ui

n)

+ ‖xn – p‖ · ∥∥�g(ui

n)

– �g(xn)∥∥.

In view of (.) and (.), we obtain

D(p, xn) – D(p, ui

n) → , n → ∞. (.)

Combining (.) and (.), limn→+∞ ζ in = , and the assumption lim infn→∞( –αn)βn( –

βn) > , we have

ρ∗r(∥∥�g(xn) – �g

(Tn

i(xn + ei

n))∥∥) → , n → ∞.


It follows from the property of ρ∗r (·) that

limn→+∞

∥∥�g(xn) – �g(Tn

i(xn + ei

n))∥∥ = . (.)

Since xn → p as n → ∞ and �g(·) is uniformly norm-to-norm continuous on anybounded sets, we obtain

∥∥�g(xn) – �g(p)

∥∥ → as n → . (.)

Note that

∥∥�g(Tn

i(xn + ei

n))

– �g(p)∥∥ ≤ ∥∥�g(p) – �g(xn)

∥∥ +∥∥�g(xn) – �g

(Tn

i(xn + ei

n))∥∥.

From (.) and (.), we see that

limn→+∞

∥∥�g(Tn

i(xn + ei

n))

– �g(p)∥∥ = . (.)

By Lemma ., note that �g∗(·) is also uniformly norm-to-norm continuous on anybounded sets. It follows from (.) that

limn→+∞

∥∥Tni(xn + ei

n)

– p∥∥ = . (.)

Noting that ‖Tn+i (xn + ei

n) – p‖ ≤ ‖Tn+i (xn + ei

n) – Tni (xn + ei

n)‖ + ‖Tni (xn + ei

n) – p‖, theuniformly asymptotic regularity of T and (.), we have limn→+∞ ‖Tn+

i (xn + ein) – p‖ = .

That is, Ti(Tni (xn + ei

n)) → p as n → ∞, and it follows from the closeness of Ti that Tip = p,∀i ≥ , i.e. p ∈ ⋂+∞

i= F(Ti).(b) Now we prove that p ∈ ⋂m

k= GMEP(fk ,ϕk) =⋂m

k= EP(Gk).In fact, in view of ui

n = θmn yi

n, (.), and Lemma ., for each q ∈ F(θ kn ), we have

≤ D(ui

n, yin)

= D(θm

n yin, yi

n) ≤ D

(p, yi

n)

– D(p, θm

n yin) ≤ D(p, xn) – D

(p, ui

n)

+ ζ in.

It follows from (.) and limn→+∞ ζ in = that D(ui

n, yin) → as n → ∞. Using Lem-

ma ., we see that ‖uin – yi

n‖ → as n → ∞. Furthermore, ‖xn – yin‖ ≤ ‖xn – ui

n‖ + ‖uin –

yin‖ → as n → ∞. Since xn → p, n → ∞ and ‖xn – yi

n‖ → , n → ∞, then yin → p,

n → ∞. By the fact that θ kn , k = , , . . . , m is Bregman relatively nonexpansive and using

Lemma . again, we have

≤ D(θ k

n yin, yi

n) ≤ D

(p, yi

n)

– D(p, θ k

n yin) ≤ D(p, xn) – D

(p, θ k

n yin)

+ ζ in. (.)

Observe that

D(p, ui

n)

= D(p, θm

n yin)

= D(p, TGm

rm,n TGm–rm–,n · · ·TG

r,n TGr,n yi

n)

= D(p, TGm

rm,n TGm–rm–,n · · · θ k

n yin) ≤ D

(p, θ k

n yin). (.)

Using (.) and (.), we obtain ≤ D(θ kn yi

n, yin) ≤ D(p, xn) – D(p, ui

n) + ζ in → , n → ∞.

Then Lemma . implies that limn→∞ ‖θ kn yi

n – yin‖ = , k = , , . . . , m. Now ‖θ k

n yin – p‖ ≤


‖θ kn yi

n – yin‖ + ‖yi

n – p‖ → , n → ∞, k = , , . . . , m. Similarly, limn→+∞ ‖θ k–n yi

n – p‖ = ,k = , , . . . , m. This further implies that

limn→+∞

∥∥θ k–

n yin – θ k

n yin∥∥ = . (.)

Also, since �g(·) is uniformly norm-to-norm continuous on any bounded sets and using(.), we obtain limn→+∞ ‖�g(θ k

n yin) – �g(θ k–

n yin)‖ = . From the assumption {rk,n}+∞

n= ⊂(, +∞) satisfying lim infn→+∞ rk,n > for each k = , , . . . , m, we see that

limn→∞

‖�g(θ kn yi

n) – �g(θ k–n yi

n)‖rk,n

= . (.)

By Lemma ., we have, for each k = , , . . . , m, Gk(θ kn yi

n, y) + rk,n

〈y – θ kn yi

n,�g(θ kn yi

n) –�g(θ k–

n yin)〉 ≥ , ∀y ∈ C. Furthermore, replacing n by nj in the last inequality and using

condition (C), we obtain

∥∥y – θ knj

yinj

∥∥ ·‖�g(θ k

njyi

nj) – �g(θ k–

njyi

nj)‖

rk,nj

≥ rk,nj

⟨y – θ k

njyi

nj,�g

(θ k

njyi

nj

)– �g

(θ k–

njyi

nj

)⟩

≥ –Gk(θ k

njyi

nj, y

) ≥ Gk(y, θ k

njyi

nj

), ∀y ∈ C.

By taking the limit as j → +∞ in the above inequality, for each k = , , . . . , m, we have fromthe condition (C), (.), and θ k

njyi

nj→ p that Gk(y, p) ≤ , ∀y ∈ C.

For < t ≤ and y ∈ C, define yt = ty + ( – t)p. It follows from y, p ∈ C that yt ∈ C, whichyields Gk(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) ≥ .

Letting t → +, from the condition (C), we obtain Gk(p, y) ≥ , ∀y ∈ C. This implies thatp ∈ EP(Gk), k = , , . . . , m, i.e., p ∈ ⋂m

k= GMEP(fk ,ϕk) =⋂m

k= EP(Gk). Thus we have p ∈ F .Step . Finally, we prove that p = Projg

F (x).From Lemma . and the definition of xn+ = Projg

Cn+(x), we see that 〈xn+ – z,�g(x) –

�g(xn+)〉 ≥ , ∀z ∈ Cn+. Since F ⊂ Cn for each n ≥ , we have

⟨xn+ – w,�g(x) – �g(xn+)

⟩ ≥ , ∀w ∈ F .

Let n → +∞ in the last inequality, we see that 〈p – w,�g(x) –�g(p)〉 ≥ , ∀w ∈ F . In viewof Lemma ., we can obtain p = Projg

F (x). This completes the proof of Theorem .. �

Remark . () If we suppose that Ti is uniformly Li-Lipschitz continuous on E for eachi ∈ N+, then the assumption that Ti is closed and uniformly asymptotically regular on Ecan be removed in Theorem ..


() If we set αn = βn = , uin = yi

n, and Ti = Resgλi

n(i = , , . . . , m) in (.), then (.) can be

rewritten as (.), hence, Theorem . improves and generalizes Theorem RS [].() For the mappings, Theorem . extends the mapping in Theorem RS [] from a fi-

nite family of relatively nonexpansive mapping to a countable family of Bregman totallyquasi-D-asymptotically nonexpansive mappings. Theorem . also removes the assump-tion F̂(T) = F(T) on the mapping T .

Setting ein ≡ for each i ≥ and ∀n ≥ in Theorem ., we have the following.

Corollary . Let E be a reflexive Banach space and g : E → R be a strongly coerciveBregman 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 subsetof 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 bi-function satisfying (C)-(C) and Ti : C → C, ∀i ∈ N be an infinite family of closed and uni-formly Bregman totally quasi-D-asymptotically nonexpansive mappings with nonnegativereal sequences {v(i)

n }, {u(i)n } and a strictly increasing and continuous function ζ : R+ → R+

with ζ () = . If limn→+∞ supi∈N∗{v(i)n } = and limn→+∞ supi∈N∗{u(i)

n } = . Assume that Ti isuniformly asymptotically regular on C for all i ≥ , i.e., limn→+∞ supx∈K ‖Tn+

i x – Tni x‖ =

holds for any bounded subset K of C and F = [⋂+∞

i= F(Ti)]∩ [⋂m

k= GMEP(fk ,ϕk)] �= ∅. For allz, y ∈ C, Gk(z, y) = fk(z, y) + ϕk(y) – ϕk(z) + 〈Akz, y – z〉, TGk

rk,n (x) = {z ∈ C : Gk(z, y) + rk,n

〈y –z,�g(z) – �g(x)〉 ≥ ,∀y ∈ C}. For an initial point x ∈ E, let Ci

= C for each i ≥ andC =

⋂∞i= Ci

and define the sequence {xn} by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩


n)],zi

n = �g∗[βn�g(xn) + ( – βn)�g(Tni xn)],

uin = TGm


r,n TGr,n yi

n,Ci

n+ = {z ∈ Cn : D(z, uin) ≤ αnD(z, xn) + ( – αn)D(z, zi

n) ≤ D(z, xn) + ζ in},

Cn+ =⋂+∞

i= Cin+,

xn+ = ProjgCn+

(x),

where the sequences {ζ in}, {rk,n}, {αn}, {βn} satisfy the following conditions:

() ζ in = v(i)

n · supp∈C ζ [D(p, xn)] + u(i)n for each n ≥ and i ≥ .

() For each k = , , . . . , m, {rk,n}+∞n= ⊂ (, +∞) satisfying lim infn→+∞ rk,n > .

() {αn}, {βn} are real sequences in [, ] such that lim infn→∞( – αn)( – βn)βn > .Then the sequence {xn} converges strongly to Projg

F (x).

Remark . Corollary . improves Theorem . in [], in the following aspects:() For the structure of Banach spaces, we extend the normalized duality mapping to a

more general case, that is, a convex, continuous, and strongly coercive Bregmanfunction which is bounded on bounded subsets and uniformly convex anduniformly smooth on bounded subsets.

() For the mappings, we extend the mapping from two quasi-nonexpansive mappingsto a countable family of Bregman totally quasi-D-asymptotically nonexpansivemappings.


() For generalized mixed equilibrium problems, we extend the problems from one to afinite family.

Setting ζ (t) = t, v(i)n = ki

n – , limn→+∞ kin = , u(i)

n = ein ≡ for each i ≥ in Theorem .,

we have the following.

Corollary . Let E be a reflexive Banach space and g : E → R be a strongly coerciveBregman 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 subsetof 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 bi-function satisfying (C)-(C) and Ti : C → C, ∀i ∈ N be an infinite family of closed andBregman quasi-D-asymptotically nonexpansive mappings with nonnegative real sequences{ki

n}. If limn→+∞ supi∈N∗{kin} = . Assume that Ti is uniformly asymptotically regular on C

for all i ≥ , i.e., limn→+∞ supx∈K ‖Tn+i x – Tn

i x‖ = holds for any bounded subset K of Cand F = [

⋂+∞i= F(Ti)] ∩ [

⋂mk= GMEP(fk ,ϕk)] �= ∅. For all z, y ∈ C, Gk(z, y) = fk(z, y) + ϕk(y) –

ϕk(z) + 〈Akz, y – z〉, TGkrk,n (x) = {z ∈ C : Gk(z, y) +

rk,n〈y – z,�g(z) – �g(x)〉 ≥ ,∀y ∈ C}. For

an initial point x ∈ E, let Ci = C for each i ≥ and C =

⋂∞i= Ci

and define the sequence{xn} by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩


n)],zi


uin = TGm


r,n TGr,n yi

n,Ci


n) ≤ D(z, xn) + ζ in},

Cn+ =⋂+∞

i= Cin+,

xn+ = ProjgCn+

(x),


() ζ in = (ki

n – ) · supp∈C D(p, xn).() For each k = , , . . . , m, {rk,n}+∞

n= ⊂ (, +∞) satisfying lim infn→+∞ rk,n > .() {αn}, {βn} are real sequences in [, ] such that lim infn→∞( – αn)( – βn)βn > .

Then the sequence {xn} converges strongly to ProjgF (x).

Setting v(i)n = u(i)

n = ein ≡ for each i ≥ in Theorem ., we have Corollary ..

Corollary . Let E be a reflexive Banach space and g : E → R be a strongly coerciveBregman 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 subsetof 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 bifunc-tion satisfying (C)-(C) and Ti : C → C, ∀i ∈ N be an infinite family of closed and Bregmanquasi-D-nonexpansive mappings. Assume that Ti is uniformly asymptotically regular onC for all i ≥ , i.e., limn→+∞ supx∈K ‖Tn+

i x – Tni x‖ = holds for any bounded subset K of C

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〉, TGk

rk,n (x) = {z ∈ C : Gk(z, y) + rk,n

〈y – z,�g(z) – �g(x)〉 ≥ ,∀y ∈ C}. Foran initial point x ∈ E, let Ci

= C for each i ≥ and C =⋂∞

i= Ci and define the sequence


{xn} by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩


n)],zi


uin = TGm


r,n TGr,n yi

n,Ci


n) ≤ D(z, xn)},Cn+ =

⋂+∞i= Ci

n+,xn+ = Projg

Cn+(x).

For each k = , , . . . , m, {rk,n}+∞n= ⊂ (, +∞) satisfying lim infn→+∞ rk,n > . {αn}, {βn} are

real sequences in [, ] such that lim infn→∞( – αn)( – βn)βn > . Then the sequence {xn}converges strongly to Projg

F (x).

Setting i = , Ti = T , ein = en, and g(x) = ‖x‖ in Theorem ., we have Corollary ..

Corollary . Let C be a nonempty, closed, and convex subset of a uniformly smoothand uniformly convex Banach space E. For each k = , , . . . , m, let Ak : C → E∗ be acontinuous and monotone mapping, ϕk : C → R be a lower semicontinuous and convexfunctional, let fk : C × C → R be a bifunction satisfying (C)-(C) and T : E → C be aclosed and totally quasi-φ-asymptotically nonexpansive mappings with nonnegative realsequences {vn}, {un} and a strictly increasing and continuous function ζ : R+ → R+ withζ () = . If vn, un → (as n → +∞). Assume that T is uniformly asymptotically regularon E, i.e., limn→+∞ supx∈K ‖Tn+x – Tnx‖ = holds for any bounded subset K of E and F =F(T)∩[

⋂mk= GMEP(fk ,ϕk)] �= ∅. For all z, y ∈ C, Gk(z, y) = fk(z, y)+ϕk(y)–ϕk(z)+〈Akz, y–z〉,

TGkrk,n (x) = {z ∈ C : Gk(z, y) +

rk,n〈y – z, Jz – Jx〉 ≥ ,∀y ∈ C}. For an initial point x ∈ E, let

C = C and define the sequence {xn} by

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

yn = J–[αnJxn + ( – αn)Jzn],zn = J–[βnJxn + ( – βn)J(Tn(xn + en))],un = TGm


r,n TGr,n yn,

Cn+ = {z ∈ Cn : φ(z, un) ≤ αnφ(z, xn) + ( – αn)φ(z, zn) ≤ φ(z, xn) + ζn},xn+ = �Cn+ (x),


() ζn = φ(xn, xn + en) + supp∈C〈xn – p, J(xn + en) – Jxn〉 + vn · supp∈C ζ [φ(p, xn + en)] + un,en ∈ E satisfying limn→+∞ ‖en‖ = for each n ≥ .


() {αn}, {βn} are real sequences in [, ] such that lim infn→∞( – αn)( – βn)βn > .Then the sequence {xn} converges strongly to �C(x).

Letting E be a Hilbert space in Theorem ., we have Corollary ..

Corollary . Let C be a nonempty, closed, and convex subset of real Hilbert spaceE. 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 abifunction satisfying (C)-(C) and Ti : E → C, ∀i ∈ N be an infinite family of closedand totally quasi-asymptotically nonexpansive mappings with nonnegative real sequences


{v(i)n }, {u(i)

n } and a strictly increasing and continuous function ζ : R+ → R+ with ζ () = .If limn→+∞ supi∈N∗{v(i)

n } = and limn→+∞ supi∈N∗{u(i)n } = . Assume that Ti is uniformly

asymptotically regular on E for all i ≥ , i.e., limn→+∞ supx∈K ‖Tn+i x – Tn

i x‖ = holds forany bounded subset K of E and F = [

⋂+∞i= F(Ti)] ∩ [

⋂mk= GMEP(fk ,ϕk)] �= ∅. For all z, y ∈ C,

Gk(z, y) = fk(z, y) + ϕk(y) – ϕk(z) + 〈Akz, y – z〉, TGkrk,n (x) = {z ∈ C : Gk(z, y) +

rk,n〈y – z, z – x〉 ≥

,∀y ∈ C}. For an initial point x ∈ E, let Ci = C for each i ≥ and C =

⋂∞i= Ci

and definethe sequence {xn} by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

yin = αnxn + ( – αn)zi

n,zi

n = βnxn + ( – βn)Tni (xn + ei

n),ui

n = TGmrm,n TGm–

rm–,n · · ·TGr,n TG

r,n yin,

Cin+ = {z ∈ Cn : ‖z – ui

n‖ ≤ αn‖z – xn‖ + ( – αn)‖z – zin‖ ≤ ‖z – xn‖ + ζ i

n},Cn+ =

⋂+∞i= Ci

n+,xn+ = PCn+ (x),


() ζ in = ‖ei

n‖ + supp∈C〈xn – p, ein〉 + v(i)

n · supp∈C ζ [‖p – (xn + ein)‖] + u(i)

n , ein ∈ E

satisfying limn→+∞ supi∈N∗{‖ein‖} = for each n ≥ and i ≥ .


() {αn}, {βn} are real sequences in [, ] such that lim infn→∞( – αn)( – βn)βn > .Then the sequence {xn} converges strongly to PF (x).

Remark . Corollary . improves Theorem . of Martinez-Yanes and Xu [] in thefollowing aspects:

() From a nonexpansive mapping to a countable family of Bregman totallyquasi-D-asymptotically nonexpansive mappings.

() Our algorithms take into account computational errors term.() Considering the generalized mixed equilibrium problems from zero to a finite

family.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors read and approved the final manuscript.

Author details1Department of Mathematics, Shaoxing University, Zhejiang, 312000, China. 2Center for General Education, ChinaMedical University, Taichung, 40402, Taiwan. 3Department of Mathematics, King Abdulaziz University, P.O. Box 80203,Jeddah, 21589, Saudi Arabia.

AcknowledgementsThe research of the first author was partially supported by the National Natural Science Foundation of China (GrantNo. 10971194) and Scientific Research Project of Educational Commission of Zhejiang Province of China (GrantNo. Y20112300). The second author was partially supported by the Grant MOST 103-2923-E-037-001-MY3.

Received: 10 August 2014 Accepted: 10 February 2015

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