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Strong convergence of hybrid Halpern iteration for Bregman totally quasiasymptotically nonexpansive multivalued mappings in reflexive Banach spaces with application
Fixed Point Theory and Applications volume 2014, Article number: 186 (2014)
Abstract
In this paper, Bregman totally quasiasymptotically nonexpansive multivalued mappings in the framework of reflexive Banach spaces are established. Under suitable limit conditions, by using the shrinking projection method introduced by Takahashi, Kubota and Takeuchi, some strong convergence theorems for hybrid Halpern’s iteration for a countable family of Bregman totally quasiasymptotically nonexpansive multivalued mappings are proved. We apply our main results to solve classical equilibrium problems in the framework of reflexive Banach spaces. The main result presented in the paper improves and extends the corresponding result in the work by Chang (Appl. Math. Comput. 2013, doi:10.1016/j.amc.2013.11.074; Appl. Math. Comput. 228:3848, 2014), Suthep (Comput. Math. Appl., 64:489499, 2012), Yi Li (Fixed Point Theory Appl. 2013:197, 2013), Reich and Sabach (Nonlinear Anal. 73:122135, 2010), Nilsrakoo and Saejung (Appl. Math. Comput. 217(14):65776586, 2011), Qin et al. (Appl. Math. Lett. 22:10511055, 2009), Wang et al. (J. Comput. Appl. Math. 235:23642371, 2011), Su et al. (Nonlinear Anal. 73:38903906, 2010) and others.
MSC:47J05, 47H09, 49J25.
1 Introduction
Throughout this paper, we denote by ℕ and ℝ the sets of positive integers and real numbers, respectively. Let D be a nonempty and closed subset of a real Banach space X. Let N(D) and CB(D) denote the family of nonempty subsets and nonempty, closed, and bounded subsets of D, respectively. The Hausdorff metric on CB(D) is defined by
for all {A}_{1},{A}_{2}\in CB(D), where d(x,{A}_{1})=inf\{\parallel xy\parallel ,y\in {A}_{1}\}. The multivalued mapping T:D\to CB(D) is called nonexpansive, if
An element p\in D is called a fixed point of the multivalued mapping T:D\to N(D) if p\in T(p). The set of fixed points of T is denoted by F(T).
In recent years, several types of iterative schemes have been constructed and proposed in order to get strong convergence results for finding fixed points of nonexpansive mappings in various settings. One classical and effective iteration process is defined by
where {\alpha}_{n}\in (0,1). This method was introduced in 1967 by Halpern [10] and is often called Halpern’s iteration. In fact, he proved, in a real Hilbert space, the strong convergence of \{{x}_{n}\} to a fixed point of the nonexpansive mapping T, where {\alpha}_{n}={n}^{a}, a\in (0,1).
Because of the simple construction, Halpern’s iteration is widely used to approximate fixed points of nonexpansive mappings and other classes of nonlinear mappings by mathematicians in different styles [3–42]. In particular, some strong convergence theorems for resolvents of accretive operators in Banach spaces were proved by Reich [11], and he also extended the result of Halpern from Hilbert spaces to uniformly smooth Banach spaces in [12]. In 2012, Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces was introduced and a strong convergence theorem for Bregman strongly nonexpansive mappings by Halpern’s iteration in the framework of reflexive Banach spaces was proved. Recently, a strong convergence theorem for Bregman strongly multivalued nonexpansive mappings as regards Halpern’s iteration in the framework of reflexive Banach spaces was proved by Chang [1, 2], Suthep [3] and Li [4].
The purpose of our work is to introduce a modified Halpern iteration for a countable family of Bregman totally quasiasymptotically nonexpansive multivalued mappings in the framework of reflexive Banach spaces, and to prove strong convergence theorems for these iterations under suitable limit conditions by using the shrinking projection method. We use our results to solve equilibrium problems in the framework of reflexive Banach spaces. The main results presented in the paper improve and extend the corresponding results in the work by Chang [1, 2], Suthep [3], Li [4], and others.
2 Preliminaries
In this section, we recall some basic definitions and results which will be used in the following.
Let X be a real reflexive Banach space with a norm \parallel \cdot \parallel, and let {X}^{\ast} be the dual space of X. Let f:X\to (\mathrm{\infty},+\mathrm{\infty}] be a proper, lower semicontinuous, and convex function. We denote by domf=\{x\in X:f(x)<+\mathrm{\infty}\} the domain of f.
Let x\in intdomf. The subdifferential of f at x is the convex set defined by
The Fenchel conjugate of f is the function {f}^{\ast}:{X}^{\ast}\to (\mathrm{\infty},+\mathrm{\infty}] defined by
We know that the YoungFenchel inequality holds, that is,
Furthermore, equality holds if {x}^{\ast}\in \partial f(x) (see [13]). The set {lev}_{\le}^{f}(r):=\{x\in X:f(x)\le r\} for some r\in \mathbb{R} is called a sublevel of f.
A function f on X is called coercive [14], if the sublevel sets of f are bounded, or equivalently,
A function f on X is said to be strongly coercive [15], if
For any x\in intdomf and y\in X, the righthand derivative of f at x in the direction y is defined by
The function f is said to be Gâteaux differentiable at x, if {lim}_{t\to {0}^{+}}\frac{f(x+ty)f(x)}{t} exists for any y. In this case, {f}^{\circ}(x,y) coincides with \mathrm{\nabla}f(x), the value of the gradient \mathrm{\nabla}f(x) of f at x. The function f is said to be Gâteaux differentiable, if it is Gâteaux differentiable for any x\in intdomf. The function f is said to be Fréchet differentiable at x, if this limit is attained uniformly in \parallel y\parallel =1. Finally, f is said to be uniformly Fréchet differentiable on a subset D of X, if the limit is attained uniformly, for x\in D and \parallel y\parallel =1. It is well known that if f is Gâteaux differentiable (resp. Fréchet differentiable) on intdomf, then f is continuous and its Gâteaux derivative ∇f is normtoweak*, continuous (resp. continuous) on intdomf (see [16, 17]).
Definition 2.1 (cf. [18])
The function f is said to be

(i)
essentially smooth, if ∂f is both locally bounded and singlevalued on its domain;

(ii)
essentially strictly convex, if {(\partial f)}^{1} is locally bounded on its domain and f is strictly convex on every convex subset of dom\partial f;

(iii)
Legendre, if it is both essentially smooth and essentially strictly convex.
Remark 2.1 (cf. [19])
Let X be a reflexive Banach space. Then we have

(a)
f is essentially smooth if and only if {f}^{\ast} is essentially strictly convex;

(b)
{(\partial f)}^{1}=\partial {f}^{\ast};

(c)
f is Legendre if and only if {f}^{\ast} is Legendre;

(d)
If f is Legendre, then ∂f is a bijection which satisfies \mathrm{\nabla}f={(\mathrm{\nabla}{f}^{\ast})}^{1}, ran\mathrm{\nabla}f=dom\mathrm{\nabla}{f}^{\ast}=intdom{f}^{\ast} and ran\mathrm{\nabla}{f}^{\ast}=dom\mathrm{\nabla}f=intdomf.
Examples of Legendre functions can be found in [30]. One important and interesting Legendre function is \frac{1}{p}{\parallel \cdot \parallel}^{p} (0<p<+\mathrm{\infty}) when X is a smooth and strictly convex Banach space. In this case the gradient ∇f of f is coincident with the generalized duality mapping of X, i.e., \mathrm{\nabla}f={J}_{p}. In particular, \mathrm{\nabla}f=I, the identity mapping in Hilbert spaces. In this paper, we always assume that f is Legendre.
The following crucial lemma was proved by Reich and Sabach [20].
Lemma 2.1 (cf. [20])
If f:X\to R is uniformly Fréchet differentiable and bounded on bounded subsets of X, then ∇f is uniformly continuous on bounded subsets of X from the strong topology of X to the strong topology of {X}^{\ast}.
Let f:X\to (\mathrm{\infty},+\mathrm{\infty}] be a convex and Gâteaux differentiable function. The function {D}_{f}:domf\times intdomf\to [0,+\mathrm{\infty}) defined by
is called the Bregman distance with respect to f.
Recall that the Bregman projection [21] of x\in intdomf onto a nonempty, closed, and convex set D\subset domf is the necessarily unique vector {proj}_{D}^{f}(x)\in D (for convenience, here we use {P}_{D}^{f}(x) for {proj}_{D}^{f}(x)) satisfying
The modulus of the total convexity of f at x\in intdomf is the function {v}_{f}(x,t):[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) defined by
The function f is called totally convex at x, if {v}_{f}(x,t)>0 whenever t>0. The function f is called totally convex, if it is totally convex at any point x\in intdomf, and it is said to be totally convex on bounded sets, if {v}_{f}(B,t)>0, for any nonempty bounded subset B of and t>0, where the modulus of the total convexity of the function f on the set B is the function {v}_{f}:intdomf\times [0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) defined by
We know that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets (cf. [22]).
Recall that the function f is said to be sequentially consistent [22], if for any two sequences \{{x}_{n}\} and \{{y}_{n}\} in X such that the first sequence is bounded, the following implication holds:
Recall that the function f is called sequentially consistent, if for any two sequences \{{x}_{n}\} and \{{y}_{n}\} in intdomf and domf, respectively, and \{{x}_{n}\} is bounded, {D}_{f}({y}_{n},x)\to 0, then \parallel {y}_{n}x\parallel \to 0.
The following crucial lemma was proved by Butnariu and Iusem [23].
Lemma 2.2 (cf. [23])
If x\in intdomf, then the following statements are equivalent:

(i)
The function f is totally convex at x.

(ii)
For any sequence \{{y}_{n}\}\subset domf, {D}_{f}({y}_{n},x)\to 0, then \parallel {y}_{n}x\parallel \to 0.
Definition 2.2 (cf. [24])
Let D be a convex subset of intdomf and let T be a multivalued mapping of D. A point p\in D is called an asymptotic fixed point of T if D contains a sequence \{{x}_{n}\} which converges weakly to p such that d({x}_{n},T{x}_{n})\to 0 (as n\to \mathrm{\infty}).
We denote by \stackrel{\u02c6}{F}(T) the set of asymptotic fixed points of T.
Definition 2.3 A multivalued mapping T:D\to N(D) with a nonempty fixed point set is said to be:

(i)
Bregman strongly nonexpansive with respect to a nonempty \stackrel{\u02c6}{F}(T), if
{D}_{f}(p,z)\le {D}_{f}(p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in \stackrel{\u02c6}{F}(T),z\in T(x)and if, whenever \{{x}_{n}\}\subset D is bounded, p\in \stackrel{\u02c6}{F}(T), and {lim}_{n\to \mathrm{\infty}}[{D}_{f}(p,{x}_{n}){D}_{f}(p,{z}_{n})]=0, then {lim}_{n\to \mathrm{\infty}}{D}_{f}({x}_{n},{z}_{n})=0, where {z}_{n}\in T{x}_{n}.

(ii)
Bregman firmly nonexpansive if
\begin{array}{r}\u3008\mathrm{\nabla}f\left({x}^{\ast}\right)\mathrm{\nabla}f\left({y}^{\ast}\right),{x}^{\ast}{y}^{\ast}\u3009\\ \phantom{\rule{1em}{0ex}}\le \u3008\mathrm{\nabla}f(x)\mathrm{\nabla}f(y),{x}^{\ast}{y}^{\ast}\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D,{x}^{\ast}\in Tx,{y}^{\ast}\in Ty.\end{array} 
(iii)
Bregman quasiasymptotically nonexpansive mapping with sequence \{{k}_{n}\}\subset [1,+\mathrm{\infty}), {k}_{n}\to 1 (as n\to \mathrm{\infty}), if \stackrel{\u02c6}{F}(T)=F(T)\ne \mathrm{\varnothing} and
{D}_{f}(p,z)\le {k}_{n}{D}_{f}(p,x),\phantom{\rule{1em}{0ex}}p\in F(T),\mathrm{\forall}z\in {T}^{n}x,x\in D. 
(iv)
Bregman totally quasiasymptotically nonexpansive mapping with nonnegative real sequence \{{v}_{n}\}, \{{\mu}_{n}\}, {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0, if \stackrel{\u02c6}{F}(T)=F(T)\ne \mathrm{\varnothing} and
{D}_{f}(p,z)\le {D}_{f}(p,x)+{\nu}_{n}\zeta ({D}_{f}(p,x))+{\mu}_{n},\phantom{\rule{1em}{0ex}}p\in F(T),\mathrm{\forall}z\in {T}^{n}x,x\in D. 
(v)
Closed, if for any sequence \{{x}_{n}\}\subset D with {x}_{n}\to x\in N(D) and d(T{x}_{n},,y)\to 0 (y\in D), then y\in Tx.
Remark 2.2 (cf. [1])
From these definitions, it is obvious that if \stackrel{\u02c6}{F}(T)=F(T)\ne \mathrm{\varnothing}, then a Bregman strongly nonexpansive multivalued mapping is a Bregman relatively nonexpansive mapping; a Bregman relatively nonexpansive multivalued mapping is a Bregman quasinonexpansive multivalued mapping; a Bregman quasinonexpansive multivalued mapping is a Bregman quasiasymptotically nonexpansive multivalued mapping; a Bregman quasiasymptotically nonexpansive multivalued mapping must be a Bregman totally quasiasymptotically nonexpansive multivalued mapping. However, converses of these statements are not true.
In particular, the existence and approximation of Bregman firmly nonexpansive single value mappings was studied in [24]. It is also known that if T is Bregman firmly nonexpansive and f is Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of X, then F(T)=\stackrel{\u02c6}{F}(T) and F(T) is closed and convex (cf. [24]). It also follows that every Bregman firmly nonexpansive mapping is Bregman strongly nonexpansive with respect to F(T)=\stackrel{\u02c6}{F}(T). The class of singlevalued Bregman totally quasiasymptotically nonexpansive mappings was introduced first in [1]. For a wealth of results concerning this class of mappings (for example, see [1], Examples 2.112.15 and the references therein).
Remark 2.3 Let X be a uniformly smooth and uniformly convex Banach space, and D is nonempty, closed, and convex subset. An operator T:C\to N(D) is called a strongly relatively nonexpansive multivalued mapping on X, if \stackrel{\u02c6}{F}(T)\ne \mathrm{\Phi} and
and, if whenever \{{x}_{n}\}\subset D is bounded, p\in \stackrel{\u02c6}{F}(T), and {lim}_{n\to \mathrm{\infty}}[\varphi (p,{x}_{n})\varphi (p,{z}_{n})]=0, then {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{z}_{n})=0, where {z}_{n}\in T{x}_{n} and \varphi (x,y)={\parallel x\parallel}^{2}2\u3008x,Jy\u3009+{\parallel y\parallel}^{2}.
Now, we give an example of Bregman totally quasiasymptotically nonexpansive multivalued mapping.
Example 2.1 (see [1], Example 2.11)
Let D be a unit ball in a real Hilbert space {l}^{2}, f(x)={\parallel x\parallel}^{2}. Since \mathrm{\nabla}f(y)=2y, the Bregman distance with respect to f
Let T:D\to N(D) be a multivalued mapping defined by
where any \{{a}_{i}\} is a sequence in (0,1) such that {\prod}_{i=2}^{\mathrm{\infty}}{a}_{i}=\frac{1}{2}.
It is proved in Goebal and Kirk [25] that

(i)
\parallel TxTy\parallel \le 2\parallel xy\parallel, \mathrm{\forall}x,y\in D;

(ii)
\parallel {T}^{n}x{T}^{n}y\parallel \le 2{\prod}_{j=2}^{n}{a}_{j}\parallel xy\parallel, \mathrm{\forall}x,y\in D, n\ge 2.
Let \sqrt{{k}_{1}}=2, \sqrt{{k}_{n}}=2{\prod}_{j=2}^{n}{a}_{j}, n\ge 2, then {lim}_{n\to \mathrm{\infty}}{k}_{n}=1. Letting {\nu}_{n}={k}_{n}1 (n\ge 2), \zeta (t)=t (t\ge 0), and \{{\mu}_{n}\} be a nonnegative real sequence with {\mu}_{n}\to 0, then from (i) and (ii) we have
Since D is a unit ball in a real Hilbert space {l}^{2}, it follows from (2.2) that {D}_{f}(x,y)={\parallel xy\parallel}^{2}, \mathrm{\forall}x,y\in D. Above inequality can be written as
Again since 0\in D and 0\in F(T), this implies that F(T)\ne \mathrm{\Phi}. From the above inequality we get
This shows that the mapping T defined as above is a Bregman total quasiasymptotically nonexpansive multivalued mapping.
Let D be a nonempty, closed, and convex subset of X. Let f:X\to \mathbb{R} be a Gâteaux differentiable and totally convex function and x\in X. It is well known from [22] that z={P}_{D}^{f}(x) if and only if
We also know the following characterization:
Let f:X\to \mathbb{R} be a convex, Legendre and Gâteaux differentiable function. Following [31] and [32], we make use of the function {V}_{f}:X\times {X}^{\ast}\to [0,+\mathrm{\infty}) associated with f, which is defined by
Then {V}_{f} is nonnegative and {V}_{f}(x,{x}^{\ast})={D}_{f}(x,\mathrm{\nabla}{f}^{\ast}({x}^{\ast})) for all x\in X and {x}^{\ast}\in {X}^{\ast}. Moreover, by the subdifferential inequality (see [26], Proposition 1(iii), p.1047),
In addition, if f:X\to (\mathrm{\infty},+\mathrm{\infty}] is a proper and lower semicontinuous function, then {f}^{\ast}:{X}^{\ast}\to (\mathrm{\infty},+\mathrm{\infty}] is a proper, weak^{∗} lower semicontinuous and convex function (see [33]). Hence {V}_{f} is convex in the second variable (see [26], Proposition 1(i), p.1047). Thus,
The properties of the Bregman projection and the relative projection operators were studied in [22] and [27].
In 2013, Yi Li and Jinhua Zhu proved the following result, respectively.
Let X be a real reflexive Banach space and let f:X\to (\mathrm{\infty},+\mathrm{\infty}] be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of X. Let D be a nonempty, closed, and convex subset of intdomf. {\alpha}_{n}\in (0,1), {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, and 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1.
(1) (see [28]) Let T:D\to N(D) be a Bregman strongly nonexpansive mapping on X such that F(T)=\stackrel{\u02c6}{F}(T)\ne \mathrm{\varnothing}. Suppose that u\in X and define the sequence \{{x}_{n}\} by
Then \{{x}_{n}\} strongly converges to {P}_{F(T)}^{f}(u).
(2) (see [4]) Let T:D\to N(D) be a Bregman strongly nonexpansive multivalued mapping on X such that F(T)=\stackrel{\u02c6}{F}(T)\ne \mathrm{\varnothing}. Suppose that u\in X and define the sequence \{{x}_{n}\} by
Then \{{x}_{n}\} strongly converges to {P}_{F(T)}^{f}(u).
In 2014, Chang SS proved the following result.
Theorem 2.1 ([1])
Let X be a real uniformly smooth, uniformly convex, and reflexive Banach space, D be a nonempty, closed, and convex subset of X. Let f:D\to (\mathrm{\infty},+\mathrm{\infty}] be a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of D and let \{{T}_{i}\}:D\to D be a family of closed and uniformly Bregman total quasiasymptotically nonexpansive mappings with sequence \{{v}_{n}\}, \{{\mu}_{n}\}, {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and let there be a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0 such that, for each i\ge 1, \{{T}_{i}\} is uniformly {L}_{i}Lipschitz continuous. Let \{{\alpha}_{n}\} be a sequence in [0,1] such that {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0. Let {x}_{n} be a sequence generated by
where {\xi}_{n}={v}_{n}{sup}_{p\in \mathcal{F}}\zeta ({D}_{f}(p,{x}_{n}))+{\mu}_{n}, \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}), {P}_{{D}_{n+1}}^{f} is the Bregman projection of X onto {D}_{n+1}. If ℱ is nonempty and bounded, then \{{x}_{n}\} converges strongly to {P}_{\mathcal{F}}^{f}{x}_{1}.
Definition 2.4

(1)
A countable family of multivalued mappings {\{{T}_{i}:D\to N(D)\}}_{i=1}^{\mathrm{\infty}} is said to be uniformly Bregman totally quasiasymptotically nonexpansive, if \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing} and there exist nonnegative real sequences \{{v}_{n}\}, \{{\mu}_{n}\}, {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0, such that
{D}_{f}(p,{z}_{n,i})\le {D}_{f}(p,x)+{\nu}_{n}\zeta ({D}_{f}(p,x))+{\mu}_{n},\phantom{\rule{1em}{0ex}}p\in F(T),\mathrm{\forall}{z}_{n,i}\in {T}_{i}^{n}x,x\in D.(2.6) 
(2)
A countable family of multivalued mappings {\{{T}_{i}:D\to N(D)\}}_{i=1}^{\mathrm{\infty}} is said to be uniformly Bregman quasiasymptotically nonexpansive, if \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing} and there exist nonnegative real sequences \{{k}_{n}\}\in [1,+\mathrm{\infty}), {k}_{n}\to 1 (as n\to \mathrm{\infty}), such that
{D}_{f}(p,{z}_{n,i})\le {D}_{f}(p,x)+{\nu}_{n}\zeta ({D}_{f}(p,x))+{\mu}_{n},\phantom{\rule{1em}{0ex}}p\in F(T),\mathrm{\forall}{z}_{n,i}\in {T}_{i}^{n}x,x\in D.(2.7) 
(3)
A multivalued mapping T:D\to N(D) is said to be uniformly LLipschitz continuous, if there exists a constant L>0 such that
H({T}^{n}x,{T}^{n}y)\le L\parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D.(2.8)
Now, we improve the above results, and the following main results are obtained.
3 Main results
To prove our main result, we first give the following propositions.
The proof of the following result in the case of singlevalued Bregman totally quasiasymptotically nonexpansive mappings was done in ([1], Lemma 2.6, and [24], Lemma 15.5). In the multivalued case the proof is identical and therefore we will omit the exact details. The interesting reader will consult [1, 24].
Proposition 3.1 Let f:X\to (\mathrm{\infty},+\mathrm{\infty}] be a Legendre function and let D be a nonempty, closed, and convex subset of intdomf. Let T:D\to N(D) be a Bregman totally quasiasymptotically nonexpansive multivalued mapping with respect to f. Then F(T) is closed and convex.
Theorem 3.1 Let X be a real uniformly smooth, uniformly convex, and reflexive Banach space, D be a nonempty, closed, and convex subset of X. Let f:D\to (\mathrm{\infty},+\mathrm{\infty}] be a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of D and let \{{T}_{i}\}:D\to N(D) be a family of closed and uniformly Bregman totally quasiasymptotically nonexpansive multivalued mappings with sequence \{{v}_{n}\}, \{{\mu}_{n}\}, {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}), and let there be a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0 such that, for each i\ge 1, \{{T}_{i}\} are uniformly {L}_{i}Lipschitz continuous. Let \{{\alpha}_{n}\} be a sequence in [0,1] and \{{\beta}_{n}\} be a sequence in (0,1) satisfying the following conditions:

(i)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0;

(ii)
0<{lim}_{n\to \mathrm{\infty}}inf{\beta}_{n}\le {lim}_{n\to \mathrm{\infty}}sup{\beta}_{n}<1.
Let {x}_{n} be a sequence generated by
where {\xi}_{n}={v}_{n}{sup}_{p\in \mathcal{F}}\zeta ({D}_{f}(p,{x}_{n}))+{\mu}_{n}, \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}), {P}_{{D}_{n+1}}^{f} is the Bregman projection of X onto {D}_{n+1}. If ℱ is nonempty and bounded, then \{{x}_{n}\} converges strongly to {P}_{\mathcal{F}}^{f}{x}_{1}.
Proof (I) First, we prove that ℱ and {D}_{n} are closed and convex subsets in D.
In fact, by Proposition 3.1 for each i\ge 1, F({T}_{i}) is closed and convex in D. Therefore ℱ is a closed and convex subset in D. We use the assumption that {D}_{1}=D is closed and convex. Suppose that {D}_{n} is closed and convex for some n\ge 1. In view of the definition of {D}_{f}(\cdot ,\cdot ), we have
This shows that {D}_{n+1} is closed and convex. The conclusions are proved.
(II) Next, we prove that \mathcal{F}\subset {D}_{n}, for all n\ge 1.
In fact, it is obvious that \mathcal{F}\subset {D}_{1}. Suppose that \mathcal{F}\subset {D}_{n} for some n\ge 1.
Letting {\omega}_{n,i}=\mathrm{\nabla}{f}^{\ast}({\beta}_{n}\mathrm{\nabla}f({x}_{n})+(1{\beta}_{n})\mathrm{\nabla}f({w}_{n,i})). Hence for any u\in \mathcal{F}\subset {D}_{n}, by (3.1), we have
and
Therefore, we have
where {\xi}_{n}={v}_{n}{sup}_{p\in \mathcal{F}}\zeta ({D}_{f}(p,{x}_{n}))+{\mu}_{n}. This shows that u\in \mathcal{F}\subset {D}_{n+1} and so \mathcal{F}\subset {D}_{n}. The conclusion is proved.

(III)
Now we prove that \{{x}_{n}\} converges strongly to some point {p}^{\ast}.
Since {x}_{n}={P}_{{D}_{n}}^{f}{x}_{1}, from (2.3), we have
Again since \mathcal{F}\subset {D}_{n}, we have
It follows from (2.4) that, for each u\in \mathcal{F} and for each n\ge 1,
Therefore \{{D}_{f}({x}_{n},{x}_{1})\} is bounded, and so is \{{x}_{n}\}. Since {x}_{n}={P}_{{D}_{n}}^{f}{x}_{1} and {x}_{n+1}={P}_{{D}_{n+1}}^{f}{x}_{1}\in {D}_{n+1}\subset {D}_{n}, we have {D}_{f}({x}_{n},{x}_{1})\le {D}_{f}({x}_{n+1},{x}_{1}). This implies that \{{D}_{f}({x}_{n},{x}_{1})\} is nondecreasing. Hence {lim}_{n\to \mathrm{\infty}}{D}_{f}({x}_{n},{x}_{1}) exists.
By the construction of \{{D}_{n}\}, for any m\ge n, we have {D}_{m}\subset {D}_{n} and {x}_{m}={P}_{{D}_{m}}^{f}{x}_{1}\in {D}_{n}. This shows that
It follows from Lemma 2.2 that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{m}{x}_{n}\parallel =0. Hence \{{x}_{n}\}is a Cauchy sequence in D. Since D is complete, without loss of generality, we can assume that {lim}_{n\to \mathrm{\infty}}{x}_{n}={p}^{\ast} (some point in D).
By the assumption, it is easy to see that

(IV)
Now we prove that {p}^{\ast}\in \mathcal{F}.
Since {x}_{n+1}\in {D}_{n+1}, from (3.1), (3.5), and (3.6), we have
Since {x}_{n}\to {p}^{\ast}, it follows from (2.6) and Lemma 2.2 that for all i\ge 1
Since \{{x}_{n}\} is bounded and \{{T}_{i}\} is a family of uniformly Bregman totally quasiasymptotically nonexpansive multivalued mappings, we have
This implies that \{{w}_{n,i}\} is uniformly bounded.
We have
and this implies that \{{\omega}_{n,i}\} is also uniformly bounded.
In view of {\alpha}_{n}\to 0, from (3.1), we have
for each i\ge 1.
Since \mathrm{\nabla}{f}^{\ast} is uniformly continuous on each bounded subset of {X}^{\ast}, it follows from (3.8) and (3.9) that
for each i\ge 1. Since ∇f is uniformly continuous on each bounded subset of X, we have
By condition (ii), we have
Since J is uniformly continuous, this shows that
for each i\ge 1. Again by the assumptions that \{{T}_{i}\}:D\to D be uniformly {L}_{i}Lipschitz continuous for each i\ge 1, thus we have
for each i\ge 1.
We get {lim}_{n\to \mathrm{\infty}}\parallel H({T}_{i}^{n+1}{x}_{n})H({T}_{i}^{n}{x}_{n})\parallel =0. Since {lim}_{n\to \mathrm{\infty}}{w}_{n,i}={p}^{\ast} and {lim}_{n\to \mathrm{\infty}}{x}_{n}={p}^{\ast}, we have {lim}_{n\to \mathrm{\infty}}d(H({T}_{i}{T}_{i}^{n}{x}_{n}),{p}^{\ast})=0.
In view of the closedness of {T}_{i}, it yields d({T}_{i}{p}^{\ast},{p}^{\ast})=0. Since {p}^{\ast}\in C, {p}^{\ast}\in {T}_{i}{p}^{\ast}, i.e., for each i\ge 1, {p}^{\ast}\in F({T}_{i}). By the arbitrariness of i\ge 1, we have {p}^{\ast}\in \mathcal{F}.

(V)
Finally we prove that {p}^{\ast}={P}_{\mathcal{F}}^{f}{x}_{1} and so {x}_{n}\to {P}_{\mathcal{F}}^{f}{x}_{1}={p}^{\ast}.
Let u={P}_{\mathcal{F}}^{f}{x}_{1}. Since u\in \mathcal{F}\subset {D}_{n} and {x}_{n}={P}_{{D}_{n}}^{f}{x}_{1}, we have {D}_{f}({x}_{n},{x}_{1})\le {D}_{f}(w,{x}_{1}). This implies that
which yields {p}^{\ast}=u={P}_{\mathcal{F}}^{f}{x}_{1}. Therefore, {x}_{n}\to {P}_{\mathcal{F}}^{f}{x}_{1}. The proof of Theorem 3.1 is completed. □
By Remark 2.2, the following corollary is obtained.
Theorem 3.2 Let D, X, \{{\alpha}_{n}\}, \{{\beta}_{n}\}, and f be the same as in Theorem 3.1, Let \{{T}_{i}\}:D\to N(D) be a family of closed and uniformly Bregman quasiasymptotically nonexpansive multivalued mappings with sequence \{{k}_{n}\}\subset [1,+\mathrm{\infty}), {k}_{n}\to 1 (as n\to \mathrm{\infty}) such that, for each i\ge 1, \{{T}_{i}\} be uniformly {L}_{i}Lipschitz continuous. Let \{{\alpha}_{n}\} be a sequence in [0,1] and \{{\beta}_{n}\} be a sequence in (0,1) satisfying the following conditions:

(i)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0;

(ii)
0<{lim}_{n\to \mathrm{\infty}}inf{\beta}_{n}\le {lim}_{n\to \mathrm{\infty}}sup{\beta}_{n}<1.
Let {x}_{n} be a sequence generated by
where {\xi}_{n}=({k}_{n}1){sup}_{p\in \mathcal{F}}\zeta ({D}_{f}(p,{x}_{n})), \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}), {P}_{{D}_{n+1}}^{f} is the Bregman projection of X onto {D}_{n+1}. If ℱ is nonempty, then \{{x}_{n}\} converges strongly to {P}_{\mathcal{F}}^{f}{x}_{1}.
As a direct consequence of Theorem 3.1 and Remark 2.3, we obtain the convergence result concerning Bregman totally quasiasymptotically nonexpansive multivalued mappings in a uniformly smooth and uniformly convex Banach space.
Theorem 3.3 Let X be a uniformly smooth and uniformly convex Banach space and J:X\to {2}^{{X}^{\ast}} is the normalized duality mapping. Let D be a nonempty, closed, and convex subset on X and let T:D\to N(D) be a family of closed and uniformly Bregman totally quasiasymptotically nonexpansive multivalued mappings with sequence \{{v}_{n}\}, \{{\mu}_{n}\}, {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0 such that, for each i\ge 1, \{{T}_{i}\} be uniformly {L}_{i}Lipschitz continuous. Let \{{\alpha}_{n}\} be a sequence in [0,1] and \{{\beta}_{n}\} be a sequence in (0,1) satisfying the following conditions:
(C1) {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty},
(C2) 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1.
Let {x}_{n} be a sequence generated by
where {\xi}_{n}={\nu}_{n}{sup}_{p\in \mathcal{F}}\zeta (\varphi (p,{x}_{n}))+{\mu}_{n}, \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}), {\mathrm{\Pi}}_{{D}_{n+1}} is a projection of X onto {D}_{n+1}. If ℱ is nonempty and bounded, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}.
Now, we provide examples of multivalued mappings to which the results of the paper can be applied.
Example 3.1 Let D be a unit ball in a real Hilbert space {l}^{2}, f(x)={\parallel x\parallel}^{2}. Since \mathrm{\nabla}f(y)=2y, the Bregman distance with respect to f
Let {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}}:D\to N(D) be a family multivalued mapping defined by
where any {\{{a}_{j}\}}_{j=1}^{\mathrm{\infty}} is a sequence in (0,1) such that {\prod}_{j=2}^{\mathrm{\infty}}{a}_{j}=\frac{1}{2}.
From Example 2.1, we know that {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} is a family of closed and uniformly Bregman totally quasiasymptotically nonexpansive multivalued mappings with sequence \{{v}_{n}\}, \{{\mu}_{n}\}, {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0 such that, for each i\ge 1, {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} is uniformly {L}_{i}Lipschitz continuous. \{{\alpha}_{n}\}, \{{\beta}_{n}\} and f are the same as in Theorem 3.1. Let \{{x}_{n}\} be a sequence generated by (3.1), then \{{x}_{n}\} converges strongly to {P}_{\mathcal{F}}^{f}{x}_{1}, where \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}) is nonempty, {P}_{\mathcal{F}}^{f} is the Bregman projection of X onto ℱ.
4 Application
In order to emphasize the importance of Theorem 3.1, we illustrate an application with the following important example, which entails equilibrium problems in the framework of reflexive Banach spaces.
Let X be a smooth, strictly convex, and reflexive Banach space, let D be a nonempty, closed, and convex subset of X and let G:D\times D\to R be a bifunction satisfying the conditions: (A1) G(x,x)=0, for all x\in D; (A2) G(x,y)+G(y,x)\le 0, for any x,y\in D; (A3) for each x,y,z\in D, {lim}_{t\to 0}G(tz+(1t)x,y)\le G(x,y); (A4) for each given x\in D, the function y\mapsto f(x,y) is convex and lower semicontinuous. The ‘socalled’ equilibrium problem for G is to find a {x}^{\ast}\in D such that G({x}^{\ast},y)\ge 0, for each y\in D. The set of its solutions is denoted by EP(G).
The resolvent of a bifunction G [5] is the operator Re{s}_{G}^{f}:X\to {2}^{D} defined by
If f:X\to (\mathrm{\infty},+\mathrm{\infty}] is a strongly coercive and Gâteaux differentiable function, and G satisfies conditions (A1)(A4), then dom(Re{s}_{G}^{f})=X (see [5]). We also know:

(1)
Re{s}_{G}^{f} is singlevalued;

(2)
Re{s}_{G}^{f} is a Bregman firmly nonexpansive mapping, so it is a closed Bregman total quasiasymptotically nonexpansive mapping;

(3)
F(Re{s}_{G}^{f})=EP(G);

(4)
EP(G) is a closed and convex subset of D;

(5)
for all x\in X and for all p\in F(Re{s}_{G}^{f}), we have
{D}_{f}(p,Re{s}_{G}^{f}(x))+{D}_{f}(Re{s}_{G}^{f}(x),x)\le {D}_{f}(p,x).(4.2)
In addition, by Reich and Sabach [24], if f is uniformly Fréchet differentiable and bounded on bounded subsets of X, then we see that F(Re{s}_{G}^{f})=\stackrel{\u02c6}{F}(Re{s}_{G}^{f})=EP(G) is closed and convex. Hence, by replacing T=Re{s}_{G}^{f} in Theorem 3.1, we obtain the following result.
Theorem 4.1 Let D, X, \{{\alpha}_{n}\}, \{{\beta}_{n}\}, and f be the same as in Theorem 3.1. Let \{{G}_{i}:D\times D\to R\} be a countable families of bifunction which satisfies the conditions (A1)(A4) such that EP({G}_{i})\ne \mathrm{\varnothing}. Let Re{s}_{{G}_{i}}^{f}(x):D\to {2}^{D}, i=1,2,\dots , be the family of mappings defined by
Let the sequence \{{x}_{n}\} be defined by
where {\xi}_{n}=({k}_{n}1){sup}_{p\in \mathcal{F}}\zeta ({D}_{f}(p,{x}_{n})), \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F(Re{s}_{{G}_{i}}^{f}), {P}_{{D}_{n+1}}^{f} is the Bregman projection of X onto {D}_{n+1}. If ℱ is nonempty, then \{{x}_{n}\} converges strongly to {P}_{\mathcal{F}}^{f}{x}_{1}, which is a common solution of the system of equilibrium problems for {G}_{m}, m=1,2,\dots .
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The authors are very grateful to both reviewers for careful reading of this paper and for their comments.
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Li, Y., Liu, H. Strong convergence of hybrid Halpern iteration for Bregman totally quasiasymptotically nonexpansive multivalued mappings in reflexive Banach spaces with application. Fixed Point Theory Appl 2014, 186 (2014). https://doi.org/10.1186/168718122014186
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DOI: https://doi.org/10.1186/168718122014186