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Strong convergence results of twosteps modifying Halpern’s iteration for Bregman strongly nonexpansive multivalued mappings in reflexive Banach spaces with application
Journal of Inequalities and Applications volume 2014, Article number: 412 (2014)
Abstract
In this paper, a twosteps modifying Halpern iteration for Bregman strongly nonexpansive multivalued mappings in the framework of reflexive Banach spaces is established. Under suitable limit conditions, some strong convergence theorems for this iteration are proved. We apply our main results to solve classical 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 Suthep et al. (Comput. Math. Appl. 64:489499, 2012), Li et al. (Fixed Point Theory Appl. 2013:197, 2013) and Chang and Wang (Appl. Math. Comput. 228:3848, 2014).
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 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). Such a method was introduced in 1967 by Halpern [1] and is often called Halpern’s iteration. In fact, he proved, in a real Hilbert space, 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 a 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 [1–25]. Reich [6] also extended the result of Halpern from Hilbert spaces to uniformly smooth Banach spaces. In 2012, Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces was introduced and strong convergence theorem for Bregman strongly nonexpansive mappings by Halpern’s iteration in the framework of reflexive Banach spaces were proved. Recently, strong convergence theorem for Bregman strongly multivalued nonexpansive mappings about Halpern’s iteration in the framework of reflexive Banach spaces was proved by Suthep et al. [2], Li et al. [3], and Chang and Wang [4].
The purpose of our work is continue to introduce twosteps modifying Halpern’s iteration for Bregman strongly nonexpansive multivalued mappings in the framework of reflexive Banach spaces and to prove strong convergence theorems for this iterations. 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 Suthep et al. [2], Li et al. [3], and Chang and Wang [4].
2 Preliminaries
In the sequel, we begin by recalling some preliminaries and lemmas which will be used in our proofs. Now we first collect some basic concepts. 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 domf=\{x\in X:f(x)<+\mathrm{\infty}\} as 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). 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 [8], if the sublevel sets of f are bounded, equivalently,
A function f on X is said to be strongly coercive [9], 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 known that if f is Gâteaux differentiable (resp. Fréchet differentiable) on intdomf, then f is continuous and its Gâteaux derivative ∇f is normtoweak^{∗}, continuous on intdomf (see [10] and [11]).
Definition 2.1 (cf. [12])
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. [26])
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 [13]. 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 ReichSabach [14].
Lemma 2.1 (cf. [14])
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 [15] 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 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 and t>0, where the modulus of totally 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 (see [16]).
Recall that the function f is said to be sequentially consistent [16], if for any two sequences \{{x}_{n}\} and \{{y}_{n}\} in X such that the first sequence is bounded, the following implication holds:
The following crucial lemma was proved by ButnariuIusem [17].
Lemma 2.2 (cf. [17])
The function f is totally convex on bounded sets if and only if it is sequentially consistent.
Definition 2.2 (cf. [25])
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
\u3008\mathrm{\nabla}f\left({x}^{\ast}\right)\mathrm{\nabla}f\left({y}^{\ast}\right),{x}^{\ast}{y}^{\ast}\u3009\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.
In particular, the existence and approximation of Bregman firmly nonexpansive single value mappings was studied in [25]. 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 (see [25]). 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 strongly nonexpansive mappings was introduced first in [27]. There is a wealth of results concerning this class of mappings (for example, see [28–32] and the references therein).
Remark 2.2 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}.
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 known from [16] 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 [18] and [19], 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 [32], 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 [21]). Hence {V}_{f} is convex in the second variable (see [32], Proposition 1(i), p.1047). Thus,
The properties of the Bregman projection and the relative projection operators were studied in [16] and [20].
The following lemmas give some nice properties of sequences of real numbers which will be useful for the forthcoming analysis.
Lemma 2.3 (cf. [23], Lemma 2.1, p.76)
Let \{{\alpha}_{n}\} be a sequence of real numbers such that there exists a nondecreasing subsequence {\alpha}_{{n}_{i}} of {\alpha}_{n}, that is, {\alpha}_{{n}_{i}}\le {\alpha}_{{n}_{i}+1} for all i\in N. Then there exists a nondecreasing subsequence \{{m}_{k}\}\subset N, such that {m}_{k}\to \mathrm{\infty} and the following properties are satisfied for all (sufficiently large number sequences, k\subset N):
In fact, {m}_{k}=max\{j\le k:{\alpha}_{j}\le {\alpha}_{j+1}\}.
Lemma 2.4 (see [2], Lemma 2.5, p.493)
Assume that \{{\alpha}_{n}\} is a sequence of nonnegative real numbers such that
where \{{\gamma}_{n}\} is a sequence in (0,1) and \{{\delta}_{n}\} is a sequence such that

(a)
{lim}_{n\to \mathrm{\infty}}{\gamma}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty};

(b)
lim{sup}_{n\to \mathrm{\infty}}{\delta}_{n}\le 0.
Then {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0.
In 2013, Yi Li and Jinhua Zhu proved the following results, 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 [5]) 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
{x}_{1}\in D,\phantom{\rule{1em}{0ex}}{x}_{n+1}=\mathrm{\nabla}{f}^{\ast}({\alpha}_{n}\mathrm{\nabla}f(u)+(1{\alpha}_{n})({\beta}_{n}\mathrm{\nabla}f({x}_{n})+(1{\beta}_{n})\mathrm{\nabla}f(T{x}_{n}))),\phantom{\rule{1em}{0ex}}n\ge 1.Then \{{x}_{n}\} strongly converges to {P}_{F(T)}^{f}(u).

(2)
(see [3]) 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
{x}_{1}\in D,\phantom{\rule{1em}{0ex}}{x}_{n+1}=\mathrm{\nabla}{f}^{\ast}({\alpha}_{n}\mathrm{\nabla}f(u)+(1{\alpha}_{n})\mathrm{\nabla}f({z}_{n})),\phantom{\rule{1em}{0ex}}{z}_{n}\in T{x}_{n},n\ge 1.Then \{{x}_{n}\} strongly converges to {P}_{F(T)}^{f}(u).
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 two propositions.
Proposition 3.1 (see [3], Proposition 3.1)
Let D be a nonempty, closed, and convex subset of a real reflexive Banach space X. Let f:X\to R be a Gâteaux differentiable and totally convex function, and let T:D\to N(D) be a multivalued mapping such that F(T)=\stackrel{\u02c6}{F}(T) is nonempty, closed, and convex. Suppose that u\in D and \{{x}_{n}\} is a bounded sequence in D such that {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0. Then
The proof of the following result in the case of singlevalued Bregman firmly nonexpansive mappings was done in ([25], Lemma 15.5, p.305). In the multivalued case the proof is identical and therefore we will omit the exact details. The interested reader may consult [25].
Proposition 3.2 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 firmly nonexpansive multivalued mapping with respect to f. Then F(T) is closed and convex.
We are now in a position to prove our main convergence results. We modify Halpern’s iteration for finding a fixed point of a Bregman strongly nonexpansive mapping in a real reflexive Banach spaces.
Theorem 3.1 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 and 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, {x}_{1}\in D, and define the sequence \{{x}_{n}\} by
where {\alpha}_{n},{\beta}_{n}\in (0,1) satisfy
(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.
Then the sequence \{{x}_{n}\} strongly converges to {P}_{F(T)}^{f}(u).
Proof First, by Proposition 3.2, we know that F(T) is closed and convex. Let p={P}_{F(T)}^{f}(u)\in F(T)=\stackrel{\u02c6}{F}(T). Since {y}_{n}=\mathrm{\nabla}{f}^{\ast}({\beta}_{n}\mathrm{\nabla}f({x}_{n})+(1{\beta}_{n})\mathrm{\nabla}f({z}_{n})), then, by using (2.2),
Since {x}_{n+1}=\mathrm{\nabla}{f}^{\ast}({\alpha}_{n}\mathrm{\nabla}f(u)+(1{\alpha}_{n})\mathrm{\nabla}f({y}_{n})), we obtain
By the induction, the sequence {D}_{f}(p,{x}_{n}) is bounded and hence {D}_{f}(p,{y}_{n}) is bounded.
Next, we show that the sequence \{{x}_{n}\} is also bounded. We follow the proof in [25]. Since {D}_{f}(p,{x}_{n}) is bounded, there exists L>0 such that
Hence \{\mathrm{\nabla}f({x}_{n})\} is contained in the sublevel set {lev}_{\le}^{\psi}(Lf(p)), where \psi ={f}^{\ast}\u3008\cdot ,p\u3009. Since f is lower semicontinuous, {f}^{\ast} is weak^{∗} lower semicontinuous. Hence the function ψ is coercive. This shows that \{\mathrm{\nabla}f({x}_{n})\} is bounded. Since f is strongly coercive, {f}^{\ast} is bounded on bounded sets (see [12]). Hence \mathrm{\nabla}{f}^{\ast} is also bounded on bounded subsets of {E}^{\ast} (see [17]). Since f is a Legendre function, it follows that {x}_{n}=\mathrm{\nabla}{f}^{\ast}(\mathrm{\nabla}f({x}_{n})), n\in \mathbb{N}, is bounded. Therefore \{{x}_{n}\} is bounded. So are \{{y}_{n}\}, \{{z}_{n}\}, \{\mathrm{\nabla}f({y}_{n})\}, and \{\mathrm{\nabla}f({z}_{n})\}.
We next show that if there exists a subsequence \{{x}_{{n}_{k}}\}\subset \{{x}_{n}\} such that
then
where {z}_{{n}_{k}}\in T{x}_{{n}_{k}}.
Since \{\mathrm{\nabla}f({y}_{n})\} is bounded, we have, from (3.2),
Since f is strongly coercive and uniformly convex on bounded subsets of X, {f}^{\ast} is uniformly Fréchet differentiable on bounded subsets of {X}^{\ast} (see [9]). Moreover, {f}^{\ast} is bounded on bounded sets. Since f is Legendre, by Lemma 2.1, we obtain
On the other hand, since f is uniformly Fréchet differentiable on bounded subsets of X, f is uniformly continuous on bounded subsets of X (see [33]). It follows that
The following equality holds:
It follows from (3.3), (3.4), and (3.5) that
Now, by virtue of condition (C2), we have
Because T is a Bregman strongly nonexpansive multivalued mapping,
The rest of the proof will be divided into two parts.
Case 1. Suppose \{{D}_{f}(p,{x}_{n})\} is eventually decreasing, i.e. there exists a sufficiently large k>0 such that {D}_{f}(p,{x}_{n})>{D}_{f}(p,{x}_{n+1}) for all n>k. In this case {lim}_{n\to \mathrm{\infty}}{D}_{f}(p,{x}_{n}) exists. In this situation, we see that {lim}_{n\to \mathrm{\infty}}{D}_{f}(p,{x}_{n}) exists. This shows that {lim}_{n\to \mathrm{\infty}}({D}_{f}(p,{x}_{n}){D}_{f}(p,{x}_{n+1}))=0 and hence {lim}_{n\to \mathrm{\infty}}({D}_{f}(p,{z}_{n}){D}_{f}(p,{x}_{n}))=0.
Since T is a Bregman strongly nonexpansive multivalued mapping, then
Since f is totally convex on bounded subsets of E, by Lemma 2.2, we have
From (2.2), we have
and
So are {lim}_{n\to \mathrm{\infty}}\parallel {z}_{n}{x}_{n}\parallel =0 and {lim}_{n\to \mathrm{\infty}}\parallel {y}_{n}{x}_{n+1}\parallel =0.
By Proposition 3.1, we obtain
Finally, we show that {x}_{n}\to p as n\to \mathrm{\infty}. Indeed
By Lemma 2.4, we conclude that {lim}_{n\to \mathrm{\infty}}{D}_{f}(p,{x}_{n})=0. Therefore, by Lemma 2.2, since f is totally convex on bounded subsets of X, we obtain {x}_{n}\to p as n\to \mathrm{\infty}.
Case 2. If \{{D}_{f}(p,{x}_{n})\} is not eventually decreasing, there exists a subsequence \{{D}_{f}(p,{x}_{{n}_{j}})\}\subset \{{D}_{f}(p,{x}_{n})\} such that {D}_{f}(p,{x}_{{n}_{j}})<{D}_{f}(p,{x}_{{n}_{j}+1}) for all j\in N. By Lemma 2.3, there exists a strictly increasing sequence \{{m}_{k}\} of positive integers such that the following properties by all k\in N:
Since inequality {D}_{f}(p,{z}_{n}){D}_{f}(p,{x}_{n})\le 0 holds by Definition 2.3, hence, by Lemma 2.3, we have
This implies that
Following the proof of Case 1, we have
and
This implies that
Hence
Using this and (3.5) together, we conclude that
The proof of Theorem 3.1 is now completed. □
As a direct consequence of Theorem 3.1 and Remark 2.2, we obtain the convergence result concerning strongly relatively nonexpansive multivalued mappings in a uniformly smooth and uniformly convex Banach space.
Corollary 3.1 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 strongly relatively nonexpansive multivalued mapping on X such that F(T)=\stackrel{\u02c6}{F}(T)\ne \mathrm{\varnothing}. Suppose that u\in D and define the sequence \{{x}_{n}\} as follows: {x}_{1}\in D and
where {\alpha}_{n},{\beta}_{n}\in (0,1) satisfy
(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.
Then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)}u, where {\mathrm{\Pi}}_{F(T)} is the generalized projection onto F(T).
4 Application
In order to emphasize the importance of Theorem 3.1, we illustrate an application with the following important example, which concerns the 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 [28] is the operator {Res}_{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({Res}_{G}^{f})=X (see [28]). We also know:

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

(2)
{Res}_{G}^{f} is a Bregman firmly nonexpansive mapping;

(3)
F({Res}_{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({Res}_{G}^{f}), we have
{D}_{f}(p,{Res}_{G}^{f}(x))+{D}_{f}({Res}_{G}^{f}(x),x)\le {D}_{f}(p,x).(4.2)
In addition, by ReichSabach [25], if f is uniformly Fréchet differentiable and bounded on bounded subsets of X, then we find that F({Res}_{G}^{f})=\stackrel{\u02c6}{F}({Res}_{G}^{f})=EP(G) is closed and convex. Hence, by replacing T={Res}_{G}^{f} in Theorem 3.1, we obtain the following result.
Theorem 4.1 Let D be a nonempty, closed, and convex subset of a real reflexive Banach space X. Let f be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of X. Let G:D\times D\to R be a bifunction which satisfies the conditions (A1)(A4) such that EP(G)\ne \mathrm{\varnothing}. Suppose that u\in X and define the sequence \{{x}_{n}\} by
with {\alpha}_{n},{\beta}_{n}\in (0,1) satisfying
(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.
Then \{{x}_{n}\} converges strongly to {P}_{EP(G)}^{f}u. Here {P}_{EP(G)}^{f}u is the Bregman projection of X onto EP(G).
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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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Liu, H., Li, Y. Strong convergence results of twosteps modifying Halpern’s iteration for Bregman strongly nonexpansive multivalued mappings in reflexive Banach spaces with application. J Inequal Appl 2014, 412 (2014). https://doi.org/10.1186/1029242X2014412
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DOI: https://doi.org/10.1186/1029242X2014412
Keywords
 Bregman strongly nonexpansive multivalued mapping
 Legendre functions
 Bregman projection
 fixed point
 twosteps modifying Halpern’s iteration