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Hybrid methods for a mixed equilibrium problem and fixed points of a countable family of multivalued nonexpansive mappings
Fixed Point Theory and Applications volume 2013, Article number: 236 (2013)
Abstract
In this paper, we prove a strong convergence theorem for a new hybrid method, using shrinking projection method introduced by Takahashi and a fixed point method for finding a common element of the set of solutions of mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We also apply our main result to the convex minimization problem and the fixed point problem of a countable family of multivalued nonexpansive mappings.
MSC:47H09, 47H10.
1 Introduction
The mixed equilibrium problem (MEP) includes several important problems arising in optimization, economics, physics, engineering, transportation, network, Nash equilibrium problems in noncooperative games, and others. Variational inequalities and mathematical programming problems are also viewed as the abstract equilibrium problems (EP) (e.g., [1, 2]). Many authors have proposed several methods to solve the EP and MEP, see, for instance, [1–9] and the references therein.
Fixed point problems for multivalued mappings are more difficult than those of singlevalued mappings and play very important role in applied science and economics. Recently, many authors have proposed their fixed point methods for finding a fixed point of both multivalued mapping and a family of multivalued mappings. All of those methods have only weak convergence.
It is known that Mann’s iterations have only weak convergence even in the Hilbert spaces. To overcome this problem, Takahashi [10] introduced a new method, known as shrinking projection method, which is a hybrid method of Mann’s iteration, and the projection method, and obtained strong convergence results of such method. In this paper, we use the shrinking projection method defined by Takahashi [10] and our new method to define a new hybrid method for MEP and a fixed point problem for a family of nonexpansive multivalued mappings.
An element p\in K is called a fixed point of a singlevalued mapping T if p=Tp and of a multivalued mapping T if p\in Tp. The set of fixed points of T is denoted by F(T).
Let X be a real Banach space. A subset K of X is called proximinal if for each x\in X, there exists an element k\in K such that
where d(x,K)=inf\{\parallel xy\parallel :y\in K\} is the distance from the point x to the set K.
Let X be a uniformly convex real Banach space, and let K be a nonempty closed convex subset of X, and let CB(K) be a family of nonempty closed bounded subsets of K, and let P(K) be a nonempty proximinal bounded subsets of K.
For multivalued mappings T:K\to P(K), define {P}_{T}(x):=\{y\in T(x):\parallel xy\parallel =d(x,T(x))\} for all x\in K.
The Hausdorff metric on CB(X) is defined by
for all A,B\in CB(X).
A multivalued mapping T:K\to CB(K) is said to be nonexpansive if H(Tx,Ty)\le \parallel xy\parallel for all x,y\in K.
Let H be a real Hilbert space with the inner product \u3008\cdot ,\cdot \u3009 and the norm \parallel \cdot \parallel. Let D be a nonempty closed convex subset of H. Let F:D\times D\to \mathbb{R} be a bifunction, and let \phi :D\to \mathbb{R}\cup \{+\mathrm{\infty}\} be a function such that D\cap dom\phi \ne \mathrm{\varnothing}, where ℝ is the set of real numbers and dom\phi =\{x\in H:\phi (x)<+\mathrm{\infty}\}.
FloresBazán [11] introduced the following mixed equilibrium problem:
The set of solutions of (1.1) is denoted by MEP(F,\phi ).
If \phi \equiv 0, then the mixed equilibrium problem (1.1) reduces to the following equilibrium problem:
The set of solutions of (1.2) is denoted by EP(F) (see Combettes and Hirstoaga [12]).
If F\equiv 0, then the mixed equilibrium problem (1.1) reduces to the following convex minimization problem:
The set of solutions of (1.3) is denoted by CMP(\phi ).
In an infinitedimensional Hilbert space, the Mann iteration algorithms have only a weak convergence. In 2003, Nakajo and Takahashi [13] introduced the method, called CQ method, to modify Mann’s iteration to obtain the strong convergence theorem for nonexpansive mapping in a Hilbert space. The CQ method has been studied extensively by many authors, for instance, Marino and Xu [14]; Zhou [15]; Zhang and Cheng [16].
In 2008, Takahashi et al. [10] introduced the following iteration scheme, which is usually called the shrinking projection method. Let \{{\alpha}_{n}\} be a sequence in (0,1) and {x}_{0}\in H. For {C}_{1}=C and {x}_{1}={P}_{{C}_{1}}{x}_{0}, define a sequence \{{x}_{n}\} of D as follows:
where {P}_{{C}_{n}} is the metric projection of H onto {C}_{n} and \{{T}_{n}\} is a family of nonexpansive mappings. They proved that the sequence \{{x}_{n}\} converges strongly to z={P}_{F(T)}{x}_{0}, where F(T)={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}). The shrinking projection method has been studied widely by many authors, for example, Tada and Takahashi [17]; Aoyama et al. [18]; Yao et al. [19]; Kang et al. [20]; Cholamjiak and Suantai [21]; Ceng et al. [22]; Tang et al. [23]; Cai and Bu [24]; Kumam et al. [25]; Kimura et al. [26]; Shehu [27, 28]; Wang et al. [29].
In 2009, Wangkeeree and Wangkeeree [30] proved a strong convergence theorem of an iterative algorithm based on extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of common fixed points of a family of infinitely nonexpansive mappings and the set of the variational inequality for a monotone Lipschitz continuous mapping in a Hilbert space.
In 2011, Rodjanadid [31] introduced another iterative method modified from an iterative scheme of Klineam and Suantai [32] for finding a common element of the set of solutions of mixed equilibrium problems and the set of common fixed points of countable family of nonexpansive mappings in real Hilbert spaces. The mixed equilibrium problems have been studied by many authors, for instance, Peng and Yao [33]; Zeng et al. [34]; Peng et al. [35]; Wangkeeree and Kamraksa [36]; Jaiboon and Kumam [37]; Chamnarnpan and Kumam [38]; Cholamjiak et al. [39].
Nadler [40] started to study fixed points of multivalued contractions and nonexpansive mapping by using the Hausdorff metric.
Sastry and Babu [41] defined Mann and Ishikawa iterates for a multivalued map T with a fixed point p, and proved that these iterates converge strongly to a fixed point q of T under the compact domain in a real Hilbert space. Moreover, they illustrated that fixed point q may be different from p.
Panyanak [42] generalized results of Sastry and Babu [41] to uniformly convex Banach spaces and proved a strong convergence theorem of Mann iterates for a mapping defined on a noncompact domain and satisfying some conditions. He also obtained a strong convergence result of Ishikawa iterates for a mapping defined on a compact domain.
Hussain and Khan [43], in 2003, introduced the best approximation operator {P}_{T} to find fixed points of ∗nonexpansive multivalued mapping and proved strong convergence of its iterates on a closed convex unbounded subset of a Hilbert space, which is not necessarily compact.
Hu et al. [44] obtained common fixed point of two nonexpansive multivalued mappings satisfying certain contractive conditions.
Cholamjiak and Suantai [45] proved strong convergence theorems of two new iterative procedures with errors for two quasinonexpansive multivalued mappings by using the best approximation operator and the end point condition in uniformly convex Banach spaces. Later, Cholamjiak et al. [46] introduced a modified Mann iteration and obtained weak and strong convergence theorems for a countable family of nonexpansive multivalued mappings by using the best approximation operator in a Banach space. They also gave some examples of multivalued mappings T such that {P}_{T} are nonexpansive.
Later, Eslamian and Abkar [47] generalized and modified the iteration of Abbas et al. [48] from two mappings to the infinite family of multivalued mappings \{{T}_{i}\} such that each {P}_{{T}_{i}} satisfies the condition (C).
In this paper, we introduce a new hybrid method for finding a common element of the set of solutions of a mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by the proposed method without the assumption of compactness of the domain and other conditions imposing on the mappings.
In Section 2, we give some preliminaries and lemmas, which will be used in proving the main results. In Section 3, we introduce a new hybrid method and a fixed point method defined by (3.1) and prove strong convergence theorem for finding a common element of the set of solutions between mixed equilibrium problem and common fixed point problems of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We also give examples of the control sequences satisfying the control conditions in main results. In Section 4, we summarize the main results of this paper.
2 Preliminaries
Let D be a closed convex subset of H. For every point x\in H, there exists a unique nearest point in D, denoted by {P}_{D}x, such that
{P}_{D} is called the metric projection of H onto D. It is known that {P}_{D} is a nonexpansive mapping of H onto D. It is also know that {P}_{D} satisfies \u3008xy,{P}_{D}x{P}_{D}y\u3009\ge {\parallel {P}_{D}x{P}_{D}y\parallel}^{2} for every x,y\in H. Moreover, {P}_{D}x is characterized by the properties: {P}_{D}x\in D and \u3008x{P}_{D}x,{P}_{D}xy\u3009\ge 0 for all y\in D.
Lemma 2.1 [13]
Let D be a nonempty closed convex subset of a real Hilbert space H and {P}_{D}:H\to D be the metric projection from H onto D. Then the following inequality holds:
Lemma 2.2 [14]
Let H be a real Hilbert space. Then the following equations hold:

(i)
{\parallel xy\parallel}^{2}={\parallel x\parallel}^{2}{\parallel y\parallel}^{2}2\u3008xy,y\u3009, \mathrm{\forall}x,y\in H;

(ii)
{\parallel tx+(1t)y\parallel}^{2}=t{\parallel x\parallel}^{2}+(1t){\parallel y\parallel}^{2}t(1t){\parallel xy\parallel}^{2}, \mathrm{\forall}t\in [0,1] and x,y\in H.
Lemma 2.3 [21]
Let H be a real Hilbert space. Then for each m\in \mathbb{N}
{x}_{i}\in H and {t}_{i},{t}_{j}\in [0,1] for all i,j=1,2,\dots ,m with {\sum}_{i=1}^{m}{t}_{i}=1.
Lemma 2.4 [49]
Let D be a nonempty closed and convex subset of a real Hilbert space H. Given x,y,z\in H and also given a\in \mathbb{R}, the set
is convex and closed.
For solving the mixed equilibrium problem, we assume the bifunction F, φ and the set D satisfy the following conditions:

(A1)
F(x,x)=0 for all x\in D;

(A2)
F is monotone, that is, F(x,y)+F(y,x)\le 0 for all x,y\in D;

(A3)
for each x,y,z\in D, {lim\hspace{0.17em}sup}_{t\downarrow 0}F(tz+(1t)x,y)\le F(x,y);

(A4)
F(x,\cdot ) is convex and lower semicontinuous for each x\in D;

(B1)
for each x\in H and r>0, there exist a bounded subset {D}_{x}\subseteq D and {y}_{x}\in D\cap dom\phi such that for any z\in D\setminus {D}_{x},
F(z,{y}_{x})+\phi ({y}_{x})+\frac{1}{r}\u3008{y}_{x}z,zx\u3009<\phi (z); 
(B2)
D is a bounded set.
Lemma 2.5 [35]
Let D be a nonempty closed and convex subset of a real Hilbert space H. Let F:D\times D\to \mathbb{R} be a bifunction satisfying conditions (A1)(A4) and \phi :D\to \mathbb{R}\cup \{+\mathrm{\infty}\} be a proper lower semicontinuous and convex function such that D\cap dom\phi \ne \mathrm{\varnothing}. For r>0 and x\in D, define a mapping {T}_{r}:H\to D as follows:
for all x\in H. Assume that either (B1) or (B2) holds. Then the following conclusions hold:

(1)
for each x\in H, {T}_{r}(x)\ne \mathrm{\varnothing};

(2)
{T}_{r} is singlevalued;

(3)
{T}_{r} is firmly nonexpansive, that is, for any x,y\in H,
{\parallel {T}_{r}(x){T}_{r}(y)\parallel}^{2}\le \u3008{T}_{r}(x){T}_{r}(y),xy\u3009; 
(4)
F({T}_{r})=MEP(F,\phi );

(5)
MEP(F,\phi ) is closed and convex.
As in ([21], Lemma 2.7), the following lemma holds true for multivalued mapping. To avoid repetition, we omit the details of proof.
Lemma 2.6 Let D be a closed and convex subset of a real Hilbert space H. Let T:D\to P(D) be a multivalued nonexpansive mapping with F(T)\ne \mathrm{\varnothing} such that {P}_{T} is nonexpansive. Then F(T) is a closed and convex subset of D.
3 Main results
In the following theorem, we prove strong convergence of the sequence \{{x}_{n}\} defined by (3.1) to a common element of the set of solutions of a mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings.
Theorem 3.1 Let D be a nonempty closed and convex subset of a real Hilbert space H. Let F be a bifunction from D\times D to ℝ satisfying (A1)(A4), and let φ be a proper lower semicontinuous and convex function from D to \mathbb{R}\cup \{+\mathrm{\infty}\} such that D\cap dom\phi \ne \mathrm{\varnothing}. Let {T}_{i}:D\to P(D) be multivalued nonexpansive mappings for all i\in \mathbb{N} with \mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap MEP(F,\phi )\ne \mathrm{\varnothing} such that all {P}_{{T}_{i}} are nonexpansive. Assume that either (B1) or (B2) holds and \{{\alpha}_{n,i}\}\subset [0,1) satisfies the condition {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,i}{\alpha}_{n,0}>0 for all i\in \mathbb{N}. Define the sequence \{{x}_{n}\} as follows: {x}_{1}\in D={C}_{1},
where the sequences {r}_{n}\in (0,\mathrm{\infty}) with {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0 and \{{\alpha}_{n,i}\}\subset [0,1) satisfying {\sum}_{i=0}^{n}{\alpha}_{n,i}=1 and {x}_{n,i}\in {P}_{{T}_{i}}{u}_{n} for i\in \mathbb{N}. Then the sequence \{{x}_{n}\} converges strongly to {P}_{\mathrm{\Omega}}{x}_{0}.
Proof We split the proof into six steps.
Step 1. Show that {P}_{{C}_{n+1}}{x}_{0} is well defined for every {x}_{0}\in D.
By Lemmas 2.52.6, we obtain that MEP(F,\phi ) and {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}) is a closed and convex subset of D. Hence Ω is a closed and convex subset of D. It follows from Lemma 2.4 that {C}_{n+1} is a closed and convex for each n\ge 0. Let v\in \mathrm{\Omega}. Then {P}_{{T}_{i}}(v)=\{v\} for all i\in \mathbb{N}. Since {u}_{n}={T}_{{r}_{n}}{x}_{n}\in dom\phi, we have
for every n\ge 0. Then
Hence v\in {C}_{n+1}, so that \mathrm{\Omega}\subset {C}_{n+1}. Therefore, {P}_{{C}_{n+1}}{x}_{0} is well defined.
Step 2. Show that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}_{0}\parallel exists.
Since Ω is a nonempty closed convex subset of H, there exists a unique v\in \mathrm{\Omega} such that v={P}_{\mathrm{\Omega}}{x}_{0}. Since {x}_{n}={P}_{{C}_{n}}{x}_{0} and {x}_{n+1}\in {C}_{n+1}\subset {C}_{n}, \mathrm{\forall}n\ge 0, we have
On the other hand, as v\in \mathrm{\Omega}\subset {C}_{n}, we obtain
It follows that the sequence \{{x}_{n}\} is bounded and nondecreasing. Therefore, {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}_{0}\parallel exists.
Step 3. Show that {lim}_{n\to \mathrm{\infty}}{x}_{n}=w\in D.
For m>n, by the definition of {C}_{n}, we get {x}_{m}={P}_{{C}_{m}}{x}_{0}\in {C}_{m}\subset {C}_{n}. By applying Lemma 2.1, we have
Since {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}_{0}\parallel exists, it follows that \{{x}_{n}\} is Cauchy. Hence there exists w\in D such that {lim}_{n\to \mathrm{\infty}}{x}_{n}=w.
Step 4. Show that \parallel {x}_{n,i}{x}_{n}\parallel \to 0 as n\to \mathrm{\infty} for every i\in \mathbb{N}.
From {x}_{n+1}\in {C}_{n+1}, we have
For v\in \mathrm{\Omega}, by Lemma 2.3 and (3.2), we get
This implies that
where M={sup}_{n\ge 0}\{\parallel {x}_{n}v\parallel +\parallel {y}_{n}v\parallel \}. By the given control condition on \{{\alpha}_{n,i}\} and (3.3), we obtain
By Lemma 2.5, we have
Hence {\parallel {u}_{n}v\parallel}^{2}\le {\parallel {x}_{n}v\parallel}^{2}{\parallel {x}_{n}{u}_{n}\parallel}^{2}. By Lemma 2.3, we get
This implies that
where M={sup}_{n\ge 0}\{\parallel {x}_{n}v\parallel +\parallel {y}_{n}v\parallel \}. From (3.3), we get {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{u}_{n}\parallel =0. It follows that
Step 5. Show that w\in \mathrm{\Omega}.
By {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0, we have
From {lim}_{n\to \mathrm{\infty}}{x}_{n}=w, we obtain {lim}_{n\to \mathrm{\infty}}{u}_{n}=w.
We will show that w\in MEP(F,\phi ). Since {u}_{n}={T}_{{r}_{n}}{x}_{n}\in dom\phi, we have
It follows by (A2) that
Hence
It follows from (3.4), (A4) and the lower semicontinuous of φ that
For t with 0<t\le 1 and y\in D, let {y}_{t}=ty+(1t)w. Since y,w\in D and D is convex, then {y}_{t}\in D and hence
This implies by (A1), (A4) and the convexity of φ, that
Dividing by t, we have
Letting t\to 0, it follows from the weakly semicontinuity of φ that
Hence w\in MEP(F,\phi ). Next, we will show that w\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}). For each i=1,2,\dots ,n, we have
By Steps 34, we have d(w,{T}_{i}w)=0. Hence w\in {T}_{i}w for all i=1,2,\dots ,n.
Step 6. Show that w={P}_{\mathrm{\Omega}}{x}_{0}.
Since {x}_{n}={P}_{{C}_{n}}{x}_{0}, we get
Since w\in \mathrm{\Omega}\subset {C}_{n}, we have
Now, we obtain that w={P}_{\mathrm{\Omega}}{x}_{0}.
This completes the proof. □
Setting \phi \equiv 0 in Theorem 3.1, we have the following result.
Corollary 3.2 Let D be a nonempty closed and convex subset of a real Hilbert space H. Let F be a bifunction from D\times D to ℝ satisfying (A1)(A4). Let {T}_{i}:D\to P(D) be multivalued nonexpansive mappings for all i\in \mathbb{N} with \mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap EP(F)\ne \mathrm{\varnothing} such that all {P}_{{T}_{i}} are nonexpansive. Assume that \{{\alpha}_{n,i}\}\subset [0,1) satisfies the condition {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,i}{\alpha}_{n,0}>0 for all i\in \mathbb{N}. Define the sequence \{{x}_{n}\} as follows: {x}_{1}\in D={C}_{1},
where the sequences {r}_{n}\in (0,\mathrm{\infty}) with {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0 and \{{\alpha}_{n,i}\}\subset [0,1) satisfying {\sum}_{i=0}^{n}{\alpha}_{n,i}=1 and {x}_{n,i}\in {P}_{{T}_{i}}{u}_{n} for i\in \mathbb{N}. Then the sequence \{{x}_{n}\} converges strongly to {P}_{\mathrm{\Omega}}{x}_{0}.
Setting F\equiv 0 in Theorem 3.1, we have the following result.
Corollary 3.3 Let D be a nonempty closed and convex subset of a real Hilbert space H. Let φ be a proper lower semicontinuous and convex function from D to \mathbb{R}\cup \{+\mathrm{\infty}\} such that D\cap dom\phi \ne \mathrm{\varnothing}. Let {T}_{i}:D\to P(D) be multivalued nonexpansive mappings for all i\in \mathbb{N} with \mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap CMP(\phi )\ne \mathrm{\varnothing} such that all {P}_{{T}_{i}} are nonexpansive. Assume that either (B1) or (B2) holds, and \{{\alpha}_{n,i}\}\subset [0,1) satisfies the condition {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,i}{\alpha}_{n,0}>0 for all i\in \mathbb{N}. Define the sequence \{{x}_{n}\} as follows: {x}_{1}\in D={C}_{1},
where the sequences {r}_{n}\in (0,\mathrm{\infty}) with {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0 and \{{\alpha}_{n,i}\}\subset [0,1) satisfying {\sum}_{i=0}^{n}{\alpha}_{n,i}=1 and {x}_{n,i}\in {P}_{{T}_{i}}{u}_{n} for i\in \mathbb{N}. Then the sequence \{{x}_{n}\} converges strongly to {P}_{\mathrm{\Omega}}{x}_{0}.
Remark 3.4

(i)
Let \{{\alpha}_{n,i}\} be double sequence in (0,1]. Let (a) and (b) be the following conditions:

(a)
{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,i}{\alpha}_{n,0}>0 for all i\in \mathbb{N},

(b)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n,i} exist and lie in (0,1] for all i=0,1,2,\dots .
It is easy to see that if \{{\alpha}_{n,i}\} satisfies the condition (a), then it satisfies the condition (b). So Theorem 3.1 and Corollaries 3.23.3 hold true when the control double sequence \{{\alpha}_{n,i}\} satisfies the condition (a).

(a)

(ii)
The following double sequences are examples of the control sequences in Theorem 3.1 and Corollaries 3.23.3:

(1)
{\alpha}_{n,k}=\{\begin{array}{ll}\frac{1}{{2}^{k}}(\frac{n}{n+1}),& n\ge k;\\ 1\frac{n}{n+1}({\sum}_{k=1}^{n}\frac{1}{{2}^{k}}),& n=k1;\\ 0,& n<k1,\end{array}
that is,
{\alpha}_{n,k}=\left(\begin{array}{ccccccccc}\frac{1}{4}& \frac{3}{4}& 0& 0& 0& 0& \cdots & 0& \cdots \\ \frac{1}{3}& \frac{1}{6}& \frac{1}{2}& 0& 0& 0& \cdots & 0& \cdots \\ \frac{3}{8}& \frac{3}{16}& \frac{3}{32}& \frac{11}{32}& 0& 0& \cdots & 0& \cdots \\ \frac{2}{5}& \frac{1}{5}& \frac{1}{10}& \frac{1}{20}& \frac{1}{4}& 0& \cdots & 0& \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \frac{n}{2(n+1)}& \frac{n}{4(n+1)}& \frac{n}{8(n+1)}& \frac{n}{16(n+1)}& \frac{n}{32(n+1)}& \frac{n}{64(n+1)}& \cdots & \frac{n}{{2}^{k}(n+1)}& \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \end{array}\right).We see that {lim}_{n\to \mathrm{\infty}}{\alpha}_{n,k}=\frac{1}{{2}^{k}} and {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,k}=\frac{1}{{2}^{k+1}} for k=1,2,3,\dots .

(2)
{\alpha}_{n,k}=\{\begin{array}{ll}\frac{1}{{2}^{k}}(\frac{n}{n+1}),& n\ge k\text{and}n\text{is odd};\\ \frac{1}{{2}^{k+1}}(\frac{n}{n+1}),& n\ge k\text{and}n\text{is even};\\ 1\frac{n}{n+1}({\sum}_{k=1}^{n}\frac{1}{{2}^{k}}),& n=k1\text{and}n\text{is odd};\\ 1\frac{n}{n+1}({\sum}_{k=1}^{n}\frac{1}{{2}^{k+1}}),& n=k1\text{and}n\text{is even};\\ 0,& nk1,\end{array}
that is,
{\alpha}_{n,k}=\left(\begin{array}{ccccccccc}\frac{1}{4}& \frac{3}{4}& 0& 0& 0& 0& \cdots & 0& \cdots \\ \frac{1}{6}& \frac{1}{12}& \frac{3}{4}& 0& 0& 0& \cdots & 0& \cdots \\ \frac{3}{8}& \frac{3}{16}& \frac{3}{32}& \frac{11}{32}& 0& 0& \cdots & 0& \cdots \\ \frac{1}{5}& \frac{1}{10}& \frac{1}{20}& \frac{1}{40}& \frac{5}{8}& 0& \cdots & 0& \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \frac{2n1}{2(2n)}& \frac{2n1}{4(2n)}& \frac{2n1}{8(2n)}& \frac{2n1}{16(2n)}& \frac{2n1}{32(2n)}& \frac{2n1}{64(2n)}& \cdots & \frac{2n1}{{2}^{k}(2n)}& \cdots \\ \frac{2n}{4(2n+1)}& \frac{2n}{8(2n+1)}& \frac{2n}{16(2n+1)}& \frac{2n}{32(2n+1)}& \frac{2n}{64(2n+1)}& \frac{2n}{128(2n+1)}& \cdots & \frac{2n}{{2}^{k+1}(2n+1)}& \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \end{array}\right).
We see that {lim}_{n\to \mathrm{\infty}}{\alpha}_{n,k} does not exist and {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,k}=\frac{1}{{2}^{k+3}} for k=1,2,3,\dots .

(1)
4 Conclusions
We use the shrinking projection method defined by Takahashi [10] together with our method for finding a common element of the set of solutions of mixed equilibrium problem and common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. The main results of paper can be applied for solving convex minimization problems and fixed point problems.
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The authors would like to thank Chiang Mai University, Chiang Mai, Thailand for the financial support of this paper.
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AB studied and researched a nonlinear analysis and also wrote this article. SS participated in the process of the study and helped to draft the manuscript. All authors read and approved the final manuscript.
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Bunyawat, A., Suantai, S. Hybrid methods for a mixed equilibrium problem and fixed points of a countable family of multivalued nonexpansive mappings. Fixed Point Theory Appl 2013, 236 (2013). https://doi.org/10.1186/168718122013236
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DOI: https://doi.org/10.1186/168718122013236
Keywords
 multivalued nonexpansive mappings
 mixed equilibrium problem
 shrinking projection method