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Iterative algorithms based on the viscosity approximation method for equilibrium and constrained convex minimization problem
Fixed Point Theory and Applications volume 2012, Article number: 201 (2012)
Abstract
The gradientprojection algorithm (GPA) plays an important role in solving constrained convex minimization problems. Based on the viscosity approximation method, we combine the GPA and averaged mapping approach to propose implicit and explicit composite iterative algorithms for finding a common solution of an equilibrium and a constrained convex minimization problem for the first time in this paper. Under suitable conditions, strong convergence theorems are obtained.
MSC:46N10, 47J20, 74G60.
1 Introduction
Let H be a real Hilbert space with inner product \u3008\cdot ,\cdot \u3009 and norm \parallel \cdot \parallel. Let C be a nonempty closed convex subset of H. Let T:C\to C be a nonexpansive mapping, namely \parallel TxTy\parallel \le \parallel xy\parallel, for all x,y\in C. The set of fixed points of T is denoted by F(T).
Let ϕ be a bifunction of C\times C into ℝ, where ℝ is the set of real numbers. Consider the equilibrium problem (EP) which is to find z\in C such that
We denoted the set of solutions of EP by EP(\varphi ). Given a mapping F:C\to H, let \varphi (x,y)=\u3008Fx,yx\u3009 for all x,y\in C, then z\in EP(\varphi ) if and only if \u3008Fz,yz\u3009\ge 0 for all y\in C, that is, z is a solution of the variational inequality. Numerous problems in physics, optimizations, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [1–3] and the references therein.
Composite iterative algorithms were proposed by many authors for finding a common solution of an equilibrium problem and a fixed point problem (see [4–18]).
On the other hand, consider the constrained convex minimization problem as follows:
where g:C\to \mathbb{R} is a realvalued convex function. It is well known that the gradientprojection algorithm (GPA) plays an important role in solving constrained convex minimization problems. If g is (Fréchet) differentiable, then the GPA generates a sequence \{{x}_{n}\} using the following recursive formula:
or more generally,
where in both (1.3) and (1.4) the initial guess {x}_{0} is taken from C arbitrarily, and the parameters, λ or {\lambda}_{n}, are positive real numbers satisfying certain conditions. The convergence of the algorithms (1.3) and (1.4) depends on the behavior of the gradient ∇g. As a matter of fact, it is known that if ∇g is αstrongly monotone and LLipschitzian with constants \alpha ,L\ge 0, then the operator
is a contraction; hence the sequence \{{x}_{n}\} defined by the algorithm (1.3) converges in norm to the unique minimizer of (1.2). However, if the gradient ∇g fails to be strongly monotone, the operator W defined by (1.5) would fail to be contractive; consequently, the sequence \{{x}_{n}\} generated by the algorithm (1.3) may fail to converge strongly (see [19]). If ∇g is Lipschitzian, then the algorithms (1.3) and (1.4) can still converge in the weak topology under certain conditions.
Recently, Xu [19] proposed an explicit operatororiented approach to the algorithm (1.4); that is, an averaged mapping approach. He gave his averaged mapping approach to the GPA (1.4) and the relaxed gradientprojection algorithm. Moreover, he constructed a counterexample which shows that the algorithm (1.3) does not converge in norm in an infinitedimensional space and also presented two modifications of GPA which are shown to have strong convergence [20, 21].
In 2011, Ceng et al. [22] proposed the following explicit iterative scheme:
where {s}_{n}=\frac{2{\lambda}_{n}L}{4} and {P}_{C}(I{\lambda}_{n}\mathrm{\nabla}g)={s}_{n}I+(1{s}_{n}){T}_{n} for each n\ge 0. He proved that the sequences \{{x}_{n}\} converge strongly to a minimizer of the constrained convex minimization problem, which also solves a certain variational inequality.
In 2000, Moudafi [2] introduced the viscosity approximation method for nonexpansive mappings, extended in [23]. Let f be a contraction on H, starting with an arbitrary initial {x}_{0}\in H, define a sequence \{{x}_{n}\} recursively by
where \{{\alpha}_{n}\} is a sequence in (0,1). Xu [24] proved that if \{{\alpha}_{n}\} satisfies certain conditions, the sequence \{{x}_{n}\} generated by (1.6) converges strongly to the unique solution {x}^{\ast}\in F(T) of the variational inequality
The purpose of the paper is to study the iterative method for finding the common solution of an equilibrium problem and a constrained convex minimization problem. Based on the viscosity approximation method, we combine the GPA and averaged mapping approach to propose implicit and explicit composite iterative method for finding the common element of the set of solutions of an equilibrium problem and the solution set of a constrained convex minimization problem. We also prove some strong convergence theorems.
2 Preliminaries
Throughout this paper, we always assume that C is a nonempty closed convex subset of a Hilbert space H. We use ‘⇀’ for weak convergence and ‘→’ for strong convergence.
It is widely known that H satisfies Opial’s condition [25]; that is, for any sequence \{{x}_{n}\} with {x}_{n}\rightharpoonup x, the inequality
holds for every y\in H with y\ne x.
In order to solve the equilibrium problem for a bifunction \varphi :C\times C\to \mathbb{R}, let us assume that ϕ satisfies the following conditions:
(A1) \varphi (x,x)=0, for all x\in C;
(A2) ϕ is monotone, that is, \varphi (x,y)+\varphi (y,x)\le 0 for all x,y\in C;
(A3) for all x,y,z\in C, {lim}_{t\downarrow 0}\varphi (tz+(1t)x,y)\le \varphi (x,y);
(A4) for each fixed x\in C, the function y\mapsto \varphi (x,y) is convex and lower semicontinuous.
Let us recall the following lemmas which will be useful for our paper.
Lemma 2.1 [26]
Let ϕ be a bifunction from C\times C into ℝ satisfying (A1), (A2), (A3), and (A4), then for any r>0 and x\in H, there exists z\in C such that
Further, if
then the following hold:

(1)
{Q}_{r} is singlevalued;

(2)
{Q}_{r} is firmly nonexpansive; that is,
{\parallel {Q}_{r}x{Q}_{r}y\parallel}^{2}\le \u3008{Q}_{r}x{Q}_{r}y,xy\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in H; 
(3)
F({Q}_{r})=EP(\varphi );

(4)
EP(\varphi ) is closed and convex.
Definition 2.1 A mapping T:H\to H is said to be firmly nonexpansive if and only if 2TI is nonexpansive, or equivalently,
Alternatively, T is firmly nonexpansive if and only if T can be expressed as
where S:H\to H is nonexpansive. Obviously, projections are firmly nonexpansive.
Definition 2.2 A mapping T:H\to H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping; that is,
where \alpha \in (0,1) and S:H\to H is nonexpansive. More precisely, when (2.1) holds, we say that T is αaveraged.
Clearly, a firmly nonexpansive mapping is a \frac{1}{2}averaged map.
Proposition 2.1 [27]
For given operators S,T,V:H\to H:

(i)
If T=(1\alpha )S+\alpha V for some \alpha \in (0,1) and if U is averaged and V is nonexpansive, then T is averaged.

(ii)
T is firmly nonexpansive if and only if the complement IT is firmly nonexpansive.

(iii)
If T=(1\alpha )S+\alpha V for some \alpha \in (0,1), U is firmly nonexpansive and V is nonexpansive, then T is averaged.

(iv)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings {\{{T}_{i}\}}_{i=1}^{N} is averaged, then so is the composite {T}_{1}\cdots {T}_{N}. In particular, if {T}_{1} is {\alpha}_{1}averaged, and {T}_{2} is {\alpha}_{2}averaged, where {\alpha}_{1},{\alpha}_{2}\in (0,1), then the composite {T}_{1}{T}_{2} is αaveraged, where \alpha ={\alpha}_{1}+{\alpha}_{2}{\alpha}_{1}{\alpha}_{2}.
Recall that the metric projection from H onto C is the mapping {P}_{C}:H\to C which assigns, to each point x\in H, the unique point {P}_{C}x\in C satisfying the property
Lemma 2.2 For a given x\in H:

(a)
z={P}_{C}x if and only if \u3008xz,yz\u3009\le 0, \mathrm{\forall}y\in C.

(b)
z={P}_{C}x if and only if {\parallel xz\parallel}^{2}\le {\parallel xy\parallel}^{2}{\parallel yz\parallel}^{2}, \mathrm{\forall}y\in C.

(c)
\u3008{P}_{C}x{P}_{C}y,xy\u3009\ge {\parallel {P}_{C}x{P}_{C}y\parallel}^{2}, \mathrm{\forall}x,y\in H.
Consequently, {P}_{C} is nonexpansive and monotone.
Lemma 2.3 The following inequality holds in an inner product space X:
The socalled demiclosedness principle for nonexpansive mappings will be used.
Lemma 2.4 (Demiclosedness principle [28])
Let T:C\to C be a nonexpansive mapping with Fix(T)\ne \mathrm{\varnothing}. If \{{x}_{n}\} is a sequence in C that converges weakly to x and if \{(IT){x}_{n}\} converges strongly to y, then (IT)x=y. In particular, if y=0, then x\in Fix(T).
Next, we introduce monotonicity of a nonlinear operator.
Definition 2.3 A nonlinear operator G whose domain D(G)\subseteq H and range R(G)\subseteq H is said to be:

(a)
monotone if
\u3008xy,GxGy\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D(G), 
(b)
βstrongly monotone if there exists \beta >0 such that
\u3008xy,GxGy\u3009\ge \beta {\parallel xy\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D(G), 
(c)
νinverse strongly monotone (for short, νism) if there exists \nu >0 such that
\u3008xy,GxGy\u3009\ge \nu {\parallel GxGy\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D(G).
It can be easily seen that if G is nonexpansive, then IG is monotone; and the projection map {P}_{C} is a 1ism.
The inverse strongly monotone (also referred to as cocoercive) operators have been widely used to solve practical problems in various fields, for instance, in traffic assignment problems; see, for example, [29, 30] and reference therein.
The following proposition summarizes some results on the relationship between averaged mappings and inverse strongly monotone operators.
Proposition 2.2 [27]
Let T:H\to H be an operator from H to itself.

(a)
T is nonexpansive if and only if the complement IT is \frac{1}{2}ism.

(b)
If T is νism, then for \gamma >0, γT is \frac{\nu}{\gamma}ism.

(c)
T is averaged if and only if the complement IT is νism for some \nu >\frac{1}{2}. Indeed, for \alpha \in (0,1), T is αaveraged if and only if IT is \frac{1}{2\alpha}ism.
Lemma 2.5 [24]
Let \{{a}_{n}\} be a sequence of nonnegative numbers satisfying the condition
where \{{\gamma}_{n}\}, \{{\delta}_{n}\} are sequences of real numbers such that:

(i)
\{{\gamma}_{n}\}\subset (0,1) and {\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty},

(ii)
{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\delta \le 0 or {\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}{\delta}_{n}<\mathrm{\infty}.
Then {lim}_{n\to \mathrm{\infty}}{a}_{n}=0.
3 Main results
In this paper, we always assume that g:C\to \mathbb{R} is a realvalued convex function and ∇g is an LLipschitzian mapping with L\ge 0. Since the Lipschitz continuity of ∇g implies that it is indeed inverse strongly monotone, its complement can be an averaged mapping. Consequently, the GPA can be rewritten as the composite of a projection and an averaged mapping, which is again an averaged mapping. This shows that an averaged mapping plays an important role in the gradientprojection algorithm.
Note that ∇g is LLipschitzian. This implies that ∇g is (1/L)ism, which then implies that \lambda \mathrm{\nabla}g is (1/\lambda L)ism. So, by Proposition 2.2, I\lambda \mathrm{\nabla}g is (\lambda L/2)averaged. Now since the projection {P}_{C} is (1/2)averaged, we see from Proposition 2.1 that the composite {P}_{C}(I\lambda \mathrm{\nabla}g) is ((2+\lambda L)/4)averaged for 0<\lambda <2/L. Hence, we have that for each n, {P}_{C}(I{\lambda}_{n}\mathrm{\nabla}g) is ((2+{\lambda}_{n}L)/4)averaged. Therefore, we can write
where {T}_{n} is nonexpansive and {s}_{n}=\frac{2{\lambda}_{n}L}{4}.
Let f:C\to C be a contraction with the constant \rho \in (0,1). Suppose that the minimization problem (1.2) is consistent, and let U denote its solution set. Let \{{Q}_{{\beta}_{n}}\} be a sequence of mappings defined as in Lemma 2.1. Consider the following mapping {G}_{n} on C defined by
where {\alpha}_{n}\in (0,1). By Lemma 2.1, we have
Since 0<1{\alpha}_{n}(1\rho )<1, it follows that {G}_{n} is a contraction. Therefore, by the Banach contraction principle, {G}_{n} has a unique fixed point {x}_{n}^{f}\in C such that
For simplicity, we will write {x}_{n} for {x}_{n}^{f} provided no confusion occurs. Next, we prove the convergence of \{{x}_{n}\}, while we claim the existence of the q\in U\cap EP(\varphi ), which solves the variational inequality
Equivalently, q={P}_{U\cap EP(\varphi )}f(q).
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H and ϕ be a bifunction from C\times C into ℝ satisfying (A1), (A2), (A3), and (A4). Let g:C\to \mathbb{R} be a realvalued convex function, and assume that ∇g is an LLipschitzian mapping with L\ge 0 and f:C\to C is a contraction with the constant \rho \in (0,1). Assume that U\cap EP(\varphi )\ne \mathrm{\varnothing}. Let \{{x}_{n}\} be a sequence generated by
where {u}_{n}={Q}_{{\beta}_{n}}{x}_{n}, {P}_{C}(I{\lambda}_{n}\mathrm{\nabla}g)={s}_{n}I+(1{s}_{n}){T}_{n}, {s}_{n}=\frac{2{\lambda}_{n}L}{4} and \{{\lambda}_{n}\}\subset (0,\frac{2}{L}). Let \{{\beta}_{n}\} and \{{\alpha}_{n}\} satisfy the following conditions:

(i)
\{{\beta}_{n}\}\subset (0,\mathrm{\infty}), {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}>0;

(ii)
\{{\alpha}_{n}\}\subset (0,1), {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0.
Then \{{x}_{n}\} converges strongly, as {s}_{n}\to 0 (\iff {\lambda}_{n}\to \frac{2}{L}), to a point q\in U\cap EP(\varphi ) which solves the variational inequality (3.1).
Proof First, we claim that \{{x}_{n}\} is bounded. Indeed, pick any p\in U\cap EP(\varphi ), since {u}_{n}={Q}_{{\beta}_{n}}{x}_{n} and p={Q}_{{\beta}_{n}}p, then we know that for any n\in \mathbb{N},
Thus, we derive that (noting {T}_{n}p=p and {T}_{n} is nonexpansive)
Then we have
and hence \{{x}_{n}\} is bounded. From (3.2), we also derive that \{{u}_{n}\} is bounded.
Next, we claim that \parallel {x}_{n}{u}_{n}\parallel \to 0. Indeed, for any p\in U\cap EP(\varphi ), by Lemma 2.1, we have
This implies that
Then from (3.3), we derive that
Since {\alpha}_{n}\to 0, it follows that
Then we show that \parallel {x}_{n}{T}_{n}{x}_{n}\parallel \to 0. Indeed,
Since {\alpha}_{n}\to 0 and \parallel {x}_{n}{u}_{n}\parallel \to 0, we obtain that
Thus,
and
we have
Observe that
where {s}_{n}=\frac{2{\lambda}_{n}L}{4}\in (0,\frac{1}{2}). Hence, we have
From the boundedness of \{{u}_{n}\}, {s}_{n}\to 0 (\iff {\lambda}_{n}\to \frac{2}{L}) and \parallel {u}_{n}{T}_{n}{u}_{n}\parallel \to 0, we conclude that
Since ∇g is LLipschitzian, ∇g is \frac{1}{L}ism. Consequently, {P}_{C}(I\frac{2}{L}\mathrm{\nabla}g) is a nonexpansive selfmapping on C. As a matter of fact, we have for each x,y\in C
Consider a subsequence \{{u}_{{n}_{i}}\} of \{{u}_{n}\}. Since \{{u}_{{n}_{i}}\} is bounded, there exists a subsequence \{{u}_{{n}_{{i}_{j}}}\} of \{{u}_{{n}_{i}}\} which converges weakly to q. Next, we show that q\in U\cap EP(\varphi ). Without loss of generality, we can assume that {u}_{{n}_{i}}\rightharpoonup q. Then, by Lemma 2.4, we obtain
This shows that q\in U.
Next, we show that q\in EP(\varphi ). Since {u}_{n}={Q}_{{\beta}_{n}}{x}_{n}, for any y\in C, we obtain
From (A2), we have
Replacing n by {n}_{i}, we have
Since \frac{{u}_{{n}_{i}}{x}_{{n}_{i}}}{{\beta}_{{n}_{i}}}\to 0 and {u}_{{n}_{i}}\rightharpoonup q, it follows from (A4) that 0\ge \varphi (y,q) for all y\in C. Let
then we have {z}_{t}\in C and hence \varphi ({z}_{t},q)\le 0. Thus, from (A1) and (A4), we have
and hence 0\le \varphi ({z}_{t},y). From (A3), we have 0\le \varphi (q,y) for all y\in C and hence q\in EP(\varphi ). Therefore, q\in EP(\varphi )\cap U.
On the other hand, we note that
Hence, we obtain
It follows that
In particular,
Since {x}_{{n}_{i}}\rightharpoonup q, it follows from (3.4) that {x}_{{n}_{i}}\to q as i\to \mathrm{\infty}.
Next, we show that q solves the variational inequality (3.1). Observe that
Hence, we conclude that
Since {T}_{n} is nonexpansive, we have that I{T}_{n}{Q}_{{\beta}_{n}} is monotone. Note that for any given z\in U\cap EP(\varphi ),
Now, replacing n with {n}_{i} in the above inequality, and letting i\to \mathrm{\infty}, we have
From the arbitrariness of z\in U\cap EP(\varphi ), it follows that q\in U\cap EP(\varphi ) is a solution of the variational inequality (3.1). Further, by the uniqueness of solution of the variational inequality (3.1), we conclude that {x}_{n}\to q as n\to \mathrm{\infty}. The variational inequality (3.1) can be written as
So, in terms of Lemma 2.2, it is equivalent to the following equality:
This completes the proof. □
Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H and ϕ be a bifunction from C\times C into ℝ satisfying (A1), (A2), (A3), and (A4). Let g:C\to \mathbb{R} be a realvalued convex function, and assume that ∇g is an LLipschitzian mapping with L\ge 0 and f:C\to C is a contraction with the constant \rho \in (0,1). Assume that U\cap EP(\varphi )\ne \mathrm{\varnothing}. Let \{{x}_{n}\} be a sequence generated by {x}_{1}\in C and
where {u}_{n}={Q}_{{\beta}_{n}}{x}_{n}, {P}_{C}(I{\lambda}_{n}\mathrm{\nabla}g)={s}_{n}I+(1{s}_{n}){T}_{n}, {s}_{n}=\frac{2{\lambda}_{n}L}{4} and \{{\lambda}_{n}\}\subset (0,\frac{2}{L}). Let \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{s}_{n}\} satisfy the following conditions:

(i)
\{{\beta}_{n}\}\subset (0,\mathrm{\infty}), lim\hspace{0.17em}inf{\beta}_{n}>0, {\sum}_{n=1}^{\mathrm{\infty}}{\beta}_{n+1}{\beta}_{n}<\mathrm{\infty};

(ii)
\{{\alpha}_{n}\}\subset (0,1), {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n+1}{\alpha}_{n}<\mathrm{\infty};

(iii)
\{{s}_{n}\}\subset (0,\frac{1}{2}), {lim}_{n\to \mathrm{\infty}}{s}_{n}=0 (\iff {lim}_{n\to \mathrm{\infty}}{\lambda}_{n}=\frac{2}{L}), {\sum}_{n=1}^{\mathrm{\infty}}{s}_{n+1}{s}_{n}<\mathrm{\infty}.
Then \{{x}_{n}\} converges strongly to a point q\in U\cap EP(\varphi ) which solves the variational inequality (3.1).
Proof First, we show that \{{x}_{n}\} is bounded. Indeed, pick any p\in U\cap EP(\varphi ), since {u}_{n}={Q}_{{\beta}_{n}}{x}_{n} and p={Q}_{{\beta}_{n}}p, then we know that for any n\in \mathbb{N},
Thus, we derive that (noting {T}_{n}p=p and {T}_{n} is nonexpansive)
By induction, we have
and hence \{{x}_{n}\} is bounded. From (3.5), we also derive that \{{u}_{n}\} is bounded.
Next, we show that \parallel {x}_{n+1}{x}_{n}\parallel \to 0. Indeed, since ∇g is \frac{1}{L}ism, {P}_{C}(I{\lambda}_{n}\mathrm{\nabla}g) is nonexpansive. It follows that for any given p\in S,
This together with the boundedness of \{{u}_{n}\} implies that \{{P}_{C}(I{\lambda}_{n}\mathrm{\nabla}g){u}_{n1}\} is bounded.
Also, observe that
for some appropriate constant {M}_{1}>0 such that
Thus, we get
for some appropriate constant {M}_{2}>0 such that
From {u}_{n+1}={Q}_{{\beta}_{n+1}}{x}_{n+1} and {u}_{n}={Q}_{{\beta}_{n}}{x}_{n}, we note that
and
Putting y={u}_{n} in (3.7) and y={u}_{n+1} in (3.8), we have
and
So, from (A2), we have
and hence
Since {lim}_{n\to \mathrm{\infty}}{\beta}_{n}>0, without loss of generality, let us assume that there exists a real number a such that {\beta}_{n}>a>0 for all n\in \mathbb{N}. Thus, we have
thus,
where {M}_{3}=sup\{\parallel {u}_{n}{x}_{n}\parallel :n\in \mathbb{N}\}.
From (3.6) and (3.9), we obtain
where M=max[{M}_{2},\frac{{M}_{3}}{a}]. Hence, by Lemma 2.5, we have
Then, from (3.9) and (3.10), and {\beta}_{n+1}{\beta}_{n}\to 0, we have
For any p\in U\cap EP(\varphi ), as in the proof of Theorem 3.1, we have
Then from (3.11), we derive that
Since {\alpha}_{n}\to 0 and \parallel {x}_{n}{x}_{n+1}\parallel \to 0, we have
Next, we have
Then, \parallel {x}_{n}{T}_{n}{x}_{n}\parallel \to 0, it follows that \parallel {u}_{n}{T}_{n}{u}_{n}\parallel \to 0.
Now, we show that
where q={P}_{U\cap EP(\varphi )}f(q) is a unique solution of the variational inequality (3.1). Indeed, take a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} such that
Since \{{x}_{n}\} is bounded, without loss of generality, we may assume that {x}_{{n}_{k}}\rightharpoonup \tilde{x}. By the same argument as in the proof of Theorem 3.1, we have \tilde{x}\in U\cap EP(\varphi ).
Since q={P}_{U\cap EP(\varphi )}f(q), it follows that
From
we have
This implies that
Then, we have
where {M}^{\ast}=sup\{{\parallel {x}_{n}q\parallel}^{2}:n\in \mathbb{N}\}, and {\delta}_{n}=\frac{{\alpha}_{n}}{2(1\rho )(1{\alpha}_{n}\rho )}{M}^{\ast}+\frac{1}{(1\rho )(1{\alpha}_{n}\rho )}\u3008(If)q,{x}_{n+1}q\u3009.
It is easy to see that {lim}_{n\to \mathrm{\infty}}2(1\rho ){\alpha}_{n}=0, {\sum}_{n=1}^{\mathrm{\infty}}2(1\rho ){\alpha}_{n}=\mathrm{\infty}, and {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}\le 0 by (3.12). Hence, by Lemma 2.5, the sequence \{{x}_{n}\} converges strongly to q. This completes the proof. □
4 Conclusions
Methods for solving the equilibrium problem and the constrained convex minimization problem have extensively been studied respectively in a Hilbert space. But to the best of our knowledge, it would probably be the first time in the literature that we introduce implicit and explicit algorithms for finding the common element of the set of solutions of an equilibrium problem and the set of solutions of a constrained convex minimization problem, which also solves a certain variational inequality.
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Acknowledgements
The authors wish to thank the referees for their helpful comments, which notably improved the presentation of this manuscript. This work was supported in part by The Fundamental Research Funds for the Central Universities (the Special Fund of Science in Civil Aviation University of China: No. ZXH2012K001), and by the Science Research Foundation of Civil Aviation University of China (No. 2012KYM03).
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Tian, M., Liu, L. Iterative algorithms based on the viscosity approximation method for equilibrium and constrained convex minimization problem. Fixed Point Theory Appl 2012, 201 (2012). https://doi.org/10.1186/168718122012201
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DOI: https://doi.org/10.1186/168718122012201
Keywords
 iterative algorithm
 equilibrium problem
 constrained convex minimization
 variational inequality