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Multistep hybrid viscosity method for systems of variational inequalities defined over sets of solutions of an equilibrium problem and fixed point problems
Fixed Point Theory and Applications volume 2012, Article number: 186 (2012)
Abstract
In this paper, we consider a system of variational inequalities defined over the intersection of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings, and the solution set of a nonexpansive mapping. We also consider a triple hierarchical variational inequality problem, that is, a variational inequality problem defined over a set of solutions of another variational inequality problem which is defined over the intersection of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings, and the solution set of a nonexpansive mapping. These two problems are very general and include, as special cases, several problems studied in the literature. We propose a multistep hybrid viscosity method to compute the approximate solutions of our system of variational inequalities and a triple hierarchical variational inequality problem. The convergence analysis of the sequences generated by the proposed method is also studied. In addition, the nontrivial examples of two systems are presented and our results are applied to these examples.
MSC:49J40, 47H05, 47H19.
1 Introduction and formulations
Let H be a real Hilbert space whose inner product and norm are denoted by \u3008\cdot ,\cdot \u3009 and \parallel \cdot \parallel, respectively. Let C be a nonempty, closed and convex subset of H and A:C\to H be a nonlinear mapping. The variational inequality problem (VIP) associated with the set C and the mapping A is stated as follows:
In particular, if C is the set of fixed points of a nonexpansive mapping T, denoted by Fix(T), and if S is another nonexpansive mapping (not necessarily with fixed points), then VIP (1.1) becomes the following problem:
It is called a hierarchical variational inequality problem, also known as a hierarchical fixed point problem, and it is studied in [1, 2]. Observe that if S has fixed points, then they are the solutions of VIP (1.2). It is worth mentioning that many practical problems can be written in the form of a hierarchical variational inequality problem; see for example [3–5] and the references therein. Such a problem is also a bilevel problem, in which we find a solution of the first problem subject to the condition that its solution is also a fixed point of a mapping. For further details on hierarchical fixed point problems and their applications, we refer to [1–3, 6–18] and the references therein. The solution methods presented in [1, 12] may not be unique. Therefore, it would be reasonable to identify the unique minimizer of an appropriate objective function over the hierarchical fixed point constraint.
Mainge and Moudafi [2] introduced a hierarchical fixed point approach to finding a solution of VIP (1.2). Subsequently, Moudafi and Mainge [1] studied the explicit scheme for computing a solution of VIP (1.2) by introducing the following iterative algorithm:
where f:C\to C and \{{\alpha}_{n}\}, \{{\lambda}_{n}\}\subset (0,1). They also proved the strong convergence of the sequence \{{x}_{n}\} generalized by (1.3) to a solution of VIP (1.2).
Yao et al. [19] introduced and analyzed the following twostep iterative algorithm that generates a sequence \{{x}_{n}\} by the following explicit scheme:
It is easy to see that if C=Fix(T) and \Theta (x,y)=\u3008(IS)x,yx\u3009, VIP (1.2) can be reformulated as follows:
It is known as an equilibrium problem. In [20, 21], it is shown that the formulation (1.5) covers monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, Nash equilibria in noncooperative games, vector equilibrium problems and certain fixed point problems.
Recently, many authors have generalized the classical equilibrium problem introduced in [22] by introducing ‘perturbation’ to the function Θ. For example, Moudafi [23] studied the following equilibrium problem:
where A is an αinverse strongly monotone operator. In [24–26], the following mixed equilibrium problem is studied:
where \phi :C\to \mathbb{R} is a functional on C. Very recently, Marino et al. [27] studied the following equilibrium problem:
It includes all previous equilibrium problems as special cases. The set of solutions of (1.6) is denoted by \mathit{EP}(\Theta ,h).
Lemma 1.1 ([[27], Lemma 2.2])
Let C be a nonempty, closed and convex subset of a Hilbert space H. Let \Theta :C\times C\to \mathbb{R} be a bifunction such that
(Θ 1) \Theta (x,x)=0, \mathrm{\forall}x\in C;
(Θ 2) Θ is monotone (that is, \Theta (x,y)+\Theta (y,x)\le 0, \mathrm{\forall}(x,y)\in C\times C) and upper hemicontinuous in the first variable (that is, {lim\hspace{0.17em}sup}_{t\to 0}\Theta (tz+(1t)x,y)\le \Theta (x,y), \mathrm{\forall}x,y,z\in C);
(Θ 3) Θ is lower semicontinuous and convex in the second variable.
Let h:C\times C\to \mathbb{R} be a bifunction such that
(h 1) h(x,x)=0, \mathrm{\forall}x\in C;
(h 2) h is monotone and weakly upper semicontinuous in the first variable;
(h 3) h is convex in the second variable.
Moreover, suppose that
(H) for fixed r>0 and x\in C, there exists a bounded K\subset C and a\in K such that for all z\in C\setminus K, \Theta (a,z)+h(z,a)+\frac{1}{r}\u3008az,zx\u3009<0.
For r>0 and x\in H, let {T}_{r}:H\to {2}^{C} be a mapping defined by
(is called the resolvent of Θ and h). Then

(i)
{T}_{r}x\ne \mathrm{\varnothing};

(ii)
{T}_{r}x is a singleton;

(iii)
{T}_{r} is firmly nonexpansive;

(iv)
\mathit{EP}(\Theta ,h)=Fix({T}_{r}) and it is closed and convex;

(v)
(see [28])
\parallel {T}_{{r}_{2}}y{T}_{{r}_{1}}x\parallel \le \parallel yx\parallel +\left\frac{{r}_{2}{r}_{1}}{{r}_{2}}\right\parallel {T}_{{r}_{2}}yy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in H,\mathrm{\forall}{r}_{1},{r}_{2}>0.(1.8)
Marino et al. [27] introduced a multistep iterative method that generalizes the twostep method studied in [19] from two nonexpansive mappings to a finite family of nonexpansive mappings, and proved that the sequence generated by this method converges strongly to a common fixed point of the mappings which is also a solution of the equilibrium problem (1.6). The multistep iterative method in [27] involves the Ishikawatype iterative method and the viscosity approximation method.
On the other hand, by combining the regularization method, the hybrid steepestdescent method, and the projection method, Ceng et al. [29] proposed an iterative algorithm that generates a sequence via the explicit scheme and proved that this sequence converges strongly to a unique solution of the following problem.
Problem 1.1 Let F:C\to H be κLipschitzian and ηstrongly monotone on the nonempty, closed and convex subset C of H, where κ and η are positive constants, that is,
Let f:C\to H be a ρcontraction with a coefficient \rho \in [0,1), and S,T:C\to C be two nonexpansive mappings with Fix(T)\ne \mathrm{\varnothing}. Let 0<\mu <2\eta /{\kappa}^{2} and 0<\gamma \le \tau, where \tau =1\sqrt{1\mu (2\eta \mu {\kappa}^{2})}. Then the objective is to find {x}^{\ast}\in \Xi such that
where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP): find {z}^{\ast}\in Fix(T) such that
Since Problem 1.1 has a triple hierarchical structure in contrast with bilevel programming problems (see [30, 31]), that is, a variational inequality problem with a variational inequality constraint over the fixed point set Fix(T), we also call it a triple hierarchical variational inequality problem, which is a generalization of the triple hierarchical constrained optimization problem (THCOP) considered by Iiduka [32, 33].
In this paper, we consider the following system of variational inequalities defined over the set consisting of the set of solutions of an equilibrium problem, the set of common fixed points of nonexpansive mappings, and the set of fixed points of a mapping.
Problem 1.2 Let F:C\to H be κLipschitzian and ηstrongly monotone on the nonempty, closed and convex subset C of H, f:C\to H be a ρcontraction with a coefficient \rho \in [0,1) and {S}_{i},S,T:C\to C be nonexpansive mappings for all i\in \{1,\dots ,N\}. Assume that \Theta ,h:C\times C\to \mathbb{R} are two bifunctions. Let 0<\mu <2\eta /{\kappa}^{2} and 0<\gamma \le \tau, where \tau =1\sqrt{1\mu (2\eta \mu {\kappa}^{2})}. Then the objective is to find {x}^{\ast}\in \Omega such that
where \Omega =Fix(T)\cap ({\bigcap}_{i}Fix({S}_{i}))\cap \mathit{EP}(\Theta ,h)\ne \mathrm{\varnothing}.
We propose the following multistep hybrid viscosity method for solving Problem 1.2.
Algorithm 1.1 Let F:C\to H be κLipschitzian and ηstrongly monotone on the nonempty, closed and convex subset C of H, f:C\to H be a ρcontraction with a coefficient \rho \in [0,1) and {S}_{i},S,T:C\to C be nonexpansive mappings for all i\in \{1,\dots ,N\}. Assume that \Theta ,h:C\times C\to \mathbb{R} are two bifunctions satisfying the hypotheses of Lemma 1.1. Let \{{\lambda}_{n}\}, \{{\alpha}_{n}\}, \{{\beta}_{n,i}\}, i=1,\dots ,N be sequences in (0,1) and \{{r}_{n}\} be a sequence in (0,\mathrm{\infty}) with {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0. Let 0<\mu <2\eta /{\kappa}^{2} and 0<\gamma \le \tau, where \tau =1\sqrt{1\mu (2\eta \mu {\kappa}^{2})}. Then the sequence \{{x}_{n}\} is generated from an arbitrary initial point {x}_{1}\in C by the following iterative scheme:
In particular, if f\equiv 0, then (1.13) reduces to the following iterative scheme:
We prove that, under appropriate conditions, the sequence \{{x}_{n}\} generated by Algorithm 1.1 converges strongly to a unique solution of Problem 1.2. In addition, we also consider and study the application of Algorithm 1.1 to solve the following triple hierarchical variational inequality problem (THVIP):
Problem 1.3 Let F:C\to H be κLipschitzian and ηstrongly monotone on the nonempty, closed and convex subset C of H, f:C\to H be a ρcontraction with a coefficient \rho \in [0,1) and {S}_{i},S,T:C\to C be nonexpansive mappings for all i\in \{1,\dots ,N\}. Assume that \Theta ,h:C\times C\to \mathbb{R} are two bifunctions. Let 0<\mu <2\eta /{\kappa}^{2} and 0<\gamma \le \tau, where \tau =1\sqrt{1\mu (2\eta \mu {\kappa}^{2})}. Then the objective is to find {x}^{\ast}\in \Xi such that
where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding {z}^{\ast}\in \Omega such that
where \Omega =Fix(T)\cap ({\bigcap}_{i}Fix({S}_{i}))\cap \mathit{EP}(\Theta ,h)\ne \mathrm{\varnothing}.
Moreover, we prove that, under very mild conditions, the sequence \{{x}_{n}\} generated by Algorithm 1.1 converges strongly to a unique solution of Problem 1.3. It is worth pointing out that Problem 1.1 is a special case of Problem 1.3 whenever \Theta \equiv h\equiv 0 and {S}_{i}\equiv I for all i\in \{1,\dots ,N\}.
2 Some basic results
We present here some basic facts and results that are needed in the sequel.
Lemma 2.1 ([[34], Lemma 2.1])
Let \{{s}_{n}\} be a sequence of nonnegative numbers satisfying the condition
where \{{\gamma}_{n}\}, \{{\delta}_{n}\} are the sequences of real numbers such that

(i)
\{{\gamma}_{n}\}\subset [0,1] and {\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty}, or equivalently,
\prod _{n=1}^{\mathrm{\infty}}(1{\gamma}_{n}):=\underset{n\to \mathrm{\infty}}{lim}\prod _{k=1}^{n}(1{\gamma}_{k})=0; 
(ii)
{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}\le 0, or
(ii)′ {\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}{\delta}_{n} is convergent.
Then {lim}_{n\to \mathrm{\infty}}{s}_{n}=0.
Lemma 2.2 ([[34], Lemma 3.1])
Let λ be a number in (0,1], and let \mu >0. Let F:C\to H be an operator on C such that for some constants \kappa ,\eta >0, F is κLipschitzian and ηstrongly monotone. Associating with a nonexpansive mapping T:C\to C, we define the mapping {T}^{\lambda}:C\to H by
Then {T}^{\lambda} is a contraction provided \mu <2\eta /{\kappa}^{2}, that is,
where \tau =1\sqrt{1\mu (2\eta \mu {\kappa}^{2})}\in (0,1].
In the sequel, given a sequence \{{z}_{n}\}, we will denote with {\omega}_{w}({z}_{n}) the set of cluster points of \{{z}_{n}\} with respect to the weak topology, that is,
Analogously, we denote by {\omega}_{s}({z}_{n}) the set of cluster points of \{{z}_{n}\} with respect to the norm topology, that is,
Lemma 2.3 ([[27], Lemma 2.5])
Suppose that the hypotheses of Lemma 1.1 are satisfied. Let \{{r}_{n}\} be a sequence in (0,\mathrm{\infty}) with {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0. Suppose that \{{x}_{n}\} is a bounded sequence. Then the following equivalent statements hold:

(a)
If \parallel {x}_{n}{T}_{{r}_{n}}{x}_{n}\parallel \to 0, as n\to \mathrm{\infty}, every weak cluster point of \{{x}_{n}\} solves the problem
\Theta (x,y)+h(x,y)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C,
that is, {\omega}_{w}({x}_{n})\subseteq \mathit{EP}(\Theta ,h).

(b)
(Demiclosedness principle) If {x}_{n}\rightharpoonup {x}^{\ast} and \parallel {x}_{n}{T}_{{r}_{n}}{x}_{n}\parallel \to 0, as n\to \mathrm{\infty}, then (I{T}_{{r}_{k}}){x}^{\ast}=0 for all k\ge 1.
Lemma 2.4 ([35])
Let \{{\alpha}_{n}\} be a sequence of nonnegative real numbers with {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<\mathrm{\infty} and \{{\beta}_{n}\} be a sequence of real numbers with {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}\le 0. Then {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}{\beta}_{n}\le 0.
3 Main results
Now we present the convergence analysis of Algorithm 1.1 for solving Problem 1.2.
Theorem 3.1 Assume that Problem 1.2 has a solution. Let \{{\lambda}_{n}\}, \{{\alpha}_{n}\}, \{{\beta}_{n,i}\}, i=1,\dots ,N be sequences in (0,1) such that {\beta}_{n,i}\to {\beta}_{i}\in (0,1) as n\to \mathrm{\infty} for all i\in \{1,\dots ,N\}. Suppose that the following conditions hold:
(C1) 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;
(C2) {lim}_{n\to \mathrm{\infty}}{\lambda}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\lambda}_{n}=\mathrm{\infty};
(C3) {\sum}_{n=2}^{\mathrm{\infty}}{\alpha}_{n}{\lambda}_{n}{\alpha}_{n1}{\lambda}_{n1}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{\alpha}_{n}{\lambda}_{n}{\alpha}_{n1}{\lambda}_{n1}}{{\lambda}_{n}}=0;
(C4) {\sum}_{n=2}^{\mathrm{\infty}}{\lambda}_{n}{\lambda}_{n1}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{\lambda}_{n}{\lambda}_{n1}}{{\lambda}_{n}}=0;
(C5) {\sum}_{n=2}^{\mathrm{\infty}}{\beta}_{n,i}{\beta}_{n1,i}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{\beta}_{n,i}{\beta}_{n1,i}}{{\lambda}_{n}}=0 for all i\in \{1,\dots ,N\};
(C6) {\sum}_{n=2}^{\mathrm{\infty}}{r}_{n}{r}_{n1}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{r}_{n}{r}_{n1}}{{\lambda}_{n}}=0.
Then the following assertions hold.

(a)
Let \{{x}_{n}\} be a sequence generated by the scheme (1.13), then \{{x}_{n}\} converges strongly to a unique solution {x}^{\ast}\in \Omega of Problem 1.2.

(b)
Let \{{x}_{n}\} be a sequence generated by the scheme (1.14), then \{{x}_{n}\} converges strongly to a unique solution {x}^{\ast}\in \Omega of the following system of variational inequalities:
\{\begin{array}{cc}\u3008F{x}^{\ast},x{x}^{\ast}\u3009\ge 0,\hfill & \mathrm{\forall}x\in \Omega ,\hfill \\ \u3008(\mu F\gamma S){x}^{\ast},y{x}^{\ast}\u3009\ge 0,\hfill & \mathrm{\forall}y\in \Omega .\hfill \end{array}
Proof We prove only part (a) since part (b) is a straightforward consequence of part (a). Let \{{x}_{n}\} be a sequence generated by the scheme (1.13). First of all, note that 0<\gamma \le \tau and
Then it follows from the ρcontractiveness of f that
Hence, from \gamma \rho <\gamma \le \tau \le \mu \eta we deduce that \mu F\gamma f is (\mu \eta \gamma \rho )strongly monotone. Since it is clear that \mu F\gamma f is Lipschitz continuous, there exists a unique solution of the following VIP:
Also, since Problem 1.2 has a solution, it is easy to see that Problem 1.2 has a unique solution. In addition, taking into account condition (C1), without loss of generality, we may assume that \{{\alpha}_{n}\}\subset [a,b] for some a,b\in (0,1).
The rest of the proof is divided into several steps.
Step 1. The sequences \{{x}_{n}\}, \{{y}_{n,i}\} for all i, \{{u}_{n}\} are bounded.
Indeed, take a point v\in \Omega arbitrarily. Then by Lemma 1.1, we have from (1.13)
For all from i=2 to i=N, by induction, one proves that
Hence, we obtain that for all i\in \{1,\dots ,N\}
Also, utilizing Lemma 2.2 and (1.13), we have
due to 0<\gamma \le \tau. So, calling
by induction we derive \parallel {x}_{n}v\parallel \le M for all n\ge 1. We obtain the claim.
Step 2. {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n+1}{x}_{n}\parallel =0, that is, \{{x}_{n}\} is asymptotically regular.
Indeed, for each n\ge 1, we set
Then we observe that
Let {M}_{0}>0 be a constant such that
It follows from (1.13) and (3.2) that
By the definition of {y}_{n,i}, we obtain that for all i=N,\dots ,2
In the case i=1, we have
Substituting (3.5) in all (3.4)type inequalities, we obtain that for i=2,\dots ,N
So, we conclude that
By Lemma 1.1(v), we know that
where L={sup}_{n\ge 1}\parallel {u}_{n}{x}_{n}\parallel. So, substituting (3.6) in the last inequality, we obtain
If we call
and c>0 a minorant for \{{r}_{n}\}, we have
due to 0<\gamma \le \tau. By conditions (C2)(C6) and Lemma 2.1, we obtain the claim.
Step 3. \parallel {x}_{n}{u}_{n}\parallel =\parallel {x}_{n}{T}_{{r}_{n}}{x}_{n}\parallel \to 0 as n\to \mathrm{\infty}.
Indeed, by the firm nonexpansivity of {T}_{{r}_{n}}, a standard calculation (see [36]) shows that for all p\in \mathit{EP}(\Theta ,h)
So, take a point v\in \mathrm{\Omega} arbitrarily; then utilizing Lemma 2.2 and (1.13), we have
This implies that
Since \parallel {x}_{n+1}{x}_{n}\parallel \to 0 and {\lambda}_{n}\to 0 as n\to \mathrm{\infty}, by the boundedness of \{{x}_{n}\} we conclude that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{u}_{n}\parallel =0.
Step 4. For all i\in \{1,\dots ,N\}, \parallel {S}_{i}{u}_{n}{u}_{n}\parallel \to 0 as n\to \mathrm{\infty}.
Indeed, let us show that for every i\in \{1,\dots ,N\} one has \parallel {S}_{i}{u}_{n}{y}_{n,i1}\parallel \to 0 as n\to \mathrm{\infty}. Take a point v\in \mathrm{\Omega} arbitrarily. When i=N, utilizing Lemmas 2.3 and 2.4, and (1.13), we have
So, we have
Since {\beta}_{n,N}\to {\beta}_{N}\in (0,1), \parallel {x}_{n+1}{x}_{n}\parallel \to 0 and {\lambda}_{n}\to 0 as n\to \mathrm{\infty}, by the boundedness of \{{x}_{n}\} we conclude that {lim}_{n\to \mathrm{\infty}}\parallel {S}_{N}{u}_{n}{y}_{n,N1}\parallel =0.
Take i\in \{1,\dots ,N1\} arbitrarily. Then, we have
and so, after (Ni+1)iterations,
Again, we obtain
Since for all k\in \{1,\dots ,N\}, {\beta}_{n,k}\to {\beta}_{k}\in (0,1), \parallel {x}_{n+1}{x}_{n}\parallel \to 0 and {\lambda}_{n}\to 0 as n\to \mathrm{\infty}, by the boundedness of \{{x}_{n}\} we conclude that
Obviously, for i=1 we have {lim}_{n\to \mathrm{\infty}}\parallel {S}_{1}{u}_{n}{u}_{n}\parallel =0. To conclude, we have that
which hence implies that {lim}_{n\to \mathrm{\infty}}\parallel {S}_{2}{u}_{n}{u}_{n}\parallel =0. Consequently, by induction, we get {lim}_{n\to \mathrm{\infty}}\parallel {S}_{i}{u}_{n}{u}_{n}\parallel =0 for all i=2,\dots ,N since it is enough to observe that
Step 5. {lim}_{n\to \mathrm{\infty}}\parallel {y}_{n,N}{x}_{n}\parallel ={lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0 and {\omega}_{w}({x}_{n})\subset \Omega.
Indeed, since \parallel {x}_{n}{u}_{n}\parallel \to 0 as n\to \mathrm{\infty}, we have {\omega}_{w}({x}_{n})={\omega}_{w}({u}_{n}) and {\omega}_{s}({x}_{n})={\omega}_{s}({u}_{n}).
Now, we observe that
By Step 4, \parallel {S}_{1}{u}_{n}{u}_{n}\parallel \to 0 as n\to \mathrm{\infty}. Hence, we get
This implies that {\omega}_{w}({x}_{n})={\omega}_{w}({y}_{n,1}) and {\omega}_{s}({x}_{n})={\omega}_{s}({y}_{n,1}).
Take a point q\in {\omega}_{w}({x}_{n}) arbitrarily. Since q\in {\omega}_{w}({u}_{n}), by Step 4 and the demiclosedness principle, we have q\in Fix({S}_{i}) for all i\in \{1,\dots ,N\}, that is, q\in {\bigcap}_{i}Fix({S}_{i}). Moreover, note that
and hence,
So, it is easy to see that {lim}_{n\to \mathrm{\infty}}\parallel {y}_{n,N}{x}_{n}\parallel =0 and {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0 since \parallel {x}_{n+1}{x}_{n}\parallel \to 0, {\lambda}_{n}\to 0, \parallel {y}_{n,1}{x}_{n}\parallel \to 0, {\beta}_{n,k}\to {\beta}_{k} and \parallel {S}_{k}{u}_{n}{y}_{n,k1}\parallel \to 0 for all k\in \{1,\dots ,N\}. Thus, by the demiclosedness principle, we have q\in Fix(T). Since \{{x}_{n}\} is bounded and {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{T}_{{r}_{n}}{x}_{n}\parallel =0 (due to Step 3), by Lemma 2.3, we derive q\in \mathit{EP}(\Theta ,h). This shows that q\in \Omega. Therefore, we obtain the claim.
Step 6. \{{x}_{n}\} converges strongly to a unique solution {x}^{\ast} of Problem 1.2.
Indeed, according to \parallel {x}_{n+1}{x}_{n}\parallel \to 0, we can take a subsequence \{{x}_{{n}_{j}}\} of \{{x}_{n}\} satisfying
Without loss of generality, we may further assume that {x}_{{n}_{j}}\rightharpoonup \tilde{x}; then \tilde{x}\in \Omega as we just proved. Since {x}^{\ast} is a solution of Problem 1.2, we get
Repeating the same argument as that of (3.10), we have
From (1.13) and (3.1), it follows that (noticing that {x}_{n+1}={P}_{C}{z}_{n} and 0<\gamma \le \tau)
It turns out that
Put {s}_{n}={\parallel {x}_{n}{x}^{\ast}\parallel}^{2}, {\gamma}_{n}={\alpha}_{n}{\lambda}_{n}\gamma (1\rho ) and
Then (3.12) can be rewritten as
From conditions (C1) and (C2), we conclude from 0<1\rho \le 1 that
Note that
and
Consequently, utilizing Lemma 2.1, we obtain that
So, this together with Lemma 2.1 leads to {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}^{\ast}\parallel =0. The proof is complete. □
We now derive the following strong convergence result for a sequence generated by Algorithm 1.1 to a unique solution of Problem 1.3.
Theorem 3.2 Let \{{\lambda}_{n}\}, \{{\alpha}_{n}\}, \{{\beta}_{n,i}\}, i=1,\dots ,N be sequences in (0,1) such that {\beta}_{n,i}\to {\beta}_{i}\in (0,1) as n\to \mathrm{\infty} for all i\in \{1,\dots ,N\}. Assume that the solution set Ξ of HVIP (1.16) is nonempty and that the following conditions hold:
(C1) 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;
(C2) {lim}_{n\to \mathrm{\infty}}{\lambda}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\lambda}_{n}=\mathrm{\infty};
(C3) {\sum}_{n=2}^{\mathrm{\infty}}{\alpha}_{n}{\lambda}_{n}{\alpha}_{n1}{\lambda}_{n1}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{\alpha}_{n}{\lambda}_{n}{\alpha}_{n1}{\lambda}_{n1}}{{\lambda}_{n}}=0;
(C4) {\sum}_{n=2}^{\mathrm{\infty}}{\lambda}_{n}{\lambda}_{n1}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{\lambda}_{n}{\lambda}_{n1}}{{\lambda}_{n}}=0;
(C5) {\sum}_{n=2}^{\mathrm{\infty}}{\beta}_{n,i}{\beta}_{n1,i}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{\beta}_{n,i}{\beta}_{n1,i}}{{\lambda}_{n}}=0 for all i\in \{1,\dots ,N\};
(C6) {\sum}_{n=2}^{\mathrm{\infty}}{r}_{n}{r}_{n1}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{r}_{n}{r}_{n1}}{{\lambda}_{n}}=0;
(C7) there are constants \overline{k},\theta >0 satisfying \parallel xTx\parallel \ge \overline{k}{[d(x,\Omega )]}^{\theta} for all x\in C.
Then the following assertions hold.

(a)
Let \{{x}_{n}\} be a sequence generated by the scheme (1.13), then \{{x}_{n}\} converges strongly to a unique solution {x}^{\ast}\in \Omega of Problem 1.3 provided {\omega}_{w}({x}_{n})\subset \Xi.

(b)
Let \{{x}_{n}\} be a sequence generated by the scheme (1.14), then \{{x}_{n}\} converges strongly to a unique solution {x}^{\ast} of the following VIP provided {\omega}_{w}({x}_{n})\subset \Xi:
\mathit{\text{find}}{x}^{\ast}\in \Xi \mathit{\text{such that}}\u3008F{x}^{\ast},x{x}^{\ast}\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in \Xi .
Proof For part (a), suppose that the sequence \{{x}_{n}\} is generated by the scheme (1.13). First of all, from the condition \Xi \ne \mathrm{\varnothing} it follows that \Omega =Fix(T)\cap ({\bigcap}_{i}Fix({S}_{i}))\cap \mathit{EP}(\Theta ,h)\ne \mathrm{\varnothing}. Note that 0<\gamma \le \tau and \kappa \ge \eta \iff \mu \eta \ge \tau. Hence, it follows from the ρcontractiveness of f and \gamma \rho <\gamma \le \tau \le \mu \eta that \mu F\gamma f is (\mu \eta \gamma \rho )strongly monotone and Lipschitz continuous. So, there exists a unique solution {x}^{\ast} of the following VIP:
Consequently, it is easy to see that Problem 1.3 has a unique solution {x}^{\ast}\in \Xi. In addition, taking into account condition (C1), without loss of generality, we may assume that \{{\alpha}_{n}\}\subset [a,b] for some a,b\in (0,1).
The rest of the proof is divided into several steps.
Step 1. The sequences \{{x}_{n}\}, \{{y}_{n,i}\} for all i, \{{u}_{n}\} are bounded.
Indeed, repeating the same argument as in Step 1 of the proof of Theorem 3.1, we can derive the claim.
Step 2. {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n+1}{x}_{n}\parallel =0, that is, \{{x}_{n}\} is asymptotically regular.
Indeed, repeating the same argument as in Step 2 of the proof of Theorem 3.1, we can derive the claim.
Step 3. \parallel {x}_{n}{u}_{n}\parallel =\parallel {x}_{n}{T}_{{r}_{n}}{x}_{n}\parallel \to 0 as n\to \mathrm{\infty}.
Indeed, repeating the same argument as in Step 3 of the proof of Theorem 3.1, we can derive the claim.
Step 4. For all i\in \{1,\dots ,N\}, \parallel {S}_{i}{u}_{n}{u}_{n}\parallel \to 0 as n\to \mathrm{\infty}.
Indeed, repeating the same argument as in Step 4 of the proof of Theorem 3.1, we can derive the claim.
Step 5. {lim}_{n\to \mathrm{\infty}}\parallel {y}_{n,N}{x}_{n}\parallel ={lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0 and {\omega}_{w}({x}_{n})\subset \Omega.
Indeed, repeating the same argument as in Step 5 of the proof of Theorem 3.1, we can derive the claim.
Step 6. \{{x}_{n}\} converges strongly to a unique solution {x}^{\ast} of Problem 1.3.
Indeed, we now take a subsequence \{{x}_{{n}_{j}}\} of \{{x}_{n}\} satisfying
Without loss of generality, we may further assume that {x}_{{n}_{j}}\rightharpoonup \tilde{x}; then \tilde{x}\in \Xi according to the assumption {\omega}_{w}({x}_{n})\subset \Xi. Since {x}^{\ast} is a solution of THVIP (1.15), we get
From (1.13) and (3.1), it follows that (noticing that {x}_{n+1}={P}_{C}{z}_{n} and 0<\gamma \le \tau)
It turns out that
However, from {x}^{\ast}\in \Xi and condition (C7), we obtain
On the other hand, we also have
Hence, for a big enough constant {\overline{k}}_{1}>0, we have
Combining (3.14)(3.17), we get
where {\gamma}_{n}={\alpha}_{n}{\lambda}_{n}\gamma (1\rho ) and
Now, conditions (C1) and (C2) imply that {\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty}. Moreover, since \parallel {x}_{n+1}{x}_{n}\parallel \to 0 (due to Step 2), \parallel {y}_{n,N}{x}_{n}\parallel \to 0 (due to Step 5) and {\lambda}_{n}\to 0, we obtain from (3.13) that
which together with Lemma 2.4 leads to
Therefore, we can apply Lemma 2.1 to (3.18) to conclude that {x}_{n}\to {x}^{\ast}. The proof of part (a) is complete. It is easy to see that part (b) now becomes a straightforward consequence of part (a) since, if f=0, THVIP (1.15) reduces to the VIP in part (b). This completes the proof. □
Utilizing Theorem 3.2, we immediately derive the following result.
Corollary 3.1 Let F:C\to H be a κLipschitzian and ηstrongly monotone operator with constants \kappa ,\eta >0, respectively, f:C\to H be a ρcontraction with a coefficient \rho \in [0,1) and S,T:C\to C be nonexpansive mappings with Fix(T)\ne \mathrm{\varnothing}. Let 0<\mu <2\eta /{\kappa}^{2} and 0<\gamma \le \tau, where \tau =1\sqrt{1\mu (2\eta \mu {\kappa}^{2})}. Assume that the solution set Ξ of HVIP (1.11) is nonempty and the following conditions hold for two sequences \{{\lambda}_{n}\},\{{\alpha}_{n}\}\subset (0,1):

(i)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;

(ii)
{lim}_{n\to \mathrm{\infty}}{\lambda}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\lambda}_{n}=\mathrm{\infty};

(iii)
{\sum}_{n=2}^{\mathrm{\infty}}{\alpha}_{n}{\lambda}_{n}{\alpha}_{n1}{\lambda}_{n1}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{\alpha}_{n}{\lambda}_{n}{\alpha}_{n1}{\lambda}_{n1}}{{\lambda}_{n}}=0;

(iv)
{\sum}_{n=2}^{\mathrm{\infty}}{\lambda}_{n}{\lambda}_{n1}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{\lambda}_{n}{\lambda}_{n1}}{{\lambda}_{n}}=0;

(v)
there are constants \overline{k},\theta >0 satisfying \parallel xTx\parallel \ge \overline{k}{[d(x,\Omega )]}^{\theta} for all x\in C.
Then the following assertions hold.

(a)
Let \{{x}_{n}\} be a sequence generated from an arbitrary initial {x}_{1}\in C by the iterative scheme
{x}_{n+1}={P}_{C}[{\lambda}_{n}\gamma ({\alpha}_{n}f({x}_{n})+(1{\alpha}_{n})S{x}_{n})+(I{\lambda}_{n}\mu F)T{x}_{n}],\phantom{\rule{1em}{0ex}}n\ge 1,(3.19)
such that {\omega}_{w}({x}_{n})\subset \Xi, then \{{x}_{n}\} converges in norm to the point {x}^{\ast}\in Fix(T) which is a unique solution of Problem 1.1.

(b)
Let \{{x}_{n}\} be a sequence generated from an arbitrary initial {x}_{1}\in C by the iterative scheme
{x}_{n+1}={P}_{C}[{\lambda}_{n}(1{\alpha}_{n})\gamma S{x}_{n}+(I{\lambda}_{n}\mu F)T{x}_{n}],\phantom{\rule{1em}{0ex}}n\ge 1,
such that {\omega}_{w}({x}_{n})\subset \Xi, then \{{x}_{n}\} converges in norm to a unique solution {x}^{\ast} of the VIP of finding {x}^{\ast}\in \Xi such that
Proof In Theorem 3.2, putting \Theta =h=0, {S}_{i}=I, i=1,\dots ,N, from (1.13) we obtain that {x}_{n}={u}_{n}={y}_{n,i}, i=1,\dots ,N. In this case, \Omega =Fix(T) and (1.13) reduces to (3.19). Moreover, it is easy to see that Problem 1.3 reduces to Problem 1.1. Thus, by Theorem 3.2 we obtain the desired results. □
Remark 3.1 Corollary 3.1 improves and extends [[29], Theorem 4.1] in the following aspects:

(a)
The restriction {lim}_{n\to 0}{\alpha}_{n}=0 in [[29], Theorem 4.1] is replaced by
0<\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}inf}{\alpha}_{n}\le \underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}{\alpha}_{n}<1; 
(b)
The condition {lim}_{n\to 0}{\lambda}_{n}^{1/\theta}/{\alpha}_{n}=0 is not assumed in Corollary 3.1;

(c)
The boundedness of the sequence \{{x}_{n}\} is not assumed in Corollary 3.1.
Very recently, Yao et al. [18] considered the following HVIP of finding {x}^{\ast}\in Fix(T) such that
where T,S:C\to C are two nonexpansive mappings and Fix(T) is the fixed point set of T. Let Ω denote the solution set of HVIP (3.20) and assume that Ω is nonempty; consequently, the metric projection {P}_{\Omega} is well defined. It is interesting to find the minimumnorm solution {x}^{\ast} of HVIP (3.20) which exists uniquely and is exactly the nearest point projection of the origin to Ω, that is, {x}^{\ast}={P}_{\Omega}(0). Alternatively, {x}^{\ast} is the unique solution of the quadratic minimization problem:
They used the contractions to regularize the nonexpansive mapping S to introduce an explicit scheme that generates a sequence \{{x}_{n}\} via an iterative algorithm and proved that this sequence converges strongly to the minimumnorm solution {x}^{\ast} of HVIP (3.20).
In Corollary 3.1, if we put \mu =2, F=\frac{1}{2}I and \gamma =\tau =1, then HVIP (1.11) reduces to HVIP (3.20) and \Xi =\Omega. In this case, THVIP (1.10) reduces to the quadratic minimization problem (3.21). In terms of Corollary 3.1(a), \{{x}_{n}\} converges in norm to the point {x}^{\ast}\in Fix(T) which is a unique solution of VIP (1.2); see (3.22). Also, by Corollary 3.1(b), \{{x}_{n}\} converges in norm to the minimumnorm solution of HVIP (3.20). Therefore, we get the following conclusions.
Corollary 3.2 Let f:C\to H be a ρcontraction with a coefficient \rho \in [0,1) and S,T:C\to C be two nonexpansive mappings with Fix(T)\ne \mathrm{\varnothing}. Assume that the solution set Ω of HVIP (3.20) is nonempty and that the following conditions hold for two sequences \{{\lambda}_{n}\},\{{\alpha}_{n}\}\subset (0,1):

(i)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;

(ii)
{lim}_{n\to \mathrm{\infty}}{\lambda}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\lambda}_{n}=\mathrm{\infty};

(iii)
{\sum}_{n=2}^{\mathrm{\infty}}{\alpha}_{n}{\lambda}_{n}{\alpha}_{n1}{\lambda}_{n1}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{\alpha}_{n}{\lambda}_{n}{\alpha}_{n1}{\lambda}_{n1}}{{\lambda}_{n}}=0;

(iv)
{\sum}_{n=2}^{\mathrm{\infty}}{\lambda}_{n}{\lambda}_{n1}<\mathrm{\infty} or {lim}_{n\to \mathrm{\infty}}\frac{{\lambda}_{n}{\lambda}_{n1}}{{\lambda}_{n}}=0;

(v)
there are constants \overline{k},\theta >0 satisfying \parallel xTx\parallel \ge \overline{k}{[d(x,\Omega )]}^{\theta} for all x\in C.
Then the following assertions hold.

(a)
Let \{{x}_{n}\} be a sequence generated from an arbitrary initial {x}_{1}\in C by the iterative scheme
{x}_{n+1}={P}_{C}[{\lambda}_{n}({\alpha}_{n}f({x}_{n})+(1{\alpha}_{n})S{x}_{n})+(1{\lambda}_{n})T{x}_{n}],\phantom{\rule{1em}{0ex}}n\ge 1,
such that {\omega}_{w}({x}_{n})\subset \Omega, then \{{x}_{n}\} converges in norm to the point {x}^{\ast}\in Fix(T) which is a unique solution of the VIP of finding {x}^{\ast}\in \Omega such that

(b)
Let \{{x}_{n}\} be a sequence generated from an arbitrary initial {x}_{1}\in C by the iterative scheme
{x}_{n+1}={P}_{C}[{\lambda}_{n}(1{\alpha}_{n})S{x}_{n}+(1{\lambda}_{n})T{x}_{n}],\phantom{\rule{1em}{0ex}}n\ge 1,
such that {\omega}_{w}({x}_{n})\subset \Omega, then \{{x}_{n}\} converges in norm to a minimumnorm solution of HVIP (3.20).
4 Applications
Let C be a nonempty, closed and convex subset of a real Hilbert space H. Recall that a point u\in C is a solution to VIP (1.1) if and only if
Definition 4.1 An operator A:C\to H is said to be an αinverse strongly monotone operator if there exists a constant \alpha >0 such that
As an example, we recall that the αinverse strongly monotone operators are firmly nonexpansive mappings if \alpha \ge 1 and that every αinverse strongly monotone operator is also \frac{1}{\alpha}Lipschitz continuous (see [37]).
Let us observe also that, if A is αinverse strongly monotone, the mappings {P}_{C}(I\lambda A) are nonexpansive for all \lambda \in (0,2\alpha ] since they are compositions of nonexpansive mappings (see [[37], pp.419]).
Let us consider {\tilde{S}}_{1},\dots ,{\tilde{S}}_{M} to be a finite number of nonexpansive selfmappings on C and {A}_{1},\dots ,{A}_{N} to be a finite number of αinverse strongly monotone operators. Let T be a nonexpansive selfmapping on C. Very recently, Marino, Muglia and Yao [27] considered an application of Theorem 3.12 of [27], to solve the following mixed problem.
To find {x}^{\ast}\in Fix(T)\cap \mathit{EP}(\Theta ,h) such that
Let us call (SVIP) the set of solutions of the (N+M)system. This problem is equivalent to finding a common fixed point of T, {\{{P}_{Fix(T)\cap \mathit{EP}(\Theta ,h)}{\tilde{S}}_{i}\}}_{i=1}^{M}, {\{{P}_{C}(I\lambda {A}_{i})\}}_{i=1}^{N}.
Based on the above mixed problem, in this section we first consider the following more general mixed problem.
Problem 4.1 Let F:C\to H be κLipschitzian and ηstrongly monotone on C, f:C\to H be a ρcontraction with a coefficient \rho \in [0,1) and S, T be nonexpansive selfmappings on C. Let 0<\mu <2\eta /{\kappa}^{2} and 0<\gamma \le \tau, where \tau =1\sqrt{1\mu (2\eta \mu {\kappa}^{2})}. Assume that \Theta ,h:C\times C\to \mathbb{R} are two bifunctions. Then the objective is to find {x}^{\ast}\in \Omega such that
where \Omega =Fix(T)\cap (\mathit{SVIP})\cap \mathit{EP}(\Theta ,h)\ne \mathrm{\varnothing}.
Utilizing Theorem 3.1 we obtain the following result.
Theorem 4.1 Assume that Problem 4.1 has a solution. Let 0<\lambda \le 2\alpha. Let \{{\lambda}_{n}\}, \{{\alpha}_{n}\}, \{{\beta}_{n,i}\}, i=1,\dots ,(N+M) be sequences in (0,1) such that {\beta}_{n,i}\to {\beta}_{i}\in (0,1) as n\to \mathrm{\infty} for all i\in \{1,\dots ,(N+M)\}. Suppose that the conditions (C1)(C6) in Theorem 3.1 hold and that the sequence \{{x}_{n}\} is defined explicitly by the following iterative scheme:
where {\overline{y}}_{n,0}={y}_{n,M} and {\overline{\beta}}_{n,j}={\beta}_{n,M+j}, j=1,\dots ,N. In particular, if f\equiv 0, then (4.4) reduces to the following iterative scheme:
Then the following assertions hold.

(a)
Let \{{x}_{n}\} be a sequence generated by the scheme (4.4), then \{{x}_{n}\} converges strongly to the point {x}^{\ast}\in \Omega which is a unique solution of Problem 4.1.

(b)
Let \{{x}_{n}\} be a sequence generated by the scheme (4.5), then \{{x}_{n}\} converges strongly to a unique solution {x}^{\ast}\in \Omega of the following system of variational inequalities:
\{\begin{array}{cc}\u3008F{x}^{\ast},x{x}^{\ast}\u3009\ge 0,\hfill & \mathrm{\forall}x\in \Omega ,\hfill \\ \u3008(\mu F\gamma S){x}^{\ast},y{x}^{\ast}\u3009\ge 0,\hfill & \mathrm{\forall}y\in \Omega .\hfill \end{array}(4.6)
Now, we consider another more general mixed problem.
Problem 4.2 Let F:C\to H be κLipschitzian and ηstrongly monotone on C, f:C\to H be a ρcontraction with a coefficient \rho \in [0,1) and S, T be nonexpansive selfmappings on C. Let 0<\mu <2\eta /{\kappa}^{2} and 0<\gamma \le \tau, where \tau =1\sqrt{1\mu (2\eta \mu {\kappa}^{2})}. Assume that are two bifunctions. Then the objective is to find {x}^{\ast}\in \Xi such that
where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding {z}^{\ast}\in \Omega such that
with \Omega =Fix(T)\cap (\mathit{SVIP})\cap \mathit{EP}(\Theta ,h)\ne \mathrm{\varnothing}.
Utilizing Theorem 3.2, we get the following result.
Theorem 4.2 Let 0<\lambda \le 2\alpha. Let \{{\lambda}_{n}\}, \{{\alpha}_{n}\}, \{{\beta}_{n,i}\}, i=1,\dots ,(N+M) be sequences in (0,1) such that {\beta}_{n,i}\to {\beta}_{i}\in (0,1) as n\to \mathrm{\infty} for all i\in \{1,\dots ,(N+M)\}. Assume that the solution set Ξ of HVIP (4.8) is nonempty and that the conditions (C1)(C7) in Theorem 3.2 hold. Then the following assertions hold.

(a)
Let \{{x}_{n}\} be a sequence generated by the scheme (4.4) such that {\omega}_{w}({x}_{n})\subset \Xi, then \{{x}_{n}\} converges strongly to the point {x}^{\ast}\in \Omega which is a unique solution of Problem 4.2.

(b)
Let \{{x}_{n}\} be a sequence generated by the scheme (4.5) such that {\omega}_{w}({x}_{n})\subset \Xi, then \{{x}_{n}\} converges strongly to a unique solution {x}^{\ast} of the following VIP:
\mathit{\text{find}}{x}^{\ast}\in \Xi \mathit{\text{such that}}\u3008F{x}^{\ast},x{x}^{\ast}\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in \Xi .(4.9)
5 Concluding remarks
We considered a system of variational inequalities defined over the intersection of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings, and the solution set of a nonexpansive mapping (Problem 1.2). We also considered a triple hierarchical variational inequality problem, that is, a variational inequality problem defined over a set of solutions of another variational inequality problem which is defined over the intersection of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings, and the solution set of a nonexpansive mapping (Problem 1.3). The nontrivial examples of Problems 1.2 and 1.3 are also given in Section 4. We combined the onestep iterative method proposed in [29] and the multistep iterative method given in [27] to propose a multistep hybrid viscosity method that generates a sequence via an explicit iterative algorithm. It is worth pointing out that the onestep iterative method given in [29] combines the regularization method, the hybrid steepestdescent method and the projection method, and that the multistep iterative method given in [27] involves the Ishikawatype iterative method and the viscosity approximation method. Moreover, it is also proven that under two different pools of suitable conditions such a sequence converges strongly to a unique solution of Problem 1.2 and to a unique solution of Problem 1.3, respectively.
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Acknowledgements
This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah under the grant No. 107130D1432. The authors, therefore, acknowledge with thanks technical and financial support of the DSR. The authors would also like to thank the referees for careful reading of the manuscript.
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Latif, A., Ceng, LC. & Ansari, Q.H. Multistep hybrid viscosity method for systems of variational inequalities defined over sets of solutions of an equilibrium problem and fixed point problems. Fixed Point Theory Appl 2012, 186 (2012). https://doi.org/10.1186/168718122012186
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DOI: https://doi.org/10.1186/168718122012186