Abstract
In this paper, we suggest and analyze an iterative algorithm to approximate a common solution of a hierarchical fixed point problem for nonexpansive mappings, a system of variational inequalities, and a split equilibrium problem in Hilbert spaces. Under some suitable conditions imposed on the sequences of parameters, we prove that the sequence generated by the proposed iterative method converges strongly to a common element of the solution set of these three kinds of problems. The results obtained here extend and improve the corresponding results of the relevant literature.
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1 Introduction
Let \(H_{1}\) and \(H_{2}\) be two real Hilbert spaces, whose inner product and norm are denoted by \(\langle \cdot, \cdot \rangle \) and \(\Vert \cdot \Vert \). And let \(C_{1}\) and \(C_{2}\) be two nonempty closed convex subsets of \(H_{1}\) and \(H_{2}\), respectively. Recall that the mapping \(T:C_{1} \to C_{1}\) is nonexpansive if \(\Vert Tx - Ty \Vert \le \Vert x - y \Vert \) for all \(x,y \in C_{1}\). We denote the fixed point set of T by \(\mathrm{Fix}(T) = \{ x \in C_{1}:x = Tx \} \). If T is nonexpansive, then \(\mathrm{Fix}(T)\) is nonempty, closed, and convex. Next, we consider the following three kinds of problems, which are paid attention to in our paper.
Problem 1
(Hierarchical fixed point problem (HFPP))
In 2006, Moudafi and Mainge [23] introduced and studied the following hierarchical fixed point problem (in short HFPP) for a nonexpansive mapping T with respect to another nonexpansive mapping S on \(C_{1}\): Find \(x \in \mathrm{Fix}(T)\) such that
which amounts to saying that \(x \in \mathrm{Fix}(T)\) satisfies the variational inequality depending on a given criterion S, namely, find \(x \in C_{1}\) such that
where I is the identity mapping on \(C_{1}\) and \(N_{\mathrm{Fix}(T)}\) is the normal cone to \(\mathrm{Fix}(T)\) at x defined by
We know that the hierarchical fixed point problem links with some monotone variational inequalities and convex programming problems, see [39] and the references therein. In 2007, Moudafi [22] introduced the following Krasnoselski–Mann algorithm for solving HFPP (1):
where \(\{ \alpha _{n} \} \) and \(\{ \sigma _{n} \} \) are two real sequences in (0,1).
On the other hand, in 2011, Ceng, Anasri, and Yao [8] proposed the following iterative method:
where U is a Lipschitzian mapping, and F is a Lipschitzian and strongly monotone mapping. Under some approximate assumptions, they proved that the sequence \(\{ x_{n} \} \) generated by the above iterative algorithm converges strongly to the unique solution of the variational inequality
Note that HFPP (2) is more general than HFPP (1).
Problem 2
(Split equilibrium problem (SEP))
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Let F be a bifunction of \(C \times C\) into R, where R is the set of real numbers. The equilibrium problem(in short, EP) for \(F:C \times C \to R\) is to find \(x \in C\) such that
which was introduced and studied by Blum and Oettli [3]. It contains many problems, such as fixed point problem, variational inequality problem, Nash equilibrium problem, optimization problem, and complementarity problem as special cases, see, e.g., [1, 2, 20, 31] and the references therein. In 1997, Combettes and Hirstoaga [15] introduced an iterative scheme of finding the best approximation to the initial data when a set of solutions (3) is nonempty and proved a strong convergence theorem. We denote the solution set of EP (3) by \(EP(F) = \{ x \in C:F(x,y) \ge 0,\forall y \in C \} \).
Recently, Kazmi and Rizvi [21] considered the following split equilibrium problem (in short, SEP): Let \(F_{1}:C_{1} \times C_{1} \to R\) and \(F_{2}:C_{2} \times C_{2} \to R\) be two nonlinear bifunctions and \(A:H_{1} \to H_{2}\) be a bounded linear operator, then the SEP is to find \(x^{*} \in C_{1}\) such that
and
where \(y^{*} = Ax^{*} \in C_{2}\). The solution set of SEP (4)–(5) is denoted by \(\Gamma = \{ p \in EP(F_{1}):Ap \in EP(F_{2}) \} \). This formalism is also the core of modeling of many inverse problems arising in phase retrieval and other real word problems, for example, in sensor networks in computerized tomography, in intensity-modulated radiation therapy treatment planning, and data compression, see, e.g., [5, 6, 12–14] and the references therein.
Problem 3
(System of variational inequalities (SVI))
Let \(C_{1}\) be a nonempty closed convex subset of \(H_{1}\) and \(A,B:C_{1} \to H_{1}\) be two mappings. Ceng, Wang, and Yao [11] considered the following problem which finds \((x^{*},y^{*}) \in C_{1} \times C_{1}\) such that
Problem (6) is called a general system of variational inequalities, where \(\lambda _{1} > 0\) and \(\lambda _{2} > 0\) are constants. In 2015, Jitsupa et al. [19] introduced the following system of variational inequalities in a Hilbert space \(H_{1}\), that is, finding \(x_{i}^{*} \in C_{1}(i = 1,2, \ldots,N)\) such that
which is called a more general system of variational inequalities, where \(\lambda _{i} > 0\) and \(B_{i}:C_{1} \to H_{1}\) is a nonlinear mapping for all \(i \in \{ 1,2, \ldots,N \} \). The solution set of SVI (7) is denoted by \(GSVI(C_{1},B_{i})\).
In view of these different three kinds of problems, there are some new research results on numerical algorithm in the recent literature. Under the setting of uniformly convex Banach spaces, in [27–30], the Thakur three-step iterative process in the context of Suzuki-type nonexpansive mappings or generalized nonexpansive mappings enriched with property (E) was studied, and a comparative numerical experiment was performed with the visualization of some convergence behaviors. In [25], an S-iteration technique for finding common fixed points for nonself quasi-nonexpansive mappings was developed, and convergence properties of the proposed algorithm were analyzed. And in [17], a hybrid projection algorithm for a countable family of mappings was considered, and the strong convergence of the algorithm converging to the common fixed point of the mappings was given. Very recently, Dadashi and Postolache [18] constructed a forward–backward splitting algorithm for approximating a zero of the sum of an α-inverse strongly monotone operator and a maximal monotone operator. They proved the strong convergence theorem under mild conditions. Especially, they added a nonexpansive mapping in the algorithm and proved that the generated sequence converged strongly to a common element of the fixed point set of a nonexpansive mapping and the zero point set of the sum of monotone operators. They also applied their main result both to equilibrium problems and convex programming.
On the other hand, Ceng et al. [9] introduced a hybrid viscosity extragradient method for finding the common elements of the solution set of a general system of variational inequalities and the common fixed point set of a countable family of nonexpansive mappings and zero points of an accretive operator in real smooth Banach spaces. Moreover, they [10] proposed an implicit composite extragradient-like method based on the Mann iteration method, the viscosity approximation method, and the Korpelevich extragradient method for solving a general system of variational inequalities with a hierarchical variational inequality constraint for countably many uniformly Lipschitzian pseudocontractive mappings and an accretive operator in a real Banach space. In [36, 38], Yao, Postolache, and Yao suggested a projected type algorithm and an extragradient algorithm for finding the common solutions of two variational inequalities and the common element of the set of fixed points of a pseudocontractive operator and the set of solutions of the variational inequality problem in Hilbert spaces, respectively. In [35, 37], Yao et al. introduced iterative algorithms for solving a split variational inequality and a fixed point problem that requires finding a solution of a generalized variational inequality whose image is a fixed point of a pseudocontractive operator or a fixed point of two quasi-pseudocontractive operators under a nonlinear transformation in Hilbert spaces. In [33, 34], Yao et al. constructed iterative algorithms for solving the split feasibility problem and the fixed point problem, the split equilibrium problems and fixed point problems involved in the pseudocontractive mappings in Hilbert spaces and proved their strong convergence.
Inspired and motivated by the above research work, we suggest an iterative approximation method for finding an element of the common solution set of HFPP (2), SEP (4)–(5), and SVI (7) involved in nonexpansive mappings. To our best knowledge, there is no further study on finding the element of the common solution set of HFPP (2), SEP (4)–(5), and SVI (7). When the mappings take different types of cases, we can obtain a corollary on the common element of the set of fixed points of a nonexpansive mapping, the solution set of a variational inequality and an equilibrium problem. So, our results presented here are new and very interesting.
The paper is organized as follows. In Sect. 2, we recall some concepts and lemmas which are needed in proving our main results. In Sect. 3, we suggest an iterative algorithm for solving the three different kinds of problems and prove its strong convergence. At last, the conclusion is given.
2 Preliminaries
In this section, we list some fundamental results that are useful in the consequent analysis.
Let H be a real Hilbert space, C be a nonempty closed and convex subset of H.
Then, for all \(x,y \in H\), the following inequalities hold:
A function \(F:C \times C \to R\) is called an equilibrium function if it satisfies the following conditions:
-
(A1)
\(F(x,x) = 0\) for all \(x \in C\);
-
(A2)
F is monotone, i.e., \(F(x,y) + F(y,x) \le 0\) for all \(x,y \in C\);
-
(A3)
\(\mathop{\lim \sup}_{t \downarrow 0}F(tz + (1 - t)x,y) \le F(x,y)\) for all \(x,y,z \in C\);
-
(A4)
for each \(x \in C\), \(y \mapsto F(x,y)\) is convex and lower semi-continuous;
-
(A5)
Fix \(r > 0\) and \(z \in C\), there exists a nonempty compact convex subset K of H and \(x \in C \cap K\) such that
$$\begin{aligned} F(y,x) + \frac{1}{r} \langle y - x,x - z \rangle < 0, \quad\forall y \in C \backslash K. \end{aligned}$$
Lemma 2.1
([16])
Assume that \(F:C \times C \to R\) is an equilibrium function. For \(r > 0\), define a mapping \(R_{r,F}:H \to C\) as follows:
for all \(x \in H\). Then the following hold:
-
(B1)
\(R_{r,F}\) is single-valued;
-
(B2)
Fix \((R_{r,F}) = EP(F)\) and \(EP(F)\) is a nonempty closed and convex subset of C;
-
(B3)
\(R_{r,F}\) is a firmly nonexpansive mapping, i.e.,
$$\begin{aligned} \bigl\Vert R_{r,F}(x) - R_{r,F}(y) \bigr\Vert ^{2} \le \bigl\langle R_{r,F}(x) - R_{r,F}(y),x - y \bigr\rangle , \quad\forall x,y \in H.C. \end{aligned}$$
Lemma 2.2
Let \(F:C \times C \to R\) be an equilibrium function, and let \(R_{r,F}\) be defined as in Lemma 2.1for \(r > 0\). Let \(x,y \in H\) and \(r_{1},r_{2} > 0\), then
Lemma 2.3
([32])
Let \(\{ a_{n} \} \) be a sequence of nonnegative real numbers such that
where \(\{ \alpha _{n} \} \) is a sequence in \(( 0,1 )\) and \(\{ \delta _{n} \} \) is a sequence in R such that
Then \(\lim_{n \to \infty } a_{n} = 0\).
Lemma 2.4
Let \(P_{C}\) denote the projection of H onto C. It is known that \(P_{C}\) is nonexpansive and the following inequalities hold:
Lemma 2.5
If B is an α-inverse-strongly monotone mapping of C into H, and \(\lambda \in [0,2\alpha ]\), then \(I - \lambda B\) is a nonexpansive mapping.
Proof
For any \(w,u \in C_{1}\), we have
which implies that \(I - \lambda B\) is nonexpansive, completing the proof. □
Lemma 2.6
([7])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let \(B_{i}:C \to H\) be an \(\alpha _{i}\)-inverse-strongly monotone mapping, where \(i \in \{ 1,2, \ldots, N \} \). Let \(G:C \to C\) be a mapping defined by
If \(\lambda _{i} \in [0,2\alpha _{i}]\), \(i = 1,2, \ldots,N\), then \(G:C \to C\) is nonexpansive.
Proof
Putting \(T^{i} = P_{C}(I - \lambda _{i}B_{i})P_{C}(I - \lambda _{i - 1}B_{i - 1}) \cdots P_{C}(I - \lambda _{2}B_{2})P_{C}(I - \lambda _{1}B_{1}), i = 1,2, \ldots,N\), and \(T^{0} = I\), where I is an identity mapping on C. Then \(G = T^{N}\). For all \(x,y \in C\), we have
Then G is nonexpansive, which completes the proof. □
Lemma 2.7
([8])
Let \(U:C \to H\) be a τ-Lipschitzian mapping, and let \(F:C \to H\) be a k-Lipschitzian mapping and η-strongly monotone mapping, then, for \(0 \le \rho \tau < \mu \eta \), \(\mu F - \rho U\) is \((\mu \eta - \rho \tau )\)-strongly monotone, i.e.,
Lemma 2.8
([26])
Suppose that \(\lambda \in (0,1)\) and \(\mu > 0\). Let \(F:C \to H\) be a k-Lipschitzian and η-strongly monotone mapping. In association with a nonexpansive mapping \(T:C \to C\), define the mapping \(T^{\lambda }:C \to H\) by
Then \(T^{\lambda }\) is a contractive mapping with \(\mu < \frac{2\eta }{k^{2}}\), that is,
where \(\nu = 1 - \sqrt{1 - \mu (2\eta - \mu k^{2})}\).
Lemma 2.9
([24])
Each Hilbert space H satisfies the Opial condition, that is, for any sequence \(\{ x_{n}\}\) with \(x_{n}\) converging weakly to x, the inequality \(\mathop{\lim \inf}_{n \to \infty } \Vert x_{n} - x \Vert < \mathop{\lim \inf}_{n \to \infty } \Vert x_{n} - y \Vert \) holds for every \(y \in H\) with \(y \ne x\).
Lemma 2.10
([4] Demiclosedness principle)
Let C be a closed convex subset of a real Hilbert space H, and let \(T:C \to C\) be a nonexpansive mapping. Then \(I - T\) is demiclosed at zero, that is, \(x_{n}\) converges weakly to \(x,x_{n} - Tx_{n} \to 0\) implies \(x = Tx\).
3 Main results
Theorem 3.1
For \(i \in \{ 1,2 \} \), let \(H_{i}\) be a real Hilbert space, \(C_{i}\) be a nonempty closed convex subset of \(H_{i}\), let \(F_{i}:C_{i} \times C_{i} \to R\) be an equilibrium function. Let \(A:H_{1} \to H_{2}\) be bounded linear operators with their adjoint operators \(A^{*}\). Let \(B_{i}\) be \(\xi _{i}\)-inverse-strongly monotone, respectively, where \(i \in \{ 1,2, \ldots, N \} \). Let \(F:C_{1} \to C_{1}\) be a k-Lipschitzian mapping and η-strongly monotone, and let \(U:C_{1} \to C_{1}\) be a τ-Lipschitzian mapping. Let \(S,T:C_{1} \to C_{1}\) be two nonexpansive mappings such that \(\Theta = \Gamma \cap \mathrm{Fix}(G) \cap \mathrm{Fix}(T) \ne \emptyset \). For a given \(x_{0} \in C_{1}\) arbitrarily, let the iterative sequences \(\{ u_{n} \} \), \(\{ y_{n} \} \), and \(\{ x_{n} \} \) be generated by
where \(\{ r_{n} \} \subset (0,\infty ),\gamma \in (0,1 / L_{A}), L_{A}\) is the spectral radius of the operators \(A^{*}A\). Suppose that the parameters satisfy \(0 < \mu < \frac{2\eta }{k^{2}}\), \(k \ge \eta \), \(0 \le \rho \tau < \nu \), where \(\nu = 1 - \sqrt{1 - \mu (2\eta - \mu k)^{2}}\), and \(\{ \alpha _{n} \} \), \(\{ \beta _{n} \} \) are the sequences in \((0,1)\) satisfying the following conditions:
-
(i)
\(\lim_{n \to \infty } \alpha _{n} = 0\) and \(\sum_{n = 0}^{\infty } \alpha _{n} = \infty \), \(\sum_{n = 1}^{\infty } \vert \alpha _{n - 1} - \alpha _{n} \vert < \infty \);
-
(ii)
\(\mathop{\lim \sup}_{n \to \infty } \frac{\beta _{n}}{\alpha _{n}} = 0\), \(\beta _{n} \le \alpha _{n} ( n \ge 1 )\) and \(\sum_{n = 1}^{\infty } \vert \beta _{n - 1} - \beta _{n} \vert < \infty \);
-
(iii)
\(\mathop{\lim \inf}_{n \to \infty } r_{n} > 0\), \(\sum_{n = 1}^{\infty } \vert r_{n - 1} - r_{n} \vert < \infty\).
Then the sequence \(\{ x_{n} \} \) generated by (8) converges strongly to \(w \in \Theta \).
Proof
Let \(p \in \Theta \), i.e., \(p \in \Gamma \), that is, \(p = R_{r_{n},F_{1}}(p)\) and \(Ap = R_{r_{n},F_{2}}(Ap)\). For convenience, we split the proof into several steps.
Step 1. We show that \(\{ x_{n} \} \), \(\{ u_{n} \} \), \(\{ y_{n} \} \), \(\{ z_{n} \} \) are bounded.
First, by (8) and the expansiveness of \(R_{r_{n},F_{1}}\), we estimate
It follows from the definition of \(L_{A}\) that
By using Lemma 2.4, we have
From (9)–(11) and \(\gamma \in (0,1 / L_{A})\) it follows that
It follows from (8), (12), and Lemma 2.6 that we have
Next, we prove that the sequence \(\{ x_{n} \} \) is bounded. Note \(\beta _{n} \le \alpha _{n}\) for all \(n \ge 1\). Put \(V_{n} = \alpha _{n}\rho U(x_{n}) + (I - \alpha _{n}\mu F)(T(z_{n}))\),
from (8), we get
So \(\{ x_{n} \} \) is bounded, and consequently we can deduce that \(\{ u_{n} \} \), \(\{ y_{n} \}, \{ z_{n} \} \) are also bounded.
Step 2. We will show the following:
Noting \(u_{n} = R_{r_{n},F_{1}}(x_{n} + \gamma A^{*}(R_{r_{n},F_{2}} - I)Ax_{n})\) and \(u_{n - 1} = R_{r_{n - 1},F_{1}}(x_{n - 1} + \gamma A^{*}(R_{r_{n - 1},F_{2}} - I)Ax_{n - 1})\), from Lemma 2.2, we have
where
So, from Lemma 2.6, we have
Then from (16) we get
Next, by Lemma 2.8, we estimate
where \(M = \max \{ \sup_{n \ge 1}( \Vert \rho U(x_{n - 1}) \Vert + \Vert \mu F(T(z_{n - 1})) \Vert ), \sup_{n \ge 1}(\gamma \Vert A \Vert \sigma _{n - 1} + \delta _{n - 1}), \sup_{n \ge 1}( \Vert Sx_{n - 1} \Vert + \Vert z_{n - 1} \Vert ) \} \). And ε is a real number such that \(0 < \varepsilon < r_{n}\). So, it follows from Conditions (i)–(iii) and Lemma 2.3 that
Next, we show that \(\lim_{n \to \infty } \Vert u_{n} - x_{n} \Vert = 0\). In view of (8), (9), (12), and (13), we obtain
From the above inequality and (12), (13), we get
which means that
Since \(\alpha _{n} \to 0\), \(\beta _{n} \to 0\) and \(\lim_{n \to \infty } \Vert x_{n + 1} - x_{n} \Vert = 0\), we obtain
And since \(R_{r_{n},F_{1}}\) is firmly nonexpansive, from (8) we get
which implies that
So, from (21) and (25) we have
which implies that
Hence
Since \(\lim_{n \to \infty } \alpha _{n} = 0,\lim_{n \to \infty } \beta _{n} = 0,\lim_{n \to \infty } \Vert x_{n + 1} - x_{n} \Vert = 0,\text{and} \lim_{n \to \infty } \Vert (R_{r_{n},F_{2}} - I)Ax_{n} \Vert = 0\), we have
Then, by Lemma 2.5 and Lemma 2.6, we obtain
From (21), we obtain
which implies that
Since \(\lim_{n \to \infty } \alpha _{n} = 0,\lim_{n \to \infty } \beta _{n} = 0\) and \(\lim_{n \to \infty } \Vert x_{n + 1} - x_{n} \Vert = 0\), we have
By Lemma 2.4, we obtain
which implies
By induction and (12), we have
It follows from (21) and (34) that we have
which implies
Since \(\lim_{n \to \infty } \alpha _{n} = 0\), \(\lim_{n \to \infty } \beta _{n} = 0\) and \(\lim_{n \to \infty } \Vert B_{i}T^{i - 1}u_{n} - B_{i}T^{i - 1}p \Vert ^{2} = 0\), we have
From (37), we obtain
which means \(\lim_{n \to \infty } \Vert u_{n} - y_{n} \Vert = 0\). Note \(\lim_{n \to \infty } \Vert u_{n} - x_{n} \Vert = 0,\lim_{n \to \infty } \Vert u_{n} - y_{n} \Vert = 0\), then we have \(\lim_{n \to \infty } \Vert x_{n} - y_{n} \Vert = 0\). Since \(T(x_{n}) \in C_{1}\), we have
Noting that \(\lim_{n \to \infty } \alpha _{n} = 0\), \(\lim_{n \to \infty } \beta _{n} = 0\), \(\lim_{n \to \infty } \Vert x_{n} - y_{n} \Vert = 0\),and \(\lim_{n \to \infty } \Vert x_{n + 1} - x_{n} \Vert = 0\), we have \(\lim_{n \to \infty } \Vert x_{n} - T(x_{n}) \Vert = 0\).
Step 3. We show that \(z \in F(T)\). Assume that \(z \notin F(T)\). Since \(x_{n_{i}}\) converges weakly to z and \(Tz \ne z\), by Lemma 2.9, we have
which is a contradiction. Thus, we obtain \(z \in F(T)\). To prove the convergence of the sequence \(\{ x_{n} \} \), we need to prove the following conclusion, that is, the sequence \(\{ x_{n} \} \) generated by (8) converges strongly to w, which is the unique solution of the variational inequality
In fact, noting that \(u_{n} = R_{r_{n,F_{1}}}(x_{n} + \gamma A^{*}(R_{r_{n,F_{2}}} - I)Ax_{n}\) and
From the monotonicity of \(F_{1}\), we have
and
Since \(\Vert u_{n} - x_{n} \Vert \to 0\), \(\Vert (R_{r_{n},F_{2}} - I)Ax_{n} \Vert \to 0\), we get \(\{ u_{n_{i}} \} \) converges weakly to z. By (A4), we know \(F_{1}(y,z) \le 0\), \(\forall y \in C_{1}\). Let \(y_{t} = ty + (1 - t)z\), \(t \in (0,1]\), it follows from \(y \in C_{1}\), \(z \in C_{1}\) and the convexity of \(C_{1}\) that \(F_{1}(y_{t},z) \le 0\). So, from (A1), (A3), and (A4), we have
Therefore \(F_{1}(z,y) \ge 0, \forall y \in C_{1}\). This is \(z \in EP(F_{1})\).
Next we show that \(Az \in EP(F_{2})\), since \(\Vert u_{n} - x_{n} \Vert \to 0\), there exists a subsequence \(\{ x_{n_{k}} \} \) of \(\{ x_{n} \} \) such that \(\{ x_{n_{k}} \} \) converges weakly to z, and since A is a bounded linear operator, \(\{ Ax_{n_{k}} \} \) converges weakly to Az. Setting \(\varpi _{n_{k}} = Ax_{n_{k}} - R_{r_{n_{k}},F_{2}}Ax_{n_{k}}\), it follows tfrom \(\lim_{n \to \infty } \Vert (R_{r_{n},F_{2}} - I)Ax_{n} \Vert = 0\) that \(\lim_{k \to \infty } \varpi _{n_{k}} = 0\). By Lemma 2.1, we have
Since \(F_{2}\) is upper semicontinuous in the first argument, taking limsup to the above inequality as \(k \to \infty \), we have \(F_{2}(Az,y) \ge 0\), \(\forall y \in C_{2}\), which means that \(Az \in EP(F_{2})\), so \(z \in \Gamma \). Next, we claim that \(z \in \mathrm{Fix}(G)\). From Lemma 2.6, we know \(G = T^{N}\) is nonexpansive, and
It follows from \(\lim_{n \to \infty } \Vert u_{n} - x_{n} \Vert = 0\) and \(\lim_{n \to \infty } \Vert x_{n} - y_{n} \Vert = 0\) that \(\lim_{n \to \infty } \Vert y_{n} - Gy_{n} \Vert = 0\). Furthermore, we get
which implies \(\lim_{n \to \infty } \Vert x_{n} - Gx_{n} \Vert = 0\). Then, by Lemma 2.10, we obtain \(z \in \mathrm{Fix}(G)\). Thus, we have \(z \in \Theta \). Observe that the constants satisfy \(0 \le \rho \tau < v\) and \(k \ge \eta \), from Lemma 2.7, the operator \(\mu F - \rho U\) is \(\mu \eta - \rho \tau \) strongly monotone. Then we get the uniqueness of the solution of the variational inequality and denote it by \(w \in \Theta \).
Last, we show that \(x_{n} \to w\). Note that
and
which implies that
Let \(\sigma _{n} = \Vert x_{n} - w \Vert ^{2},\phi _{n} = \alpha _{n}(\nu - \rho \tau )\) and
Then the above inequality turns into the following:
From Conditions (i) and (ii) of Theorem 3.1, we have
Then all conditions in Lemma 2.3 are satisfied, thus we can get \(\sigma _{n} \to 0\ (n \to \infty )\), that is, \(x_{n} \to w\ (n \to \infty )\). This completes the proof. □
Corollary 3.1
For \(i \in \{ 1,2 \} \), let \(H_{i}\) be a real Hilbert space, \(C_{i}\) be a nonempty closed convex subset of \(H_{i}\), let \(F_{i}:C_{i} \times C_{i} \to R\) be an equilibrium function. Let \(A:H_{1} \to H_{2}\) be bounded linear operators with their adjoint operators \(A^{*}\). Let \(B_{1}\) be \(\xi _{1}\)-inverse-strongly monotone. Let \(F:C_{1} \to C_{1}\) be a k-Lipschitzian mapping and be η-strongly monotone, and let \(U:C_{1} \to C_{1}\) be a τ-Lipschitzian mapping. Let \(S,T:C_{1} \to C_{1}\) be two nonexpansive mappings such that \(\Theta = \Gamma \cap \mathrm{Fix}(G) \cap \mathrm{Fix}(T) \ne \emptyset \). For given \(x_{0} \in C_{1}\) arbitrarily, let the iterative sequences \(\{ u_{n} \} \), \(\{ y_{n} \} \), and \(\{ x_{n} \} \) be generated by
where \(\{ r_{n} \} \subset (0,\infty ),\gamma \in (0,1 / L_{A}), L_{A}\) is the spectral radius of the operators \(A^{*}A\). Suppose that the parameters satisfy \(0 < \mu < \frac{2\eta }{k^{2}}\), \(k \ge \eta \), \(0 \le \rho \tau < \nu \), where \(\nu = 1 - \sqrt{1 - \mu (2\eta - \mu k)^{2}}\), and \(\{ \alpha _{n} \} \), \(\{ \beta _{n} \} \) are the sequences in \((0,1)\) satisfying the following conditions:
-
(i)
\(\lim_{n \to \infty } \alpha _{n} = 0\) and \(\sum_{n = 0}^{\infty } \alpha _{n} = \infty, \sum_{n = 1}^{\infty } \vert \alpha _{n - 1} - \alpha _{n} \vert < \infty \);
-
(ii)
\(\mathop{\lim \sup}_{n \to \infty } \frac{\beta _{n}}{\alpha _{n}} = 0\), and \(\beta _{n} \le \alpha _{n} ( n \ge 1 )\), \(\sum_{n = 1}^{\infty } \vert \beta _{n - 1} - \beta _{n} \vert < \infty \);
-
(iii)
\(\mathop{\lim \inf}_{n \to \infty } r_{n} > 0\), \(\sum_{n = 1}^{\infty } \vert r_{n - 1} - r_{n} \vert < \infty \).
Then the sequence \(\{ x_{n} \} \) generated by (39) converges strongly to \(w \in \Theta \).
Proof
Putting \(N = 1\) in Theorem 3.1, we can conclude the desired conclusion directly. □
4 Conclusion
In this paper, we considered a hierarchical fixed point problem (2), a split equilibrium problem (4)–(5), and a system of variational inequalities (7) in Hilbert spaces. An iterative algorithm for finding the common element of the solution sets of the three kinds of problems is presented. Strong convergence of the proposed algorithm is proved. The results presented here are new and very interesting.
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The authors would like to thank the reviewers for their valuable comments, which have helped to improve the quality of this paper.
Funding
This research was supported by the National Natural Science Foundation of China, Liaoning Provincial Department of Education, and Liaoning Natural Fund Guidance Plan under project No. 11371070, No. LJ2019011, No. 2019-ZD-0502.
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Zhao, Y., Liu, X. & Sun, R. Iterative algorithms of common solutions for a hierarchical fixed point problem, a system of variational inequalities, and a split equilibrium problem in Hilbert spaces. J Inequal Appl 2021, 111 (2021). https://doi.org/10.1186/s13660-021-02645-4
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DOI: https://doi.org/10.1186/s13660-021-02645-4