Abstract
In this paper, we introduce and analyze a multi-step hybrid steepest-descent extragradient algorithm and multi-step composite Mann-type viscosity iterative algorithm for finding a solution of triple hierarchical variational inequalities defined over the common set of solutions of mixed equilibrium problems, variational inclusions, variational inequalities, and fixed point problems. Under appropriate assumptions, we prove that the proposed algorithms converge strongly to a common element of the fixed point set of a strict pseudocontractive mapping, a solution set of finitely many generalized mixed equilibrium problems, a solution set of finitely many variational inclusions, and a solution set of a general system of variational inequalities. Such an element is a unique solution of a triple hierarchical variational inequality problem. In addition, we also consider as an application the proposed algorithm to solve a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, and a general system of variational inequalities. The results obtained in this paper improve and extend the corresponding results announced by many other authors.
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1 Introduction and formulations
Let C be a nonempty, closed, and convex subset of a real Hilbert space H and \(A:C\to H\) be a nonlinear mapping on C. The variational inequality problem (VIP) defined by C and A is to find \(x\in C\) such that
The solution set of VIP (1.1) is denoted by \({\operatorname{VI}}(C,A)\). The theory of variational inequalities is well established area in nonlinear analysis and optimization. For further details on this topic, we refer to [1–5] and the references therein.
It is well known that if A is a strongly monotone and Lipschitz-continuous mapping on C, then VIP (1.1) has a unique solution. Several iterative methods have been proposed in the literature to compute the approximate solutions of a VIP. Korpelevich’s extragradient method is one of them which was proposed by Korpelevich [6]. During the last two decades, this method received much attention from many authors, who improved and generalized it in various directions and ways; see, for example, [7–18] and the references therein.
Let \(\varphi:C\to{\mathbf{R}}\) be a real-valued function, \(A : C \to H\) be a nonlinear mapping and \({\varTheta } : C \times C \to\mathbb{R}\) be a bifunction. The generalized mixed equilibrium problem (GMEP) is to find \(x\in C\) such that
We denote the set of solutions of GMEP (1.2) by \({\operatorname{GMEP}}({\varTheta },\varphi,A)\). The GMEP (1.2) is very general in the sense that it includes many problems as special cases, namely, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, etc. For different aspects and solution methods, we refer to [7, 9, 10, 14, 18–20] and the references therein.
If \(\varphi=0\), then GMEP (1.2) reduces to the generalized equilibrium problem (GEP) of finding \(x\in C\) such that
It was considered and studied in [21, 22] and further studied in [23]. The set of solutions of GEP is denoted by \({\operatorname{GEP}} ({\varTheta },A)\).
If \(A \equiv0\), then GMEP (1.2) reduces to the mixed equilibrium problem (MEP) which is to find \(x\in C\) such that
It was considered and studied in [24]. The set of solutions of MEP is denoted by \({\operatorname{MEP}}({\varTheta },\varphi)\).
If \(\varphi\equiv0\), \(A \equiv0\), then GMEP reduces to the equilibrium problem (EP) which is to find \(x\in C\) such that
The set of solutions of EP is denoted by \({\operatorname{EP}}({\varTheta })\). It is worth to mention that the EP is an unified model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, etc.
The common assumptions on a bifunction \({\varTheta } : C \times C \to \mathbb{R}\) are the following:
-
(A1)
\({\varTheta }(x,x)=0\) for all \(x\in C\);
-
(A2)
Θ is monotone, i.e., \({\varTheta }(x,y)+{\varTheta }(y,x)\leq0\) for any \(x,y\in C\);
-
(A3)
Θ is upper-hemicontinuous, i.e., for each \(x,y,z\in C\),
$$\limsup_{t\to0^{+}}{\varTheta }\bigl(tz+(1-t)x,y\bigr)\leq{ \varTheta }(x,y); $$ -
(A4)
\({\varTheta }(x,\cdot)\) is convex and lower semicontinuous for each \(x\in C\).
We also consider the assumptions (B0) and (B1) or (B2) on the function \(\varphi:C\to\mathbb{R}\):
-
(B0)
φ is lower semicontinuous and convex.
-
(B1)
For each \(x\in H\) and \(r>0\), there exist a bounded subset \(D_{x}\subset C\) and \(y_{x}\in C\) such that, for any \(z\in C\setminus D_{x}\),
$${\varTheta }(z,y_{x})+\varphi(y_{x})-\varphi(z)+ \frac{1}{r}\langle y_{x}-z,z-x\rangle<0; $$or
-
(B2)
C is a bounded set.
On the other hand, let \(B : C \to H\) be a single-valued mapping and R be a multi-valued mapping with \(D(R)=C\). Consider the following variational inclusion: find \(x\in C\) such that
We denote by \({\mathrm{I}}(B,R)\) the solution set of the variational inclusion (1.3). In particular, if \(B \equiv R \equiv0\), then \({\mathrm{I}}(B,R)=C\). If \(B \equiv0\), then problem (1.3) becomes the inclusion problem introduced by Rockafellar [25]. It is well known that problem (1.3) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, etc. Let a set-valued mapping \(R:D(R)\subset H\to2^{H}\) be maximal monotone. We define the resolvent operator \(J_{R,\lambda}:H\to\overline{D(R)}\) associated with R and λ by
where λ is a positive number.
Huang [26] studied problem (1.3) in the case where R is maximal monotone and B is strongly monotone and Lipschitz continuous with \(D(R)=C=H\). Subsequently, Zeng et al. [27] further studied this problem in a more general setting than in [26]. Moreover, Zeng et al. [27] obtained the same strong convergence result as in [26]. In addition, Zeng et al. [27] also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, we refer to [12, 28, 29] and the references therein.
Let \(F_{1},F_{2}:C\to H\) be two mappings. Consider the general system of variational inequalities (GSVI) of finding \((x^{*},y^{*})\in C\times C\) such that
where \(\nu_{1}>0\) and \(\nu_{2}>0\) are two constants. It was considered and studied in [11, 15–17, 30]. In particular, if \(F_{1} \equiv F_{2} \equiv A\), then the GSVI (1.4) reduces to the problem of finding \((x^{*},y^{*})\in C\times C\) such that
which is studied by Verma [31] and is called a new system of variational inequalities (NSVI). Further, if \(x^{*} = y^{*}\) additionally, then the NSVI reduces to the classical VIP (1.1). By considering \(G:=P_{C}(I-\nu_{1}F_{1})P_{C}(I-\nu_{2}F_{2})\) and \(y^{*}=P_{C}(I-\nu_{2}F_{2})x^{*}\), where \(P_{C}\) denotes the metric projection of H onto C, Ceng et al. [16] transformed GSVI (1.4) into the following fixed point equation:
A variational inequality problem defined over the set of fixed points of a mapping is called a hierarchical variational inequality problem.
Let S and T be two nonexpansive mappings. Yao et al. [32] considered the following hierarchical variational inequality problem (HVIP): find hierarchically a fixed point of T which is a solution to the VIP for the monotone mapping \(I-S\), namely, find \(\tilde{x}\in{\operatorname{Fix}}(T)\) such that
The solution set of HVIP (1.7) is denoted by Λ. It is easy to check that solving the HVIP (1.7) is equivalent to solving the fixed point problem of the composite mapping \(P_{{\operatorname{Fix}}(T)}S\), that is, find \(\tilde{x}\in C\) such that \(\tilde{x}=P_{{\operatorname{Fix}}(T)}S\tilde{x}\). Ceng et al. [33] introduced and analyzed an iterative algorithm for solving HVIP (1.7). They also studied the strong convergence of the sequences generated by their algorithm.
A variational inequality problem defined over the set of solutions of a hierarchical variational inequality problem is called a triple hierarchical variational inequality problem. For further details of triple hierarchical variational inequalities, we refer to [10, 33–36] and the references therein. Very recently, Kong et al. [7] introduced and studied the following triple hierarchical variational inequality problem (THVIP) (over the fixed point set of a strictly pseudocontractive mapping) with a variational inequality constraint.
Problem 1.1
[7, Problem II]
Let \(F:C\to H\) be κ-Lipschitzian and η-strongly monotone on the nonempty, closed, and convex subset C of H, where κ and η are positive constants. Let \(A:C\to H\) be a monotone and L-Lipschitzian mapping, \(V:C\to H\) be a ρ-contraction with coefficient \(\rho\in[0,1)\), \(S:C\to C\) be a nonexpansive mapping, and \(T:C\to C\) be a ξ-strictly pseudocontractive mapping with \({\operatorname{Fix}}(T)\cap{\operatorname{VI}}(C,A)\neq\emptyset\). Let \(0<\mu<\frac{2\eta}{\kappa^{2}}\) and \(0<\gamma\leq\tau\), where \(\tau=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\). Then the objective is to find \(x^{*}\in{ \varXi }\) such that
where Ξ denotes the solution set of the hierarchical variational inequality problem (HVIP) of finding \(z^{*} \in{\operatorname{Fix}}(T)\cap{\operatorname{VI}}(C,A)\) such that
They proposed an algorithm for solving Problem 1.1 and studied the convergence analysis for the sequences generated by the proposed algorithm.
In this paper, we introduce and study the following triple hierarchical variational inequality problem (THVIP) (defined over the fixed point set of a strictly pseudocontractive mapping) with constraints of finitely many GMEPs, finitely many variational inclusions, and a general system of variational inequalities.
Throughout the paper, M and N are assumed to be positive integers.
Problem 1.2
Assume that
-
(i)
for each \(j=1,2\), \(F_{j}:C\to H\) is \(\zeta _{j}\)-inverse-strongly monotone and \(F:H\to H\) is κ-Lipschitzian and η-strongly monotone with positive constants \(\kappa,\eta>0\) such that \(0<\gamma\leq\tau\) and \(0<\mu<\frac {2\eta}{\kappa^{2}}\) where \(\tau=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\);
-
(ii)
for each \(k\in\{1,2,\ldots,M\}\), \({\varTheta }_{k} : C\times C \to\mathbb{R}\) is a bifunction satisfying conditions (A1)-(A4) and \(\varphi_{k}:C\to{\mathbf{R}}\cup\{+ \infty\}\) is a proper lower semicontinuous and convex function with restriction (B1) or (B2);
-
(iii)
for \(k\in\{1,2,\ldots,M\}\) and \(i\in\{1,2,\ldots,N\}\), \(R_{i}:C\to2^{H}\) is a maximal monotone mapping, and \(A_{k}:H\to H\) and \(B_{i}:C\to H\) are \(\mu_{k}\)-inverse-strongly monotone and \(\eta_{i}\)-inverse-strongly monotone, respectively;
-
(iv)
\(T:H\to H\) is a ξ-strictly pseudocontractive mapping, \(S:H\to H\) is a nonexpansive mapping and \(V:H\to H\) is a ρ-contraction with coefficient \(\rho\in[0,1)\);
-
(v)
\({\operatorname{VI}}({\varOmega },\mu F-\gamma S)\neq\emptyset\) where \({\varOmega } := \bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi _{k},A_{k}) \cap \bigcap^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i}) \cap{\operatorname{GSVI}}(G) \cap{\operatorname{Fix}}(T)\).
Then the objective is to find \(x^{*}\in{ \varXi }\) such that
where \({\varXi }:={\operatorname{VI}}({\varOmega },\mu F-\gamma S)\), that is, the solution set of the hierarchical variational inequality problem (HVIP) of finding \(z^{*}\in{ \varOmega }\) such that
Motivated and inspired by the above facts, we introduce and analyze two iterative methods for solving Problem 1.2, that is, THVIP 1.2. By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, and Mann’s iteration method, we first propose a multi-step hybrid steepest-descent extragradient method. However, by combining Mann’s iteration method, Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, and the projection method, we propose a multi-step composite Mann-type viscosity iterative algorithm. We prove the strong convergence results for these methods. In particular, we prove that the proposed algorithms converge strongly to a common element \(x^{*}\in{ \varOmega } := \bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k}, \varphi_{k},A_{k}) \cap\bigcap^{N}_{i=1}{\mathrm{I}}(B_{i}, R_{i}) \cap{\operatorname{GSVI}}(G) \cap {\operatorname{Fix}}(T)\) which is a unique solution of the THVIP 1.2. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions and GSVI (1.4). The results obtained in this paper improve and extend the corresponding results announced by many others.
2 Preliminaries
Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by \(\langle\cdot, \cdot\rangle\) and \(\|\cdot\| \), respectively. Let C be a nonempty, closed, and convex subset of H. We write \(x_{n}\rightharpoonup x\) to indicate that the sequence \(\{x_{n}\}\) converges weakly to x and \(x_{n}\to x\) to indicate that the sequence \(\{x_{n}\}\) converges strongly to x. Moreover, we use \(\omega_{w}(x_{n})\) to denote the weak ω-limit set of the sequence \(\{x_{n}\}\), that is,
Definition 2.1
A mapping \(A:C\to H\) is called
-
(i)
monotone if
$$\langle Ax-Ay,x-y\rangle\geq0,\quad \forall x,y\in C; $$ -
(ii)
η-strongly monotone if there exists a constant \(\eta >0\) such that
$$\langle Ax-Ay,x-y\rangle\geq\eta\|x-y\|^{2},\quad \forall x,y\in C; $$ -
(iii)
ζ-inverse-strongly monotone if there exists a constant \(\zeta>0\) such that
$$\langle Ax-Ay,x-y\rangle\geq\zeta\|Ax-Ay\|^{2}, \quad \forall x,y\in C. $$
It is easy to see that the projection \(P_{C}\) is 1-inverse-strongly monotone. Inverse-strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields. It is obvious that if A is ζ-inverse-strongly monotone, then A is monotone and \(\frac{1}{\zeta}\)-Lipschitz continuous. Moreover, we also have for all \(u,v\in C\) and \(\lambda>0\),
So, if \(\lambda\leq2\zeta\), then \(I-\lambda A\) is a nonexpansive mapping from C to H.
The metric (or nearest point) 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
Some important properties of projections are gathered in the following proposition.
Proposition 2.1
For given \(x\in H\) and \(z\in C\):
-
(i)
\(z=P_{C}x \Leftrightarrow\langle x-z,y-z\rangle\leq0\), \(\forall y\in C\);
-
(ii)
\(z=P_{C}x \Leftrightarrow\|x-z\|^{2}\leq\|x-y\|^{2}-\|y-z\|^{2}\), \(\forall y\in C\);
-
(iii)
\(\langle P_{C}x-P_{C}y,x-y\rangle\geq\|P_{C}x-P_{C}y\|^{2}\), \(\forall y\in H\). Consequently, \(P_{C}\) is nonexpansive and monotone.
Definition 2.2
A mapping \(T:H\to H\) is said to be
-
(a)
nonexpansive if
$$\|Tx-Ty\|\leq\|x-y\|,\quad \forall x,y\in H; $$ -
(b)
firmly nonexpansive if \(2T-I\) is nonexpansive, or equivalently, if T is 1-inverse-strongly monotone (1-ism),
$$\langle x-y,Tx-Ty\rangle\geq\|Tx-Ty\|^{2},\quad \forall x,y\in H; $$alternatively, T is firmly nonexpansive if and only if T can be expressed as
$$T=\frac{1}{2}(I+S), $$where \(S: H\to H\) is nonexpansive; projections are firmly nonexpansive.
It can easily be seen that if T is nonexpansive, then \(I-T\) is monotone.
Next we list some elementary conclusions for the MEP.
Proposition 2.2
[24]
Assume that \({\varTheta }:C\times C \to\mathbb{R}\) satisfies (A1)-(A4) and let \(\varphi: C\to\mathbb{R}\) be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For \(r>0\) and \(x\in H\), define a mapping \(T^{({\varTheta },\varphi )}_{r}:H\to C\) as follows:
for all \(x\in H\). Then the following statements hold:
-
(i)
For each \(x\in H\), \(T^{({\varTheta },\varphi)}_{r}(x)\) is nonempty and single-valued;
-
(ii)
\(T^{({\varTheta },\varphi)}_{r}\) is firmly nonexpansive, that is, for any \(x,y\in H\),
$$\bigl\Vert T^{({\varTheta },\varphi)}_{r}x-T^{({\varTheta },\varphi)}_{r}y \bigr\Vert ^{2}\leq\bigl\langle T^{({\varTheta }, \varphi)}_{r}x-T^{({\varTheta },\varphi)}_{r}y,x-y \bigr\rangle ; $$ -
(iii)
\({\operatorname{Fix}}(T^{({\varTheta },\varphi)}_{r})={\operatorname{MEP}}({ \varTheta },\varphi)\);
-
(iv)
\({\operatorname{MEP}}({\varTheta },\varphi)\) is closed and convex;
-
(v)
\(\|T^{({\varTheta },\varphi)}_{s}x-T^{({\varTheta },\varphi)}_{t}x\|^{2} \leq\frac{s-t}{s}\langle T^{({\varTheta },\varphi)}_{s}x -T^{({\varTheta },\varphi)}_{t}x,T^{({\varTheta },\varphi )}_{s}x-x\rangle\) for all \(s,t>0\) and \(x\in H\).
Ceng et al. [16] transformed the GSVI (1.4) into a fixed point problem in the following way.
Proposition 2.3
[16]
For given \(\bar{x},\bar{y}\in C\), \((\bar{x},\bar{y})\) is a solution of the GSVI (1.4) if and only if \(\bar{x}\) is a fixed point of the mapping \(G:C\to C\) defined by
where \(\bar{y}=P_{C}(I-\nu_{2}F_{2})\bar{x}\).
In particular, if the mapping \(F_{j}:C\to H\) is \(\zeta _{j}\)-inverse-strongly monotone for \(j=1,2\), then the mapping G is nonexpansive provided \(\nu_{j}\in(0,2\zeta_{j}]\) for \(j=1,2\). We denote by \({\operatorname{GSVI}}(G)\) the fixed point set of the mapping G.
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma 2.1
Let X be a real inner product space. Then we have the following inequality:
Lemma 2.2
Let H be a real Hilbert space. Then the following hold:
-
(a)
\(\|x-y\|^{2}=\|x\|^{2}-\|y\|^{2}-2\langle x-y,y\rangle\) for all \(x,y\in H\).
-
(b)
\(\|\lambda x+\mu y\|^{2}=\lambda\|x\|^{2}+\mu\|y\|^{2}-\lambda \mu\|x-y\|^{2}\) for all \(x,y\in H\) and \(\lambda,\mu\in[0,1]\) with \(\lambda+\mu=1\).
-
(c)
If \(\{x_{n}\}\) is a sequence in H such that \(x_{n}\rightharpoonup x\), it follows that
$$\limsup_{n\to\infty}\|x_{n}-y\|^{2}=\limsup _{n\to\infty}\|x_{n}-x\|^{2}+\| x-y \|^{2},\quad \forall y\in H. $$
It is clear that, in a real Hilbert space H, \(T:C\to C\) is ξ-strictly pseudocontractive if and only if the following inequality holds:
This immediately implies that if T is a ξ-strictly pseudocontractive mapping, then \(I-T\) is \(\frac{1-\xi}{2}\)-inverse-strongly monotone; for further details, we refer to [37] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings and that the class of pseudocontractions strictly includes the class of strict pseudocontractions.
Lemma 2.3
[37, Proposition 2.1]
Let C be a nonempty, closed, and convex subset of a real Hilbert space H and \(T: C\to C\) be a mapping.
-
(i)
If T is a ξ-strictly pseudocontractive mapping, then T satisfies the Lipschitzian condition
$$\|Tx-Ty\|\leq\frac{1+\xi}{1-\xi}\|x-y\|,\quad \forall x,y\in C. $$ -
(ii)
If T is a ξ-strictly pseudocontractive mapping, then the mapping \(I-T\) is semiclosed at 0, that is, if \(\{x_{n}\}\) is a sequence in C such that \(x_{n}\rightharpoonup\tilde{x}\) and \((I-T)x_{n}\to0\), then \((I-T)\tilde{x}=0\).
-
(iii)
If T is ξ-(quasi-)strict pseudocontraction, then the fixed-point set \({\operatorname{Fix}}(T)\) of T is closed and convex so that the projection \(P_{{\operatorname{Fix}}(T)}\) is well defined.
Lemma 2.4
[11]
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(T:C\to C\) be a ξ-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that \((\gamma+\delta)\xi\leq\gamma\). Then
Lemma 2.5
(Demiclosedness principle [38])
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let S be a nonexpansive self-mapping on C with \({\operatorname{Fix}}(S)\neq \emptyset\). Then \(I-S\) is demiclosed. That is, whenever \(\{x_{n}\}\) is a sequence in C weakly converging to some \(x\in C\) and the sequence \(\{(I-S)x_{n}\}\) strongly converges to some y, it follows that \((I-S)x=y\). Here I is the identity operator of H.
Lemma 2.6
Let \(A:C\to H\) be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 2.1(i)) implies
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let λ be a number in \((0,1]\) and let \(\mu>0\). Associated with a nonexpansive mapping \(T:C\to H\), we define the mapping \(T^{\lambda}:C\to H\) by
where \(F:H\to H\) is an operator such that, for some positive constants \(\kappa,\eta>0\), F is κ-Lipschitzian and η-strongly monotone on H; that is, F satisfies the conditions:
for all \(x,y\in H\).
Lemma 2.7
[39, Lemma 3.1]
\(T^{\lambda}\) is a contraction provided \(0<\mu<\frac{2\eta}{\kappa ^{2}}\); that is,
where \(\tau=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\in(0,1]\).
Remark 2.1
-
(i)
Since F is κ-Lipschitzian and η-strongly monotone on H, we get \(0<\eta\leq\kappa\). Hence, whenever \(0<\mu<\frac{2\eta}{\kappa^{2}}\), we have
$$\begin{aligned} 0&\leq(1-\mu\eta)^{2}=1-2\mu\eta+\mu^{2}\eta^{2} \\ &\leq1-2\mu\eta+\mu^{2}\kappa^{2} \\ &<1-2\mu\eta+\frac{2\eta}{\kappa^{2}}\mu\kappa^{2}=1, \end{aligned}$$which implies
$$0<1-\sqrt{1-2\mu\eta+\mu^{2}\kappa^{2}}\leq1. $$So, \(\tau=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\in(0,1]\).
-
(ii)
In Lemma 2.7, put \(F=\frac{1}{2}I\) and \(\mu=2\). Then we know that \(\kappa=\eta=\frac{1}{2}\), \(0<\mu=2<\frac{2\eta}{ \kappa^{2}}=4\), and
$$\tau=1-\sqrt{1-\mu\bigl(2\eta-\mu\kappa^{2}\bigr)}=1-\sqrt{1-2\biggl(2 \times\frac {1}{2}-2\times\biggl(\frac{1}{2}\biggr)^{2} \biggr)}=1. $$
Lemma 2.8
[39]
Let \(\{s_{n}\}\) be a sequence of nonnegative numbers satisfying the conditions
where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are sequences of real numbers such that
-
(i)
\(\{\alpha_{n}\}\subset[0,1]\) and \(\sum^{\infty}_{n=1}\alpha _{n}=\infty\), or equivalently,
$$\prod^{\infty}_{n=1}(1-\alpha_{n}):= \lim_{n\to\infty}\prod^{n}_{k=1}(1- \alpha_{k})=0; $$ -
(ii)
\(\limsup_{n\to\infty}\beta_{n}\leq0\), or \(\sum^{\infty}_{n=1}|\alpha_{n}\beta_{n}|<\infty\).
Then \(\lim_{n\to\infty}s_{n}=0\).
Finally, recall that a set-valued mapping \(T:D(T)\subset H\to2^{H}\) is called monotone if for all \(x,y\in D(T)\), \(f\in Tx\) and \(g\in Ty\) imply
A set-valued mapping T is called maximal monotone if T is monotone and \((I+\lambda T)D(T)=H\) for each \(\lambda>0\), where I is the identity mapping of H. We denote by \(G(T)\) the graph of T. It is well known that a monotone mapping T is maximal if and only if, for \((x,f)\in H\times H\), \(\langle f-g,x-y\rangle\geq0\) for every \((y,g)\in G(T)\) implies \(f\in Tx\).
Next we provide an example to illustrate the concept of maximal monotone mapping.
Let \(A:C\to H\) be a monotone, k-Lipschitz-continuous mapping and let \(N_{C}v\) be the normal cone to C at \(v\in C\), that is,
Define
Then \(\widetilde{T}\) is maximal monotone (see [25]) such that
Let \(R:D(R)\subset H\to2^{H}\) be a maximal monotone mapping. Let \(\lambda,\mu>0\) be two positive numbers.
Lemma 2.9
[40]
We have the resolvent identity
Remark 2.2
For \(\lambda,\mu>0\), we have the following relation:
Indeed, whenever \(\lambda\geq\mu\), utilizing Lemma 2.9 we deduce that
Similarly, whenever \(\lambda<\mu\), we get
Combining the above two cases we conclude that (2.3) holds.
In terms of Huang [26] (see also [27]), we have the following property for the resolvent operator \(J_{R,\lambda }:H\to\overline{D(R)}\).
Lemma 2.10
\(J_{R,\lambda}\) is single-valued and firmly nonexpansive, that is,
Consequently, \(J_{R,\lambda}\) is nonexpansive and monotone.
Lemma 2.11
[12]
Let R be a maximal monotone mapping with \(D(R)=C\). Then for any given \(\lambda>0\), \(u\in C\) is a solution of problem (1.5) if and only if \(u\in C\) satisfies
Lemma 2.12
[27]
Let R be a maximal monotone mapping with \(D(R)=C\) and let \(B:C\to H\) be a strongly monotone, continuous, and single-valued mapping. Then for each \(z\in H\), the equation \(z\in(B+\lambda R)x\) has a unique solution \(x_{\lambda}\) for \(\lambda>0\).
Lemma 2.13
[12]
Let R be a maximal monotone mapping with \(D(R)=C\) and \(B:C\to H\) be a monotone, continuous, and single-valued mapping. Then \((I+\lambda(R+B))C=H\) for each \(\lambda>0\). In this case, \(R+B\) is maximal monotone.
3 Hybrid steepest-descent extragradient method and convergence results
In this section, we introduce and analyze a multi-step hybrid steepest-descent extragradient algorithm for finding a solution of THVIP 1.2. This algorithm is based on Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, and Mann’s iteration method. We prove the strong convergence of the proposed algorithm to a unique solution of THVIP 1.2 under suitable conditions. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem (HVIP).
In the rest of the paper, unless otherwise specified, we assume that M and N are positive integers, and C is a nonempty, closed, and convex subset of a real Hilbert space H.
Assumption 3.1
-
(a)
For each \(k\in\{1, 2, \ldots,M\}\), let \({\varTheta }_{k} : C\times C \to\mathbb{R}\) be a bifunction satisfying (A1)-(A4) and \(\varphi_{k}:C\to\mathbb{R}\cup\{+\infty\}\) be a proper lower semicontinuous and convex function with restriction (B1) or (B2).
-
(b)
For each \(k\in\{1,2, \ldots,M\}\) and each \(i\in\{1,2, \ldots,N\}\), let \(R_{i}:C\to2^{H}\) be a maximal monotone mapping, and \(A_{k}:H\to H\) and \(B_{i}:C\to H\) be \(\mu_{k}\)-inverse-strongly monotone and \(\eta_{i}\)-inverse-strongly monotone, respectively.
-
(c)
Let \(T:H\to H\) be a ξ-strictly pseudocontractive mapping, \(S:H\to H\) be a nonexpansive mapping and \(V:H\to H\) be a ρ-contraction with coefficient \(\rho\in[0,1)\).
-
(d)
For \(j=1,2\), let \(F_{j}:C\to H\) be \(\zeta _{j}\)-inverse-strongly monotone, and \(F:H\to H\) be κ-Lipschitzian and η-strongly monotone with positive constants \(\kappa,\eta>0\) such that \(0<\mu<\frac{2\eta}{\kappa^{2}}\) and \(0<\gamma\leq\tau\), where \(\tau=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\).
-
(e)
\({\varOmega } := \bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k}) \cap\bigcap^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i}) \cap{\operatorname{GSVI}}(G) \cap {\operatorname{Fix}}(T)\).
-
(f)
The solution set Ξ of HVIP (1.11) is nonempty.
-
(g)
Let \(\{\alpha_{n}\}, \{\lambda_{n}\}\subset(0,1]\), \(\{\beta_{n}\}, \{\gamma_{n}\}, \{\delta_{n}\}\subset[0,1]\), \(\{\lambda_{i,n}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\), and \(\{r_{k,n}\} \subset[c_{k},d_{k}]\subset(0,2\mu_{k})\), where \(i\in\{1,2, \ldots,N\}\) and \(k\in\{1,2,\ldots,M\}\).
We propose the following multi-step hybrid steepest-descent extragradient algorithm for finding a solution of THVIP 1.2.
Algorithm 3.1
For given arbitrarily \(x_{0}\in H\), let \(\{x_{n}\}\) be the sequence generated by
where \(G:=P_{C}(I-\nu_{1}F_{1})P_{C}(I-\nu_{2}F_{2})\) with \(\nu_{j}\in(0,2\zeta _{j})\) for \(j=1,2\).
If \(V\equiv0\), then Algorithm 3.1 reduces to the following algorithm.
Algorithm 3.2
For given arbitrarily \(x_{0}\in H\), let \(\{x_{n}\}\) be the sequence generated by
The following result provides the convergence of the sequences generated by the above algorithm.
Theorem 3.1
In addition to Assumption 3.1, suppose that
-
(i)
\(\lim_{n\to\infty}\lambda_{n}=0\), \(\sum^{\infty}_{n=0}\lambda_{n}\alpha_{n}=\infty\) and \(\lim_{n\to\infty}\frac{1}{\lambda_{n}\alpha_{n}}|1-\frac{\alpha _{n-1}}{\alpha_{n}}|=0\);
-
(ii)
\(\limsup_{n\to\infty}\frac{\alpha_{n}}{\lambda _{n}}<\infty\), \(\lim_{n\to\infty}\frac{1}{\lambda_{n}\alpha_{n}}|\frac{1}{\alpha _{n}}-\frac{1}{\alpha_{n-1}}|=0\) and \(\lim_{n\to\infty}\frac{1}{\alpha^{2}_{n}}|1-\frac{\lambda _{n-1}}{\lambda_{n}}|=0\);
-
(iii)
\(\lim_{n\to\infty}\frac{|\beta_{n}-\beta _{n-1}|}{\lambda_{n}\alpha^{2}_{n}}=0\) and \(\lim_{n\to\infty}\frac{|\gamma_{n}-\gamma_{n-1}|}{\lambda _{n}\alpha^{2}_{n}}=0\);
-
(iv)
\(\lim_{n\to\infty}\frac{|\lambda_{i,n}-\lambda _{i,n-1}|}{\lambda_{n}\alpha^{2}_{n}}=0\) and \(\lim_{n\to\infty}\frac{|r_{k,n}-r_{k,n-1}|}{\lambda_{n}\alpha^{2}_{n}}=0\) for \(i=1,2,\ldots,N\) and \(k=1,2,\ldots,M\);
-
(v)
\(\beta_{n}+\gamma_{n}+\delta_{n}=1\), \((\gamma_{n}+\delta_{n})\xi\leq\gamma_{n}\) (\(\forall n\geq0\)), \(\{\beta_{n}\} \subset[a,b]\subset(0,1)\), and \(\liminf_{n\to\infty}\delta_{n}>0\);
-
(vi)
\(\|x-Tx\|\geq\bar{k}[d(x,{\varOmega })]^{\theta}\) (\(\forall x\in C\)) and \(\lim_{n\to\infty}(\lambda^{1/\theta}_{n}/\alpha_{n})=0\) for some \(\bar{k},\theta>0\).
Then the following conclusions hold.
-
(a)
If \(\{Sx_{n}\}\) is bounded, then \(\{x_{n}\}\) converges strongly to a point \(x^{*}\in{ \varOmega }\) which is a unique solution of Problem 1.2 provided that \(\|y_{n}-Tx_{n}\|=o(\alpha_{n})\).
-
(b)
If \(\{Sx_{n}\}\) is bounded, where \(\{x_{n}\}\) is the sequence generated by (3.2), then \(\{x_{n}\}\) converges strongly to a unique solution \(x^{*}\in{ \varXi }\) of the following VIP provided that \(\|y_{n}-Tx_{n}\|=o(\alpha_{n})\):
$$\bigl\langle Fx^{*},x-x^{*}\bigr\rangle \geq0,\quad \forall x\in{ \varXi }. $$
Proof
Put
for all \(k\in\{1,2,\ldots,M\}\) and \(n\geq1\),
for all \(i\in\{1,2,\ldots,N\}\), \({\varDelta }^{0}_{n}=I\), and \({\varLambda }^{0}_{n}=I\), where I is the identity mapping on H. Then we have \(u_{n}={\varDelta }^{M}_{n}x_{n}\) and \(v_{n}={\varLambda }^{N}_{n}u_{n}\).
We divide the rest of the proof into several steps.
Step 1. We prove that \(\{x_{n}\}\) is bounded.
Indeed, take a fixed \(p\in{ \varOmega }\) arbitrarily. Utilizing (2.1) and Proposition 2.2(ii), we have
Utilizing (2.1) and Lemma 2.10, we have
which together with the last inequality implies that
Since \(p=Gp=P_{C}(I-\nu_{1}F_{1})P_{C}(I-\nu_{2}F_{2})p\), \(F_{j}\) is \(\zeta _{j}\)-inverse-strongly monotone for \(j=1,2\), and \(0<\nu_{j}\leq2\zeta_{j}\) for \(j=1,2\), we deduce that, for any \(n\geq0\),
(This shows that \(G:C\to C\) is a nonexpansive mapping.) Since \((\gamma_{n}+\delta_{n})\xi\leq\gamma_{n}\) for all \(n\geq0\) and T is ξ-strictly pseudocontractive, utilizing Lemma 2.4, we obtain from (3.1), (3.4), and (3.5)
Noticing the boundedness of \(\{Sx_{n}\}\), we get \(\sup_{n\geq0}\|\gamma Sx_{n}-\mu Fp\|\leq\widehat{M}\) for some \(\widehat{M}>0\). Moreover, utilizing Lemma 2.7, we deduce from (3.1), (3.6), and \(0<\gamma\leq\tau\) that, for all \(n\geq0\),
By induction, we obtain
Thus, \(\{x_{n}\}\) is bounded and so are the sequences \(\{u_{n}\}\), \(\{v_{n}\}\) and \(\{y_{n}\}\).
Step 2. We prove that \(\lim_{n\to\infty}\frac{\|x_{n+1}-x_{n}\| }{\alpha_{n}}=0\).
Indeed, utilizing (2.1) and (2.3), we obtain
where
for some \(\widetilde{M}>0\) and \(\sup_{n\geq0}\{\sum^{N}_{i=1}\|B_{i}{ \varLambda }^{i-1}_{n+1}u_{n+1}\| +\widetilde{M}\}\leq\widetilde{M}_{0}\) for some \(\widetilde{M}_{0}>0\). Utilizing Proposition 2.2(ii), (v), we have
where \(\widetilde{M}_{1}>0\) is a constant such that, for each \(n\geq0\),
Furthermore, define \(y_{n}=\beta_{n}x_{n}+(1-\beta_{n})w_{n}\) for all \(n\geq 0\). It follows that
Since \((\gamma_{n}+\delta_{n})\xi\leq\gamma_{n}\) for all \(n\geq0\), utilizing Lemma 2.4, we have
Hence, it follows from (3.7)-(3.10) that
In the meantime, simple calculation shows that
So, it follows from (3.11) that
where \(\sup_{n\geq0}\{\|x_{n+1}-w_{n+1}\|+\frac{\|Gv_{n}\|+\|TGv_{n}\| }{1-b}+\widetilde{M}_{0} +\widetilde{M}_{1}\}\leq\widetilde{M}_{2}\) for some \(\widetilde{M}_{2}>0\).
On the other hand, define \(z_{n}:=\alpha_{n}Vx_{n}+(1-\alpha_{n})Sx_{n}\) for all \(n\geq0\). Then as well known \(x_{n+1}=\lambda_{n}\gamma z_{n}+(I-\lambda_{n}\mu F)y_{n}\) for all \(n\geq0\). Simple calculations show that
Since V is a ρ-contraction with coefficient \(\rho\in[0,1)\) and S is a nonexpansive mapping, we conclude that
which, together with (3.12) and \(0<\gamma\leq\tau\), implies that
where \(\sup_{n\geq0}\{\|\gamma z_{n}-\mu Fy_{n}\|+\|Vx_{n}-Sx_{n}\| +\widetilde{M}_{2}\}\leq\widetilde{M}_{3}\) for some \(\widetilde{M}_{3}>0\). Consequently,
where \(\sup_{n\geq1}\{\|x_{n}-x_{n-1}\|+\widetilde{M}_{3}\}\leq\widetilde{M}_{4}\) for some \(\widetilde{M}_{4}>0\). From conditions (i)-(iv), it follows that \(\sum^{\infty}_{n=0}\lambda _{n}\alpha_{n}\gamma(1-\rho)=\infty\) and
Thus, utilizing Lemma 2.8, we immediately conclude that
So, from \(\alpha_{n}\to0\), it follows that
Step 3. We prove that \(\lim_{n\to\infty}\|x_{n}-u_{n}\|=0\), \(\lim_{n\to\infty}\|x_{n}-v_{n}\|=0\), \(\lim_{n\to\infty}\|v_{n}-Gv_{n}\|=0\), and \(\lim_{n\to\infty}\|v_{n}-Tv_{n}\|=0\).
Indeed, utilizing Lemmas 2.1 and 2.2(b), from (3.1), (3.4)-(3.5), and \(0<\gamma\leq\tau\), we deduce that
and hence
which together with \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), immediately yields
Since \(\lambda_{n}\to0\), \(\alpha_{n}\to0\), \(\|x_{n+1}-x_{n}\|\to0\), and \(\{ x_{n}\}\) is bounded, we have
Observe that
and
for \(i\in\{1,2,\ldots,N\}\) and \(k\in\{1,2,\ldots,M\}\). Combining (3.14), (3.17), and (3.18), we get
which immediately leads to
Since \(\|x_{n}-y_{n}\|\to0\), \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\{\lambda_{i,n}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\), \(\{r_{k,n}\}\subset[c_{k},d_{k}]\subset(0,2\mu_{k})\), \(i\in\{1,2,\ldots,N\} \), \(k\in\{1,2,\ldots,M\}\), and \(\{x_{n}\}\), \(\{y_{n}\}\) are bounded sequences, we have
for all \(k\in\{1,2,\ldots,M\}\) and \(i\in\{1,2,\ldots,N\}\).
Furthermore, by Proposition 2.2(ii) and Lemma 2.2(a), we have
which implies that
By Lemma 2.2(a) and Lemma 2.10, we obtain
which immediately leads to
Combining (3.14) and (3.21), we conclude that
which yields
Since \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\{\lambda_{i,n}\} \subset[a_{i},b_{i}]\subset(0,2\eta_{i})\), \(i=1,2,\ldots,N\), and \(\{u_{n}\}\), \(\{x_{n}\}\), and \(\{y_{n}\}\) are bounded sequences, we deduce from (3.19) and \(\|x_{n}-y_{n}\|\to0\) that
Also, combining (3.3), (3.14), and (3.20), we deduce that
which yields
Since \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\{r_{k,n}\}\subset [c_{k},d_{k}]\subset(0,2\mu_{k})\) for \(k=1,2,\ldots,M\), and \(\{x_{n}\}\), \(\{y_{n}\}\) are bounded sequences, we deduce from (3.19) and \(\| x_{n}-y_{n}\|\to0\) that
Hence, from (3.22) and (3.23), we get
and
respectively. Thus, from (3.24) and (3.25), we obtain
On the other hand, for simplicity, we write \(\tilde{p}=P_{C}(I-\nu_{2}F_{2})p\), \(\tilde{v}_{n}=P_{C}(I-\nu_{2}F_{2})v_{n}\), and \(k_{n}=Gv_{n}= P_{C}(I-\nu_{1}F_{1})\tilde{v}_{n}\) for all \(n\geq1\). Then
We now show that \(\lim_{n\to\infty}\|Gv_{n}-v_{n}\|=0\), that is, \(\lim_{n\to\infty}\|k_{n}-v_{n}\|=0\). As a matter of fact, for \(p\in{ \varOmega }\), it follows from (3.4), (3.5), and (3.14) that
which immediately yields
Since \(\|x_{n}-y_{n}\|\to0\), \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\nu _{j}\in(0,2\zeta_{j})\), \(j=1,2\), and \(\{x_{n}\}\), \(\{y_{n}\}\) are bounded sequences, we have
Also, in terms of the firm nonexpansivity of \(P_{C}\) and the \(\zeta _{j}\)-inverse-strong monotonicity of \(F_{j}\) for \(j=1,2\), we obtain from \(\nu_{j}\in(0,2\zeta_{j})\), \(j=1,2\), and (3.5)
and
Thus, we have
and
Consequently, from (3.4), (3.27), and (3.29), it follows that
which hence leads to
Since \(\|x_{n}-y_{n}\|\to0\), \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\nu_{2}\in(0,2\zeta_{2})\), and \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{v_{n}\}\), \(\{\tilde{v}_{n}\}\) are bounded sequences, we obtain from (3.28)
Furthermore, from (3.4), (3.27), and (3.30), it follows that
which hence yields
Since \(\|x_{n}-y_{n}\|\to0\), \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\nu_{1}\in(0,2\zeta_{1})\), and \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{k_{n}\}\), \(\{\tilde{v}_{n}\}\) are bounded sequences, we obtain from (3.28)
Note that
Hence, from (3.31) and (3.32), we get
Also, observe that
Hence, we obtain
So, from \(\liminf_{n\to\infty}\delta_{n}>0\), (3.16), (3.26), and (3.33), it follows that
In addition, noticing that
we know from (3.33) and (3.34) that
Step 4. We prove that \(\omega_{w}(x_{n})\subset{ \varOmega }\).
Indeed, since H is reflexive and \(\{x_{n}\}\) is bounded, there exists at least a weak convergence subsequence of \(\{x_{n}\}\). Hence it is well known that \(\omega_{w}(x_{n})\neq\emptyset\). Now, take an arbitrary \(w\in\omega_{w}(x_{n})\). Then there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}}\rightharpoonup w\). From (3.22)-(3.24) and (3.26) we have \(u_{n_{i}}\rightharpoonup w\), \(v_{n_{i}}\rightharpoonup w\), \({\varLambda }^{m}_{n_{i}}u_{n_{i}}\rightharpoonup w\), and \({\varDelta }^{k}_{n_{i}} x_{n_{i}}\rightharpoonup w\), where \(m\in\{ 1,2,\ldots,N\}\) and \(k\in\{1,2,\ldots,M\}\). Utilizing Lemma 2.3(ii), we deduce from \(v_{n_{i}}\rightharpoonup w\) and (3.35) that \(w\in{\operatorname{Fix}}(T)\). In the meantime, utilizing Lemma 2.5, we obtain from \(v_{n_{i}} \rightharpoonup w\) and (3.33) \(w\in{\operatorname{GSVI}}(G)\). Next, we prove that \(w\in\bigcap^{N}_{m=1}{\mathrm{I}}(B_{m},R_{m})\). As a matter of fact, since \(B_{m}\) is \(\eta_{m}\)-inverse-strongly monotone, \(B_{m}\) is a monotone and Lipschitz continuous mapping. It follows from Lemma 2.13 that \(R_{m}+B_{m}\) is maximal monotone. Let \((v,g)\in G(R_{m}+B_{m})\), that is, \(g-B_{m}v\in R_{m}v\). Again, since \({\varLambda }^{m}_{n}u_{n}=J_{R_{m},\lambda_{m,n}}(I-\lambda_{m,n}B_{m}){ \varLambda }^{m-1}_{n}u_{n}\), \(n\geq1\), \(m\in\{1,2,\ldots,N\}\), we have
that is,
In terms of the monotonicity of \(R_{m}\), we get
and hence
In particular,
Since \(\|{\varLambda }^{m}_{n}u_{n}-{\varLambda }^{m-1}_{n}u_{n}\|\to0\) (due to (3.22)) and \(\|B_{m}{\varLambda }^{m}_{n}u_{n}-B_{m}{\varLambda }^{m-1}_{n}u_{n}\|\to0\) (due to the Lipschitz continuity of \(B_{m}\)), we conclude from \({\varLambda }^{m}_{n_{i}}u_{n_{i}}\rightharpoonup w\) and \(\{\lambda_{i,n}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\) that
It follows from the maximal monotonicity of \(B_{m}+R_{m}\) that \(0\in (R_{m}+B_{m})w\), that is, \(w\in{\mathrm{I}}(B_{m},R_{m})\). Therefore, \(w \in\bigcap^{N}_{m=1}{\mathrm{I}}(B_{m},R_{m})\). Next we prove that \(w \in\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\). Since \({\varDelta }^{k}_{n}x_{n}=T^{({\varTheta }_{k},\varphi_{k})}_{r_{k,n}} (I-r_{k,n}A_{k}){\varDelta }^{k-1}_{n}x_{n}\), \(n\geq1\), \(k\in\{1,2,\ldots,M\}\), we have
By (A2), we have
Let \(z_{t}=ty+(1-t)w\) for all \(t\in(0,1]\) and \(y\in C\). This implies that \(z_{t}\in C\). Then we have
By (3.23), we have \(\|A_{k}{\varDelta }^{k}_{n}x_{n}-A_{k}{\varDelta }^{k-1}_{n}x_{n}\|\to0\) as \(n\to\infty\). Furthermore, by the monotonicity of \(A_{k}\), we obtain \(\langle z_{t}-{\varDelta }^{k}_{n}x_{n},A_{k}z_{t}-A_{k}{\varDelta }^{k}_{n}x_{n}\rangle \geq0\). Then, by (A4), we obtain
Utilizing (A1), (A4), and (3.37), we have
and hence
Letting \(t\to0\), we have, for each \(y\in C\),
This implies that \(w\in{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\), and hence, \(w \in\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\). Thus, \(w\in{ \varOmega } = \bigcap^{\infty}_{n=1}{\operatorname{Fix}}(T_{n}) \cap \bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k}) \cap\bigcap^{N}_{m=1}{\mathrm{I}}(B_{m},R_{m})\). Consequently, \(w \in\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k}) \cap\bigcap^{N}_{m=1}{\mathrm{I}} (B_{m},R_{m}) \cap{\operatorname{GSVI}}(G) \cap {\operatorname{Fix}}(T)=:{\varOmega }\). This shows that \(\omega_{w}(x_{n})\subset{ \varOmega }\).
Step 5. We prove that \(\omega_{w}(x_{n})\subset{ \varXi }\).
Indeed, take an arbitrary \(w\in\omega_{w}(x_{n})\). Then there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}}\rightharpoonup w\). Utilizing (3.15), we obtain for all \(p\in{ \varOmega }\),
which implies that
Since \(\alpha_{n}\to0\), \(\|x_{n}-x_{n+1}\|\to0\), and
from (3.38), we conclude that
that is,
In addition, observe that
and
Since \(0<\gamma\leq\tau\) and \(\kappa\geq\eta\), we know that \(\mu \eta\geq\tau\geq\gamma\) and hence the mapping \(\mu F-\gamma S\) is monotone. Moreover, it is clear that the mapping \(\mu F-\gamma S\) is \((\mu\kappa+\gamma)\)-Lipschitzian. So, by Minty’s lemma [38], we know that (3.40) is equivalent to the VIP
This shows that \(w\in{\operatorname{VI}}({\varOmega },\mu F-\gamma S)=:{\varXi }\). Thus, we derive \(\omega_{w}(x_{n})\subset{ \varXi }\) according to the arbitrariness of w.
Step 6. We prove that \(\lim_{n\to\infty}\|x_{n}-x^{*}\|=0\) provided \(\|y_{n}-Tx_{n}\|=o(\alpha_{n})\), where \(\{x^{*}\}={\operatorname{VI}}({\varXi },\mu F-\gamma V)\).
Indeed, it is clear that \(\mu F-\gamma V\) is \((\mu\eta-\gamma\rho )\)-strongly monotone and \((\mu\kappa+\gamma\rho)\)-Lipschitzian. Then it is well known that \({\operatorname{VI}}({\varXi },\mu F-\gamma V)\) is a singleton and hence we write \({\operatorname{VI}}({\varXi },\mu F-\gamma V) = \{x^{*}\}\), that is, \({\operatorname{VI}}({\operatorname{VI}}({\varOmega },\mu F-\gamma S),\mu F-\gamma V)=\{x^{*}\}\).
Utilizing (3.6) and (3.15) with \(p=x^{*}\), we get
Now let us show that
In fact, we may assume, without loss of generality, that there exists a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{j}} \rightharpoonup\hat{x}\) and
In terms of the fact that \(\omega_{w}(x_{n})\subset{ \varXi }\), we get \(\hat{x}\in{ \varXi }\). Since \({\operatorname{VI}}({\varXi },\mu F-\gamma V)=\{x^{*}\}\), it is easy to see from (3.45) that
that is, (3.44) holds.
In addition, from \(x^{*}\in{ \varXi }\) and condition (vi) we obtain
Utilizing Lemma 2.3(i) we have
where \(\sup_{n\geq0}\|\gamma\alpha_{n}(Vx_{n}-Sx_{n})+\gamma Sx_{n}-\mu Fy_{n}\|\leq\widetilde{M}_{5}\) for some \(\widetilde{M}_{5}>0\). Hence, for a big enough constant \(\bar{k}_{1}>0\), from (3.46), we have
Combining (3.43) and (3.48), we get
Since \(\sum^{\infty}_{n=0}\lambda_{n}\alpha_{n}=\infty\), \(\limsup_{n\to \infty}\frac{\alpha_{n}}{\lambda_{n}}<\infty\), \(\|x_{n}-x_{n+1}\|=o(\alpha_{n})\), and \(\frac{\lambda^{1/\theta }_{n}}{\alpha_{n}}\to0\), we conclude from (3.44) and the assumption \(\|y_{n}-Tx_{n}\| =o(\alpha_{n})\) that \(\sum^{\infty}_{n=0}\lambda_{n}\alpha_{n}\gamma(1-\rho)=\infty\),
and
Therefore, applying Lemma 2.8 to (3.49) we infer that \(\lim_{n\to\infty}\|x_{n}-x^{*}\|=0\). 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 \(V\equiv0\), THVIP (1.10) reduces to the VIP in part (b). This completes the proof. □
Remark 3.1
It is clear that the iterative scheme (3.1) is different from the one considered in [7, 32]. We extend the three-step iterative scheme in [7, Algorithm I] to the four-step iterative scheme for THVIP (1.10) by combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method [41] and Mann’s iteration method. It is worth pointing out that under the lack of assumptions similar to those in [32, Theorem 3.2], for example, \(\{x_{n}\}\) is bounded and \({\operatorname{Fix}}(T) \cap{\operatorname{int}}C\neq\emptyset\), the sequence \(\{x_{n}\}\) generated by (3.1) converges strongly to a point \(x^{*}\in\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k}) \cap\bigcap^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i}) \cap{\operatorname{GSVI}}(G) \cap {\operatorname{Fix}}(T) =: {\varOmega }\), which is a unique solution \(x^{*}\in{ \varXi }\) of THVIP (1.10) (over the fixed point set of a strictly pseudocontractive mapping T), that is, \(\langle(\mu F-\gamma V)x^{*},p-x^{*}\rangle\geq0\), \(\forall p\in{ \varXi }\). We note that the nonexpansive mapping T in [32] is extended to a strictly pseudocontractive mapping T in (3.1).
Remark 3.2
Theorem 3.1 improves and extends Theorems 3.1 and 3.2 in [32] and Theorem 14 in [7] in the following aspects:
-
(a)
THVIP (1.10) with the unique solution \(x^{*}\in \varOmega \) satisfying
$$x^{*}=P_{\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\cap\bigcap ^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i}) \cap{\operatorname{GSVI}}(G)\cap{\operatorname{Fix}}(T)} \bigl(I-(\mu F-\gamma S)\bigr)x^{*} $$is more general than the problem of finding a point \(\tilde{x}\in C\) satisfying \(\tilde{x}=P_{{\operatorname{Fix}}(T)}S\tilde{x}\) in [32] and than the problem of finding a point \(x^{*}\in{\operatorname{Fix}}(T)\cap{\operatorname{VI}}(C,A)\) satisfying \(x^{*}=P_{{\operatorname{Fix}}(T)\cap{\operatorname{VI}}(C,A)} (I-(\mu F-\gamma S))x^{*}\) in [7, Theorem 14]. It is worth to point out that S is nonexpansive if and only if the complement \(I-S\) is \(\frac{1}{2}\)-inverse-strongly monotone; see [15].
-
(b)
The four-step iterative scheme (3.1) for THVIP (1.10) is more flexible and subtle than the three-step iterative scheme considered in [7, Algorithm I] and than the two-step iterative scheme studied in [32] because it can be used to solve several kinds of problems, for example, THVIP, HVIP, and the problem of finding a common point of four sets: \(\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\), \(\bigcap^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i})\), \({\operatorname{GSVI}}(G)\), and \({\operatorname{Fix}} (T)\). In addition, Theorem 3.1 drops the crucial requirements in [32, Theorem 3.2] that \(\lim_{n\to\infty}\frac{\alpha_{n}}{\beta_{n}}=0\), \(\lim_{n\to \infty}\frac{\beta^{2}_{n}}{\alpha_{n}}=0\), \({\operatorname{Fix}}(T)\cap{\operatorname{int}}C\neq\emptyset\), and \(\{x_{n}\}\) is bounded, and also removes the crucial ones in [7, Theorem 14] that \(\sum^{\infty}_{n=0}\alpha_{n}<\infty\), \(\lim_{n\to\infty}(\alpha_{n}/\lambda^{2}_{n})=0\), and \(\|x_{n+1}-x_{n}\| +\|x_{n}-z_{n}\|=o(\lambda^{2}_{n})\). In the meantime, Problem 1.1 (that is, Problem II in [7]) is extended and generalized to the setting of GSVI (1.4), finitely many GMEPs, and finitely many variational inclusions in Problem 1.2.
-
(c)
The technique used in Theorem 3.1 is different from the one used in [32, Theorems 3.1 and 3.2] and in [7, Theorem 14] because we used the properties of strictly pseudocontractive mappings (see Lemmas 2.3 and 2.4), the properties of resolvent operators and maximal monotone mappings (see Proposition 2.2, Remark 2.2 and Lemmas 2.9-2.13), the fixed point equation \(x^{*}=P_{C}(I-\nu_{1}F_{1})P_{C}(I-\nu_{2}F_{2})x^{*}\) equivalent to GSVI (1.4) (see Proposition 2.3), and contractive coefficient estimates for contractions associated with nonexpansive mappings (see Lemma 2.7).
-
(d)
Compared with the restrictions on the parameter sequences in [32, Theorem 3.2] and [7, Theorem 14], respectively, the hypotheses (iii)-(iv) in Theorem 3.1 are additionally added because Theorem 3.1 involves a quite complex problem, that is, THVIP (1.10) (over the fixed point set \({\operatorname{Fix}}(T)\) of a strictly pseudocontractive mapping T) with constraints of several problems: GSVI (1.4), finitely many GMEPs, and finitely many variational inclusions.
We prove the strong convergence of the proposed algorithm to a unique solution of THVI Problem 1.2.
Theorem 3.2
In addition to Assumption 3.1, suppose that
-
(i)
\(\lim_{n\to\infty}\lambda_{n}=0\), \(\lim_{n\to\infty }\alpha_{n}=0\) and \(\sum^{\infty}_{n=0}\lambda_{n}\alpha_{n}=\infty\);
-
(ii)
\(\beta_{n}+\gamma_{n}+\delta_{n}=1\) and \((\gamma_{n}+\delta _{n})\xi\leq\gamma_{n}\) for all \(n\geq0\);
-
(iii)
\(\liminf_{n\to\infty}\delta_{n}>0\) and \(\{\beta_{n}\} \subset[a,b]\subset(0,1)\);
-
(iv)
\(\lim_{n\to\infty}(\lambda_{n}/\alpha^{2}_{n})=0\) and \(\|x-Tx\|\geq\bar{k}\cdot d(x,{\varOmega })\), \(\forall x\in C\), for some \(\bar{k}\).
If \(\{Sx_{n}\}\) is bounded, then
-
(a)
\(\omega_{w}(x_{n})\subset{ \varOmega }\) provided \(\| x_{n}-x_{n+1}\|\to0\) (\(n\to\infty\));
-
(b)
\(\omega_{w}(x_{n})\subset{ \varXi }\) provided \(\|x_{n}-x_{n+1}\| =o(\lambda_{n})\);
-
(c)
\(\{x_{n}\}\) converges strongly to a unique solution of Problem 1.2 provided \(\|x_{n}-x_{n+1}\|+\|x_{n}-Tx_{n}\|=o(\lambda_{n}\alpha^{2}_{n})\).
Proof
Since the solution set Ξ of the HVIP (1.11) is nonempty, it is well known that \({\varOmega }\neq\emptyset\). As in the proof of Theorem 3.1, put
for all \(k\in\{1,2,\ldots,M\}\) and \(n\geq1\),
for all \(i\in\{1,2,\ldots,N\}\), \({\varDelta }^{0}_{n}=I\), and \({\varLambda }^{0}_{n}=I\), where I is the identity mapping on H. Then we have \(u_{n}={\varDelta }^{M}_{n}x_{n}\) and \(v_{n}={\varLambda }^{N}_{n}u_{n}\).
The rest of the proof is divided into several steps.
Step 1. As in the proof (Step 1) of Theorem 3.1, \(\{ x_{n}\}\) is bounded.
Step 2. We prove that \(\omega_{w}(x_{n})\subset{ \varOmega }\) provided \(\|x_{n}-x_{n+1}\|\to0\) (\(n\to\infty\)).
Indeed, we first show that \(\lim_{n\to\infty}\|x_{n}-u_{n}\|=0\), \(\lim_{n\to\infty}\|x_{n}-v_{n}\|=0\), \(\lim_{n\to\infty}\|v_{n}-Gv_{n}\|=0\), and \(\lim_{n\to\infty}\|v_{n}-Tv_{n}\|=0\).
As a matter of fact, utilizing Lemmas 2.1 and 2.2(b), from (3.1), (3.4), (3.5), and \(0<\gamma \leq\tau\), we deduce
and hence
which together with \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\) yields
Since \(\alpha_{n}\to0\), \(\lambda_{n}\to0\), \(\|x_{n+1}-x_{n}\|\to0\), and \(\{x_{n}\}\) is bounded, we have
Observe that
and
for \(i\in\{1,2,\ldots,N\}\) and \(k\in\{1,2,\ldots,M\}\). Combining (3.50), (3.53), and (3.11), we get
which immediately leads to
Since \(\|x_{n}-y_{n}\|\to0\), \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\{\lambda_{i,n}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\), \(\{r_{k,n}\}\subset[c_{k},d_{k}]\subset(0,2\mu_{k})\), \(i\in\{1,2,\ldots,N\}\), \(k\in\{1,2,\ldots,M\}\), and \(\{x_{n}\}\), \(\{y_{n}\}\) are bounded sequences, we have
for all \(k\in\{1,2,\ldots,M\}\) and \(i\in\{1,2,\ldots,N\}\).
Furthermore, by Proposition 2.2(ii) and Lemma 2.2(a), we have
which implies that
By Lemma 2.2(a) and Lemma 2.10, we obtain
which immediately leads to
Combining (3.50) and (3.57), we have
which yields
Since \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\{\lambda_{i,n}\}\subset[a_{i},b_{i}] \subset(0,2\eta_{i})\), \(i=1,2,\ldots,N\), and \(\{u_{n}\}\), \(\{x_{n}\}\), and \(\{y_{n}\}\) are bounded sequences, we deduce from (3.55) and \(\|x_{n}-y_{n}\|\to0\) that
Also, combining (3.3), (3.57), and (3.20), we deduce that
which yields
Since \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\{r_{k,n}\}\subset[c_{k},d_{k}]\subset(0,2\mu_{k})\) for \(k=1,2,\ldots,M\), and \(\{x_{n}\}\), \(\{y_{n}\}\) are bounded sequences, we deduce from (3.55) and \(\|x_{n}-y_{n}\|\to0\) that
Hence, from (3.58) and (3.16), we get
and
respectively. Thus, from (3.60) and (3.18), we obtain
On the other hand, for simplicity, we write \(\tilde{p}=P_{C}(I-\nu_{2}F_{2})p\), \(\tilde{v}_{n}=P_{C}(I-\nu_{2}F_{2})v_{n}\), and \(k_{n}=Gv_{n}= P_{C}(I-\nu_{1}F_{1})\tilde{v}_{n}\) for all \(n\geq1\). Then
We now show that \(\lim_{n\to\infty}\|Gv_{n}-v_{n}\|=0\), that is, \(\lim_{n\to\infty}\|k_{n}-v_{n}\|=0\).
As a matter of fact, for \(p\in{ \varOmega }\), it follows from (3.4), (3.5), and (3.50) that
which immediately yields
Since \(\|x_{n}-y_{n}\|\to0\), \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\nu_{j}\in(0,2\zeta_{j})\), \(j=1,2\), and \(\{x_{n}\}\), \(\{y_{n}\}\) are bounded sequences, we have
Also, in terms of the firm nonexpansivity of \(P_{C}\) and the \(\zeta _{j}\)-inverse-strong monotonicity of \(F_{j}\) for \(j=1,2\), we obtain from \(\nu_{j}\in(0,2\zeta_{j})\), \(j=1,2\), and (3.5)
and
Thus, we have
and
Consequently, from (3.4), (3.63), and (3.65) it follows that
which hence leads to
Since \(\|x_{n}-y_{n}\|\to0\), \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\nu _{2}\in(0,2\zeta_{2})\), and \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{v_{n}\}\), \(\{\tilde{v}_{n}\}\) are bounded sequences, we obtain from (3.64)
Furthermore, from (3.4), (3.63), and (3.66), it follows that
which hence yields
Since \(\|x_{n}-y_{n}\|\to0\), \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\), \(\nu_{1}\in(0,2\zeta_{1})\), and \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{k_{n}\}\), \(\{\tilde{v}_{n}\}\) are bounded sequences, we obtain from (3.64)
Note that
Hence, from (3.67) and (3.68), we get
Also, observe that
Hence, we find
So, from \(\liminf_{n\to\infty}\delta_{n}>0\), (3.52), (3.62), and (3.69), it follows that
In addition, noticing that
we know from (3.69) and (3.70) that
Secondly, we show that \(\omega_{w}(x_{n})\subset{ \varOmega }\).
In fact, since H is reflexive and \(\{x_{n}\}\) is bounded, there exists at least a weak convergence subsequence of \(\{x_{n}\}\). Hence, it is well known that \(\omega_{w}(x_{n})\neq\emptyset\). Now, take an arbitrary \(w\in\omega_{w}(x_{n})\). Then there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}}\rightharpoonup w\). From (3.58)-(3.60) and (3.62), we have \(u_{n_{i}}\rightharpoonup w\), \(v_{n_{i}}\rightharpoonup w\), \({\varLambda }^{m}_{n_{i}}u_{n_{i}}\rightharpoonup w\), and \({\varDelta }^{k}_{n_{i}} x_{n_{i}}\rightharpoonup w\), where \(m\in\{ 1,2,\ldots,N\}\) and \(k\in\{1,2,\ldots,M\}\). Utilizing Lemma 2.3(ii), we deduce from \(v_{n_{i}}\rightharpoonup w\) and (3.71) that \(w\in{\operatorname{Fix}}(T)\). In the meantime, utilizing Lemma 2.5, we obtain from \(v_{n_{i}} \rightharpoonup w\) and (3.69) \(w\in{\operatorname{GSVI}}(G)\).
Next, we prove that \(w\in\bigcap^{N}_{m=1}{\mathrm{I}}(B_{m},R_{m})\). As a matter of fact, since \(B_{m}\) is \(\eta_{m}\)-inverse-strongly monotone, \(B_{m}\) is a monotone and Lipschitz continuous mapping. It follows from Lemma 2.13 that \(R_{m}+B_{m}\) is maximal monotone. Let \((v,g)\in G(R_{m}+B_{m})\), that is, \(g-B_{m}v\in R_{m}v\). Again, since \({\varLambda }^{m}_{n}u_{n}=J_{R_{m},\lambda_{m,n}}(I-\lambda _{m,n}B_{m}){\varLambda }^{m-1}_{n}u_{n}\), \(n\geq1\), \(m\in\{1,2,\ldots,N\}\), we have
that is,
In terms of the monotonicity of \(R_{m}\), we get
and hence,
In particular,
Since \(\|{\varLambda }^{m}_{n}u_{n}-{\varLambda }^{m-1}_{n}u_{n}\|\to0\) (due to (3.58)) and \(\|B_{m}{\varLambda }^{m}_{n}u_{n}-B_{m}{\varLambda }^{m-1}_{n}u_{n}\|\to0\) (due to the Lipschitz continuity of \(B_{m}\)), we conclude from \({\varLambda }^{m}_{n_{i}}u_{n_{i}}\rightharpoonup w\) and \(\{\lambda_{i,n}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\) that
It follows from the maximal monotonicity of \(B_{m}+R_{m}\) that \(0\in(R_{m}+B_{m})w\), that is, \(w\in{\mathrm{I}}(B_{m},R_{m})\). Therefore, \(w\in\bigcap^{N}_{m=1}{\mathrm{I}}(B_{m},R_{m})\). Next we prove that \(w\in\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\). Since \({\varDelta }^{k}_{n}x_{n}=T^{({\varTheta }_{k},\varphi _{k})}_{r_{k,n}}(I-r_{k,n}A_{k}){\varDelta }^{k-1}_{n}x_{n}\), \(n \geq1\), \(k\in\{1,2,\ldots,M\}\), we have
By (A2), we have
Let \(z_{t}=ty+(1-t)w\) for all \(t\in(0,1]\) and \(y\in C\). This implies that \(z_{t}\in C\). Then we have
By (3.59), we have \(\|A_{k}{\varDelta }^{k}_{n}x_{n}-A_{k}{\varDelta }^{k-1}_{n}x_{n}\|\to0\) as \(n\to\infty\). Furthermore, by the monotonicity of \(A_{k}\), we obtain \(\langle z_{t}-{\varDelta }^{k}_{n}x_{n},A_{k}z_{t}-A_{k}{\varDelta }^{k}_{n}x_{n}\rangle \geq0\). Then, by (A4), we get
Utilizing (A1), (A4), and (3.73), we obtain
and hence
Letting \(t\to0\), we have, for each \(y\in C\),
This implies that \(w\in{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\), and hence, \(w\in\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\). Thus, \(w\in{ \varOmega }=\bigcap^{\infty}_{n=1}{\operatorname{Fix}}(T_{n})\cap\bigcap^{M}_{k=1} {\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\cap\bigcap^{N}_{m=1}{\mathrm{I}}(B_{m},R_{m})\). Consequently, \(w\in\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi _{k},A_{k})\cap \bigcap^{N}_{m=1}{\mathrm{I}} (B_{m},R_{m})\cap{\operatorname{GSVI}}(G)\cap{\operatorname{Fix}}(T)=:{ \varOmega }\). This shows that \(\omega_{w}(x_{n})\subset{ \varOmega }\).
Step 3. We prove that \(\omega_{w}(x_{n})\subset{ \varXi }\) provided \(\|x_{n}-x_{n+1}\|=o(\lambda_{n})\).
Indeed, we first note that \(0<\gamma\leq\tau\) and \(\mu\eta\geq\tau \Leftrightarrow\kappa\geq\eta\) by (3.41). It is clear that
Hence, it follows from \(0<\gamma\leq\tau\leq\mu\eta\) that \(\mu F-\gamma S\) is monotone. Moreover, it is also obvious that \(\mu F-\gamma S\) is \((\mu\kappa +\gamma)\)-Lipschitzian.
Now, take an arbitrary \(w\in\omega_{w}(x_{n})\). Then there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}}\rightharpoonup w\). Utilizing (3.51), we obtain, for each fixed \(p\in{ \varOmega }\),
which implies that
Since \(\alpha_{n}\to0\) and \(\|x_{n}-x_{n+1}\|=o(\lambda_{n})\), from (3.74), we conclude that
that is,
Since \(\mu F-\gamma S\) is monotone and \((\mu\kappa+\gamma)\)-Lipschitzian, by Minty’s lemma [38], we know that (3.75) is equivalent to the VIP
This shows that \(w\in{\operatorname{VI}}({\varOmega },\mu F-\gamma S)=:{\varXi }\), and hence, \({\varOmega }\subset{ \varXi }\).
Step 4. We prove that \(\{x_{n}\}\) converges strongly to a unique solution of Problem 1.2 provided \(\|x_{n}-x_{n+1}\|+\|x_{n}-Tx_{n}\|=o(\lambda_{n}\alpha^{2}_{n})\).
Indeed, it is easy from (3.51) to see that, for each fixed \(p\in{ \varOmega }\),
which, together with \(\|x_{n}-x_{n+1}\|=o(\alpha^{2}_{n})\) and \(\lambda _{n}=o(\alpha^{2}_{n})\), implies that
In the meantime, it is clear that
Hence, it follows from \(0<\gamma\leq\tau\leq\mu\eta\) and \(\rho <1\) that \(\mu F-\gamma V\) is \((\mu\eta-\gamma\rho)\)-strongly monotone. Moreover, it is also obvious that \(\mu F-\gamma V\) is \((\mu\kappa +\gamma\rho)\)-Lipschitzian. So, we can write \({\operatorname{VI}}(\mu F-\gamma V,\varXi )=\{x^{*}\}\). We now take a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) satisfying
Without loss of generality, we may further assume that \(x_{n_{i}}\rightharpoonup\tilde{x}\); then \(\tilde{x}\in{ \varXi }\) as we just proved. Since \(x^{*}\) is a solution of the THVIP (1.10), we get
From (3.1), (3.6), and (3.51) with \(p=x^{*}\), it follows that
So, it follows from \(x^{*}\in{ \varXi }\) and condition (iv) that
In addition, utilizing Lemma 2.3(i), we have
where \(\sup_{n\geq0}\|\gamma\alpha_{n}(Vx_{n}-Sx_{n})+\gamma Sx_{n}-\mu Fy_{n}\|\leq\widetilde{M}_{0}\) for some \(\widetilde{M}_{0}>0\). Hence, for a big enough constant \(\bar{k}_{1}>0\), from (3.79), we have
Since \(\sum^{\infty}_{n=0}\alpha_{n}\lambda_{n}=\infty\), \(\lim_{n\to \infty}\frac{\lambda_{n}}{\alpha^{2}_{n}}=0\), \(\|x_{n}-x_{n+1}\| +\|x_{n}-Tx_{n}\|=o(\alpha_{n})\), we deduce from (3.76) and (3.78) that \(\sum^{\infty}_{n=0}\alpha_{n}\lambda_{n} (1-\rho)\gamma=\infty\) and
Applying Lemma 2.8 to (3.80), we conclude that \(x_{n}\to x^{*}\) as \(n\to\infty\). This completes the proof. □
Remark 3.3
It is obvious that iterative scheme (3.1) is different from the iterative schemes considered in [7, 32]. We extended the three-step iterative scheme in [7, Algorithm I] to our four-step iterative scheme (3.1) for the THVIP (1.10) by combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method [41], and Mann’s iteration method. It is worth pointing out that under the lack of assumptions similar to those in [32, Theorem 3.2], for example, \(\{x_{n}\}\) is bounded and \({\operatorname{Fix}}(T) \bigcap{\operatorname{int}}C \neq\emptyset\), the sequence \(\{x_{n}\} \) generated by (3.1) converges strongly to a point \(x^{*} \in\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k}) \cap\bigcap^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i}) \cap{\operatorname{GSVI}}(G) \cap{\operatorname{Fix}}(T) =: {\varOmega }\), which is a unique solution \(x^{*}\in{ \varXi }\) of THVIP (1.10) (over the fixed point set of a strictly pseudocontractive mapping T), that is, \(\langle(\mu F-\gamma V)x^{*},p-x^{*}\rangle\geq0\), \(\forall p\in{ \varXi }\).
Remark 3.4
Theorem 3.2 improves and extends [32, Theorems 3.1 and 3.2] and [7, Theorem 14] in the following aspects:
-
(a)
THVIP (1.10), with the unique solution \(x^{*}\in \varOmega \) satisfying
$$x^{*}=P_{\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\cap\bigcap ^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i}) \cap{\operatorname{GSVI}}(G)\cap{\operatorname{Fix}}(T)} \bigl(I-(\mu F-\gamma S)\bigr)x^{*}, $$is more general than the problem of finding a point \(\tilde{x}\in C\) satisfying \(\tilde{x}=P_{{\operatorname{Fix}}(T)}S\tilde{x}\) in [32] and than the problem of finding a point \(x^{*}\in{\operatorname{Fix}}(T)\cap{\operatorname{VI}}(C,A)\) satisfying \(x^{*}=P_{{\operatorname{Fix}}(T)\cap{\operatorname{VI}}(C,A)} (I-(\mu F-\gamma S))x^{*}\) in [7, Theorem 14]. It is worth pointing out that S is nonexpansive if and only if the complement \(I-S\) is \(\frac{1}{2}\)-inverse-strongly monotone.
-
(b)
Four-step iterative scheme (3.1) for THVIP (1.10) is flexible, and subtle than those considered in [7, Algorithm I] and [32] because it can be used to solve several kinds of problems, for example, the THVIP, the HVIP and the problem of finding a common point of four sets: \(\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\), \(\bigcap^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i})\), \({\operatorname{GSVI}}(G)\), and \({\operatorname{Fix}} (T)\). In addition, Theorem 3.2 drops the crucial requirements in [32, Theorem 3.2] that \(\lim_{n\to\infty}\frac{\alpha_{n}}{\beta_{n}}=0\), \(\lim_{n\to \infty}\frac{\beta^{2}_{n}}{\alpha_{n}}=0\), \({\operatorname{Fix}}(T)\cap{\operatorname{int}}C\neq\emptyset\), and \(\{x_{n}\}\) is bounded, and also it removes the crucial ones in [7, Theorem 14] that \(\sum^{\infty}_{n=0}\alpha_{n}<\infty\), \(\lim_{n\to\infty}(\alpha_{n}/\lambda^{2}_{n})=0\). In the meantime, Problem 1.1 (that is, [7, Problem II]) is extended and generalized to the setting of the GSVI (1.4), finitely many GMEPs, and finitely many variational inclusions in our Problem 1.2.
-
(c)
The argument and technique in Theorem 3.2 are different from [32, Theorems 3.1 and 3.2] and [7, Theorem 14] because we make use of the properties of strictly pseudocontractive mappings (see Lemmas 2.3 and 2.4), the properties of resolvent operators and maximal monotone mappings (see Proposition 2.2, Remark 2.2, and Lemmas 2.9-2.13), the fixed point equation \(x^{*}=P_{C}(I-\nu_{1}F_{1})P_{C}(I-\nu_{2}F_{2})x^{*}\) equivalent to the GSVI (1.4) (see Proposition 2.3) and the contractive coefficient estimates for the contractions associated with nonexpansive mappings (see Lemma 2.7).
-
(d)
Compared with the requirement \(\|x_{n}-x_{n+1}\|+\|x_{n}-z_{n}\| =o(\lambda^{2}_{n})\) in [7, Theorem 14], the one \(\|x_{n}-x_{n+1}\| +\|x_{n}-Tx_{n}\| =o(\lambda_{n}\alpha^{2}_{n})\) in conclusion (c) of Theorem 3.2 is additionally added because Theorem 3.2 involves a quite complex problem, that is, the THVIP (1.10) (over the fixed point set \({\operatorname{Fix}}(T)\) of a strictly pseudocontractive mapping T) with constraints of several problems: GSVI (1.4), finitely many GMEPs, and finitely many variational inclusions.
4 Composite Mann-type viscosity approximation method and convergence results
In this section, we introduce and analyze a multi-step composite Mann-type viscosity iterative algorithm for finding a solution of the THVIP (1.10) (over the fixed point set of a strictly pseudocontractive mapping) with constraints of several problems: finitely many GMEPs, finitely many variational inclusions and GSVI (1.4) in a real Hilbert space. This algorithm is based on Mann’s iteration method, Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, and the projection method. We prove the strong convergence of the proposed algorithm to a unique solution of THVIP (1.10) under suitable conditions. In addition, we also consider the application of the proposed algorithm to solving a hierarchical VIP with the same constraints.
Algorithm 4.1
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. For each \(k\in\{1, 2,\ldots,M\}\), let \({\varTheta }_{k} : C\times C \to \mathbb{R}\) be a bifunction satisfying (A1)-(A4) and \(\varphi_{k}:C \to\mathbb{R} \cup\{+\infty\} \) be a proper lower semicontinuous convex function with restriction (B1) or (B2). For each \(k\in\{1,2,\ldots,M\}\), \(i\in\{1,2,\ldots,N\}\), let \(R_{i}:C\to2^{H}\) be a maximal monotone mapping, and \(A_{k}:H\to H\) and \(B_{i}:C\to H\) be \(\mu_{k}\)-inverse-strongly monotone and \(\eta_{i}\)-inverse-strongly monotone, respectively. Let \(T:C\to C\) be a ξ-strictly pseudocontractive mapping, \(S:C\to C\) be a nonexpansive mapping and \(V:C\to H\) be a ρ-contraction with coefficient \(\rho\in[0,1)\). Let \(F_{j}:C\to H\) be \(\zeta_{j}\)-inverse-strongly monotone for \(j=1,2\), and \(F:C\to H\) be κ-Lipschitzian and η-strongly monotone with positive constants \(\kappa, \eta>0\) such that \(0<\mu<\frac{2\eta}{\kappa^{2}}\) and \(0<\gamma\leq\tau\) where \(\tau=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\). Assume that the solution set Ξ of HVIP (1.11) is nonempty, where \({\varOmega } := \bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k}) \cap\bigcap^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i}) \cap{\operatorname{GSVI}}(G)\cap{\operatorname{Fix}}(T)\). Let \(\{\lambda_{n}\}, \{\sigma_{n}\}, \{\alpha_{n}\} \subset(0,1]\), \(\{\beta_{n}\}, \{\gamma_{n}\}, \{\delta_{n}\}\subset[0,1]\), \(\{\lambda_{i,n}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\), and \(\{r_{k,n}\}\subset[c_{k},d_{k}]\subset(0,2\mu_{k})\), where \(i\in\{1,2,\ldots,N\}\) and \(k\in\{1,2,\ldots,M\}\). For given arbitrarily \(x_{0}\in H\), let \(\{x_{n}\}\) be the sequence generated by
where \(G:=P_{C}(I-\nu_{1}F_{1})P_{C}(I-\nu_{2}F_{2})\) with \(\nu_{j}\in(0,2\zeta _{j})\) for \(j=1,2\).
Theorem 4.1
In addition to assumptions of Algorithm 4.1, suppose that
-
(i)
\(\lim_{n\to\infty}\lambda_{n}=0\), \(\lim_{n\to\infty }\alpha_{n}=0\) and \(\sum^{\infty}_{n=0}\lambda_{n}\alpha_{n}=\infty\);
-
(ii)
\(\beta_{n}+\gamma_{n}+\delta_{n}=1\) and \((\gamma_{n}+\delta _{n})\xi\leq\gamma_{n}\) for all \(n\geq0\);
-
(iii)
\(\liminf_{n\to\infty}\delta_{n}>0\), \(\liminf_{n\to \infty}\sigma_{n}>0\) and \(\{\beta_{n}\}\subset[a,b]\subset(0,1)\);
-
(iv)
\(\lim_{n\to\infty}(\lambda^{1/\theta}_{n}/\alpha_{n})=0\) and \(\|x-Tx\|\geq\bar{k}[d(x,{\varOmega })]^{\theta}\), \(\forall x\in C\), for some \(\bar{k},\theta>0\).
If \(\{Sx_{n}\}\) is bounded, then
-
(a)
\(\omega_{w}(x_{n})\subset{ \varOmega }\) provided \(\| x_{n}-x_{n+1}\|\to0\) (\(n\to\infty\));
-
(b)
\(\omega_{w}(x_{n})\subset{ \varXi }\) provided \(\|x_{n}-x_{n+1}\| +\|x_{n}-y_{n}\|=o(\lambda_{n})\);
-
(c)
\(\{x_{n}\}\) converges strongly to a unique solution of Problem 1.2 provided \(\|x_{n}-x_{n+1}\|+\|x_{n}-y_{n}\|+\|x_{n}-z_{n}\|=o(\lambda_{n})\).
Proof
Since the solution set Ξ of the HVIP (1.11) is nonempty, as is well known, \({\varOmega }\neq\emptyset\). As in the proof of Theorem 3.1, we put
for all \(k\in\{1,2,\ldots,M\}\) and \(n\geq1\),
for all \(i\in\{1,2,\ldots,N\}\), \({\varDelta }^{0}_{n}=I\), and \({\varLambda }^{0}_{n}=I\), where I is the identity mapping on H. Then we have \(u_{n}={\varDelta }^{M}_{n}x_{n}\) and \(v_{n}={\varLambda }^{N}_{n}u_{n}\).
We divide the rest of the proof into several steps.
Step 1. As in the proof of Theorem 3.1, \(\{x_{n}\}\) is bounded. So are the sequences \(\{u_{n}\}\), \(\{v_{n}\}\), \(\{y_{n}\}\), and \(\{z_{n}\}\).
Step 2. We prove that \(\omega_{w}(x_{n})\subset{ \varOmega }\) provided \(\|x_{n}-x_{n+1}\|\to0\) (\(n\to\infty\)).
As in the proof of Step 2 in Theorem 3.2, we have
Since \(\liminf_{n\to\infty}\sigma_{n}>0\), \(\|x_{n}-y_{n}\|\to0\), \(\{ \beta_{n}\}\subset[a,b]\subset(0,1)\), \(\{\lambda_{i,n}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\), \(\{r_{k,n}\}\subset[c_{k},d_{k}]\subset(0,2\mu_{k})\), \(i\in\{1,2,\ldots,N\} \), \(k\in\{1,2,\ldots,M\}\), and \(\{x_{n}\}\), \(\{y_{n}\}\) are bounded sequences, we have
for all \(k\in\{1,2,\ldots,M\}\) and \(i\in\{1,2,\ldots,N\}\).
As in the proof of Step 2 in Theorem 3.2, we have
Since \(\liminf_{n\to\infty}\sigma_{n}>0\), \(\|x_{n}-y_{n}\|\to0\), \(\{ \beta_{n}\}\subset[a,b]\subset(0,1)\), \(\nu_{1}\in(0,2\zeta_{1})\), and \(\{x_{n}\},\{y_{n}\}\), \(\{k_{n}\}\), \(\{\tilde{v}_{n}\}\) are bounded sequences, we obtain from (3.22)
Note that
Hence from (3.67) and (4.3), we get
Also, observe that \(z_{n}-x_{n}=\sigma_{n}(Gv_{n}-x_{n})\) and
Hence we find that
and
So, from \(\liminf_{n\to\infty}\delta_{n}>0\), (3.52), (3.62), and (4.4) it follows that
Secondly, let us show that \(\omega_{w}(x_{n})\subset{ \varOmega }\).
In fact, since H is reflexive and \(\{x_{n}\}\) is bounded, there exists at least a weak convergence subsequence of \(\{x_{n}\}\). Hence it is well known that \(\omega_{w}(x_{n})\neq\emptyset\). Now, take an arbitrary \(w\in\omega_{w}(x_{n})\). Then there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}}\rightharpoonup w\). From (3.58)-(3.60), (3.62), and (4.5), we have that \(u_{n_{i}}\rightharpoonup w\), \(v_{n_{i}}\rightharpoonup w\), \(z_{n_{i}}\rightharpoonup w\), \({\varLambda }^{m}_{n_{i}}u_{n_{i}}\rightharpoonup w\) and \({\varDelta }^{k}_{n_{i}}x_{n_{i}}\rightharpoonup w\), where \(m\in\{ 1,2,\ldots,N\}\) and \(k\in\{1,2,\ldots,M\}\). Utilizing Lemma 2.3(ii), we deduce from \(z_{n_{i}}\rightharpoonup w\) and (4.5) that \(w\in{\operatorname{Fix}}(T)\). In the meantime, utilizing Lemma 2.5, we obtain from \(v_{n_{i}}\rightharpoonup w\) and (4.4) \(w\in{\operatorname{GSVI}}(G)\).
As in the proof of Step 2 in Theorem 3.2, \(w\in\bigcap^{N}_{m=1}{\mathrm{I}}(B_{m}, R_{m})\).
Step 3. We prove that \(\omega_{w}(x_{n})\subset{ \varXi }\) provided \(\|x_{n}-x_{n+1}\|+\|x_{n}-y_{n}\|=o(\lambda_{n})\).
Indeed, as in the proof of Step 3 in Theorem 3.2, we have \(0<\gamma\leq\tau\) and \(\mu\eta\geq\tau\Leftrightarrow \kappa\geq\eta\). It is clear that
Hence, it follows from \(0<\gamma\leq\tau\leq\mu\eta\) that \(\mu F-\gamma S\) is monotone. Moreover, it is also obvious that \(\mu F-\gamma S\) is \((\mu\kappa+\gamma)\)-Lipschitzian. Noticing
we obtain
Set
It can easily be seen from (3.31) that
This yields, for all \(p\in{ \varOmega }\) (noticing \(x_{n}=P_{C}w_{n-1}\)),
In (4.9), the first term is nonnegative due to Proposition 2.1(i), and the fourth term is also nonnegative due to the monotonicity of \(\mu F-\gamma S\). We, therefore, deduce from (4.9) that (noticing again \(x_{n+1}=P_{C}w_{n}\))
Note that
Hence, it follows from \(\|x_{n}-y_{n}\|=o(\lambda_{n})\) that
Also, since \(e_{n}\to0\) (due to \(\|x_{n+1}-x_{n}\|=o(\lambda_{n})\)), \(\alpha_{n}\to0\) and \(\{x_{n}\}\) is bounded by Step 1 which implies that \(\{w_{n}\}\) is bounded, we obtain from (4.10)
This suffices to guarantee that \(\omega_{w}(x_{n})\subset{ \varXi }\); namely, every weak limit point of \(\{x_{n}\}\) solves the HVIP (1.11). As a matter of fact, if \(x_{n_{i}}\rightharpoonup\tilde{x}\in\omega _{w}(x_{n})\) for some subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\), then we deduce from (3.60) that, for all \(p\in{ \varOmega }\),
that is,
In addition, note that \(\omega_{w}(x_{n})\subset{ \varOmega }\) by Step 2. Since \(\mu F-\gamma S\) is monotone and Lipschitz continuous and Ω is nonempty, closed, and convex, by Minty’s lemma [38] the last inequality is equivalent to (1.11). Consequently, we get \(\tilde{x}\in{ \varXi }\).
Step 4. We prove that \(\{x_{n}\}\) converges strongly to a unique solution of Problem 1.2 provided \(\|x_{n}-x_{n+1}\|+\|x_{n}-y_{n}\|+\|x_{n}-z_{n}\|=o(\lambda_{n})\).
Indeed, it is clear that
Hence, it follows from \(0<\gamma\leq\tau\leq\mu\eta\) and \(\rho <1\) that \(\mu F-\gamma V\) is \((\mu\eta-\gamma\rho)\)-strongly monotone. Moreover, it is also obvious that \(\mu F-\gamma V\) is \((\mu\kappa+\gamma\rho)\)-Lipschitzian. So, we can write \({\operatorname{VI}}(\mu F-\gamma V,\varXi )=\{x^{*}\}\). We now take a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) satisfying
Without loss of generality, we may further assume that \(x_{n_{i}}\rightharpoonup\tilde{x}\); then \(\tilde{x}\in{ \varXi }\) as we just proved. Since \(x^{*}\) is a solution of the THVIP (1.10), we get
From (4.1) and (3.50), it follows that (noticing that \(x_{n+1}=P_{C}w_{n}\) and \(0<\gamma\leq\tau\))
It turns out that
However, from \(x^{*}\in{ \varXi }\) and condition (iv) we obtain
Since \((\gamma_{n}+\delta_{n})\xi\leq\gamma_{n}\), utilizing Lemmas 2.3(i) and 2.4, we get
That is, \(\|\delta_{n}(Tx_{n}-x_{n})\|=o(\lambda_{n})\). Taking into account \(\liminf_{n\to\infty}\delta_{n}>0\), we have \(\|x_{n}-Tx_{n}\| =o(\lambda_{n})\). Furthermore, utilizing Lemma 2.3(i), we have
where \(\sup_{n\geq0}\|\gamma\alpha_{n}(Vx_{n}-Sx_{n})+\gamma Sx_{n}-\mu Fy_{n}\|\leq\widetilde{M}_{0}\) for some \(\widetilde{M}_{0}>0\). Hence, for a big enough constant \(\bar{k}_{1}>0\), from (4.18), we have
Combining (4.17)-(4.19), we get
Since \(\sum^{\infty}_{n=0}\alpha_{n}\lambda_{n}=\infty\), \(\lim_{n\to\infty}\frac{\lambda^{1/\theta}_{n}}{\alpha_{n}}=0\), \(\|x_{n}-x_{n+1}\|+\|y_{n}-x_{n}\| +\|x_{n}-Tx_{n}\|=o(\lambda_{n})\), we deduce from (4.16) that \(\sum^{\infty}_{n=0}\alpha_{n}\lambda_{n}\gamma(1-\rho)=\infty\) and
Applying Lemma 2.8 to (4.19), we conclude that \(x_{n}\to x^{*}\) as \(n\to\infty\). This completes the proof. □
Remark 4.1
It is obvious that iterative scheme (4.1) is different from the iterative schemes in [7, 32] The three-step iterative scheme in [7, Algorithm I] is extended to the five-step iterative scheme (4.1) for the THVIP (1.10) by combining Mann’s iteration method, Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method [41], and the projection method. It is worth pointing out that under the lack of the assumptions similar to those in [32, Theorem 3.2], for example, \(\{x_{n}\}\) is bounded and \({\operatorname{Fix}}(T)\cap{\operatorname{int}}C\neq\emptyset\), the sequence \(\{x_{n}\}\) generated by (3.1) converges strongly to a point \(x^{*}\in\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k}) \cap\bigcap^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i}) \cap{\operatorname{GSVI}}(G) \cap {\operatorname{Fix}} (T)=:{\varOmega }\), which is a unique solution \(x^{*}\in{ \varXi }\) of the THVIP (1.10) (over the fixed point set of a strictly pseudocontractive mapping T), that is, \(\langle(\mu F-\gamma V)x^{*},p-x^{*}\rangle\geq0\), \(\forall p\in{ \varXi }\).
Remark 4.2
Theorem 4.1 improves and extends [32, Theorems 3.1 and 3.2] and [7, Theorem 14] in the following aspects:
-
(a)
THVIP (1.10) with the unique solution \(x^{*}\in \varOmega \) satisfying
$$x^{*}=P_{\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\cap\bigcap ^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i}) \cap{\operatorname{GSVI}}(G)\cap{\operatorname{Fix}}(T)} \bigl(I-(\mu F-\gamma S)\bigr)x^{*} $$is more general than the problem of finding a point \(\tilde{x}\in C\) satisfying \(\tilde{x}=P_{{\operatorname{Fix}}(T)}S\tilde{x}\) in [32] and the problem of finding a point \(x^{*}\in{\operatorname{Fix}}(T)\cap{\operatorname{VI}}(C,A)\) satisfying \(x^{*}=P_{{\operatorname{Fix}}(T)\cap{\operatorname{VI}}(C,A)} (I-(\mu F-\gamma S))x^{*}\) in [7, Theorem 14].
-
(b)
The five-step iterative scheme (4.1) for THVIP (1.10) is flexible and more advantageous than the three-step iterative scheme in [7, Algorithm I] and the two-step iterative scheme in [32, Theorems 3.1 and 3.2] because it can be used to solve several kinds of problems, for example, the THVIP, the HVIP, and the problem of finding a common point of four sets: \(\bigcap^{M}_{k=1}{\operatorname{GMEP}}({\varTheta }_{k},\varphi_{k},A_{k})\), \(\bigcap^{N}_{i=1}{\mathrm{I}}(B_{i},R_{i})\), \({\operatorname{GSVI}}(G)\), and \({\operatorname{Fix}} (T)\). In addition, Theorem 4.1 drops the crucial requirements in [32, Theorem 3.2] that \(\lim_{n\to\infty}\frac{\alpha_{n}}{\beta_{n}}=0\), \(\lim_{n\to \infty}\frac{\beta^{2}_{n}}{\alpha_{n}}=0\), \({\operatorname{Fix}}(T)\cap{\operatorname{int}}C\neq\emptyset\), and \(\{x_{n}\}\) is bounded, and also it removes the crucial ones [7, Theorem 14] that \(\sum^{\infty}_{n=0}\alpha_{n}<\infty\), \(\lim_{n\to\infty}(\alpha _{n}/\lambda^{2}_{n})=0\), and \(\|x_{n+1}-x_{n}\|+\|x_{n}-z_{n}\|=o(\lambda^{2}_{n})\).
-
(c)
The argument and techniques in Theorem 4.1 are different from the ones in [32, Theorems 3.1 and 3.2] and in [7, Theorem 14] because we make use of the properties of strictly pseudocontractive mappings (see Lemmas 2.3 and 2.4), the properties of resolvent operators and maximal monotone mappings (see Proposition 2.2, Remark 2.2, and Lemmas 2.9-2.13), the fixed point equation \(x^{*}=P_{C}(I-\nu_{1}F_{1})P_{C}(I-\nu_{2}F_{2})x^{*}\) equivalent to the GSVI (1.4) (see Proposition 2.3), and the contractive coefficient estimates for the contractions associated with nonexpansive mappings (see Lemma 2.7).
-
(d)
Compared with the requirement \(\|x_{n}-x_{n+1}\|+\|x_{n}-z_{n}\| =o(\lambda^{2}_{n})\) in [7, Theorem 14], the one \(\|x_{n}-x_{n+1}\| +\|x_{n}-y_{n}\|+\|x_{n}-z_{n}\|=o(\lambda_{n})\) in conclusion (c) of Theorem 4.1 is additionally added because Theorem 4.1 involves a quite complex problem, that is, the THVIP 1.10 (over the fixed point set \({\operatorname{Fix}}(T)\) of a strictly pseudocontractive mapping T) with constraints of several problems: GSVI (1.4), finitely many GMEPs and finitely many variational inclusions.
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Acknowledgements
In this research, first was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and PhD Program Foundation of Ministry of Education of China (20123127110002). In this research, second and third author were partially supported by a grant from National Science Council of Taiwan.
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Ceng, LC., Lin, YC. & Wen, CF. Iterative methods for triple hierarchical variational inequalities with mixed equilibrium problems, variational inclusions, and variational inequalities constraints. J Inequal Appl 2015, 16 (2015). https://doi.org/10.1186/s13660-014-0535-x
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DOI: https://doi.org/10.1186/s13660-014-0535-x