1 Introduction
Let
H be a real Hilbert space with inner product
\(\langle\cdot,\cdot \rangle\) and induced norm
\(\|\cdot\|\),
C be a nonempty, closed, and convex subset of
H, and
\(P_{C}\) be the metric projection of
H onto
C. Let
\(T:C\to C\) be a self-mapping on
C. We denote by
\(\operatorname{Fix}(T)\) the set of fixed points of
T and by
R the set of all real numbers. A mapping
\(A:H\to H\) is called
γ̄-strongly positive on
H if there exists a constant
\(\bar{\gamma}>0\) such that
$$\langle Ax,x\rangle\geq\bar{\gamma}\|x\|^{2},\quad \forall x\in H. $$
A mapping
\(F:C\to H\) is called
L-Lipschitz-continuous if there exists a constant
\(L\geq0\) such that
$$\|Fx-Fy\|\leq L\|x-y\|,\quad \forall x,y\in C. $$
In particular, if
\(L=1\) then
F is called a nonexpansive mapping; if
\(L\in[0,1)\) then
F is called a contraction. A mapping
\(T:C\to C\) is called
k-strictly pseudocontractive (or a
k-strict pseudocontraction) if there exists a constant
\(k\in[0,1)\) such that
$$\|Tx-Ty\|^{2}\leq\|x-y\|^{2}+k\bigl\| (I-T)x-(I-T)y \bigr\| ^{2},\quad \forall x,y\in C. $$
In particular, if
\(k=0\), then
T is a nonexpansive mapping. The mapping
T is pseudocontractive if and only if
$$\langle Tx-Ty,x-y\rangle\leq\|x-y\|^{2},\quad \forall x,y\in C. $$
T is strongly pseudocontractive if and only if there exists a constant
\(\lambda\in(0,1)\) such that
$$\langle Tx-Ty,x-y\rangle\leq\lambda\|x-y\|^{2},\quad \forall x,y\in C. $$
Note that the class of strictly pseudocontractive mappings includes the class of nonexpansive mappings as a subclass. That is,
T is nonexpansive if and only if
T is 0-strictly pseudocontractive. The mapping
T is also said to be pseudocontractive if
\(k=1\) and
T is said to be strongly pseudocontractive if there exists a positive constant
\(\lambda\in(0,1)\) such that
\(T+(1-\lambda)I\) is pseudocontractive. Clearly, the class of strictly pseudocontractive mappings falls into the one between the classes of nonexpansive mappings and of pseudocontractive mappings. Also it is clear that the class of strongly pseudocontractive mappings is independent of the class of strictly pseudocontractive mappings (see [
1]). The class of pseudocontractive mappings is one of the most important classes of mappings among nonlinear mappings. Recently, many authors have been devoting to the study of the problem of finding fixed points of pseudocontractive mappings; see
e.g., [
2‐
9] and the references therein.
Let
\({\mathcal{A}}:C\to H\) be a nonlinear mapping on
C. The variational inequality problem (VIP) associated with the set
C and the mapping
\({\mathcal{A}}\) is stated as follows: find
\(x^{*}\in C\) such that
$$ \bigl\langle {\mathcal{A}}x^{*},x-x^{*}\bigr\rangle \geq0,\quad \forall x\in C. $$
(1.1)
The solution set of VIP (
1.1) is denoted by
\(\operatorname{VI}(C,{\mathcal{A}})\).
The VIP (
1.1) was first discussed by Lions [
10]. There are many applications of VIP (
1.1) in various fields; see,
e.g., [
4,
5,
7,
11]. It is well known that, if
\({\mathcal{A}}\) is a strongly monotone and Lipschitz-continuous mapping on
C, then VIP (
1.1) has a unique solution. In 1976, Korpelevich [
12] proposed an iterative algorithm for solving VIP (
1.1) in Euclidean space
\(\mathbf{R}^{n}\):
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=P_{C}(x_{n}-\tau{\mathcal{A}}x_{n}),\\ x_{n+1}=P_{C}(x_{n}-\tau{\mathcal{A}}y_{n}), \quad\forall n\geq0, \end{array}\displaystyle \right . $$
(1.2)
with
\(\tau>0\) a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see,
e.g., [
5,
11,
13‐
29] and references therein, to name but a few.
In 2011, Ceng
et al. [
30] also introduced the following iterative method:
$$ x_{n+1}=P_{C}\bigl[\alpha_{n}\gamma Vx_{n}+(I-\alpha_{n}\mu F)Tx_{n}\bigr],\quad \forall n\geq0, $$
(1.3)
where
\(T:C\to C\) is a nonexpansive mapping such that
\(\operatorname{Fix}(T)\neq \emptyset\),
\(F:C\to H\) is a
κ-Lipschitzian and
η-strongly monotone operator with positive constants
\(\kappa,\eta>0\),
\(V:C\to H\) is an
l-Lipschitzian mapping with constant
\(l \geq0\) and
\(0<\mu<\frac{2\eta}{\kappa^{2}}\). They proved that, under mild conditions, the sequence
\(\{x_{n}\}\) generated by (
1.3) converges strongly to a point
\(\tilde{x}\in\operatorname{Fix}(T)\) which is the unique solution to the VIP
$$ \bigl\langle (\mu F-\gamma V)\tilde{x},p-\tilde{x}\bigr\rangle \geq0,\quad \forall p \in \operatorname{Fix}(T). $$
(1.4)
Their results also improve Tian’s results [
31] from the contractive mapping
f to the Lipschitzian mapping
V.
In 2011, Ceng
et al. [
32] introduced one general composite implicit scheme that generates a net
\(\{x_{t}\}_{t\in(0,\min\{1, \frac{2-\bar{\gamma}}{\tau-\gamma\alpha}\})}\) in an implicit way
$$ x_{t}=(I-\theta_{t}A)Tx_{t}+ \theta_{t}\bigl[Tx_{t}-t\bigl(\mu FTx_{t}-\gamma f(x_{t})\bigr)\bigr], $$
(1.5)
and also proposed another general composite explicit scheme that generates a sequence
\(\{x_{n}\}\) in an explicit way
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=(I-\alpha_{n}\mu F)Tx_{n}+\alpha_{n}\gamma f(x_{n}),\\ x_{n+1}=(I-\beta_{n}A)Tx_{n}+\beta_{n}y_{n}, \quad\forall n\geq0, \end{array}\displaystyle \right . $$
(1.6)
where
\(x_{0}\in H\) is an arbitrary initial guess,
\(F:H\to H\) is a
κ-Lipschitzian and
η-strongly monotone operator with positive constants
\(\kappa,\eta>0\),
\(T:H\to H\) is a nonexpansive mapping,
\(A:H\to H\) is a
γ̄-strongly positive bounded linear operator, and
\(f:H\to H\) is an
α-contractive mapping with
\(\alpha\in(0,1)\). They proved that, under appropriate conditions, the net
\(\{x_{t}\}\) and the sequence
\(\{x_{n}\}\) generated by (
1.5) and (
1.6), respectively, converge strongly to the same point
\(\tilde{x}\in\operatorname{Fix}(T)\), which is the unique solution to the VIP
$$ \bigl\langle (A-I)\tilde{x},p-\tilde{x}\bigr\rangle \geq0, \quad\forall p\in \operatorname{Fix}(T). $$
(1.7)
Their results supplement and develop the corresponding ones of Marino and Xu [
33], Yamada [
34] and Tian [
31].
Very recently, inspired by Ceng
et al. [
32], Jung [
1] introduced one general composite implicit scheme that generates a net
\(\{x_{t}\}_{t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})}\) in an implicit way
$$ x_{t}=(I-\theta_{t}A)T_{t}x_{t}+ \theta_{t}\bigl[t\gamma Vx_{t}+(I-t\mu F)T_{t}x_{t} \bigr], $$
(1.8)
and also proposed another general composite explicit scheme that generates a sequence
\(\{x_{n}\}\) in an explicit way,
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=\alpha_{n}\gamma Vx_{n}+(I-\alpha_{n}\mu F)T_{n}x_{n},\\ x_{n+1}=(I-\beta_{n}A)T_{n}x_{n}+\beta_{n}y_{n},\quad \forall n\geq0, \end{array}\displaystyle \right . $$
(1.9)
where
\(x_{0}\in H\) is an arbitrary initial guess and the following conditions are satisfied:
-
\(T:H\to H\) is a k-strictly pseudocontractive mapping with \(\operatorname{Fix}(T)\neq\emptyset\);
-
A is a γ̄-strongly positive bounded linear operator on H with \(\bar{\gamma}\in(1,2)\);
-
\(F:H\to H\) is a κ-Lipschitzian and η-strongly monotone operator with \(0<\mu<\frac{2\eta}{\kappa^{2}}\);
-
\(V:H\to H\) is an l-Lipschitzian mapping with \(0\leq\gamma l<\tau\) and \(\tau=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\);
-
\(T_{t}:H\to H\) is a mapping defined by \(T_{t}x=\lambda_{t}x+(1-\lambda _{t})Tx\), \(t\in(0,1)\), for \(0\leq k\leq\lambda_{t}\leq\lambda<1\) and \(\lim_{t\to0}\lambda_{t}=\lambda\);
-
\(T_{n}:H\to H\) is a mapping defined by \(T_{n}x=\lambda_{n}x+(1-\lambda_{n})Tx\) for \(0\leq k\leq\lambda_{n}\leq\lambda<1\) and \(\lim_{n\to\infty}\lambda_{n}=\lambda\);
-
\(\{\alpha_{n}\}\subset[0,1]\), \(\{\beta_{n}\}\subset(0,1]\) and \(\{\theta_{t}\} _{t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})} \subset(0,1)\).
The author of [
1] proved that, under weaker control conditions than the previous ones, the net
\(\{x_{t}\}\) and the sequence
\(\{x_{n}\}\) generated by (
1.8) and (
1.9), respectively, converge strongly to the same point
\(\tilde{x}\in\operatorname{Fix}(T)\), which is the unique solution to the VIP
$$ \bigl\langle (A-I)\tilde{x},p-\tilde{x}\bigr\rangle \geq0, \quad\forall p\in \operatorname{Fix}(T). $$
(1.10)
His results extend and improve Ceng
et al.’s corresponding ones [
32] from the nonexpansive mapping
T to the strictly pseudocontractive mapping
T and from the contractive mapping
f to the Lipschitzian mapping
V.
On the other hand, let
\(\varphi:C\to\mathbf{R}\) be a real-valued function,
\({\mathcal{A}}:C\to H\) be a nonlinear mapping and
\({ \varTheta }:C\times C\to\mathbf{R}\) be a bifunction. In 2008, Peng and Yao [
13] introduced the generalized mixed equilibrium problem (GMEP) of finding
\(x\in C\) such that
$$ { \varTheta }(x,y)+\varphi(y)-\varphi(x)+\langle{\mathcal{A}}x,y-x\rangle \geq0, \quad\forall y\in C. $$
(1.11)
We denote the set of solutions of GMEP (
1.11) by
\(\operatorname{GMEP}({ \varTheta },\varphi,{\mathcal{A}})\). The GMEP (
1.11) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. Recently, many authors have been devoting to the study of the GMEP (
1.11) and its special cases,
e.g., generalized equilibrium problem (GEP), mixed equilibrium problem (MEP),equilibrium problem (EP),
etc.; see,
e.g., [
15,
18,
23‐
29,
31,
35‐
38] and the references therein.
It was assumed in [
13] that
\({ \varTheta }:C\times C\to\mathbf{R}\) is a bifunction satisfying conditions (A1)-(A4) and
\(\varphi:C \to\mathbf{R}\) is a lower semicontinuous and convex function with restriction (B1) or (B2), where
(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 \(x\in C\);
(B1)
for each
\(x\in H\) and
\(r>0\), there exists a bounded subset
\(D_{x}\subset C\) and
\(y_{x}\in C\) such that for any
\(z\in C\setminus D_{x}\),
$${ \varTheta }(z,y_{x})+\varphi(y_{x})-\varphi(z)+ \frac{1}{r}\langle y_{x}-z,z-x\rangle< 0; $$
Given a positive number
\(r>0\). Let
\(T^{({ \varTheta },\varphi)}_{r}:H\to C\) be the solution set of the auxiliary mixed equilibrium problem, that is, for each
\(x\in H\),
$$T^{({ \varTheta },\varphi)}_{r}(x):=\biggl\{ y\in C:{ \varTheta }(y,z)+\varphi (z)- \varphi(y)+\frac{1}{r}\langle y-x, z-y\rangle\geq0,\forall z\in C\biggr\} . $$
In particular, if
\(\varphi\equiv0\) then
\(T^{({ \varTheta },\varphi)}_{r}\) is rewritten as
\(T^{ \varTheta }_{r}:H\to C\),
i.e.,
$$T^{ \varTheta }_{r}(x):=\biggl\{ y\in C:{ \varTheta }(y,z)+ \frac{1}{r}\langle y-x,z-y\rangle\geq0,\forall z\in C\biggr\} . $$
Let
\({ \varPhi }_{1},{ \varPhi }_{2}:C\times C\to\mathbf{R}\) be two bifunctions and
\(F_{1},F_{2}:C\to H\) be two mappings. Consider the problem of finding
\((x^{*},y^{*})\in C\times C\) such that
$$ \left \{ \textstyle\begin{array}{@{}l} { \varPhi }_{1}(x^{*},x)+\langle F_{1}y^{*},x-x^{*}\rangle+\frac {1}{\nu_{1}}\langle x^{*}-y^{*},x-x^{*}\rangle \geq0, \quad\forall x\in C,\\ { \varPhi }_{2}(y^{*},y)+\langle F_{2}x^{*},y-y^{*}\rangle+\frac{1}{\nu_{2}}\langle y^{*}-x^{*},y-y^{*}\rangle \geq0, \quad\forall y\in C, \end{array}\displaystyle \right . $$
(1.12)
which is called a system of generalized equilibrium problems (SGEP) where
\(\nu_{1}>0\) and
\(\nu_{2}>0\) are two constants. In 2010, Ceng and Yao [
23] transformed the SGEP (
1.12) into the fixed point problem of the mapping
\(G=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1} F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})\), that is,
\(Gx^{*}=x^{*}\), where
\(y^{*}=T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x^{*}\). Throughout this paper, the fixed point set of the mapping
G is denoted by
Ξ.
In particular, if
\({ \varPhi }_{1}\equiv{ \varPhi }_{2}\equiv0\), then problem (
1.12) reduces to the system of variational inequalities (SVI) of finding
\((x^{*},y^{*})\in C\times C\) such that
$$ \left \{ \textstyle\begin{array}{@{}l} \langle\nu_{1}F_{1}y^{*}+x^{*}-y^{*},x-x^{*}\rangle\geq0,\quad \forall x\in C,\\ \langle\nu_{2}F_{2}x^{*}+y^{*}-x^{*},y-y^{*}\rangle\geq0,\quad \forall y\in C, \end{array}\displaystyle \right . $$
(1.13)
where
\(\nu_{1}>0\) and
\(\nu_{2}>0\) are two constants. Recently, many authors have addressed the study of the SVI (
1.13); see,
e.g., [
11,
14,
15,
17‐
20,
39‐
41] and the references therein.
Let
\(T:C\to C\) be a
k-strictly pseudocontractive mapping. In 2010, Ceng and Yao [
23] proposed and analyzed the following relaxed extragradient-like iterative scheme for finding a common solution
\(x^{*}\in{ \varOmega }:=\operatorname{Fix}(T)\cap\operatorname{GMEP}({ \varTheta },\varphi,{\mathcal{A}})\cap{ \varXi }\) of the GMEP (
1.11), the SGEP (
1.12), and the fixed point problem of
T:
$$ \left \{ \textstyle\begin{array}{@{}l} z_{n}=T^{({ \varTheta },\varphi)}_{\lambda_{n}}(I-\lambda _{n}{\mathcal{A}})x_{n},\\ y_{n}=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu _{2}F_{2})z_{n},\\ x_{n+1}=\alpha_{n}u+\beta_{n}x_{n}+\gamma_{n}y_{n}+\delta_{n}Ty_{n}, \quad\forall n\geq0, \end{array}\displaystyle \right . $$
(1.14)
where
\(0<\nu_{j}<2\zeta_{j}\) for
\(j=1,2\), and
\(\{\lambda_{n}\}\subset[0,2\eta ]\),
\(\{\alpha_{n}\},\{\beta_{n}\},\{\gamma_{n}\},\{\delta_{n}\}\subset [0,1]\) such that
\(\alpha_{n}+\beta_{n}+\gamma_{n}+\delta_{n}=1\) and
\((\gamma _{n}+\delta_{n})k\leq\gamma_{n}\),
\(\forall n\geq0\). Under some mild assumptions, the authors [
23] proved that
\(\{x_{n}\} \) converges strongly to
\(x^{*}=P_{ \varOmega }u\) and
\((x^{*},y^{*})\) is a solution of the SGEP (
1.12), where
\(y^{*}=T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x^{*}\).
In this paper, we introduce one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEP) with the constraints of the SGEP (
1.12) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping in a real Hilbert space. We establish the strong convergence of these two composite relaxed extragradient-like schemes to the same common solution of finitely many GMEPs and the SGEP (
1.12), which is the unique solution of the HFPP for a strictly pseudocontractive mapping. In particular, we make use of weaker control conditions than the previous ones for the sake of proving strong convergence. Utilizing these results, we first propose the composite implicit and explicit relaxed extragradient-like schemes for finding a common fixed point of a finite family of strictly pseudocontractive mappings, and then derive their strong convergence to the unique common solution of the SGEP (
1.12) and some HFPP. Our results complement, develop, improve, and extend the corresponding ones given by some authors recently in this area. See,
e.g., Ceng
et al. [
32], Jung [
1], and Ceng and Yao [
23].
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
\(\{x_{n}\}\),
i.e.,
$$\omega_{w}(x_{n}):=\bigl\{ x\in H:x_{n_{i}} \rightharpoonup x \mbox{ for some subsequence }\{x_{n_{i}}\} \mbox{ of } \{x_{n}\}\bigr\} . $$
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
$$\|x-P_{C}x\|=\inf_{y\in C}\|x-y\|=:d(x,C). $$
The following properties of projections are useful and pertinent to our purpose.
It can easily be seen that if T is nonexpansive, then \(I-T\) is monotone. It is also easy to see that the projection \(P_{C}\) is 1-ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.
On the other hand, it is obvious that if
\(F:C\to H\) is
α-inverse strongly monotone, then
F is monotone and
\(\frac{1}{\alpha}\)-Lipschitz-continuous. Moreover, we also have, for all
\(u,v\in C\) and
\(\lambda>0\),
$$ \bigl\| (I-\lambda F)u-(I-\lambda F)v\bigr\| ^{2}\leq\|u-v\|^{2}+ \lambda(\lambda-2\alpha )\|Fu-Fv\|^{2}. $$
(2.1)
Consequently, if
\(\lambda\leq2\alpha\), then
\(I-\lambda F\) is a nonexpansive mapping from
C to
H.
Next we list some elementary conclusions for the MEP.
In 2010, Ceng and Yao [
23] transformed the SGEP (
1.12) into a fixed point problem in the following way:
In Proposition
2.3, putting
\({ \varPhi }_{1}\equiv{ \varPhi }_{2}\equiv0\), we get the following.
We need some facts and tools in a real Hilbert space H; these are listed as lemmas below.
It is clear that, in a real Hilbert space
H,
\(T:C\to C\) is
k-strictly pseudocontractive if and only if the following inequality holds:
$$\langle Tx-Ty,x-y\rangle\leq\|x-y\|^{2}-\frac{1-k}{2} \bigl\| (I-T)x-(I-T)y\bigr\| ^{2}, \quad\forall x,y\in C. $$
This immediately implies that if
T is a
k-strictly pseudocontractive mapping, then
\(I-T\) is
\(\frac{1-k}{2}\)-inverse strongly monotone; for further detail, we refer to [
42] 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.
Let LIM be a Banach limit. According to time and circumstances, we use
\(\operatorname{LIM}_{n}a_{n}\) instead of LIM
a for every
\(a=\{a_{n}\}\in l^{\infty}\). The following properties are well known:
(i)
for all \(n\geq1\), \(a_{n}\leq c_{n}\) implies \(\operatorname{LIM}_{n}a_{n} \leq\operatorname{LIM}_{n}c_{n}\);
(ii)
\(\operatorname{LIM}_{n}a_{n+N}=\operatorname{LIM}_{n}a_{n}\) for any fixed positive integer N;
(iii)
\(\liminf_{n\to\infty}a_{n}\leq\operatorname{LIM}_{n}a_{n} \leq\limsup_{n\to\infty }a_{n}\) for all \(\{a_{n}\}\in l^{\infty}\).
The following lemma was given in [
39], Proposition 2.
Recall that a set-valued mapping
\(\widetilde{T}:D(\widetilde{T})\subset H\to2^{H}\) is called monotone if for all
\(x,y\in D (\widetilde{T})\),
\(f\in{\widetilde{T}}x\), and
\(g\in{\widetilde{T}}y\) imply
$$\langle f-g,x-y\rangle\geq0. $$
A set-valued mapping
T̃ is called maximal monotone if
T̃ is monotone and
\((I+\lambda\widetilde{T})D (\widetilde{T})=H\) for each
\(\lambda>0\), where
I is the identity mapping of
H. We denote by
\(G(\widetilde{T})\) the graph of
T̃. It is well known that a monotone mapping
T̃ is maximal if and only if, for
\((x,f)\in H\times H\),
\(\langle f-g,x-y\rangle\geq0\) for every
\((y,g)\in G(\widetilde{T})\) implies
\(f\in \widetilde{T}x\). Next we provide an example to illustrate the concept of a maximal monotone mapping.
Let
\({ \varGamma }:C\to H\) be a monotone and Lipschitz-continuous mapping and let
\(N_{C}v\) be the normal cone to
C at
\(v\in C\),
i.e.,
$$N_{C}v=\bigl\{ u\in H:\langle v-p,u\rangle\geq0, \forall p\in C\bigr\} . $$
Define
$$\widetilde{T}v= \left \{ \textstyle\begin{array}{@{}l@{\quad}ll} { \varGamma }v+N_{C}v,& \mbox{if }v\in C,\\ \emptyset, &\mbox{if }v\notin C. \end{array}\displaystyle \right . $$
Then it is well known [
27] that
T̃ is maximal monotone and
\(0\in\widetilde{T}v\) if and only if
\(v\in\operatorname{VI}(C,{ \varGamma })\).
3 Main results
Let
C be a nonempty, closed, and convex subset of a real Hilbert space
H. Throughout this section, we always assume the following:
-
\(F:C\to H\) is a κ-Lipschitzian and η-strongly monotone operator with positive constants \(\kappa,\eta>0\), and \(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse strongly monotone for \(j=1,2\);
-
\(T:C\to C\) is a k-strictly pseudocontractive mapping and \({\mathcal {A}}_{i}:C\to H\) is \(\eta_{i}\)-inverse strongly monotone for each \(i=1,\ldots,N\);
-
A is a γ̄-strongly positive bounded linear operator on H with \(\bar{\gamma}\in(1,2)\) and \(V:C\to H\) is an l-Lipschitzian mapping with \(l\geq0\);
-
\({ \varTheta }_{i},{ \varPhi }_{j}:C\times C\to\mathbf{R}\) are the bifunctions satisfying conditions (A1)-(A4) and \(\varphi_{i}:C\to \mathbf{R}\cup\{+\infty\}\) be a proper lower semicontinuous and convex function with restrictions (B1) or (B2) for each \(i =1,\ldots,N\) and \(j=1,2\);
-
\(0<\mu<\frac{2\eta}{\kappa^{2}}\) and \(0\leq\gamma l<\tau\) with \(\tau =1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\);
-
\(S:C\to C\) is a mapping defined by \(Sx=\lambda x+(1-\lambda)Tx\) for \(0\leq k\leq\lambda<1\);
-
\(G:C\to C\) is a mapping defined by \(Gx=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu _{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x\) with \(0< \nu_{j}<2\zeta_{j}\) for \(j=1,2\);
-
\({ \varDelta }^{N}_{t}:C\to C\) is a mapping defined by \({ \varDelta }^{N}_{t}x=T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t}}(I-r_{N,t}{\mathcal {A}}_{N})\cdots T^{({ \varTheta }_{1},\varphi _{1})}_{r_{1,t}}(I-r_{1,t}{\mathcal{A}}_{1})x\), \(t\in(0,1)\), for \(\{r_{i,t}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\), \(i=1,\ldots,N\);
-
\({ \varDelta }^{N}_{n}:C\to C\) is a mapping defined by \({ \varDelta }^{N}_{n}x=T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,n}}(I-r_{N,n}{\mathcal {A}}_{N})\cdots T^{({ \varTheta }_{1},\varphi _{1})}_{r_{1,n}}(I-r_{1,n}{\mathcal{A}}_{1})x\) with \(\{r_{i,n}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\) and \(\lim_{n\to \infty}r_{i,n}=r_{i}\), for each \(i=1,\ldots,N\);
-
\({ \varOmega }:=(\bigcap^{N}_{i=1}\operatorname{GMEP}({ \varTheta }_{i},\varphi _{i},{\mathcal{A}}_{i}))\cap\operatorname{Fix}(T)\cap{ \varXi }\neq\emptyset\) and \(P_{{ \varOmega }}\) is the metric projection of H onto Ω;
-
\(\{\alpha_{n}\}\subset[0,1]\), \(\{\beta_{n}\}\subset(0,1]\) and \(\{\theta_{t}\} _{t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})} \subset(0,1)\).
Next, put
$${ \varDelta }^{i}_{t}=T^{({ \varTheta }_{i},\varphi _{i})}_{r_{i,t}}(I-r_{i,t}{ \mathcal{A}}_{i})T^{({ \varTheta }_{i-1},\varphi_{i-1})}_{r_{i-1,t}}(I-r_{i-1,t}{ \mathcal{A}}_{i-1})\cdots T^{({ \varTheta }_{1},\varphi_{1})}_{r_{1,t}}(I-r_{1,t}{ \mathcal{A}}_{1}),\quad \forall t\in(0,1), $$
and
$${ \varDelta }^{i}_{n}=T^{({ \varTheta }_{i},\varphi _{i})}_{r_{i,n}}(I-r_{i,n}{ \mathcal{A}}_{i})T^{({ \varTheta }_{i-1},\varphi _{i-1})}_{r_{i-1,n}} (I-r_{i-1,n}{ \mathcal{A}}_{i-1})\cdots T^{({ \varTheta }_{1},\varphi _{1})}_{r_{1,n}}(I-r_{1,n}{ \mathcal{A}}_{1}),\quad \forall n\geq0, $$
for all
\(i\in\{1,\ldots,N\}\), and
\({ \varDelta }^{0}_{t}={ \varDelta }^{0}_{n}=I\), where
I is the identity mapping on
H.
By Lemma
2.4, we know that
S is nonexpansive. It is clear that
\(\operatorname{Fix}(S)=\operatorname{Fix}(T)\). Since
\(\{\lambda_{i,t}\}\subset [a_{i},b_{i}]\subset(0,2\eta_{i})\), utilizing (
2.1) and Proposition
2.2(ii) we have for all
\(x,y\in C\)
$$\begin{aligned}[b] \bigl\| { \varDelta }^{N}_{t}x-{ \varDelta }^{N}_{t}y \bigr\| &=\bigl\| T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t}}(I-r_{N,t}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t}x -T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t}}(I-r_{N,t}{ \mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}y\bigr\| \\ &\leq\bigl\| (I-r_{N,t}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}x-(I-r_{N,t}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}y\bigr\| \\ &\leq\bigl\| { \varDelta }^{N-1}_{t}x-{ \varDelta }^{N-1}_{t}y \bigr\| \\ &\leq\cdots \\ &\leq\bigl\| { \varDelta }^{i}_{t}x-{ \varDelta }^{i}_{t}y \bigr\| \\ &\leq\cdots \\ &\leq\bigl\| { \varDelta }^{0}_{t}x-{ \varDelta }^{0}_{t}y \bigr\| \\ &=\|x-y\|, \end{aligned} $$
which implies that
\({ \varDelta }^{i}_{t}:C\to C\) is a nonexpansive mapping for all
\(t\in(0,1)\). Also, since
\(\{r_{i,n}\}\subset [a_{i},b_{i}]\subset(0,2\eta_{i})\), utilizing (
2.1) and Proposition
2.2(ii) we have for all
\(x,y\in C\)
$$\begin{aligned} \bigl\| { \varDelta }^{N}_{n}x-{ \varDelta }^{N}_{n}y \bigr\| &=\bigl\| T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,n}}(I-r_{N,n}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{n}x -T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,n}}(I-r_{N,n}{ \mathcal{A}}_{N}){ \varDelta }^{N-1}_{n}y\bigr\| \\ &\leq\bigl\| (I-r_{N,n}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{n}x-(I-r_{N,n}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{n}y\bigr\| \\ &\leq\bigl\| { \varDelta }^{N-1}_{n}x-{ \varDelta }^{N-1}_{n}y \bigr\| \\ &\leq\cdots \\ &\leq\bigl\| { \varDelta }^{i}_{n}x-{ \varDelta }^{i}_{n}y \bigr\| \\ &\leq\cdots \\ &\leq\bigl\| { \varDelta }^{0}_{n}x-{ \varDelta }^{0}_{n}y \bigr\| \\ &=\|x-y\|, \end{aligned}$$
which implies that
\({ \varDelta }^{i}_{n}:C\to C\) is a nonexpansive mapping for all
\(n\geq0\).
In this section, we introduce the first composite relaxed extragradient-like scheme that generates a net
\(\{x_{t}\}_{t\in(0,\min\{1, \frac{2-\bar{\gamma}}{\tau-\gamma l}\})}\) in an implicit manner:
$$ x_{t}=P_{C}\bigl[(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+\theta_{t}\bigl(t \gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t} \bigr)\bigr]. $$
(3.1)
We prove the strong convergence of
\(\{x_{t}\}\) as
\(t\to0\) to a point
\(\tilde{x}\in{ \varOmega }\) which is a unique solution to the VIP
$$ \bigl\langle (A-I)\tilde{x},p-\tilde{x}\bigr\rangle \geq0,\quad \forall p\in{ \varOmega }. $$
(3.2)
For arbitrarily given
\(x_{0}\in C\), we also propose the second composite relaxed extragradient-like scheme, which generates a sequence
\(\{x_{n}\}\) in an explicit way:
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=\alpha_{n}\gamma Vx_{n}+(I-\alpha_{n}\mu F)S{ \varDelta }^{N}_{n}Gx_{n},\\ x_{n+1}=P_{C}[(I-\beta_{n}A)S{ \varDelta }^{N}_{n}Gx_{n}+\beta_{n}y_{n}],\quad \forall n\geq0, \end{array}\displaystyle \right . $$
(3.3)
and establish the strong convergence of
\(\{x_{n}\}\) as
\(n\to\infty\) to the same point
\(\tilde{x}\in{ \varOmega }\), which is also the unique solution to VIP (
3.2).
Now, for
\(t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\), and
\(\theta_{t}\in(0,\|A\|^{-1}]\), consider a mapping
\(Q_{t} :C\to C\) defined by
$$Q_{t}x=P_{C}\bigl[(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx+\theta_{t}\bigl(t\gamma Vx+(I-t \mu F)S{ \varDelta }^{N}_{t}Gx\bigr)\bigr],\quad \forall x\in C. $$
It is easy to see that
\(Q_{t}\) is a contractive mapping with constant
\(1-\theta_{t}(\bar{\gamma}-1+t(\tau-\gamma l))\). Indeed, by Proposition
2.3 and Lemmas
2.7 and
2.9, we have
$$\begin{aligned} \bigl\| Q_{t}x-Q_{t}y\bigr\| \leq&\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx+\theta_{t}\bigl(t\gamma Vx+(I-t \mu F)S{ \varDelta }^{N}_{t}Gx\bigr) \\ &{} -(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gy- \theta_{t}\bigl(t\gamma Vx+(I-t\mu F)S{ \varDelta }^{N}_{t}Gy \bigr)\bigr\| \\ \leq&\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx-(I- \theta_{t}A)S{ \varDelta }^{N}_{t}Gy\bigr\| \\ &{} +\theta_{t}\bigl\| \bigl(t\gamma Vx+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx \bigr)-\bigl(t\gamma Vy+(I-t\mu F)S{ \varDelta }^{N}_{t}Gy\bigr) \bigr\| \\ \leq&(1-\theta_{t}\bar{\gamma})\bigl\| S{ \varDelta }^{N}_{t}Gx-S{ \varDelta }^{N}_{t}Gy\bigr\| +\theta_{t}\bigl[t\gamma \|Vx-Vy\| \\ &{} +\bigl\| (I-t\mu F)S{ \varDelta }^{N}_{t}Gx-(I-t\mu F)S{ \varDelta }^{N}_{t}Gy\bigr\| \bigr] \\ \leq&(1-\theta_{t}\bar{\gamma})\|x-y\|+\theta_{t}\bigl[t \gamma l\|x-y\|+(1-t\tau )\|x-y\|\bigr] \\ =&\bigl[1-\theta_{t}\bigl(\bar{\gamma}-1+t(\tau-\gamma l)\bigr)\bigr] \|x-y\|. \end{aligned}$$
Since
\(\bar{\gamma}\in(1,2)\),
\(\tau-\gamma l>0\), and
$$0< t< \min\biggl\{ 1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\biggr\} \leq\frac{2-\bar{\gamma}}{\tau-\gamma l}, $$
it follows that
$$0< \bar{\gamma}-1+t(\tau-\gamma l)< 1, $$
which together with
\(0<\theta_{t}\leq\|A\|^{-1}<1\) yields
$$0< 1-\theta_{t}\bigl(\bar{\gamma}-1+t(\tau-\gamma l)\bigr)< 1. $$
Hence
\(Q_{t}:C\to C\) is a contractive mapping. By the Banach contraction principle,
\(Q_{t}\) has a unique fixed point, denoted by
\(x_{t}\), which uniquely solves the fixed point equation (
3.1).
We summarize the basic properties of
\(\{x_{t}\}\). The argument techniques in [
1,
22,
32] extend to developing the new argument ones for these basic properties. We include the argument process for the sake of completeness.
We prove the following theorem for strong convergence of the net
\(\{ x_{t}\}\) as
\(t\to0\), which guarantees the existence of solutions of the variational inequality (
3.2).
Taking
\(F=\frac{1}{2}I\),
\(\mu=2\), and
\(\gamma=1\) in Theorem
3.1, we get
First, we prove the following result in order to establish the strong convergence of the sequence
\(\{x_{n}\}\) generated by the composite explicit relaxed extragradient-like scheme (
3.3).
Now, using Theorem
3.2, we establish the strong convergence of the sequence
\(\{x_{n}\}\) generated by the composite explicit relaxed extragradient-like scheme (
3.3) to a point
\(\tilde{x}\in{ \varOmega }\), which is also the unique solution of the VIP (
3.2).
Putting
\(\mu=2\),
\(F=\frac{1}{2}I\), and
\(\gamma=1\) in Theorem
3.3, we obtain the following.
Putting
\(\alpha_{n}=0\),
\(\forall n\geq0\) in Corollary
3.3, we get the following.
In view of this observation, we have the following.
4 Applications
Let
C be a nonempty, closed, and convex subset of a real Hilbert space
H. For a given nonlinear mapping
\({\mathcal{A}}:C\to H\), we consider the variational inequality problem (VIP) of finding
\(x^{*}\in C\) such that
$$ \bigl\langle {\mathcal{A}}x^{*},y-x^{*}\bigr\rangle \geq0, \quad\forall y\in C. $$
(4.1)
We will denote by
\(\operatorname{VI}(C,{\mathcal{A}})\) the set of solutions of the VIP (
4.1).
Recall that if
u is a point in
C, then the following relation holds:
$$u\in\operatorname{VI}(C,{\mathcal{A}}) \quad\Leftrightarrow\quad u=P_{C}(I- \lambda{\mathcal {A}})u,\quad \lambda>0. $$
In the meantime, it is easy to see that the following relation holds:
$$ \operatorname{SVI}\mbox{ (1.13) with }F_{2}=0 \quad \Leftrightarrow\quad \operatorname{VIP}\mbox{ (4.1) with } { \mathcal{A}}=F_{1}. $$
(4.2)
An operator
\({\mathcal{A}}:C\to H\) is said to be an
α-inverse strongly monotone operator if there exists a constant
\(\alpha>0\) such that
$$\langle{\mathcal{A}}x-{\mathcal{A}}y,x-y\rangle\geq\alpha\|{\mathcal {A}}x-{ \mathcal{A}}y\|^{2}, \quad\forall x,y\in C. $$
As an example, we recall that the
α-inverse strongly monotone operators are firmly nonexpansive mappings if
\(\alpha \geq1\) and that every
α-inverse strongly monotone operator is also
\(\frac{1}{\alpha}\)-Lipschitz-continuous (see [
17]).
Let us observe also that, if \({\mathcal{A}}\) is α-inverse strongly monotone, the mappings \(P_{C}(I-\lambda{\mathcal{A}})\) are nonexpansive for all \(\lambda\in(0,2\alpha]\) since they are compositions of nonexpansive mappings.
Throughout the rest of this paper, we always assume the following:
-
\(F:C\to H\) is a κ-Lipschitzian and η-strongly monotone operator with positive constants \(\kappa,\eta>0\);
-
\(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse strongly monotone for \(j=1,2\) and \(T_{i}:C\to C\) is a \(k_{i}\)-strictly pseudocontractive mapping for each \(i=1,\ldots,N\);
-
A is a γ̄-strongly positive bounded linear operator on H with \(\bar{\gamma}\in(1,2)\) and \(V:C\to H\) is an l-Lipschitzian mapping with \(l\geq0\);
-
\(0<\mu<\frac{2\eta}{\kappa^{2}}\) and \(0\leq\gamma l<\tau\) with \(\tau =1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\);
-
\(G:C\to C\) is a mapping defined by \(Gx=P_{C}(I-\nu_{1}F_{1})P_{C}(I-\nu _{2}F_{2})x\) with \(0<\nu_{j}<2\zeta_{j}\) for \(j=1,2\), and the fixed point set of G is denoted by Ξ;
-
\({ \varDelta }^{N}_{t}:C\to C\) is a mapping defined by \({ \varDelta }^{N}_{t}x=(I-r_{N,t}{\mathcal{A}}_{N})\cdots(I-r_{1,t}{\mathcal{A}}_{1})x\), \(t\in(0,1)\) with \({\mathcal{A}}_{i}=I-T_{i}\) and \(\{r_{i,t}\}\subset [a_{i},b_{i}]\subset(0,1-k_{i})\) for each \(i=1,\ldots,N\);
-
\({ \varDelta }^{N}_{n}:C\to C\) is a mapping defined by \({ \varDelta }^{N}_{n}x=(I-r_{N,n}{\mathcal{A}}_{N})\cdots(I-r_{1,n}{\mathcal{A}}_{1})x\) with \(\{r_{i,n}\}\subset[a_{i},b_{i}]\subset(0,1-k_{i})\) and \(\lim_{n\to \infty}r_{i,n}=r_{i}\), for each \(i=1,\ldots,N\);
-
\({ \varOmega }=\bigcap^{N}_{i=1}\operatorname{Fix}(T_{i})\cap{ \varXi }\neq\emptyset\) and \(P_{{ \varOmega }}\) is the metric projection of H onto Ω;
-
\(\{\alpha_{n}\}\subset[0,1]\), \(\{\beta_{n}\}\subset(0,1]\) and \(\{\theta_{t}\} _{t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})} \subset(0,1)\).
We now introduce the following composite implicit relaxed extragradient-like scheme that generates a net
\(\{x_{t}\}_{t\in(0,\min\{1, \frac{2-\bar{\gamma}}{\tau-\gamma l}\})}\) in one implicit manner:
$$ x_{t}=P_{C}\bigl[(I-\theta_{t}A){ \varDelta }^{N}_{t}Gx_{t}+\theta_{t}\bigl(t \gamma Vx_{t}+(I-t\mu F){ \varDelta }^{N}_{t}Gx_{t} \bigr)\bigr]. $$
(4.3)
Moreover, we also propose the following composite explicit relaxed extragradient-like scheme, which generates a sequence in another explicit way:
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=\alpha_{n}\gamma Vx_{n}+(I-\alpha_{n}\mu F){ \varDelta }^{N}_{n}Gx_{n},\\ x_{n+1}=P_{C}[(I-\beta_{n}A){ \varDelta }^{N}_{n}Gx_{n}+\beta_{n}y_{n}],\quad \forall n\geq0, \end{array}\displaystyle \right . $$
(4.4)
where
\(x_{0}\in C\) is an arbitrary initial guess.
Taking
\(F=\frac{1}{2}I\),
\(\mu=2\), and
\(\gamma=1\) in Theorem
4.1, we get
Next, by utilizing Theorem
3.2, we prove the following result in order to establish the strong convergence of the sequence
\(\{x_{n}\}\) generated by the composite explicit relaxed extragradient-like scheme (
4.4).
Now, using Theorem
4.2, we establish the strong convergence of the sequence
\(\{x_{n}\}\) generated by the composite explicit relaxed extragradient-like scheme (
4.4) to a point
\(\tilde{x}\in{ \varOmega }\), which is also the unique solution of the VIP (
4.5).
Putting
\(\mu=2\),
\(F=\frac{1}{2}I\), and
\(\gamma=1\) in Theorem
4.3, we obtain the following.
Putting
\(\alpha_{n}=0\),
\(\forall n\geq0\) in Corollary
4.3, we get the following.
In view of Remark
4.4, we have the following.
We introduced and analyzed one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEPs) with the constraints of a system of generalized equilibrium problems (SGEP) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping by virtue of the general composite implicit and explicit schemes for a nonexpansive mapping
\(T:H\to H\) (see [
32]) and the general composite implicit and explicit ones for a strict pseudocontraction
\(T:H\to H\) (see [
1]). Our Theorems
3.1-
3.3 and Corollary
3.5 improve, extend, supplement, and develop Theorems 3.1 and 3.2 of [
32], Theorems 3.1-3.3 and Corollary 3.5 of [
1] and Theorem 3.1 of [
23] in the following aspects.
(i) Ceng
et al.’s general composite implicit scheme for a nonexpansive mapping
\(T:H\to H\) (see (3.1) in [
32]) and Jung’s general composite implicit one for a strict pseudocontraction
\(T:H\to H\) (see (3.1) in [
1]) extends to developing the composite implicit relaxed extragradient-like scheme (
3.1) for a finite family of GMEPs with constraints of SGEP (
1.12) and the HFPP for a strict pseudocontraction. Moreover, Ceng
et al.’s general composite explicit scheme for a nonexpansive mapping
\(T:H\to H\) (see (3.5) in [
32]) and Jung’s general composite explicit one for a strict pseudocontraction (see (3.3) in [
1]) extends to developing the composite explicit relaxed extragradient-like one (
3.3) for a finite family of GMEPs with constraints of SGEP (
1.12) and the HFPP for a strict pseudocontraction.
(ii) The argument techniques in our Theorems
3.1-
3.3 and Corollary
3.5 are very different from those techniques in [
32] Theorems 3.1-3.2 and [
1] Theorems 3.1-3.3 and Corollary 3.5 because we make use of the properties of the resolvent
\(T^{({ \varTheta }, \varphi)}_{r}\) (see,
e.g., Proposition
2.2 and the argument of (
3.9), (
3.14), (
3.16), and (
3.20)), the ones of the strong positive bounded linear operators (see Lemma
2.9), the ones of the Banach limit LIM (see Lemma
2.10), the equivalence of the fixed point equation
\(x^{*}=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu _{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x^{*}\) to the SGEP (
1.12) for
\(\zeta_{j}\)-inverse strongly monotone mappings
\(F_{j}:C\to H\),
\(j=1,2\) (see Proposition
2.3) and the contractive coefficient estimates for the contractions
\(T^{\lambda}\) associating with nonexpansive mappings (see Lemma
2.7).
(iii) The problem of finding a common solution
\(\tilde{x}\in\bigcap^{N}_{i=1}\operatorname{GMEP}({ \varTheta }_{i},\varphi_{i},{\mathcal{A}}_{i}) \cap\operatorname{Fix}(T)\cap{ \varXi }\) of SGEP (
1.12), the fixed point problem of a
k-strict pseudocontraction
T and a finite family of GMEPs in our Theorems
3.1-
3.3 and Corollary
3.5 is more general and more flexible than the one of finding a fixed point of a nonexpansive mapping
\(T:H\to H\) in [
32] Theorems 3.1 and 3.2, the one of finding a fixed point of a strictly pseudocontractive mapping
\(T:H\to H\) in [
1] Theorems 3.1-3.3 and Corollary 3.5, and the one of finding a common solution of GMEP (
1.11), SGEP (
1.12), and the fixed point problem of a
k-strict pseudocontraction
T in [
17] Theorem 3.1. It is worth pointing out that the problem of finding
\(\tilde{x}\in(\bigcap^{N}_{i=1}\operatorname{GMEP}({ \varTheta }_{i},\varphi_{i},{\mathcal{A}}_{i}))\cap\operatorname{Fix}(T)\cap{ \varXi }\) extends the fixed point problems in [
1,
32] from the domain
H of the mapping
T to the domain
C for the one of finding
\(\tilde{x}\in(\bigcap^{N}_{i=1}\operatorname{GMEP}({ \varTheta }_{i},\varphi_{i},{\mathcal {A}}_{i}))\cap\operatorname{Fix}(T)\cap{ \varXi }\) and generalizes the fixed point problems in [
1,
32] to the setting of SGEP (
1.12) and a finite family of GMEPs. In the meantime, the problem of finding
\(\tilde{x} \in(\bigcap^{N}_{i=1}\operatorname{GMEP}({ \varTheta }_{i},\varphi_{i},{\mathcal {A}}_{i}))\cap\operatorname{Fix}(T)\cap{ \varXi }\) extends the problem of finding
\(\tilde{x} \in\operatorname{GMEP}({ \varTheta },\varphi, {\mathcal{A}})\cap\operatorname{Fix}(T)\cap{ \varXi }\) in [
23] from one GMEP to a finite family of GMEPs.
(iv) Our Theorems
3.1-
3.3 and Corollary
3.5 generalize [
32] Theorems 3.1 and 3.2 from a nonexpansive mapping
\(T:H\to H\) to a
k-strict pseudocontraction
\(T:C\to C\) and extend [
32] Theorems 3.1 and 3.2 to the setting of SGEP (
1.12) and a finite family of GMEPs. Moreover, Theorems
3.1-
3.3 and Corollary
3.5 generalize Theorems 3.1-3.3 and Corollary 3.5 of [
1] from a strict pseudocontraction
\(T:H\to H\) to the setting of SGEP (
1.12) and a finite family of GMEPs. In the meantime, the operators
\(T_{t}\) in the implicit scheme (
3.1) of Jung [
1] are replaced by the composite ones
\(S{ \varDelta }^{N}_{t}G\) in our implicit scheme (
3.1) and the operators
\(T_{n}\) in the explicit scheme (
3.3) of Jung [
1] are replaced by the composite ones
\(S{\varDelta }^{N}_{n}G\) in our explicit scheme (
3.3).
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.