1 Introduction
Let
H be a real Hilbert space with inner product
\(\langle\cdot,\cdot \rangle\) and norm
\(\|\cdot\|\),
C be a nonempty closed convex subset of
H and
\(P_{C}\) be the metric projection of
H onto
C. Let
\(S:C\to C\) be a self-mapping on
C. We denote by
\(\operatorname{Fix}(S)\) the set of fixed points of
S and by
R the set of all real numbers. A mapping
\(A:C\to H\) is called
α-inverse strongly monotone, if there exists a constant
\(\alpha>0\) such that
$$\langle Ax-Ay,x-y\rangle\geq\alpha\|Ax-Ay\|^{2}, \quad \forall x,y\in C. $$
A mapping
\(A:C\to H\) is called
L-Lipschitz continuous if there exists a constant
\(L>0\) such that
$$\|Ax-Ay\|\leq L\|x-y\|,\quad \forall x,y\in C. $$
In particular, if \(L=1\) then A is called a nonexpansive mapping; if \(L\in(0,1)\) then A is called a contraction.
Let
\(A:C\to H\) be a nonlinear mapping on
C. We consider the following variational inequality problem (VIP): find a point
\(x^{*}\in C\) such that
$$ \bigl\langle Ax^{*},x-x^{*}\bigr\rangle \geq0,\quad \forall x\in C. $$
(1.1)
The solution set of VIP (
1.1) is denoted by
\(\operatorname{VI}(C,A)\).
The VIP (
1.1) was first discussed by Lions [
1] and now is well known; there are a lot of different approaches toward solving VIP (
1.1) in finite-dimensional and infinite-dimensional spaces, and the research is intensively continued. The VIP (
1.1) has many applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and other fields; see,
e.g., [
2‐
5]. 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. Not only the existence and uniqueness of solutions are important topics in the study of VIP (
1.1), but also how to actually find a solution of VIP (
1.1) is important.
In 1976, Korpelevich [
6] proposed an iterative algorithm for solving the VIP (
1.1) in Euclidean space
\({\mathbf{R}}^{n}\):
$$\left \{ \textstyle\begin{array}{l} y_{n}=P_{C}(x_{n}-\tau Ax_{n}), \\ x_{n+1}=P_{C}(x_{n}-\tau Ay_{n}), \quad \forall n\geq0, \end{array}\displaystyle \right . $$
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 much attention by many authors, who improved it in various ways; see,
e.g., [
7‐
24] and references therein, to name but a few. In particular, motivated by the idea of Korpelevich’s extragradient method [
6], Nadezhkina and Takahashi [
11] introduced an extragradient iterative scheme:
$$ \left \{ \textstyle\begin{array}{l} x_{0}=x\in C\quad \mbox{chosen arbitrary}, \\ y_{n}=P_{C}(x_{n}-\lambda_{n}Ax_{n}), \\ x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})SP_{C}(x_{n}-\lambda_{n}Ay_{n}),\quad \forall n\geq0, \end{array}\displaystyle \right . $$
(1.2)
where
\(A:C\to H\) is a monotone,
L-Lipschitz continuous mapping,
\(S:C\to C\) is a nonexpansive mapping and
\(\{\lambda_{n}\}\subset[a,b]\) for some
\(a,b\in(0,1/L)\) and
\(\{\alpha_{n}\}\subset [c,d]\) for some
\(c,d\in(0,1)\). They proved the weak convergence of
\(\{x_{n}\}\) generated by (
1.2) to an element of
\(\operatorname{Fix}(S)\cap\operatorname{VI}(C,A)\). Subsequently, given a contractive mapping
\(f:C\to C\), an
α-inverse strongly monotone mapping
\(A:C\to H\) and a nonexpansive mapping
\(T: C\to C\), Jung ([
25], Theorem 3.1) introduced the following two-step iterative scheme by the viscosity approximation method:
$$ \left \{ \textstyle\begin{array}{l} x_{0}=x\in C\quad \mbox{chosen arbitrary}, \\ y_{n}=\alpha_{n}f(x_{n})+(1-\alpha_{n})TP_{C}(x_{n}-\lambda_{n}Ax_{n}), \\ x_{n+1}=(1-\beta_{n})y_{n}+\beta_{n}TP_{C}(y_{n}-\lambda_{n}Ay_{n}), \quad \forall n\geq0, \end{array}\displaystyle \right . $$
(1.3)
where
\(\{\lambda_{n}\}\subset(0,2\alpha)\) and
\(\{\alpha_{n}\},\{\beta_{n}\} \subset[0,1)\). It was proven in [
25] that, if
\(\operatorname{Fix} (T)\cap\operatorname{VI}(C,A)\neq\emptyset\), then the sequence
\(\{x_{n}\}\) generated by (
1.3) converges strongly to
\(q=P_{\operatorname{Fix}(T) \cap\operatorname{VI}(C,A)}f(q)\).
On the other hand, we consider the general mixed equilibrium problem (GMEP) (see also [
26,
27]) of finding
\(x \in C\) such that
$$ {\varTheta }(x,y)+h(x,y)\geq0,\quad \forall y\in C, $$
(1.4)
where
\({\varTheta },h:C\times C\to{\mathbf{R}}\) are two bi-functions. The GMEP (
1.4) has been considered and studied by many authors; see,
e.g., [
28‐
30]. We denote the set of solutions of GMEP (
1.4) by
\(\operatorname{GMEP}({\varTheta },h)\). The GMEP (
1.4) is very general, for example, it includes the following equilibrium problems as special cases.
As an example, in [
14,
15,
31], the authors considered and studied the generalized equilibrium problem (GEP) which is to find
\(x\in C\) such that
$${\varTheta }(x,y)+\langle Ax,y-x\rangle\geq0,\quad \forall y\in C. $$
The set of solutions of GEP is denoted by
\(\operatorname{GEP}({\varTheta },A)\).
In [
22,
26,
32,
33], the authors considered and studied the mixed equilibrium problem (MEP) which is to find
\(x\in C\) such that
$${\varTheta }(x,y)+\varphi(y)-\varphi(x)\geq0, \quad \forall y\in C. $$
The set of solutions of MEP is denoted by
\(\operatorname{MEP}({\varTheta },\varphi)\).
In [
34‐
37], the authors considered and studied the equilibrium problem (EP) which is to find
\(x\in C\) such that
$${\varTheta }(x,y)\geq0,\quad \forall y\in C. $$
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.
Throughout this paper, it is assumed as in [
38] that
\({\varTheta }:C\times C\to{\mathbf{R}}\) is a bi-function satisfying conditions (
θ1)-(
θ3) and
\(h:C\times C\to{\mathbf{R}}\) is a bi-function with restrictions (h1)-(h3), where
(θ1)
\({\varTheta }(x,x)=0\) for all \(x\in C\);
(θ2)
Θ is monotone (
i.e.,
\({\varTheta }(x,y)+{\varTheta }(y,x)\leq0\),
\(\forall x,y\in C\)) and upper hemicontinuous in the first variable,
i.e., for each
\(x,y,z\in C\),
$$\limsup_{t\to0^{+}}{\varTheta }\bigl(tz+(1-t)x,y\bigr)\leq{ \varTheta }(x,y); $$
(θ3)
Θ is lower semicontinuous and convex in the second variable;
(h1)
\(h(x,x)=0\) for all \(x\in C\);
(h2)
h is monotone and weakly upper semicontinuous in the first variable;
(h3)
h is convex in the second variable.
For
\(r>0\) and
\(x\in H\), let
\(T_{r}:H\to2^{C}\) be a mapping defined by
$$T_{r}x=\biggl\{ z\in C:{\varTheta }(z,y)+h(z,y)+\frac{1}{r}\langle y-z,z-x\rangle \geq0,\forall y\in C\biggr\} $$
called the resolvent of
Θ and
h.
In 2012, Marino
et al. [
30] introduced a multi-step iterative scheme
$$ \left \{ \textstyle\begin{array}{l} {\varTheta }(u_{n},y)+h(u_{n},y)+\frac{1}{r_{n}}\langle y-u_{n},u_{n}-x_{n}\rangle\geq0, \quad \forall y\in C, \\ y_{n,1}=\beta_{n,1}S_{1}u_{n}+(1-\beta_{n,1})u_{n}, \\ y_{n,i}=\beta_{n,i}S_{i}u_{n}+(1-\beta_{n,i})y_{n,i-1},\quad i=2,\ldots,N, \\ x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})Ty_{n,N}, \end{array}\displaystyle \right . $$
(1.5)
with
\(f:C\to C\) a
ρ-contraction and
\(\{\alpha_{n}\},\{\beta_{n,i}\} \subset(0,1)\),
\(\{r_{n}\}\subset(0,\infty)\), which generalizes the two-step iterative scheme in [
39] for two nonexpansive mappings to a finite family of nonexpansive mappings
\(T,S_{i}:C\to C\),
\(i=1,\ldots,N\), and proved that the proposed scheme (
1.5) converges strongly to a common fixed point of the mappings that is also an equilibrium point of the GMEP (
1.4).
More recently, Marino
et al.’s multi-step iterative scheme (
1.5) was extended to develop the following composite viscosity iterative algorithm by virtue of Jung’s two-step iterative scheme (
1.3).
It was proven in [
29] that the proposed scheme (
1.6) converges strongly to a common fixed point of the mappings
\(T,S_{i}:C\to C\),
\(i=1,\ldots,N\), that is also an equilibrium point of the GMEP (
1.4) and a solution of the VIP (
1.1).
In this paper, we introduce a new composite viscosity iterative algorithm for finding a common element of the solution set
\(\operatorname{GMEP}({\varTheta },h)\) of GMEP (
1.4), the solution set
\(\bigcap^{M}_{k=1}\operatorname{VI}(C,A_{k})\) of a finite family of variational inequalities for inverse strongly monotone mappings
\(A_{k}:C\to H\),
\(k=1,\ldots,M\), and the common fixed point set
\(\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\bigcap^{\infty}_{n=1}\operatorname{Fix}(T_{n})\) of one finite family of nonexpansive mappings
\(S_{i}:C\to C\),
\(i=1,\ldots,N\), and another infinite family of nonexpansive mappings
\(T_{n}:C\to C\),
\(n=1,2,\ldots \) , in the setting of the infinite-dimensional Hilbert space. The iterative algorithm is based on viscosity approximation method [
40] (see also [
41]), Mann’s iterative method, Korpelevich’s extragradient method and the
W-mapping approach to common fixed points of finitely many nonexpansive mappings. Our aim is to prove that the iterative algorithm converges strongly to a common fixed point of the mappings
\(S_{i},T_{n}:C\to C\),
\(i=1,\ldots,N\),
\(n=1,2,\ldots \) , which is also an equilibrium point of GMEP (
1.4) and a solution of a finite family of variational inequalities for inverse strongly monotone mappings
\(A_{k}:C\to H\),
\(k=1,\ldots,M\).
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
\(\{x_{n}\}\) and
\(\omega_{s}(x_{n})\) to denote the strong
ω-limit set of the sequence
\(\{x_{n}\}\),
i.e.,
$$\omega_{w}(x_{n}):=\bigl\{ x\in H:x_{n_{i}} \rightharpoonup x \mbox{ for some subsequence }\{x_{n_{i}}\} \mbox{ of } \{x_{n}\}\bigr\} $$
and
$$\omega_{s}(x_{n}):=\bigl\{ x\in H:x_{n_{i}}\to x \mbox{ for some subsequence }\{ x_{n_{i}}\} \mbox{ of } \{x_{n}\} \bigr\} . $$
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
A is
η-inverse strongly monotone, then
A is monotone and
\(\frac{1}{\eta} \)-Lipschitz continuous. Moreover, we also have, for all
\(u,v\in C\) and
\(\lambda>0\),
$$\begin{aligned} \bigl\Vert (I-\lambda A)u-(I-\lambda A)v\bigr\Vert ^{2}&=\bigl\Vert (u-v)-\lambda (Au-Av)\bigr\Vert ^{2} \\ &=\|u-v\|^{2}-2\lambda\langle Au-Av,u-v\rangle+\lambda^{2} \|Au-Av\|^{2} \\ &\leq\|u-v\|^{2}+\lambda(\lambda-2\eta)\|Au-Av\|^{2}. \end{aligned}$$
(2.1)
So, if
\(\lambda\leq2\eta\), then
\(I-\lambda A\) is a nonexpansive mapping from
C to
H.
We need some facts and tools in a real Hilbert space H, which are listed as lemmas below.
Let
\(\{T_{n}\}^{\infty}_{n=1}\) be an infinite family of nonexpansive self-mappings on
C and
\(\{\lambda_{n}\}^{\infty}_{n=1}\) be a sequence of nonnegative numbers in
\([0,1]\). For any
\(n\geq1\), define a mapping
\(W_{n}\) on
C as follows:
$$ \left \{ \textstyle\begin{array}{l} U_{n,n+1}=I, \\ U_{n,n}=\lambda_{n}T_{n}U_{n,n+1}+(1-\lambda_{n})I, \\ U_{n,n-1}=\lambda_{n-1}T_{n-1}U_{n,n}+(1-\lambda_{n-1})I, \\ \ldots, \\ U_{n,k}=\lambda_{k}T_{k}U_{n,k+1}+(1-\lambda_{k})I, \\ U_{n,k-1}=\lambda_{k-1}T_{k-1}U_{n,k}+(1-\lambda_{k-1})I, \\ \ldots, \\ U_{n,2}=\lambda_{2}T_{2}U_{n,3}+(1-\lambda_{2})I, \\ W_{n}=U_{n,1}=\lambda_{1}T_{1}U_{n,2}+(1-\lambda_{1})I. \end{array}\displaystyle \right . $$
(2.2)
Such a mapping
\(W_{n}\) is called the
W-mapping generated by
\(T_{n},T_{n-1},\ldots,T_{1}\) and
\(\lambda_{n},\lambda_{n-1}, \ldots,\lambda_{1}\).
In the sequel, we will denote by
\(\operatorname{GMEP}({\varTheta },h)\) the solution set of GMEP (
1.4).
Finally, recall that a set-valued mapping
\(\widetilde{T}:H\to2^{H}\) is called monotone if for all
\(x,y\in H\),
\(f\in\widetilde{T}x\) and
\(g\in \widetilde{T}y\) imply
\(\langle x-y,f-g\rangle\geq0\). A monotone mapping
\(\widetilde{T}:H\to2^{H}\) is maximal if its graph
\(G(\widetilde{T})\) is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping
\(\widetilde{T}\) is maximal if and only if for
\((x,f)\in H\times H\),
\(\langle x-y,f-g\rangle\geq0\) for all
\((y,g)\in G(\widetilde{T})\) implies
\(f\in \widetilde{T}x\). Let
\(A:C\to H\) be a monotone,
L-Lipschitz continuous mapping and let
\(N_{C}v\) be the normal cone to
C at
\(v\in C\),
i.e.,
\(N_{C}v=\{w\in H:\langle v-u,w\rangle\geq0, \forall u\in C\}\). Define
$$\widetilde{T}v=\left \{ \textstyle\begin{array}{l@{\quad}l} Av+N_{C}v, &\mbox{if }v\in C, \\ \emptyset, & \mbox{if }v\notin C. \end{array}\displaystyle \right . $$
It is well known [
45] that in this case
\(\widetilde{T}\) is maximal monotone, and
$$ 0\in\widetilde{T}v \quad \Leftrightarrow\quad v\in\operatorname{VI}(C,A). $$
(2.3)
3 Main results
Let
\(M,N\geq1\) be two integers and let us consider the following new composite viscosity iterative scheme:
$$ \left \{ \textstyle\begin{array}{l} {\varTheta }(u_{n},y)+h(u_{n},y)+\frac{1}{r_{n}}\langle y-u_{n},u_{n}-x_{n}\rangle\geq0,\quad \forall y\in C, \\ y_{n,1}=\beta_{n,1}S_{1}u_{n}+(1-\beta_{n,1})u_{n}, \\ y_{n,i}=\beta_{n,i}S_{i}u_{n}+(1-\beta_{n,i})y_{n,i-1}, \quad i=2,\ldots,N, \\ y_{n}=\alpha_{n}f(y_{n,N})+(1-\alpha_{n})W_{n}{\varLambda }^{M}_{n}y_{n,N}, \\ x_{n+1}=(1-\beta_{n})y_{n}+\beta_{n}W_{n}{\varLambda }^{M}_{n}y_{n},\quad \forall n\geq1, \end{array}\displaystyle \right . $$
(3.1)
where
-
the mapping \(f:C\to C\) is an ρ-contraction;
-
\(A_{k}:C\to H\) is \(\eta_{k}\)-inverse strongly monotone for each \(k=1,\ldots,M\);
-
\(S_{i},T_{n}:C\to C\) are nonexpansive mappings for each \(i=1,\ldots,N\) and \(n=1,2,\ldots \) ;
-
\(\{\lambda_{n}\}\) is a sequence in
\((0,b]\) for some
\(b\in(0,1)\) and
\(W_{n}\) is the
W-mapping defined by (
2.2);
-
\({\varTheta },h:C\times C\to{\mathbf{R}}\) are two bi-functions satisfying the hypotheses of Lemma
2.9;
-
\(\{\lambda_{k,n}\}\subset[a_{k},b_{k}]\subset(0,2\eta_{k})\), \(\forall k\in\{ 1,\ldots,M\}\), and \({\varLambda }^{M}_{n}:=P_{C}(I-\lambda_{M,n}A_{M})\cdots P_{C}(I-\lambda_{1,n}A_{1})\);
-
\(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) are sequences in \((0,1)\) with \(0<\liminf_{n\to\infty}\beta_{n}\leq\limsup_{n\to\infty}\beta_{n}<1\);
-
\(\{\beta_{n,i}\}^{N}_{i=1}\) are sequences in \((0,1)\) and \(\{r_{n}\}\) is a sequence in \((0,\infty)\) with \(\liminf_{n\to\infty}r_{n}>0\).
Of course, if
\(\beta_{n,i}\to\beta_{i}\neq0\) as
\(n\to\infty\), for all indices
i, the assumptions of Lemma
3.2 are enough to assure that
$$\lim_{n\to\infty}\frac{\|x_{n+1}-x_{n}\|}{\beta_{n,i}}=0,\quad \forall i\in \{1,\ldots,N \}. $$
In the next lemma, we estimate the case in which at least one sequence
\(\{\beta_{n,k_{0}}\}\) is a null sequence.
In the following, we provide a numerical example to illustrate how our main theorem, Theorem
3.1, works.
In a similar way, we can conclude to another theorem as follows.
4 Applications
For a given nonlinear mapping
\(A:C\to H\), we consider the variational inequality problem (VIP) of finding
\(\bar{x}\in C\) such that
$$ \langle A\bar{x},y-\bar{x}\rangle\geq0,\quad \forall y\in C. $$
(4.1)
We will denote by
\(\operatorname{VI}(C,A)\) the set of solutions of the VIP (
4.1).
Recall that if
u is a point in
C, then the following relation holds:
$$ u\in\operatorname{VI}(C,A) \quad \Leftrightarrow\quad u=P_{C}(I- \lambda A)u, \quad \forall \lambda>0. $$
(4.2)
An operator
\(A:C\to H\) is said to be an
α-inverse strongly monotone operator if there exists a constant
\(\alpha>0\) such that
$$\langle Ax-Ay,x-y\rangle\geq\alpha\|Ax-Ay\|^{2},\quad \forall x,y\in C. $$
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 [
46]).
Let us observe also that, if
A is
α-inverse strongly monotone, the mappings
\(P_{C}(I-\lambda A)\) are nonexpansive for all
\(\lambda\in(0,2\alpha]\) since they are compositions of nonexpansive mappings (see p.419 in [
46]).
Let us consider
\(\widetilde{S}_{1},\ldots,\widetilde{S}_{K}\) a finite number of nonexpansive self-mappings on
C and
\(\widetilde{A}_{1},\ldots,\widetilde{A}_{N}\) be a finite number of
α-inverse strongly monotone operators. Let
\(\{T_{n}\}^{\infty}_{n=1}\) be a sequence of nonexpansive self-mappings on
C. Let us consider the mixed problem of finding
\(x^{*}\in\bigcap^{\infty}_{n=1}\operatorname{Fix}(T_{n})\cap\operatorname{GMEP} ({\varTheta },h)\cap\bigcap^{M}_{k=1}\operatorname{VI}(C,A_{k})\) such that
$$ \left \{ \textstyle\begin{array}{l} \langle(I-\widetilde{S}_{1})x^{*},y-x^{*}\rangle\geq0,\quad \forall y\in\bigcap^{\infty}_{n=1}\operatorname{Fix}(T_{n})\cap\operatorname{GMEP}({\varTheta },h)\cap \bigcap^{M}_{k=1}\operatorname{VI}(C,A_{k}), \\ \langle(I-\widetilde{S}_{2})x^{*},y-x^{*}\rangle\geq0, \quad \forall y\in\bigcap^{\infty}_{n=1}\operatorname{Fix}(T_{n})\cap\operatorname{GMEP}({\varTheta },h)\cap \bigcap^{M}_{k=1}\operatorname{VI}(C,A_{k}), \\ \ldots, \\ \langle(I-\widetilde{S}_{K})x^{*},y-x^{*}\rangle\geq0,\quad \forall y\in\bigcap^{\infty}_{n=1}\operatorname{Fix}(T_{n})\cap\operatorname{GMEP}({\varTheta },h)\cap \bigcap^{M}_{k=1}\operatorname{VI}(C,A_{k}), \\ \langle\widetilde{A}_{1}x^{*},y-x^{*}\rangle\geq0,\quad \forall y\in C, \\ \langle\widetilde{A}_{2}x^{*},y-x^{*}\rangle\geq0,\quad \forall y\in C, \\ \ldots, \\ \langle\widetilde{A}_{N}x^{*},y-x^{*}\rangle\geq0, \quad \forall y\in C. \end{array}\displaystyle \right . $$
(4.3)
Let us call
\((\mathrm{SVI})\) the set of solutions of the
\((K+N)\)-system. This problem is equivalent to finding a common fixed point of
\(\{T_{n}\}^{\infty}_{n=1}\),
\(\{P_{\bigcap^{\infty}_{n=1}\operatorname{Fix}(T_{n})\cap\operatorname{GMEP}({\varTheta },h)\cap\bigcap^{M}_{k=1}\operatorname{VI}(C,A_{k})}\widetilde{S}_{i}\}^{K}_{i=1}, \{P_{C}(I-\lambda\widetilde{A}_{i})\}^{N}_{i=1}\). So we claim that the following holds.
On the other hand, recall that a mapping
\({\varGamma }:C\to C\) is called
κ-strictly pseudocontractive if there exists a constant
\(\kappa\in[0,1)\) such that
$$\|{\varGamma }x-{\varGamma }y\|^{2}\leq\|x-y\|^{2}+\kappa\bigl\Vert (I-{\varGamma })x-(I-{\varGamma })y\bigr\Vert ^{2},\quad \forall x,y \in C. $$
If
\(\kappa=0\), then
Γ is nonexpansive. Put
\(A=I-{\varGamma }\), where
\({\varGamma }:C\to C\) is a
κ-strictly pseudocontractive mapping. Then
A is
\(\frac{1-\kappa}{2}\)-inverse strongly monotone; see [
25].
Utilizing Theorems
3.1 and
3.2, we first give the following strong convergence theorems for finding a common element of the solution set
\(\operatorname{GMEP}({\varTheta },h)\) of GMEP (
1.4) and the common fixed point set
\(\bigcap^{\infty}_{n=1}\operatorname{Fix}(T_{n})\cap \bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\bigcap^{M}_{k=1}\operatorname{Fix}({\varGamma }_{k})\) of a finite family of
\(\kappa_{k}\)-strictly pseudocontractive mappings
\(\{{\varGamma }_{k}\}^{M}_{k=1}\), one finite family of nonexpansive mappings
\(\{S_{i}\}^{N}_{i=1}\), and another infinite family of nonexpansive mappings
\(\{T_{n}\}^{\infty}_{n=1}\).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.