1 Introduction
Let
H be a real Hilbert space with the inner product
\(\langle\cdot ,\cdot\rangle\) and the 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
\({\mathcal{A}}:C\to H\) is called
L-Lipschitz continuous if there exists a constant
\(L\geq0\) such that
$$\|{\mathcal{A}}x-{\mathcal{A}}y\|\leq L\|x-y\|,\quad \forall x,y\in C. $$
In particular, if
\(L=1\) then
\({\mathcal{A}}\) is called a nonexpansive mapping; if
\(L\in[0,1)\) then
\({\mathcal{A}}\) is called a contraction. A mapping
\(T:C\to C\) is called
ξ-strictly pseudocontractive if there exists a constant
\(\xi\in [0,1)\) such that
$$\|Tx-Ty\|^{2}\leq\|x-y\|^{2}+\xi\bigl\Vert (I-T)x-(I-T)y \bigr\Vert ^{2},\quad \forall x,y\in C. $$
In particular, if
\(\xi=0\), then
T is a nonexpansive mapping.
Let
\({\mathcal{A}}:C\to H\) be a nonlinear mapping on
C. We consider the following variational inequality problem (VIP): find a point
\(\bar{x}\in C\) such that
$$ \langle{\mathcal{A}}\bar{x},y-\bar{x}\rangle\geq0,\quad \forall y\in C. $$
(1.1)
The solution set of VIP (
1.1) is denoted by
\({\operatorname {VI}}(C,{\mathcal{A}})\).
VIP (
1.1) was first discussed by Lions [
1] and now it is well known. Variational inequalities have extensively been investigated; see the monographs [
2‐
6]. 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 the literature, the recent research work shows that variational inequalities like VIP (
1.1) cover several topics, for example, monotone inclusions, convex optimization and quadratic minimization over fixed point sets; see [
7‐
11] for more details.
In 1976, Korpelevich [
12] proposed an iterative algorithm for solving VIP (
1.1) in the Euclidean space
\({\mathbf{R}}^{n}\):
$$\left \{ \begin{array}{l} y_{n}=P_{C}(x_{n}-\tau{\mathcal{A}}x_{n}), \\ x_{n+1}=P_{C}(x_{n}-\tau{\mathcal{A}}y_{n}), \quad \forall n\geq0, \end{array} \right . $$
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 great attention given by many authors who improved it in various ways; see,
e.g., [
11,
13‐
21] and the references therein, to name but a few.
On the other hand, let
C and
Q be nonempty closed convex subsets of infinite-dimensional real Hilbert spaces
H and
\({\mathcal{H}}\), respectively. The split feasibility problem (SFP) is to find a point
\(x^{*}\) with the property
$$ x^{*}\in C\quad \mbox{and}\quad Ax^{*}\in Q, $$
(1.2)
where
\(A\in B(H,{\mathcal{H}})\) and
\(B(H,{\mathcal{H}})\) denotes the family of all bounded linear operators from
H to
\({\mathcal{H}}\). We denote by
Γ the solution set of the SFP.
In 1994, the SFP was first introduced by Censor and Elfving [
22], in finite-dimensional Hilbert spaces, for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. A number of image reconstruction problems can be formulated as the SFP; see,
e.g., [
23] and the references therein. Recently, it has been found that the SFP can also be applied to study intensity-modulated radiation therapy (IMRT); see,
e.g., [
24,
25] and the references therein. In the recent past, a wide variety of iterative methods have been used in signal processing and image reconstruction and for solving the SFP; see,
e.g., [
13,
15,
18,
19,
23‐
28] and the references therein. A seemingly more popular algorithm that solves the SFP is the
CQ algorithm of Byrne [
23,
27] which is found to be a gradient-projection method (GPM) in convex minimization. However, it remains a challenge how to implement the
CQ algorithm in the case where the projections
\(P_{C}\) and/or
\(P_{Q}\) fail to have closed-form expressions, though theoretically we can prove the (weak) convergence of the algorithm.
Very recently, Xu [
26] gave a continuation of the study on the
CQ algorithm and its convergence. He applied Mann’s algorithm to the SFP and proposed an averaged
CQ algorithm which was proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained.
Throughout this paper, assume that the SFP is consistent, that is, the solution set
Γ of the SFP is nonempty. Let
\(f:H\to{\mathbf{R}}\) be a continuous differentiable function. The minimization problem
$$\min_{x\in C}f(x):=\frac{1}{2}\|Ax-P_{Q}Ax \|^{2} $$
is ill-posed. Therefore, Xu [
26] considered the following Tikhonov regularization problem:
$$\min_{x\in C}f_{\alpha}(x):=\frac{1}{2} \|Ax-P_{Q}Ax\|^{2}+\frac{1}{2}\alpha \|x \|^{2}, $$
where
\(\alpha>0\) is the regularization parameter.
Very recently, by combining the gradient-projection method with regularization and extragradient method due to Nadezhkina and Takahashi [
14], Ceng
et al. [
19] proposed a Mann-type extragradient-like algorithm, and proved that the sequences generated by the proposed algorithm converge weakly to a common solution of SFP (
1.2) and the fixed point problem of a nonexpansive mapping.
In this paper, we consider the following general mixed equilibrium problem (GMEP) (see also [
29,
30]) of finding
\(x\in C\) such that
$$ {\varTheta }(x,y)+h(x,y)\geq0,\quad \forall y\in C, $$
(1.3)
where
\({\varTheta },h:C\times C\to{\mathbf{R}}\) are two bi-functions. We denote the set of solutions of GMEP (
1.3) by
\({\operatorname {GMEP}}({\varTheta },h)\). GMEP (
1.3) is very general; for example, it includes the following equilibrium problems as special cases.
As an example, in [
16,
31,
32] the authors considered and studied the generalized equilibrium problem (GEP) which is to find
\(x\in C\) such that
$${\varTheta }(x,y)+\langle{\mathcal{A}}x,y-x\rangle\geq0,\quad \forall y\in C. $$
The set of solutions of GEP is denoted by
\({\operatorname{GEP}}({\varTheta },{\mathcal{A}})\).
In [
29,
33,
34], 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 [
35‐
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 a 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
$$T_{r}x=\biggl\{ z\in C:{\varTheta }(z,y)+h(z,y)+\frac{1}{r}\langle y-z,z-x\rangle\geq 0,\forall y\in C\biggr\} $$
called the resolvent of
Θ and
h.
Assume that
C is the fixed point set of a nonexpansive mapping
\(T:H\to H\),
i.e.,
\(C={\operatorname{Fix}}(T)\). Let
\(F:H\to H\) be
η-strongly monotone and
κ-Lipschitzian with positive constants
\(\eta,\kappa>0\). Let
\(u_{0}\in H\) be given arbitrarily and
\(\{\lambda_{n}\}^{\infty}_{n=1}\) be a sequence in
\([0,1]\). The hybrid steepest-descent method introduced by Yamada [
39] is the algorithm
$$ u_{n+1}:=T^{\lambda_{n+1}}u_{n}=(I-\lambda_{n+1}\mu F)Tu_{n}, \quad \forall n\geq 0, $$
(1.4)
where
I is the identity mapping on
H.
In 2003, Xu and Kim [
40] proved the following strong convergence result.
Let
\(F_{1},F_{2}:C\to H\) be two mappings. Consider the following general system of variational inequalities (GSVI) of finding
\((x^{*},y^{*})\in C\times C\) such that
$$ \left \{ \begin{array}{l} \langle\nu_{1}F_{1}y^{*}+x^{*}-y^{*},x-x^{*}\rangle\geq0, \quad \forall x\in C, \\ \langle\nu_{2}F_{2}x^{*}+y^{*}-x^{*},x-y^{*}\rangle\geq0,\quad \forall x\in C, \end{array} \right . $$
(1.6)
where
\(\nu_{1}>0\) and
\(\nu_{2}>0\) are two constants. The solution set of GSVI (
1.6) is denoted by
\({\operatorname{GSVI}}(C,F_{1},F_{2})\).
In particular, if
\(F_{1}=F_{2}={\mathcal{A}}\), then the GSVI (
1.6) reduces to the following problem of finding
\((x^{*},y^{*})\in C\times C\) such that
$$\left \{ \begin{array}{l} \langle\nu_{1}{\mathcal{A}}y^{*}+x^{*}-y^{*},x-x^{*}\rangle\geq 0, \quad \forall x\in C, \\ \langle\nu_{2}{\mathcal{A}}x^{*}+y^{*}-x^{*},x-y^{*}\rangle\geq0, \quad \forall x\in C, \end{array} \right . $$
which is defined by Verma [
41] and it is called a new system of variational inequalities (NSVI). Further, if
\(x^{*}=y^{*}\) additionally, then the NSVI reduces to the classical VIP (
1.1). In 2008, Ceng
et al. [
21] transformed GSVI (
1.6) into the fixed point problem of the mapping
\(G=P_{C}(I-\nu _{1}F_{1})P_{C}(I-\nu_{2}F_{2})\), that is,
\(Gx^{*}=x^{*}\), where
\(y^{*}= P_{C}(I-\nu_{2}F_{2})x^{*}\). Throughout this paper, the fixed point set of the mapping
G is denoted by
Ξ.
On the other hand, if
C is the fixed point set
\({\operatorname {Fix}}(T)\) of a nonexpansive mapping
T and
S is another nonexpansive mapping (not necessarily with fixed points), then VIP (
1.1) becomes the variational inequality problem of finding
\(x^{*}\in {\operatorname{Fix}}(T)\) such that
$$ \bigl\langle (I-S)x^{*},x-x^{*}\bigr\rangle \geq0,\quad \forall x\in{\operatorname {Fix}}(T). $$
(1.7)
This problem, introduced by Mainge and Moudafi [
34,
36], is called the hierarchical fixed point problem. It is clear that if
S has fixed points, then they are solutions of VIP (
1.7).
If
S is a
ρ-contraction (
i.e.,
\(\|Sx-Sy\|\leq\rho\| x-y\|\) for some
\(0\leq\rho<1\)), the solution set of VIP (
1.7) is a singleton and it is well known as the viscosity problem. This was previously introduced by Moudafi [
7] and also developed by Xu [
8]. In this case, it is easy to see that solving VIP (
1.7) is equivalent to finding a fixed point of the nonexpansive mapping
\(P_{{\operatorname{Fix}}(T)}S\), where
\(P_{{\operatorname {Fix}}(T)}\) is the metric projection on the closed and convex set
\({\operatorname{Fix}}(T)\).
In 2012, Marino
et al. [
42] introduced a multi-step iterative scheme
$$ \left \{ \begin{array}{l} {\varTheta }(u_{n},y)+h(u_{n},y)+\frac{1}{r_{n}}\langle y-u_{n},u_{n}-x_{n}\rangle\geq0,\quad \forall y\in C, \\ y_{n,1}=\beta_{n,1}S_{1}u_{n}+(1-\beta_{n,1})u_{n}, \\ y_{n,i}=\beta_{n,i}S_{i}u_{n}+(1-\beta_{n,i})y_{n,i-1},\quad i=2,\ldots,N, \\ x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})Ty_{n,N}, \end{array} \right . $$
(1.8)
with
\(f:C\to C\) a
ρ-contraction and
\(\{\alpha_{n}\},\{\beta_{n,i}\} \subset(0,1)\),
\(\{r_{n}\}\subset(0,\infty)\), that generalizes the two-step iterative scheme in [
10] 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 (
1.8) converges strongly to a common fixed point of the mappings that is also an equilibrium point of GMEP (
1.3).
More recently, Marino, Muglia and Yao’s multi-step iterative scheme (
1.8) was extended to develop the following relaxed viscosity iterative algorithm.
The authors [
43] proved that the proposed scheme (
1.9) 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 GMEP (
1.3) and a solution of GSVI (
1.6).
In this paper, we introduce a hybrid extragradient viscosity iterative algorithm for finding a common element of the solution set
\({\operatorname{GMEP}}({\varTheta },h)\) of GMEP (
1.3), the solution set
\({\operatorname{GSVI}}(C,F_{1},F_{2})\) (
i.e.,
Ξ) of GSVI (
1.6), the solution set
Γ of SFP (
1.2), and the common fixed point set
\(\bigcap^{N}_{i=1}{\operatorname{Fix}}(S_{i})\cap {\operatorname{Fix}}(T)\) of finitely many nonexpansive mappings
\(S_{i}:C\to C\),
\(i=1,\ldots,N\), and a strictly pseudocontractive mapping
\(T:C\to C\), in the setting of the infinite-dimensional Hilbert space. The iterative algorithm is based on Korpelevich’s extragradient method, viscosity approximation method [
7] (see also [
8]), Mann’s iteration method, hybrid steepest-descent method [
40] and gradient-projection method (GPM) with regularization. Our aim is to prove that the iterative algorithm converges strongly to a common element of these sets, which also solves some hierarchical variational inequality. We observe that related results have been derived say in [
10,
13,
28,
34,
36,
37,
42‐
54].
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 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 be easily 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
\({\mathcal{A}}:C\to H\) is
α-inverse-strongly monotone, then
A is monotone and
\(\frac{1}{\alpha}\)-Lipschitz continuous. Moreover, we also have that, for all
\(u,v\in C\) and
\(\lambda>0\),
$$ \bigl\Vert (I-\lambda{\mathcal{A}})u-(I-\lambda{\mathcal{A}})v\bigr\Vert ^{2}\leq\|u-v\| ^{2}+\lambda(\lambda-2\alpha)\|{ \mathcal{A}}u-{\mathcal{A}}v\|^{2}. $$
(2.1)
So, if
\(\lambda\leq2\alpha\), then
\(I-\lambda{\mathcal{A}}\) is a nonexpansive mapping from
C to
H.
In 2008, Ceng
et al. [
21] transformed problem (
1.6) into a fixed point problem in the following way.
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 Ξ the fixed point set of the mapping G.
The following result is easy to prove.
It is clear from Proposition
2.1 that
$${\varGamma }={\operatorname{Fix}}\bigl(P_{C}(I-\lambda\nabla f)\bigr)={ \operatorname {VI}}(C,\nabla f),\quad \forall\lambda>0. $$
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
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:
$$\langle Tx-Ty,x-y\rangle\leq\|x-y\|^{2}-\frac{1-\xi}{2}\bigl\Vert (I-T)x-(I-T)y\bigr\Vert ^{2}, \quad \forall x,y\in C. $$
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 [
57] 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
C be a nonempty closed convex subset of a real Hilbert space
H. We introduce some notations. Let
λ be a number in
\((0,1]\) and let
\(\mu>0\). Associating with a nonexpansive mapping
\(T:C\to C\), we define the mapping
\(T^{\lambda}:C\to H\) by
$$T^{\lambda}x:=Tx-\lambda\mu F(Tx),\quad \forall x\in C, $$
where
\(F:C\to H\) is an operator such that, for some positive constants
\(\kappa,\eta>0\),
F is
κ-Lipschitzian and
η-strongly monotone on
C; that is,
F satisfies the conditions
$$\|Fx-Fy\|\leq\kappa\|x-y\| \quad \mbox{and} \quad \langle Fx-Fy,x-y\rangle\geq \eta\| x-y\|^{2} $$
for all
\(x,y\in C\).
In the sequel, we will indicate with
\({\operatorname{GMEP}}({\varTheta },h)\) the solution set of GMEP (
1.3).
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
$$\langle f-g,x-y\rangle\geq0. $$
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 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
\({\mathcal{A}}:C\to H\) be a monotone,
k-Lipschitz-continuous mapping, and let
\(N_{C}v\) be the normal cone to
C at
\(v\in C\),
i.e.,
$$N_{C}v=\bigl\{ u\in H:\langle v-p,u\rangle\geq0, \forall p\in C\bigr\} . $$
Define
$$\widetilde{T}v=\left \{ \begin{array}{l@{\quad}l} {\mathcal{A}}v+N_{C}v, &\mbox{if }v\in C, \\ \emptyset, &\mbox{if }v\notin C. \end{array} \right . $$
Then it is known in [
60] that
\(\widetilde{T}\) is maximal monotone and
$$ 0\in\widetilde{T}v \quad \Leftrightarrow\quad v\in{\operatorname{VI}}(C,{ \mathcal {A}}). $$
(2.2)
3 Main results
We now propose the following hybrid extragradient viscosity iterative scheme:
$$ \left \{ \begin{array}{l} {\varTheta }(u_{n},y)+h(u_{n},y)+\frac{1}{r_{n}}\langle y-u_{n},u_{n}-x_{n}\rangle\geq0,\quad \forall y\in C, \\ y_{n,1}=\beta_{n,1}S_{1}u_{n}+(1-\beta_{n,1})u_{n}, \\ y_{n,i}=\beta_{n,i}S_{i}u_{n}+(1-\beta_{n,i})y_{n,i-1}, \quad i=2,\ldots,N, \\ \tilde{y}_{n,N}=P_{C}(y_{n,N}-\lambda_{n}\nabla f_{\alpha_{n}}(y_{n,N})), \\ y_{n}=P_{C}[\epsilon_{n}\gamma Vy_{n,N}+(I-\epsilon_{n}\mu F)GP_{C}(y_{n,N}-\lambda_{n}\nabla f_{\alpha_{n}}(\tilde{y}_{n,N}))], \\ x_{n+1}=\beta_{n}y_{n}+\gamma_{n}P_{C}(y_{n,N}-\lambda_{n}\nabla f_{\alpha _{n}}(\tilde{y}_{n,N})) +\delta_{n}TP_{C}(y_{n,N}-\lambda_{n}\nabla f_{\alpha_{n}}(\tilde{y}_{n,N})) \end{array} \right . $$
(3.1)
for all
\(n\geq0\), where
-
\(F:C\to H\) is a κ-Lipschitzian and η-strongly monotone operator with positive constants \(\kappa,\eta>0\) and \(V:C\to C\) is an l-Lipschitzian mapping with constant \(l\geq0\);
-
\(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse strongly monotone and \(G:=P_{C}(I-\nu _{1}F_{1})P_{C}(I-\nu_{2}F_{2})\) with \(\nu_{j}\in(0,2\zeta_{j})\) for \(j=1,2\);
-
\(T:C\to C\) is a ξ-strict pseudocontraction and \(S_{i}:C\to C\) is a nonexpansive mapping for each \(i=1,\ldots,N\);
-
\({\varTheta },h:C\times C\to{\mathbf{R}}\) are two bi-functions satisfying the hypotheses of Lemma
2.9;
-
\(\{\lambda_{n}\}\) is a sequence in \((0,\frac{1}{\|A\|^{2}})\) with \(0<\liminf_{n\to\infty}\lambda_{n}\leq\limsup_{n\to\infty}\lambda_{n}<\frac {1}{\|A\|^{2}}\);
-
\(0<\mu<2\eta/\kappa^{2}\) and \(0\leq\gamma l<\tau\) with \(\tau:=1-\sqrt {1-\mu(2\eta-\mu\kappa^{2})}\);
-
\(\{\alpha_{n}\}\) is a sequence in \((0,\infty)\) with \(\sum^{\infty}_{n=0}\alpha_{n}<\infty\);
-
\(\{\epsilon_{n}\}\), \(\{\beta_{n}\}\) are sequences in \((0,1)\) with \(0<\liminf_{n\to\infty}\beta_{n}\leq\limsup_{n\to\infty}\beta_{n}<1\);
-
\(\{\gamma_{n}\}\), \(\{\delta_{n}\}\) are sequences in \([0,1]\) with \(\beta _{n}+\gamma_{n}+\delta_{n}=1\), \(\forall n\geq0\);
-
\(\{\beta_{n,i}\}^{N}_{i=1}\) are sequences in \((0,1)\) and \((\gamma _{n}+\delta_{n})\xi\leq\gamma_{n}\), \(\forall n\geq0\);
-
\(\{r_{n}\}\) is a sequence in \((0,\infty)\) with \(\liminf_{n\to\infty }r_{n}>0\) and \(\liminf_{n\to\infty}\delta_{n}>0\).
We start our main result from the following series of propositions.
Of course, if
\(\beta_{n,i}\to\beta_{i}\neq0\) as
\(n\to\infty\), for all indices
i, the assumptions of Proposition
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 proposition, we estimate the case in which at least one sequence
\(\{\beta_{n,k_{0}}\}\) is a null sequence.
In a similar way, we can conclude another theorem as follows.
4 Applications
For a given nonlinear mapping
\({\mathcal{A}}:C\to H\), we consider the variational inequality problem (VIP) of finding
\(\bar{x}\in C\) such that
$$ \langle{\mathcal{A}}\bar{x},y-\bar{x}\rangle\geq0, \quad \forall y\in C. $$
(4.1)
We will indicate with
\({\operatorname{VI}}(C,{\mathcal{A}})\) the set of solutions of VIP (
4.1).
Recall that if
u is a point in
C, then the following relation holds:
$$u\in{\operatorname{VI}}(C,{\mathcal{A}})\quad \Leftrightarrow \quad u=P_{C}(I-\lambda {\mathcal{A}})u,\quad \forall\lambda>0. $$
In the meantime, it is easy to see that the following relation holds:
$$ \mbox{GSVI (1.6) with }F_{2}=0\quad \Leftrightarrow\quad \mbox{VIP (4.1) with } {\mathcal{A}}=F_{1}. $$
(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 [
45]).
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 (see p.419 in [
45]).
Let us consider
\(\widetilde{S}_{1},\ldots,\widetilde{S}_{M}\) be a finite number of nonexpansive self-mappings on
C and
\(A_{1},\ldots,A_{N}\) be a finite number of
α-inverse strongly monotone operators. Let
\(T:C\to C\) be a
ξ-strict pseudocontraction with fixed points. Let us consider the following mixed problem of finding
\(x^{*}\in{\operatorname{Fix}}(T)\cap{\operatorname{GMEP}}({\varTheta },h)\cap {\varXi }\cap{ \varGamma }\) such that
$$ \left \{ \begin{array}{l} \langle(I-\widetilde{S}_{1})x^{*},y-x^{*}\rangle\geq0,\quad \forall y\in{\operatorname{Fix}}(T)\cap{\operatorname{GMEP}}({\varTheta },h)\cap {\varXi }\cap{ \varGamma }, \\ \langle(I-\widetilde{S}_{2})x^{*},y-x^{*}\rangle\geq0, \quad \forall y\in{\operatorname{Fix}}(T)\cap{\operatorname{GMEP}}({\varTheta },h)\cap {\varXi }\cap{ \varGamma }, \\ \ldots, \\ \langle(I-\widetilde{S}_{M})x^{*},y-x^{*}\rangle\geq0,\quad \forall y\in{\operatorname{Fix}}(T)\cap{\operatorname{GMEP}}({\varTheta },h)\cap {\varXi }\cap{ \varGamma }, \\ \langle A_{1}x^{*},y-x^{*}\rangle\geq0, \quad \forall y\in C, \\ \langle A_{2}x^{*},y-x^{*}\rangle\geq0,\quad \forall y\in C, \\ \ldots, \\ \langle A_{N}x^{*},y-x^{*}\rangle\geq0, \quad \forall y\in C. \end{array} \right . $$
(4.3)
Let us call (SVI) the set of solutions of the
\((M+N)\)-system. This problem is equivalent to finding a common fixed point of
T,
\(\{P_{{\operatorname{Fix}}(T)\cap{\operatorname {GMEP}}({\varTheta },h)\cap{ \varXi }\cap{ \varGamma }}\widetilde{S}_{i}\}^{M}_{i=1}\),
\(\{P_{C}(I-\lambda A_{i})\}^{N}_{i=1}\). So we claim that the following holds.
On the other hand, recall that a mapping
\(S:C\to C\) is called
ζ-strictly pseudocontractive if there exists a constant
\(\zeta\in[0,1)\) such that
$$\|Sx-Sy\|^{2}\leq\|x-y\|^{2}+\zeta\bigl\Vert (I-S)x-(I-S)y \bigr\Vert ^{2}, \quad \forall x,y\in C. $$
If
\(\zeta=0\), then
S is nonexpansive. Put
\({\mathcal{A}}=I-S\), where
\(S:C\to C\) is a
ζ-strictly pseudocontractive mapping. Then
\({\mathcal{A}}\) is
\(\frac{1-\zeta}{2}\)-inverse strongly monotone; see [
57].
Utilizing Theorems
3.1 and
3.2, we also give two strong convergence theorems for finding a common element of the solution set
\({\operatorname{GMEP}}({\varTheta },h)\) of GMEP (
1.3), the solution set
Γ of SFP (
1.2) and the common fixed point set
\(\bigcap^{N}_{i=1}{\operatorname{Fix}}(S_{i})\cap{\operatorname{Fix}}(S)\) of finitely many nonexpansive mappings
\(S_{i}:C\to C\),
\(i=1,\ldots,N\), and a
ζ-strictly pseudocontractive mapping
\(S:C\to C\).