Abstract
Let X be a uniformly convex and q-uniformly smooth Banach space with \(1< q\leq 2\). In the framework of this space, we are concerned with a composite gradient-like implicit rule for solving a hierarchical monotone variational inequality with the constraints of a system of monotone variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonlinear operators \(\{S_{n}\}^{\infty }_{n=0}\). Our rule is based on the Korpelevich extragradient method, the perturbation mapping, and the W-mappings constructed by \(\{S_{n}\}^{\infty }_{n=0}\).
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1 Introduction
Throughout this work, one always supposes that C is a nonempty convex set in a Banach space X whose dual is denoted by \(X^{*}\). One denotes by the same notation, \(\|\cdot \|\), the norms of X and \(X^{*}\). A common problem in machine learning, automatic control, and utility-based bandwidth allocation problems consists of finding a solution of some equation satisfying some constraints. This common problem is called the convex feasibility problem, which can be characterized via the following model: \(x\in \bigcap_{i\in I}C_{i}\), where I denotes some index set, \(C_{i}\) is a convex set in X.
Next, one employs \(J_{q}:X\to 2^{X^{*}}\), where \(q>1\) is real number, to denote the duality mapping, which is defined by \(J_{q}(x):=\{ \phi \in X^{*}:\langle x,\phi \rangle =\|x\| ^{q},\|x\|^{q-1}=\| \phi \|\}\), \(\forall x\in X\). Let \(A_{1},A_{2}:C\to X\) be two nonlinear non-self mappings. Consider the problem of finding \((x^{*},y^{*}) \in C\times C\) such that
with two positive real constants \(\mu _{1}\) and \(\mu _{2}\). This is called a system of generalized variational inequalities (SGVIs). This is a natural extension of the generalized variational inequality considered by Aoyama, Iiduka and Takahashi [1] in uniformly convex and 2-uniformly smooth Banach spaces; see [1] for more details. In Hilbert spaces, the system is reduced to the system of variational inequalities considered by Ceng et al. [2]. Problem (1.1) and its special cases are now under the spotlight of research because of their connections to other real convex and set optimization problems; see, e.g., [3–8] and the references therein. Recently, a fixed point method has been studied for solving convex and non-convex optimization problems since the equivalence between fixed point problems and zero point problems; see, e.g., [9–13] and the references therein. Indeed, one can transfer zero point problems (inclusion problems) to some fixed point problem of nonexpansive operators. The core is the resolvent of original operators. For example, one can show that the resolvent operator of m-accretive or maximally accretive operators is nonexpansive. Hence, Mann-like algorithms are applicable, however, they are only weakly convergent. Strong convergence is desirable in lots of situations, such as, image recovery, optimal control and quantum physics since they are in infinite-dimensional spaces. In this paper, we study, in the framework of Banach spaces, a convex feasibility problem with the constraints of the generalized system of monotone variational inequalities, a variational inclusion and a countable family of nonexpansive operators. Strong convergence theorems are obtained without any compact assumption on operators. Our rule is based on the Korpelevich extragradient method, the perturbation mapping, and the W-mappings constructed by \(\{S_{n}\}^{\infty }_{n=0}\). The main results extend and improve some recent results in [14–17].
2 Preliminaries
Next, one uses \(\rho _{X}:[0,\infty )\to [0,\infty )\) to stand for the smoothness modulus of space X which is defined by \(\rho _{X}(t)= \sup \{(\|x+y\|+\|x-y\|) /2-1:x\in U, \|y\|\leq t\}\). One says that X is uniformly smooth if \(\lim_{t\to 0^{+}}\rho _{X}(t)/t=0\). Let \(q\in (1,2]\) be a fixed real number. A Banach space X is said to be q-uniformly smooth if \(\rho _{X}(t)\leq t^{q}d\), \(\forall t>0\), where d is some constant. It is well known that Hilbert spaces, \(L^{p}\) and \(\ell _{p}\) are uniformly smooth where \(p>1\). More precisely, each Hilbert space is 2-uniformly smooth, while \(L^{p}\) and \(\ell _{p}\) are \(\min \{p,2\}\)-uniformly smooth for each \(p>1\).
Let \(A:C\to 2^{X}\) be a set-valued operator with \(Ax\neq \emptyset \), \(\forall x\in C\). An operator A is said to be accretive if, \(\forall x,y\in C\), \(\langle u-v,j_{q}(x-y)\rangle \geq 0\), \(\forall u\in Ax\), \(v\in Ay\), where \(j_{q}(x-y)\in J_{q}(x-y)\). A single-valued accretive operator A is said to be α-inverse-strongly accretive of order q if, \(\forall x,y \in C\), there exist \(\alpha >0\) and \(j_{q}(x-y)\in J_{q}(x-y)\) such that \(\langle u-v,j_{q}(x-y)\rangle \geq \alpha \|Ax-Ay\|^{q}\), \(\forall u \in Ax\), \(v\in Ay\). Back to Hilbert spaces, A is called the inverse-strongly monotone. This class of mappings is a key component in projection-based approximation methods; see, e.g., [18–22]. An accretive operator A is said to be m-accretive if and only if A is accretive and satisfies the range condition: \((I+\lambda A)C=X\) for all \(\lambda >0\). For an accretive operator A, we define the mapping \(J^{A}_{\lambda }:(I+\lambda A)C \to C\) by \(J^{A}_{\lambda }= (I+\lambda A)^{-1}\) for each \(\lambda >0\). Such \(J^{A}_{\lambda }\) is called the resolvent of A; see, e.g., [23–25] and the references therein. Recall now that a single-valued mapping \(F:C\to X\) is called η-strongly accretive if \(\langle Fx-Fy,j(x-y)\rangle \geq \eta \|x-y\|^{2}\) for some \(\eta \in (0,1)\) and \(j(x-y)\in J(x-y)\). Moreover, F is called ξ-strictly pseudocontractive if, \(\forall x,y\in C\), \(\langle Fx-Fy,j(x-y) \rangle \leq \|x-y\|^{2}-\xi \|x-y-(Fx-Fy)\|^{2}\) for some \(\xi \in (0,1)\), where \(j(x-y)\in J(x-y)\).
Let \(F:C\to X\) be a mapping. Then (i) if \(F:C\to X\) is η-strongly accretive and ξ-strictly pseudocontractive with \(\eta +\xi \geq 1\), then \(I-F\) is nonexpansive, and F is Lipschitz continuous with constant \(1+\frac{1}{\xi }\); (ii) if \(F:C\to X\) is η-strongly accretive and ξ-strictly pseudocontractive with \(\eta +\xi \geq 1\), then, for any fixed \(\tau \in (0,1)\), \(I-\tau F\) is a contraction with constant \(1-\tau (1-\sqrt{\frac{1-\eta }{\xi }})\).
From now on, one employs Π to denote a mapping from C onto its subset D. One says that Π is sunny if, whenever \({\varPi }(x)+t(x- {\varPi }(x))\in C\) for \(x\in C\), \({\varPi }[{\varPi }(x)+t(x-{\varPi }(x))]= {\varPi }(x)\). A mapping Π defined on C is called a retraction if \(\varPi =\varPi ^{2}\). One says that subset D is a sunny nonexpansive retract of the set C if there exists a sunny nonexpansive retraction from C onto D.
Let \(\{S_{n}\}^{\infty }_{n=0}\) be a countable family of nonexpansive mappings defined on C, which is a convex and closed subset of a strictly convex Banach space, and let \(\{\zeta _{n}\}^{\infty }_{n=0}\) be a sequence in \([0,1]\). For any \(n\geq 0\), define a mapping \(W_{n}:C\to C\) as follows:
Lemma 2.1
Suppose that\(\{S_{n}\}^{\infty }_{n=0}\)is a countable family of nonexpansive mappings defined on a subsetCof a strictly convex spaceX. Suppose that\(\bigcap^{\infty }_{n=0}{\operatorname{Fix}}(S _{n})\neq \emptyset \), and\(\{\zeta _{n}\}^{\infty }_{n=0}\)is a real sequence such that\(0<\zeta _{n}\leq b<1\), \(\forall n\geq 0\). Then
- (i)
\(W_{n}\)is nonexpansive and\(\operatorname{Fix}(W_{n})=\bigcap^{n}_{i=0} {\operatorname{Fix}}(S_{i})\), \(\forall n\geq 0\);
- (ii)
the limit\(\lim_{n\to \infty }U_{n,k}x\)exists for all\(x\in C\)and\(k\geq 0\);
- (iii)
the mapping\(W:C\to C\)defined by\(Wx:=\lim_{n\to \infty }W_{n}x= \lim_{n\to \infty }U_{n,0}x\), \(\forall x\in C\), is a nonexpansive mapping satisfying\(\operatorname{Fix}(W)=\bigcap^{\infty }_{n=0}{\operatorname{Fix}}(S _{n})\)and it is called theW-mapping. IfDis any bounded subset ofC, then\(\lim_{n\to \infty }\sup_{x\in D}\|W_{n}x-Wx\|=0\).
For our main strong convergence theorems, the following tools are also needed.
Lemma 2.2
([27])
LetXbe smooth, Dbe a nonempty subset ofCandΠbe a retraction ofContoD. Then the following are equivalent: (i) Πis sunny and nonexpansive; (ii) \(\|{\varPi }(x)- {\varPi }(y)\|^{2}\leq \langle x-y,J({\varPi }(x)-\varPi (y))\rangle\), \(\forall x,y\in C\); (iii) \(\langle x-{\varPi }(x),J(y-{\varPi }(x))\rangle \leq 0\), \(\forall x\in C\), \(y\in D\).
Lemma 2.3
([28])
Let\(q\in (1,2]\)a given real number and letXbeq-uniformly smooth. Then\(\|x+y\|^{q}\leq q\langle y,J_{q}(x)\rangle +\|x\|^{q}+\kappa _{q}\|y\|^{q}\), \(\forall x,y\in X\), where\(\kappa _{q}\)is theq-uniformly smooth constant ofX. For any given\(x,y\in X\), one has\(\|x+y\|^{q}\leq \|x\|^{q}+q\langle y,j_{q}(x+y)\rangle \), \(\forall j_{q}(x+y)\in J_{q}(x+y)\).
Lemma 2.4
LetXbe a uniformly convex andq-uniformly, where\(1< q\leq 2\), smooth Banach space. Let\(A:C\to X\)be anα-inverse-strongly accretive mapping of orderqand\(B:C\to 2^{X}\)be anm-accretive operator. In the sequel, we will use the notation\(T_{\lambda }:=J^{B}_{\lambda }(I-\lambda A)=(I+\lambda B)^{-1}(I-\lambda A)\), \(\forall \lambda >0\). The following statements hold:
- (i)
the resolvent identity: \(J_{\lambda }x=J_{\mu }(\frac{\mu }{ \lambda }x+(1-\frac{\mu }{\lambda })J_{\lambda }x)\), \(\forall \lambda ,\mu >0\), \(x\in X\);
- (ii)
if\(J^{A}_{\lambda }\)is a resolvent ofAfor\(\lambda >0\), then\(J^{A}_{\lambda }\)is a single-valued nonexpansive mapping with\(\operatorname{Fix}(J^{A}_{\lambda })=A^{-1}0\), where\(A^{-1}0=\{x\in C:0 \in Ax\}\);
- (iii)
\(\operatorname{Fix}(T_{\lambda })=(A+B)^{-1}0\), \(\forall \lambda >0\);
- (iv)
\(\|x-T_{\lambda }x\|\leq 2\|x-T_{s}x\|\)for\(0<\lambda \leq s\)and\(x\in X\);
- (v)
\(\|T_{\lambda }x-T_{\lambda }y\|\leq \|x-y\|\);
- (vi)
\(\|(I-\lambda A)x-(I-\lambda A)y\|^{q}\leq \|x-y\|^{q}-\lambda (q \alpha -\kappa _{q}\lambda ^{q-1})\|Ax-Ay\|^{q}\), \(\forall x,y\in C\). In particular, if\(0<\lambda \leq (\frac{q\alpha }{\kappa _{q}})^{ \frac{1}{q-1}}\), then\(I-\lambda A\)is nonexpansive.
Lemma 2.5
([30])
Let\(T:C\to C\)be nonexpansive with\(\operatorname{Fix}(T) \neq \emptyset \), and let\(f:C\to C\)be a fixed contraction mapping, whereCis convex and closed set in a real reflexive Banach space with the uniformly Gâteaux differentiable norm and the normal structure. Let\(z_{t}\in C\), where\(t\in (0,1)\), be the unique fixed point of the contraction\(C\ni z\mapsto (1-t)Tz+tf(z)\)onC, that is, \(z_{t}=(1-t)Tz _{t}+tf(z_{t})\). Then\(\{z_{t}\}\)converges to\(x^{*}\in {\operatorname{Fix}}(T)\)in norm. This convergent point also solves\(\langle (f-I)x^{*}, J(p-x^{*})\rangle \leq 0\), \(\forall p\in {\operatorname{Fix}}(T)\).
Lemma 2.6
([14])
Suppose that\({\varPi }_{C}\)is a sunny nonexpansive retraction from aq-uniformly smoothXonto its convex closed subsetC. Let the mapping\(A_{i}:C\to X\)be\(\alpha _{i}\)-inverse-strongly accretive of orderqfor\(i=1,2\). Let the mapping\(G:C\to C\)be defined as\(Gx:={\varPi }_{C}(I-\mu _{1}A_{1}){\varPi }_{C}(I-\mu _{2}A_{2})\), \(\forall x\in C\). If\(0<\mu _{i}\leq (\frac{q\alpha _{i}}{\kappa _{q}})^{ \frac{1}{q-1}}\)for\(i=1,2\), then\(G:C\to C\)is a Lipschitz mapping. More precisely, it is nonexpansive. Let\(A_{1},A_{2}:C\to X\)be two nonlinear mappings. For given\((x^{*},y^{*})\in C\times C\), \((x^{*},y^{*})\)is a solution of SVIs (1.1) iff\(x^{*}={\varPi } _{C}(y^{*}-\mu _{1}A_{1}y^{*})\), where\(y^{*}={\varPi }_{C}(x^{*}-\mu _{2}A _{2}x^{*})\).
Lemma 2.7
([31])
Let\(\{a_{n}\}\)be a sequence defined by\(a_{n+1}\leq \gamma _{n}\lambda _{n}+a_{n}(1-\lambda _{n})\), \(\forall n\geq 0\), where\(\{\lambda _{n}\}\)and\(\{\gamma _{n}\}\)are sequences of real numbers such that (i) \(\limsup_{n\to \infty }\gamma _{n}\leq 0\)or\(\sum^{ \infty }_{n=0}|\lambda _{n}\gamma _{n}|<\infty \); (ii) \(\{\lambda _{n}\} \subset [0,1]\)and\(\sum^{\infty }_{n=0}\lambda _{n}=\infty \). Then\(\lim_{n\to \infty }a_{n}=0\).
Lemma 2.8
([28])
Let\(B_{r}=\{x\in X:\|x\|\leq r\}\), \(r>0\), whereXis a uniformly convex Banach space. Then there exists a continuous, strictly increasing and convex function\(g:[0,\infty )\to [0,\infty )\), \(g(0)=0\)such that, with\(p>1\),
for all\(x,y,z\in B_{r}\)and\(\alpha ,\beta ,\gamma \in [0,1]\)with\(\alpha +\beta +\gamma =1\).
Lemma 2.9
([32])
Suppose that\(\{x_{n}\}\)is a sequence defined by\(x_{n+1}=\alpha _{n}x_{n}+(1-\alpha _{n})y_{n}\), \(\forall n\geq 0\), where\(\{y_{n}\}\)is bounded sequences in Banach spaceXand let\(\{\alpha _{n}\}\)be a real sequence such that\(0<\liminf_{n\to \infty }\alpha _{n}\leq \limsup_{n\to \infty }\alpha _{n}<1\). If\(\limsup_{n\to \infty }(\|y_{n+1}-y_{n}\|- \|x_{n+1}-x_{n}\|)\leq 0\), then\(\lim_{n\to \infty }\|y_{n}-x_{n}\|=0\).
3 Iterative algorithms and convergence criteria
Theorem 3.1
LetXbe a both uniformly convex andq-uniformly smooth space with\(1< q\leq 2\)and let\(B:C\to 2^{X}\)be anm-accretive operator. Let\(A_{i}:C\to X\)be an\(\alpha _{i}\)-inverse-strongly accretive operator of orderqfor each\(i=1,2\)and\(A:C\to X\)be anα-inverse-strongly accretive of orderq. Assume that\({\varOmega }=\bigcap^{\infty }_{n=0}{\operatorname{Fix}}(S_{n})\cap {\mathrm{SVI}}(C,A _{1}, A_{2})\cap (A+B)^{-1}0\neq \emptyset \), where\(\mathrm{SVI}(C,A _{1},A_{2})\)is the fixed point set of\(G:={\varPi }_{C}(I-\mu _{1}A_{1}) {\varPi }_{C}(I-\mu _{2}A_{2})\)with\(0<\mu _{i}<(\frac{q\alpha _{i}}{\kappa _{q}})^{\frac{1}{q-1}}\)for\(i=1,2\). Let\(f:C\to C\)be aδ-contraction with constant\(\delta \in (0,1)\)and let\(F:C\to X\)beη-strongly accretive andξ-strictly pseudocontractive with\(\eta +\xi \geq 1\). For arbitrarily given\(x_{0}\in C\), let\(\{x_{n}\}\)be a sequence generated by
where\({\varPi }_{C}\)is the sunny nonexpansive retraction fromXontoC, \(\{W_{n}\}\)is the sequence defined by (2.1), \(\{\lambda _{n}\}\subset (0,(\frac{q\alpha }{\kappa _{q}})^{\frac{1}{q-1}})\), \(\{\sigma _{n}\}\subset [0,1)\)and\(\{\alpha _{n}\}\), \(\{\beta _{n}\}\), \(\{\gamma _{n}\},\{\delta _{n}\},\{t_{n}\}\subset (0,1)\)satisfy the following conditions:
- (i)
\(\alpha _{n}+\beta _{n}+\gamma _{n}=1\), \(\sum^{\infty }_{n=0}\alpha _{n}=\infty \)and\(\lim_{n\to \infty }\alpha _{n}=0\);
- (ii)
\(\lim_{n\to \infty }\frac{\sigma _{n}}{\alpha _{n}}=0\), \(\lim_{n\to \infty }|\gamma _{n}-\gamma _{n-1}|=0\)and\(\lim_{n\to \infty }|\beta _{n}-\beta _{n-1}|=0\);
- (iii)
\(\lim_{n\to \infty }|t_{n}-t_{n-1}|=0\), \(\limsup_{n\to \infty } \gamma _{n}t_{n}(1-t_{n})<1\)and\(\liminf_{n\to \infty }\gamma _{n}(1-t _{n})>0\);
- (iv)
\(\liminf_{n\to \infty }\beta _{n}\gamma _{n}>0\), \(\limsup_{n\to \infty }\delta _{n}<1\)and\(\liminf_{n\to \infty }\delta _{n}>0\);
- (v)
\(0<\bar{\lambda }\leq \lambda _{n}\), \(\forall n\geq 0\)and\(\lim_{n\to \infty }\lambda _{n}=\lambda <(\frac{q\alpha }{\kappa _{q}}) ^{\frac{1}{q-1}}\).
Then\(x_{n}\to x^{*}\in {\varOmega }\), which is a unique solution to the generalized variational inequality (GVI) \(\langle (I-f)x^{*},J(x^{*}-p) \rangle \leq 0\), \(\forall p\in {\varOmega }\).
Proof
Put \(u_{n}={\varPi }_{C}(y_{n}-\mu _{2}A_{2}y_{n})\). It is easy to see that scheme (3.1) can be rewritten as
where \(T_{n}:=J^{B}_{\lambda _{n}}(I-\lambda _{n}A)\). From \(\eta + \xi \geq 1\), \(\{\sigma _{n}\}\subset [0,1)\), one asserts that \({\varPi } _{C}(I-\sigma _{n}F):C\to C\) is a nonexpansive mapping for each \(n\geq 0\). Because of the situation \(\alpha _{n}+\beta _{n}+\gamma _{n}=1\), one knows that
One now shows that the sequence \(\{x_{n}\}\) generated by (3.2) is well defined. Define a mapping \(F_{n}:C\to C\) by \(F_{n}(x)=\beta _{n}x _{n}+\gamma _{n}{\varPi }_{C}(I-\sigma _{n}F)(t_{n}x_{n}+(1-t_{n})W_{n} Gx)+ \alpha _{n}f(x)\), \(\forall x\in C\). Then
This guarantees the result that \(F_{n}\) is a contraction mapping. Hence there is a unique fixed point \(y_{n}\in C\) satisfying
One next divides the rest of the proof into several steps.
Step 1. Show that \(\{x_{n}\}\) is bounded.
From \(\{\lambda _{n}\}\subset (0,(\frac{q\alpha }{\kappa _{q}})^{ \frac{1}{q-1}})\), one observes that \(T_{n}: C\to C\) is a nonexpansive mapping for each \(n\geq 0\). Take a fixed \(p\in {\varOmega }=\bigcap^{\infty }_{n=0}{\operatorname{Fix}}(S_{n})\cap {\mathrm{SVI}}(C,A_{1},A_{2})\cap (A+B)^{-1}0\) arbitrarily. From Lemmas 2.4 and 2.6, we know that \(W_{n}p=p\), \(Gp=p\) and \(T_{n}p=p\). Moreover, using the nonexpansivity of \(W_{n}\) and G yields
which hence implies that
Since \(\lim_{n\to \infty }\frac{\sigma _{n}}{\alpha _{n}}=0\), one may suppose \(\sigma _{n}\leq \alpha _{n}\). Thus, from (3.2), (3.3) and the nonexpansivity of \(T_{n}\), we find that
It immediately follows that \(\{x_{n}\}\) is a bounded vector in set C.
Step 2. One shows that \(\|x_{n+1}-x_{n}\|\to 0\) as \(n\to \infty \).
Indeed,
and
Utilizing Lemmas 2.1 and 2.4 yields
where \(\sup_{n\geq 1}\{\frac{1}{\bar{\lambda }}\|J^{B}_{\lambda _{n}}(I- \lambda _{n}A)y_{n-1}-(I-\lambda _{n}A)y_{n-1}\| +\|Ay_{n-1}\|\}\leq M _{1}\) for some \(M_{1}>0\). Also, it follows from the nonexpansivity of \({\varPi }_{C}\) and \((I-\sigma _{n}F)\) that
This together with (3.4) guarantees
where \(\sup_{n\geq 0}\{\|x_{n}\|+\|f(y_{n})\|+\|W_{n}Gy_{n}\|+\|Fz _{n}\|+\|{\varPi }_{C}(I-\sigma _{n}F)z_{n}\|\}\leq M_{2}\) for some \(M_{2}>0\). Then
which together with (3.5) asserts that
Since \(\lim_{n\to \infty }\sup_{x\in D}\|W_{n}x-Wx\|=0\) on bounded subset \(D=\{Gy_{n}:n\geq 0\}\) of C, one knows that
Note that \(\lim_{n\to \infty }\alpha _{n}=0\), \(\lim_{n\to \infty }\frac{ \sigma _{n}}{\alpha _{n}}=0\), \(\lim_{n\to \infty }\lambda _{n}= \lambda \) and \(\liminf_{n\to \infty }(1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n})))>0\). Thus, from \(|\beta _{n}-\beta _{n-1}|\to 0\), \(|\gamma _{n}-\gamma _{n-1}| \to 0\) and \(|t_{n}-t_{n-1}|\to 0\) as \(n\to \infty \) (due to conditions (ii), (iii)), we get
So it follows from condition (iv) and Lemma 2.9 that \(\lim_{n\to \infty }\|T_{n}y_{n}-x_{n}\|=0\). Hence
Step 3. One shows that \(\|x_{n}-y_{n}\|\to 0\) and \(\|x_{n}-Gx _{n}\|\to 0\) as \(n\to \infty \). Indeed, for simplicity, set \(\bar{p}:={\varPi }_{C} (I-\mu _{2}A_{2})p\). Note that \(u_{n}={\varPi }_{C}(I- \mu _{2}A_{2})y_{n}\) and \(v_{n}={\varPi }_{C}(I-\mu _{1}A_{1})u_{n}\). Then \(v_{n}=Gy_{n}\). An application of Lemma 2.4 yields
One also has
By using (3.7) and (3.8), one reaches
Equations (3.2) and (3.9) further guarantee that \(\|z_{n}-p\|^{q}\leq t_{n}\|x_{n}-p\|^{q}+(1-t_{n})\|v_{n}-p\|^{q}\) and
Thus
which immediately yields
On the other hand, (3.2) implies
where
for some \(M_{3}>0\). So it follows from (3.10) that
Thanks to \(0<\mu _{i}<(\frac{q\alpha _{i}}{\kappa _{q}})^{\frac{1}{q-1}}\) for \(i=1,2\), \(\liminf_{n\to \infty }\gamma _{n} (1-t_{n})>0\), \(\liminf_{n\to \infty }(1-\delta _{n})>0\) and \(\lim_{n\to \infty }\alpha _{n}=0\), one asserts
This further implies
from which one concludes
One also derives that
Employing (3.12) and (3.13), one arrives at
Utilizing Lemma 2.8, we obtain from (3.2) and (3.14)
and hence
which immediately yields
This guarantees
which immediately yields
Utilizing (3.6) and (3.11), we asserts from \(\liminf_{n\to \infty }(1-\delta _{n})>0\), \(\liminf_{n\to \infty }\gamma _{n}t_{n}(1-t_{n})>0\) and \(\liminf_{n\to \infty }\beta _{n}\gamma _{n}>0\) that \(\lim_{n\to \infty }g_{1}(\|y_{n}-u_{n}-(p-\bar{p})\|)=0\), \(\lim_{n\to \infty }g_{2}(\|u_{n}-v_{n}+(p-\bar{p})\|)=0\), \(\lim_{n\to \infty }g_{3}(\|x_{n}-W_{n}Gy_{n}\|)=0\) and \(\lim_{n\to \infty } g_{4}(\|x_{n}-{\varPi }_{C}(I-\sigma _{n}F)z_{n}\|)=0\). So, \(\lim_{n\to \infty }\|y_{n}-u_{n}-(p-\bar{p})\|=\lim_{n\to \infty }\|u_{n}-v_{n}+(p-\bar{p})\|=0\) and
Furthermore, one has
Since \(y_{n}-x_{n}=\alpha _{n}(f(y_{n})-x_{n})+\gamma _{n}({\varPi }_{C}(I- \sigma _{n}F)z_{n}-x_{n})\), we see from (3.15) that \(\|y_{n}-x _{n}\|\leq \|{\varPi }_{C}(I-\sigma _{n}F)z_{n}-x_{n}\|+\alpha _{n}\|x_{n}-f(y _{n})\|\to 0\) (\(n\to \infty \)). With the aid of (3.16), one asserts
Step 4. One shows that \(\|x_{n}-Wx_{n}\|\to 0\), \(\|x_{n}-T_{ \lambda }x_{n}\|\to 0\) and \(\|x_{n}-{\varGamma }x_{n}\|\to 0\) as \(n\to \infty \), where \(Wx=\lim_{n\to \infty }W_{n}x\), \(\forall x\in C\), \(T_{\lambda }=J^{B}_{\lambda }(I-\lambda A)\) and \({\varGamma }x= \theta _{1}Wx+\theta _{2}Gx+\theta _{3}T_{\lambda }x\), \(\forall x\in C\) for constants \(\theta _{1},\theta _{2},\theta _{3}\in (0,1)\) satisfying \(\theta _{1}+\theta _{2} +\theta _{3}=1\). Indeed, utilizing (3.15) and (3.17), one deduces that
Furthermore, since \(x_{n+1}-x_{n}+x_{n}-y_{n}=\delta _{n}(x_{n}-y_{n})+(1- \delta _{n})(T_{n}y_{n}-y_{n})\), from \(x_{n}-x_{n+1}\to 0\) and \(x_{n}-y_{n}\to 0\), we have
Also, utilizing similar arguments to those of (3.5), we obtain
Since \(\lim_{n\to \infty }\lambda _{n}=\lambda \) and the sequences \(\{y_{n}\}\), \(\{T_{n}y_{n}\}\), \(\{Ay_{n}\}\) are bounded, we get
Taking into account condition (v), i.e., \(0<\bar{\lambda }\leq \lambda _{n}\), \(\forall n\geq 0\) and \(\lim_{n\to \infty }\lambda _{n}=\lambda \), where \(\kappa _{q}\lambda ^{q-1}< q\alpha \), we know that \(0<\kappa _{q}\bar{ \lambda }^{q-1}\leq \kappa _{q}\lambda ^{q-1}< q\alpha \). So \(\operatorname{Fix}(T_{\lambda })=(A+B)^{-1}0\) and \(T_{\lambda }:C\to C\) is nonexpansive. Therefore, we infer from (3.19) and \(x_{n}-y_{n} \to 0\) that
One now defines the mapping \({\varGamma }x=\theta _{1}Wx+\theta _{2}Gx+ \theta _{3}T_{\lambda }x\), \(\forall x\in C\) with constants \(\theta _{1}, \theta _{2},\theta _{3}\in (0,1)\) satisfying \(\theta _{1}+\theta _{2}+ \theta _{3}=1\). One gets \(\operatorname{Fix}({\varGamma })=\operatorname{Fix}(W) \mathrel{\cap} \operatorname{Fix}(G)\mathrel{\cap} \operatorname{Fix}(T_{\lambda })={\varOmega }\). Observe that
From (3.17), (3.18), (3.20) and (3.21), one gets
Step 5. Letting \(x_{t}\) is the unique fixed point of \(x\mapsto (1-t){\varGamma }x+tf(x)\) for each \(t\in (0,1)\), one shows that
where \(x^{*}=\mbox{s-}\lim_{n\to \infty }x_{t}\). By Lemmas 2.3 and 2.5, one asserts
where
It follows from (3.24) that
Letting \(n\to \infty \) and employing (3.25), one derives
where \(\sup \{\|x_{t}-x_{n}\|^{2}:t\in (0,1) \text{ and } n\geq 0\} \leq M_{4}\) for some \(M_{4}>0\). Taking \(t\to 0\) in (3.27), we have
On the other hand, we have
So, it follows that
Taking into account that \(x_{t}\to x^{*}\) as \(t\to 0\), we have
Thanks to the space (q-uniformly smooth), one knows that the two limits can be interchangeable. Equation (3.23) therefore holds. Note that \(x_{n}-y_{n}\to 0\) implies \(J(y_{n}-x^{*})-J(x_{n}-x^{*})\to 0\). Thus, we conclude from (3.23) that
Step 6. One shows \(\|x_{n}-x^{*}\|\to 0\) as \(n\to \infty \).
which hence yields
Due to the convexity of \(\|\cdot \|^{2}\), and the nonexpansivity of \(T_{n}\), one asserts
Since \(\liminf_{n\to \infty }\frac{(1-\delta _{n})(1-\delta )}{1-( \alpha _{n}\delta +\gamma _{n}(1-t_{n}))}>0\), \(\{\frac{\alpha _{n}(1- \delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\}\subset (0,1)\) and \(\sum^{\infty }_{n=0}\alpha _{n}=\infty \), we know
and
Utilizing (3.29) and Lemma 2.7, we conclude from (3.30) that \(\|x_{n}-x^{*}\|\to 0\) as \(n\to \infty \). This completes the proof. □
Remark 3.1
Comparing with the corresponding results in and Chang et al. [8], we have the following aspects. The problem of solving a HVI with the constraints of SGVIs (1.1) and a countable family of nonexpansive mappings in [8, Theorem 3.1] is extended to our problem of solving a HVI with the constraints of SGVIs (1.1), a variational inclusion (VI) and a countable family of nonexpansive mappings. The modified relaxed extragradient method in[8, Theorem 3.1] is extended to our composite extragradient implicit rule (3.1). That is, two iterative steps \(y_{n}=(1-\beta _{n})x_{n}+ \beta _{n}Gx_{n}\) and \(x_{n+1}={\varPi }_{C}[\gamma _{n}x_{n}+((1-\gamma _{n})I-\alpha _{n}\rho F)S_{n}y_{n}+\alpha _{n}\gamma f(x_{n})]\) in [8, Theorem 3.1] are extended to our two iterative steps \(y_{n}=\beta _{n}x_{n}+\gamma _{n}{\varPi }_{C}(I-\sigma _{n}F)(t_{n}x_{n}+(1-t _{n})W_{n}Gy_{n})+\alpha _{n}f(y_{n})\) and \(x_{n+1}=\delta _{n}x_{n} +(1- \delta _{n})T_{n}y_{n}\), where \(T_{n}=J^{B}_{\lambda _{n}}(I-\lambda _{n}A)\).
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Ceng, LC., Shang, M. Composite extragradient implicit rule for solving a hierarch variational inequality with constraints of variational inclusion and fixed point problems. J Inequal Appl 2020, 33 (2020). https://doi.org/10.1186/s13660-020-2306-1
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DOI: https://doi.org/10.1186/s13660-020-2306-1