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A system of variational inclusions (GSVI) is considered in Banach spaces. An implicit iterative procedure is proposed for solving the GSVI. Strong convergence of the proposed algorithm is given.
Let X be a smooth Banach space and a closed convex set. Let and be nonlinear mappings. In the present article, we consider the following system of variational inclusions (GSVI, for short) which aims to seek verifying
where and are two positive constants.
Special cases: If and , then the relation (1) reduces to seek verifying
If in (2), then the relation (2) reduces to seek verifying
Especially, if , where is a proper convex lower semi-continuous function, then we have the following mixed quasi-variational inequality
Variational inequalities and variational inclusions have played vital roles in practical applications. Numerous iterative procedures for approaching variational inequalities and variational inclusions have been computed by the researchers [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29].
In [4], the authors introduced an iterative procedure for approaching GSVI (1). Qin et al. [30] suggested an extragradient algorithm for solving GSVI (1), and demonstrated the strong convergence analysis of the presented algorithm. Lan et al. [28], Buong et al. [11], Zhang et al. [13] studied iterative procedures for approaching variational inclusion (3).
On the other hand, iterative computation of zeros or fixed points of nonlinear operators has been studied extensively in the literature [14,31,32,33,34,35,36]. Zhang et al. [37] introduced an iterative procedure for approaching a solution of the inclusion problem (3) and a fixed point of a nonexpansive mapping in Hilbert spaces. Peng et al. [38] presented a viscosity algorithm for finding a solution of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem and a fixed point of a nonexpansive mapping.
Motivated by the above work, in the present paper, we consider the GSVI (1) with the hierarchical variational inequality constraint for a strict pseudocontraction T in Banach spaces. We suggest an implicit iterative procedure for solving the GSVI (1) with the HVI constraint for strict pseudocontraction T. We show the strong convergence of the suggested procedure to a solution of the GSVI (1).
2. Preliminaries
Let X be a real Banach space and a closed convex set. A mapping is said to be k-Lipschitz if for some . If , then f is said to be a k-contraction. If , then f is said to be nonexpansive.
Recall that an operator is called
(i)
accretive if
where .
(ii)
-inverse-strongly accretive if
where and .
(iii)
strictly pseudocontractive if
where and .
If X is q-uniformly smooth with , then
where is some constant.
Proposition1
([32]).In a smooth and uniformly convex Banach space X, for all , there holds
where is a strictly increasing, continuous, and convex function satisfying .
Proposition2
([35]).In a 2-uniformly smooth Banach space X, there holds
Let and be an operator. If whenever for and , we call is sunny.
Proposition3
([26]).Let X be a smooth Banach space and a closed convex set. Let be a set and be a retraction. Then the following conclusions are equivalent:
(i)
and ;
(ii)
;
(iii)
Π is sunny nonexpansive operator.
If an accretive operator M satisfies for each , then M is said to be m-accretive. Assume that an accretive M satisfies the range condition . Define the resolvent of M by . Note that is nonexpansive and [31]. If , then the inclusion is solvable.
Lemma1.
Let X be a smooth Banach space and a closed convex set. Let be an m-accretive operator. Then, for any given ,
This means that is nonexpansive.
Proof.
Put and . Then we have and . Hence, there exist and such that and . Utilizing the accretiveness of M, we obtain
□
Lemma2.
Let be two m-accretive operators and be two operators. is a solution of the GSVI (1) iff , where .
Proof.
Observe that
□
Lemma3
([3]).Let X be a strictly convex Banach space and a closed convex set. Let be a constant. Define an operator by , where be two nonexpansive mappings with . Then S is nonexpansive and .
Lemma4
([3]).Let X be a 2-uniformly smooth Banach space and a closed convex set. If the operator is α-inverse-strongly accretive, then
Lemma5
([3]).Let X be a 2-uniformly smooth Banach space and a closed convex set. Let be two m-accretive operators and be -inverse-strongly accretive operator. Define an operator by . If , then is nonexpansive.
Lemma6
([36]).Let X be a uniformly smooth Banach space and a closed convex set. Let be a nonexpansive mapping with , and be a contraction. Let . Define a net by . Then and
Lemma7
([36]).Assume the sequence satisfies where the sequences and satisfy
(i)
;
(ii)
either or .
Then .
Lemma8
([33]).Let X be a 2-uniformly smooth Banach space and a closed convex set. Let be a λ-strict pseudocontraction. Define an operator by . Then, is nonexpansive with provided .
3. Main Results
Theorem1.
Let X be a uniformly convex and 2-uniformly smooth Banach space and a closed convex set. Let be two m-accretive operators and be -inverse-strongly accretive operator. Let be a contraction with coefficient . Let be a nonexpansive operator and be a λ-strict pseudocontraction with , where the operator Q is defined as in Lemma 5. Assume that the sequences , and satisfy
Observe that . With the help of Lemma 7, we get . Moreover, putting and in (15), we obtain
Note that
So, it follows that as . Consequently, is a solution of (1) by Lemma 2. □
Corollary1.
Let X be a uniformly convex and 2-uniformly smooth Banach space and a closed convex set. Let be an m-accretive operator and be a ζ-inverse-strongly accretive operator. Let be a contraction with coefficient . Let be a nonexpansive operator and be a λ-strict pseudocontraction with , where the operator and . Assume that the sequences and satisfy
Let H be a Hilbert space and a closed convex set. Let be a maximal monotone operator and be a ζ-inverse-strongly monotone operator. Let be a contraction with coefficient . Let be a nonexpansive operator and be a λ-strict pseudocontraction with , where the operator and . Assume that the sequences and satisfy
In this paper, we consider the GSVI (1) with the hierarchical variational inequality (HVI) constraint for a strict pseudocontraction in a uniformly convex and 2-uniformly smooth Banach space. By utilizing the equivalence between the GSVI (1) and the fixed point problem, we construct an implicit composite viscosity approximation method for solving the GSVI (1) with the HVI constraint for strict pseudocontractions. We prove the strong convergence of the proposed algorithm to a solution of the GSVI (1) with the HVI constraint for strict pseudocontraction under very mild conditions. Note that our algorithm (4) is an implicit manner. This brings us a natural question: could we construct an explicit algorithm with strong convergence?
Author Contributions
All the authors have contributed equally to this paper. All the authors have read and approved the final manuscript.
Funding
This research was partially supported by the Innovation Program of Shanghai Municipal Education Commission (15ZZ068), Ph.D. Program Foundation of Ministry of Education of China (20123127110002) and Program for Outstanding Academic Leaders in Shanghai City (15XD1503100). Yonghong Yao was supported in part by the grant TD13-5033.
Conflicts of Interest
The authors declare no conflict of interest.
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Ceng, L.-C.; Postolache, M.; Yao, Y.
Iterative Algorithms for a System of Variational Inclusions in Banach Spaces. Symmetry2019, 11, 811.
https://doi.org/10.3390/sym11060811
AMA Style
Ceng L-C, Postolache M, Yao Y.
Iterative Algorithms for a System of Variational Inclusions in Banach Spaces. Symmetry. 2019; 11(6):811.
https://doi.org/10.3390/sym11060811
Chicago/Turabian Style
Ceng, Lu-Chuan, Mihai Postolache, and Yonghong Yao.
2019. "Iterative Algorithms for a System of Variational Inclusions in Banach Spaces" Symmetry 11, no. 6: 811.
https://doi.org/10.3390/sym11060811
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.
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Ceng, L.-C.; Postolache, M.; Yao, Y.
Iterative Algorithms for a System of Variational Inclusions in Banach Spaces. Symmetry2019, 11, 811.
https://doi.org/10.3390/sym11060811
AMA Style
Ceng L-C, Postolache M, Yao Y.
Iterative Algorithms for a System of Variational Inclusions in Banach Spaces. Symmetry. 2019; 11(6):811.
https://doi.org/10.3390/sym11060811
Chicago/Turabian Style
Ceng, Lu-Chuan, Mihai Postolache, and Yonghong Yao.
2019. "Iterative Algorithms for a System of Variational Inclusions in Banach Spaces" Symmetry 11, no. 6: 811.
https://doi.org/10.3390/sym11060811
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.