Abstract
In this paper, quasi-variational inclusions and fixed point problems of pseudocontractions are considered. An iterative algorithm is presented. A strong convergence theorem is demonstrated.
MSC:49J40, 47J20, 47H09, 65J15.
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1 Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a single-valued nonlinear mapping and be a multi-valued mapping. The ‘so called’ quasi-variational inclusion problem is to find an such that
The set of solutions of (1.1) is denoted by . A number of problems arising in structural analysis, mechanics, and economics can be studied in the framework of this kind of variational inclusions; see for instance [1–4]. For related work, see [5–10]. The problem (1.1) includes many problems as special cases.
-
(1)
If , where is a proper convex lower semi-continuous function and ∂ϕ is the subdifferential of ϕ, then the variational inclusion problem (1.1) is equivalent to finding such that
which is called the mixed quasi-variational inequality (see [11]).
-
(2)
If , where C is a nonempty closed convex subset of H and is the indicator function of C, i.e.,
then the variational inclusion problem (1.1) is equivalent to finding such that
This problem is called the Hartman-Stampacchia variational inequality (see [12]).
Let be a nonlinear mapping. The iterative scheme of Mann’s type for approximating fixed points of T is the following: and
for all , where is a sequence in ; see [13]. For two nonlinear mappings S and T, Takahashi and Tamura [14] considered the following iteration procedure: and
for all , where and are two sequences in . Algorithms for finding the fixed points of nonlinear mappings or for finding the zero points of maximal monotone operators have been studied by many authors. The reader can refer to [15–19]. Especially, Takahashi et al. [20] recently gave the following convergence result.
Theorem 1.1 Let C be a closed and convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. Let be the resolvent of B for and let T be a nonexpansive mapping of C into itself, such that . Let and let be a sequence generated by
for all , where , and satisfy
Then converges strongly to a point of .
Recently, Zhang et al. [21] introduced a new iterative scheme for finding a common element of the set of solutions to the inclusion problem and the set of fixed points of nonexpansive mappings in Hilbert spaces. Peng et al. [22] introduced another iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping.
Motivated and inspired by the works in this field, the purpose of this paper is to consider the quasi-variational inclusions and fixed point problems of pseudocontractions. An iterative algorithm is presented. A strong convergence theorem is demonstrated.
2 Notations and lemmas
Let H be a real Hilbert space with inner product and norm , respectively. Let C be a nonempty closed convex subset of H. It is well known that in a real Hilbert space H, the following equality holds:
for all and .
Recall that a mapping is called
(D1) L-Lipschitzian ⟹ there exists such that for all ; in the case of , T is said to be nonexpansive;
(D2) Firmly nonexpansive ⟹ for all ;
(D3) Pseudocontractive ⟹ for all ;
(D4) Strongly monotone ⟹ there exists a positive constant such that for all ;
(D5) Inverse strongly monotone for some and for all .
Let B be a mapping of H into . The effective domain of B is denoted by , that is, . A multi-valued mapping B is said to be a monotone operator on H iff
for all , , and . A monotone operator B on H is said to be maximal iff its graph is not strictly contained in the graph of any other monotone operator on H. Let B be a maximal monotone operator on H and let .
For a maximal monotone operator B on H and , we may define a single-valued operator , which is called the resolvent of B for λ. It is known that the resolvent is firmly nonexpansive, i.e.,
for all and for all .
Usually, the convergence of fixed point algorithms requires some additional smoothness properties of the mapping T such as demi-closedness.
Recall that a mapping T is said to be demiclosed if, for any sequence which weakly converges to , and if the sequence strongly converges to z, then . For the pseudocontractions, the following demiclosed principle is well known.
Lemma 2.1 ([23])
Let H be a real Hilbert space, C a closed convex subset of H. Let be a continuous pseudo-contractive mapping. Then
-
(i)
is a closed convex subset of C,
-
(ii)
is demiclosed at zero.
Lemma 2.2 ([24])
Let be a sequence of real numbers. Assume does not decrease at infinity, that is, there exists at least a subsequence of such that for all . For every , define an integer sequence as
Then as , and for all
Lemma 2.3 ([25])
Assume is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(1)
;
-
(2)
or .
Then .
In the sequel we shall use the following notations:
-
1.
denote the weak ω-limit set of ;
-
2.
stands for the weak convergence of to x;
-
3.
stands for the strong convergence of to x;
-
4.
stands for the set of fixed points of T.
3 Main results
In this section, we consider a strong convergence theorem for quasi-variational inclusions and fixed point problems of pseudocontractive mappings in a Hilbert space.
Algorithm 3.1 Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. Let be the resolvent of B for λ. Let be an -Lipschitzian and ς strongly monotone mapping and be a ρ-contraction such that . Let be an -Lipschitzian pseudocontraction. For , define a sequence as follows:
for all , where λ, ν and ζ are three constants, and are two sequences in .
Now, we demonstrate the convergence analysis of the algorithm (3.1).
Theorem 3.2 Suppose . Assume the following conditions are satisfied:
(C1) ;
(C2) and ;
(C3) and .
Then the sequence defined by (3.1) converges strongly to .
Proof Let . Then, we get . From (3.1), we have
It follows that
Since , we have from (D3) that
for all .
Thus,
By (3.4), (3.5), and (2.1), we obtain
Noting that T is -Lipschitzian and , we have
Since , we have . From (3.6), we can deduce
Hence,
By (C3) and (3.8), we obtain
Let for all . Then, we have
Since F is -Lipschitzian and ς strongly monotone, we have
Noting that and , without loss of generality, we assume that for all . Thus, . So,
We have from (3.9), (3.10), and (3.12)
From (3.1) and (3.13), we have
By the definition of , we have
Hence,
Since and , it follows from (3.16), (3.3), and (3.9) that
By (3.15), we obtain
Combining (3.17) and (3.18) to deduce
Hence, we obtain
It follows that, hence, we obtain
Next we divide our proof into two possible cases.
Case 1. There exists an integer number m such that for all . In this case, we have exists. Since and , by (3.19), we derive
This together with (3.18) implies that
Note that
So,
By (3.20) and (3.22), we obtain
From (3.2) and (3.9), we have
Hence,
Therefore,
Since is firmly nonexpansive and A is monotone, we have
It follows that
By (3.25) and (3.9), we deduce
Therefore,
Equations (3.20), (3.24), and (3.26) imply that
Notice that is strongly monotone. Thus, the variational inequality of finding such that for all has a unique solution, denoted by , that is, . Next, we prove that
Since is bounded, without loss of generality, we assume that there exists a subsequence of such that for some and
Thus, we have that and
Therefore, .
Next we show that . First, we show that . As a matter of fact, is obvious. Next, we show that .
Take any . We have . Set . We have . Write . Then, . Now we show . In fact,
Since, , we deduce . Thus, . Hence, . Therefore, .
By (3.1), (3.20), and (3.27), we deduce
Next we prove that is demiclosed at 0. Let the sequence satisfying and . Next, we will show that .
Since T is -Lipschizian, we have
It follows that
Hence,
Since is demiclosed at 0 by Lemma 2.1, we immediately deduce . Therefore, is demiclosed at 0. By (3.28), we deduce . Hence, . So,
Note that
It follows that
So,
Applying Lemma 2.3 to (3.30) we deduce .
Case 2. Assume there exists an integer such that . In this case, we set . Then, we have . Define an integer sequence for all as follows:
It is clear that is a non-decreasing sequence satisfying
and
for all . From (3.19), we get
It follows that
By a similar argument to that of (3.29) and (3.30), we can prove that
and
Since , we have from (3.33)
Combining (3.33) and (3.34), we have
and hence
From (3.33), we also obtain
This together with (3.35) imply that
Applying Lemma 2.2 to get
Therefore, . That is, . This completes the proof. □
References
Noor MA, Noor KI: Sensitivity analysis of quasi variational inclusions. J. Math. Anal. Appl. 1999, 236: 290–299. 10.1006/jmaa.1999.6424
Chang SS: Set-valued variational inclusions in Banach spaces. J. Math. Anal. Appl. 2000, 248: 438–454. 10.1006/jmaa.2000.6919
Chang SS: Existence and approximation of solutions of set-valued variational inclusions in Banach spaces. Nonlinear Anal. 2001, 47: 583–594. 10.1016/S0362-546X(01)00203-6
Demyanov VF, Stavroulakis GE, Polyakova LN, Panagiotopoulos PD: Quasidifferentiability and Nonsmooth Modeling in Mechanics, Engineering and Economics. Kluwer Academic, Dordrecht; 1996.
Peng JW, Wang Y, Shyu DS, Yao JC: Common solutions of an iterative scheme for variational inclusions, equilibrium problems and fixed point problems. J. Inequal. Appl. 2008., 2008: Article ID 720371
Yao Y, Cho YJ, Liou YC: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 2011, 212: 242–250. 10.1016/j.ejor.2011.01.042
Yao Y, Cho YJ, Liou YC: Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems. Fixed Point Theory Appl. 2011., 2011: Article ID 101
Agarwal RP, Cho YJ, Petrot N: Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 31
Cho YJ, Qin X, Shang M, Su Y: Generalized nonlinear variational inclusions involving-monotone mappings in Hilbert spaces. Fixed Point Theory Appl. 2007., 2007: Article ID 029653
Cholamjiak P, Cho YJ, Suantai S: Composite iterative schemes for maximal monotone operators in reflexive Banach spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 7
Noor MA: Generalized se-valued variational inclusions and resolvent equations. J. Math. Anal. Appl. 1998, 228: 206–220. 10.1006/jmaa.1998.6127
Hartman P, Stampacchia G: On some nonlinear elliptic differential equations. Acta Math. 1966, 115: 271–310. 10.1007/BF02392210
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3
Takahashi W, Tamura T: Convergence theorems for a pair of nonexpansive mappings. J. Convex Anal. 1998, 5: 45–56.
Fang YP, Huang NJ: H -Monotone operator resolvent operator technique for quasi-variational inclusions. Appl. Math. Comput. 2003, 145: 795–803. 10.1016/S0096-3003(03)00275-3
Ding XP: Perturbed Ishikawa type iterative algorithm for generalized quasivariational inclusions. Appl. Math. Comput. 2003, 141: 359–373. 10.1016/S0096-3003(02)00261-8
Huang NJ: Mann and Ishikawa type perturbed iteration algorithm for nonlinear generalized variational inclusions. Comput. Math. Appl. 1998, 35: 9–14.
Lin LJ: Variational inclusions problems with applications to Ekeland’s variational principle, fixed point and optimization problems. J. Glob. Optim. 2007, 39: 509–527. 10.1007/s10898-007-9153-1
Verma RU:General system of -monotone variational inclusion problems based on generalized hybrid iterative algorithm. Nonlinear Anal. Hybrid Syst. 2007, 1: 326–335. 10.1016/j.nahs.2006.07.002
Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 2010, 147: 27–41. 10.1007/s10957-010-9713-2
Zhang SS, Lee JHW, Chan CK: Algorithms of common solutions for quasi variational inclusion and fixed point problems. Appl. Math. Mech. 2008, 29: 571–581. 10.1007/s10483-008-0502-y
Peng JW, Wang Y, Shyu DS, Yao JC: Common solutions of an iterative scheme for variational inclusions, equilibrium problems and fixed point problems. J. Inequal. Appl. 2008., 2008: Article ID 720371
Zhou H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear Anal. 2009, 70: 4039–4046. 10.1016/j.na.2008.08.012
Mainge PE: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2007, 325: 469–479. 10.1016/j.jmaa.2005.12.066
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332
Acknowledgements
Yonghong Yao was supported in part by NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
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Yao, Y., Agarwal, R.P. & Liou, YC. Iterative algorithms for quasi-variational inclusions and fixed point problems of pseudocontractions. Fixed Point Theory Appl 2014, 82 (2014). https://doi.org/10.1186/1687-1812-2014-82
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DOI: https://doi.org/10.1186/1687-1812-2014-82