- Research
- Open access
- Published:
On weak convergence of an iterative algorithm for common solutions of inclusion problems and fixed point problems in Hilbert spaces
Fixed Point Theory and Applications volume 2013, Article number: 155 (2013)
Abstract
In this paper, a monotone inclusion problem and a fixed point problem of nonexpansive mappings are investigated based on a Mann-type iterative algorithm with mixed errors. Strong convergence theorems of common elements are established in the framework of Hilbert spaces.
MSC:47H05, 47H09, 47J25.
1 Introduction
Variational inclusion has become rich of inspiration in pure and applied mathematics. In recent years, classical variational inclusion problems have been extended and generalized to study a large variety of problems arising in image recovery, economics, and signal processing; for more details, see [1–14]. Based on the projection technique, it has been shown that the variational inclusion problems are equivalent to the fixed point problems. This alternative formulation has played a fundamental and significant part in developing several numerical methods for solving variational inclusion problems and related optimization problems.
The purposes of this paper is to study the zero point problem of the sum of a maximal monotone mapping and an inverse-strongly monotone mapping, and the fixed point problem of a nonexpansive mapping. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a Mann-type iterative algorithm with mixed errors is investigated. A weak convergence theorem is established. Applications of the main results are also discussed in this section.
2 Preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with the inner product and the norm , respectively. Let C be a nonempty closed convex subset of H and let be the metric projection from H onto C.
Let be a mapping. stands for the fixed point set of S; that is, .
Recall that S is said to be nonexpansive iff
If C is a bounded, closed, and convex subset of H, then is not empty, closed, and convex; see [15].
Let be a mapping. Recall that A is said to be monotone iff
A is said to be strongly monotone iff there exists a constant such that
For such a case, A is also said to be α-strongly monotone. A is said to be inverse-strongly monotone iff there exists a constant such that
For such a case, A is also said to be α-inverse-strongly monotone. It is not hard to see that inverse-strongly monotone mappings are monotone and Lipschitz continuous.
Recall that the classical variational inequality is to find an such that
In this paper, we use to denote the solution set of (2.1). It is known that is a solution to (2.1) iff ω is a fixed point of the mapping , where is a constant, and I stands for the identity mapping. If A is α-inverse-strongly monotone and , then the mapping is nonexpansive. Indeed, we have
This shows that is nonexpansive. It follows that is closed and convex.
A multivalued operator with the domain and the range is said to be monotone if for , , , and , we have . A monotone operator T is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. Let I denote the identity operator on H and let be a maximal monotone operator. Then we can define, for each , a nonexpansive single-valued mapping by . It is called the resolvent of T. We know that for all and is firmly nonexpansive, that is,
for more details, see [16–22] and the references therein.
In [19], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator based on the following algorithm:
where is a sequence in , is a positive sequence, is maximal monotone and . They showed that the sequence converges weakly to some provided that the control sequence satisfies some restrictions. Further, using this result, they also investigated the case that , where is a proper lower semicontinuous convex function. Convergence theorems are established in the framework of real Hilbert spaces.
In [16], Takahashi an Toyoda investigated the problem of finding a common solutions of the variational inequality problem (2.1) and a fixed point problem of nonexpansive mappings based on the following algorithm:
where is a sequence in , is a positive sequence, is a nonexpansive mapping and is an inverse-strongly monotone mapping. They showed that the sequence converges weakly to some provided that the control sequence satisfies some restrictions.
In [23], Tada and Takahashi investigated the problem of finding common solutions of an equilibrium problem and a fixed point problem of nonexpansive mappings based on the following algorithm: and
where is a sequence in , is a positive sequence, is a nonexpansive mapping and is a bifunction. They showed that the sequence converges weakly to some provided that the control sequence satisfies some restrictions.
Recently, fixed point and zero point problems have been studied by many authors based on iterative methods; see, for example, [23–34] and the references therein. In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problem and the fixed point problem based on Mann-type iterative methods with errors. Weak convergence theorems are established in the framework of Hilbert spaces.
To obtain our main results in this paper, we need the following lemmas.
Recall that a space is said to satisfy Opial’s condition [35] if, for any sequence with , where ⇀ denotes the weak convergence, the inequality
holds for every with . Indeed, the above inequality is equivalent to the following:
Lemma 2.1 [34]
Let C be a nonempty, closed, and convex subset of H, let be a mapping, and let be a maximal monotone operator. Then , where is the resolvent of B for .
Lemma 2.2 [36]
Let , , and be three nonnegative sequences satisfying the following condition:
where is some nonnegative integer, and . Then the limit exists.
Lemma 2.3 [37]
Suppose that H is a real Hilbert space and for all . Suppose further that and are sequences of H such that
and
hold for some . Then .
Lemma 2.4 [15]
Let C be a nonempty, closed, and convex subset of H. Let be a nonexpansive mapping. Then the mapping is demiclosed at zero, that is, if is a sequence in C such that and , then .
3 Main results
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, let be an α-inverse-strongly monotone mapping, let be a nonexpansive mapping and let B be a maximal monotone operator on H such that the domain of B is included in C. Assume that . Let , , and be real number sequences in such that . Let be a positive real number sequence and let be a bounded error sequence in C. Let be a sequence in C generated in the following iterative process:
for all , where . Assume that the sequences , , , and satisfy the following restrictions:
-
(a)
,
-
(b)
,
-
(c)
,
where a, b, c, and d are some real numbers. Then the sequence generated in (3.1) converges weakly to some point in ℱ.
Proof Notice that is nonexpansive. Indeed, we have
In view of the restriction (b), we find that is nonexpansive. Fixing , we find from Lemma 2.1 that
Put . Since and are nonexpansive, we have
On the other hand, we have
We find that exists with the aid of Lemma 2.2. This in turn implies that and are bounded. Put . Notice that
This implies from the restriction (c) that
We also have
This implies from the restriction (c) that
On the other hand, we have
It follows from Lemma 2.3 that
Notice that
This implies that
It follows that
In view of the restrictions (a), (b), and (c), we obtain that
Notice that
It follows that
On the other hand, we have
Substituting (3.8) into (3.9), we arrive at
It derives that
In view of the restrictions (a) and (c), we find from (3.7) that
Notice that
It follows from (3.4) and (3.10) that
Since is bounded, there exists a subsequence of such that , where ⇀ denotes the weak convergence. From Lemma 2.4, we find that . In view of (3.10), we can choose a subsequence of such that . Notice that
This implies that
That is,
Since B is monotone, we get for any that
Replacing n by and letting , we obtain from (3.10) that
This means , that is, . Hence, we get . This completes the proof that .
Suppose that there is another subsequence of such that . Then we can show that in exactly the same way. Assume that since exits for any . Put . Since the space satisfies Opial’s condition, we see that
This is a contradiction. This shows that . This proves that the sequence converges weakly to . This completes the proof. □
We obtain from Theorem 3.1 the following inclusion problem.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, let be an α-inverse-strongly monotone mapping, and let B be a maximal monotone operator on H such that the domain of B is included in C. Assume that . Let , , and be real number sequences in such that . Let be a positive real number sequence and let be a bounded error sequence in C. Let be a sequence in C generated in the following iterative process:
for all , where . Assume that the sequences , , , and satisfy the following restrictions:
-
(a)
,
-
(b)
,
-
(c)
,
where a, b, c, and d are some real numbers. Then the sequence converges weakly to some point in .
Let be a proper lower semicontinuous convex function. Define the subdifferential
for all . Then ∂f is a maximal monotone operator of H into itself; for more details, see [38]. Let C be a nonempty closed convex subset of H and let be the indicator function of C, that is,
Furthermore, we define the normal cone of C at v as follows:
for any . Then is a proper lower semicontinuous convex function on H and is a maximal monotone operator. Let for any and . From and , we get
where is the metric projection from H into C. Similarly, we can get that . Putting in Theorem 3.1, we find that . The following results are not hard to derive.
Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H, let be an α-inverse-strongly monotone mapping and let be a nonexpansive mapping. Assume that . Let , , and be real number sequences in such that . Let be a positive real number sequence and let be a bounded error sequence in C. Let be a sequence in C generated in the following iterative process:
for all . Assume that the sequences , , , and satisfy the following restrictions:
-
(a)
,
-
(b)
,
-
(c)
,
where a, b, c, and d are some real numbers. Then the sequence converges weakly to some point in ℱ.
In view of Theorem 3.3, we have the following result.
Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H and let be an α-inverse-strongly monotone mapping such that . Let , , and be real number sequences in such that . Let be a positive real number sequence and let be a bounded error sequence in C. Let be a sequence in C generated in the following iterative process:
for all . Assume that the sequences , , , and satisfy the following restrictions:
-
(a)
,
-
(b)
,
-
(c)
,
where a, b, c, and d are some real numbers. Then the sequence converges weakly to some point in .
Let F be a bifunction of into ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem.
In this paper, we use to denote the solution set of the equilibrium problem.
To study the equilibrium problems, we may assume that F satisfies the following conditions:
-
(A1)
for all ;
-
(A2)
F is monotone, i.e., for all ;
-
(A3)
for each ,
-
(A4)
for each , is convex and weakly lower semi-continuous.
Putting for every , we see that the equilibrium problem is reduced to the variational inequality (2.1).
The following lemma can be found in [39].
Lemma 3.5 Let C be a nonempty closed convex subset of H and let be a bifunction satisfying (A1)-(A4). Then, for any and , there exists such that
Further, define
for all and . Then the following hold:
-
(a)
is single-valued,
-
(b)
is firmly nonexpansive, i.e., for any ,
-
(c)
,
-
(d)
is closed and convex.
Lemma 3.6 [5]
Let C be a nonempty closed convex subset of a real Hilbert space H, let F a bifunction from to ℝ which satisfies (A1)-(A4) and let be a multivalued mapping of H into itself defined by
Then is a maximal monotone operator with the domain , and
where is defined as in (3.13).
Theorem 3.7 Let C be a nonempty closed convex subset of a real Hilbert space H, let be a nonexpansive mapping and let F be a bifunction from to ℝ which satisfies (A1)-(A4). Assume that . Let , , and be real number sequences in such that . Let be a positive real number sequence and let be a bounded error sequence in C. Let be a sequence in C generated in the following iterative process:
for all , where such that
Assume that the sequences , , , and satisfy the following restrictions:
-
(a)
,
-
(b)
,
-
(c)
,
where a, b, c, and d are some real numbers. Then the sequence converges weakly to some point in ℱ.
References
Douglas J, Rachford HH: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 1956, 82: 421–439. 10.1090/S0002-9947-1956-0084194-4
Shen J, Pang LP: An approximate bundle method for solving variational inequalities. Commun. Optim. Theory 2012, 1: 1–18.
Lions PL, Mercier B: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 1979, 16: 964–979. 10.1137/0716071
Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017
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
Abdel-Salam HS, Al-Khaled K: Variational iteration method for solving optimization problems. J. Math. Comput. Sci. 2012, 2: 1475–1497.
Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618.
Noor MA, Noor KI, Waseem M: Decomposition method for solving system of linear equations. Eng. Math. Lett. 2013, 2: 31–41.
Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199
Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.
Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008
He RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces. Adv. Fixed Point Theory 2012, 2: 47–57.
Kellogg RB: Nonlinear alternating direction algorithm. Math. Comput. 1969, 23: 23–27. 10.1090/S0025-5718-1969-0238507-3
Cho SY, Kang SM: Zero point theorems of m -accretive operators in a Banach space. Fixed Point Theory 2012, 13: 49–58.
Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 1976, 18: 78–81.
Takahahsi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003, 118: 417–428. 10.1023/A:1025407607560
Qin X, Cho YJ, Kang SM: Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim. 2010, 46: 75–87. 10.1007/s10898-009-9410-6
Eshita K, Takahashi W: Approximating zero points of accretive operators in general Banach spaces. Fixed Point Theory Appl. 2007, 2: 105–116.
Kammura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 2000, 106: 226–240. 10.1006/jath.2000.3493
Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl. 2007, 329: 415–424. 10.1016/j.jmaa.2006.06.067
Yuan Q, Shang M: Convergence of an extragradient-like iterative algorithm for monotone mappings and nonexpansive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 67
Luo H, Wang Y: Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci. 2012, 2: 1660–1670.
Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mappings and an equilibrium problem. J. Optim. Theory Appl. 2007, 133: 359–370. 10.1007/s10957-007-9187-z
Noor MA, Huang Z: Some resolvent iterative methods for variational inclusions and nonexpansive mappings. Appl. Math. Comput. 2007, 194: 267–275. 10.1016/j.amc.2007.04.037
Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011
Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi- ϕ -nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 2010, 234: 750–760. 10.1016/j.cam.2010.01.015
Shehu Y: An iterative method for fixed point problems, variational inclusions and generalized equilibrium problems. Math. Comput. Model. 2011, 54: 1394–1404. 10.1016/j.mcm.2011.04.008
Saejung S, Yotkae P: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 2012, 75: 742–750. 10.1016/j.na.2011.09.005
Saejung S, Wongchan K, Yotkae P: Another weak convergence theorems for accretive mappings in Banach spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 26
Cho YJ, Kang SM, Zhou H: Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces. J. Inequal. Appl. 2008., 2008: Article ID 598191
Wu C, Liu A: Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems. Fixed Point Theory Appl. 2012., 012: Article ID 90
Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031
Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10
Aoyama K, et al.: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear Convex Anal. 2007, 8: 471–489.
Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0
Tan KK, Xu HK: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 1992, 114: 399–404. 10.1090/S0002-9939-1992-1068133-2
Schu J: Weak and strong convergence to fixed points of a asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Acknowledgements
The author is grateful to the reviewers for useful suggestions which improved the contents of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Hecai, Y. On weak convergence of an iterative algorithm for common solutions of inclusion problems and fixed point problems in Hilbert spaces. Fixed Point Theory Appl 2013, 155 (2013). https://doi.org/10.1186/1687-1812-2013-155
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-155