- Research
- Open access
- Published:
Weak and strong convergence theorems for common zeros of accretive operators
Journal of Inequalities and Applications volume 2014, Article number: 282 (2014)
Abstract
In this paper, we propose two proximal point algorithms for investigating common zeros of a family of accretive operators. Weak and strong convergence of the two algorithms are obtained in a Banach space.
MSC:47H06, 47H09, 47J25.
1 Introduction and preliminaries
Let E be a real Banach space, and let be the dual space of E. Let be a positive real number set. Let be a continuous strictly increasing function such that and as . This function φ is called a gauge function. The duality mapping associated with a gauge function φ is defined by
where denotes the generalized duality pairing. In the case that , we write J for and call J the normalized duality mapping.
Following Browder [1], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping is single-valued and weak-to-weak∗ sequentially continuous (i.e., if is a sequence in E weakly convergent to a point x, then the sequence converges weakly∗ to ). It is known that has a weakly continuous duality mapping with a gauge function for all .
Let . E is said to be smooth or is said to be have a Gâteaux differentiable norm if the limit exists for each . E is said to have a uniformly Gâteaux differentiable norm if for each , the limit is attained uniformly for all . E is said to be uniformly smooth or is said to have a uniformly Fréchet differentiable norm if the limit is attained uniformly for .
It is well known that Fréchet differentiability of the norm of E implies Gâteaux differentiability of the norm of E. It is known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single-valued and uniformly norm-to-weak∗ continuous on each bounded subset of E.
Let D be a nonempty subset of a set C. Let . Q is said to be
-
(1)
a contraction if ;
-
(2)
sunny if for each and , we have ;
-
(3)
a sunny nonexpansive retraction if is sunny, nonexpansive and a contraction.
D is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from C onto D. The following result, which was established in [2–4], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Let E be a smooth Banach space, and let C be a nonempty subset of E. Let be a retraction and be the duality mapping on E. Then the following are equivalent:
-
(1)
is sunny and nonexpansive;
-
(2)
, ;
-
(3)
, , .
It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from E onto C. Let C be a nonempty closed convex subset of a smooth Banach space E, let and let . Then we have from the above that if and only if for all , where is a sunny nonexpansive retraction from E onto C.
A Banach space E is said to be strictly convex if and only if
for and implies that .
E is said to be uniformly convex if for any there exists such that for any ,
It is known that a uniformly convex Banach space is reflexive and strictly convex.
Let C be a nonempty closed convex subset of E. Let be a mapping. In this paper, we use to denote the set of fixed points of S. Recall that S is said to be nonexpansive iff , . For the existence of fixed points of a nonexpansive mapping, we refer the readers to [5]. Let x be a fixed element in C, and let S be a nonexpansive mapping with a nonempty fixed point set. For each , let be the unique solution of the equation . In the framework of uniformly smooth Banach spaces, Reich [6] proved that converges strongly to a fixed point , where is the unique sunny nonexpansive retraction from C onto , of S as . Xu [7] further extended the results to the framework of reflexive Banach spaces; for more details, see [7] and [8] and the references therein.
Let I denote the identity operator on E. An operator with domain and range is said to be accretive if for each and , , there exists such that . An accretive operator A is said to be M-accretive if for all . In this paper, we use to denote the set of zero points of A. For an accretive operator A, we can define a nonexpansive single-valued mapping by for each , which is called the resolvent of A. Set
then
where ∂ denotes the sub-differential in the sense of convex analysis.
Zero problems of accretive operators recently have been extensively studied (see [6–21] and the references therein) because of their important applications in real world. Proximal point algorithm, which was proposed by Martinet [22, 23] and generalized by Rockafellar [24, 25], is a classical method for investigating zeros of monotone operators. In this paper, we propose two proximal point algorithms for investigating common zeros of a family of m-accretive operators. Weak and strong convergence theorems are established in Banach spaces.
The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [26].
Lemma 1.1 Assume that a Banach space E has a weakly continuous duality mapping with gauge φ.
-
(i)
For all , the following inequality holds:
In particular, for all ,
-
(ii)
Assume that a sequence in E converges weakly to a point .
Then the following identity holds:
Lemma 1.2 [7]
Let E be a reflexive Banach space and have a weakly continuous duality map with gauge φ. Let C be a closed convex subset of E, and let be a nonexpansive mapping. Fix and . Let be the unique fixed point of the mapping . Then S has a fixed point if and only if remains bounded as , and in this case, converges as strongly to a fixed point of S. Define a mapping by . Then is the sunny nonexpansive retraction from C onto .
Lemma 1.3 [27]
Let C be a closed convex subset of a strictly convex Banach space E. Let be some positive integer, and let be a nonexpansive mapping. Suppose that is nonempty. Then the mapping , where is a real number sequence in such that , is nonexpansive with .
Lemma 1.4 [28]
Let , and be three nonnegative real sequences satisfying , , where is some positive integer, is a number sequence in such that , is a number sequence such that . Then .
Lemma 1.5 [29]
Let E be a uniformly convex Banach space, be a positive number, and be a closed ball of E. There exits a continuous, strictly increasing and convex function with such that
for all and such that .
Lemma 1.6 [30]
Let E be a uniformly convex Banach space. Let C be a nonempty closed convex subset of E, and let be a nonexpansive mapping. Then is demiclosed at zero.
2 Main results
Theorem 2.1 Let E be a uniformly convex Banach space which has the Opial condition. Let be some positive integer, and let be an M-accretive operator in E for each . Assume that is convex. Let and be real number sequences in , and let be a positive real number sequence. Assume that is not empty. Let be a sequence generated in the following manner: and
where . Assume that the following restrictions are satisfied:
-
(a)
;
-
(b)
and ,
where a, b and c are real numbers. Then the sequence converges weakly to .
Proof We start the proof with the boundedness of the sequence . Fixing , we find that
This shows that the limit exists. This implies that is bounded. Using Lemma 1.5, we find that
This implies that
In view of restriction (a), we find that . It follows that . Using restriction (b), we arrive at for each . Since is bounded, we see that there exists a subsequence of converging weakly to . Using Lemma 1.6, we obtain that . This proves that .
Next we show that converges weakly to . Supposing the contrary, we see that there exists some subsequence of such that converges weakly to , where . Similarly, we can show . Note that we have proved that exists for every . Assume that , where d is a nonnegative number. Since the space has the Opial condition [31], we see that
This is a contradiction. Hence . This completes the proof. □
Theorem 2.2 Let E be a strictly convex and reflexive Banach space which has a weakly continuous duality map . Let be some positive integer, and let be an M-accretive operator in E for each . Assume that is convex. Let and be real number sequences in , and let be a positive real number sequence for each . Assume that is not empty. Let be a sequence generated in the following manner: and
where x is a fixed element in and . Assume that the following restrictions are satisfied:
-
(a)
, and ;
-
(b)
, and .
Then the sequence converges strongly to , where is the unique sunny nonexpansive retract from onto .
Proof We start the proof with the boundedness of the sequence . Fixing , we find that
This implies that . This shows that is bounded. Put . It follows that
This implies that
In light of restrictions (a) and (b), we find that
Set . It follows from Lemma 1.3 that S is nonexpansive with . Note that
In view of (2.1), we find from the restrictions (a) and (b) that
Now, we are in a position to prove
By Lemma 1.2, we have the sunny nonexpansive retraction . Take a subsequence of such that
Since E is reflexive, we may further assume that for some . Since is weakly continuous, we have from Lemma 1.1
Put , . It follows that
From (2.2), we have
Using (2.5), we have
Combining (2.6) with (2.7), we obtain that
Hence ; that is, . It follows from (2.4) that
Finally, we prove that as . Using Lemma 1.1, we find that
Using Lemma 1.4, we see that . This implies that
This completes the proof. □
3 Applications
In this section, we give an application of Theorem 2.1 in the framework of Hilbert spaces.
Let C be a nonempty closed and convex subset of a Hilbert space H. Let F be a bifunction of into ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem:
To study the equilibrium problem (3.1), we may assume that F satisfies the following restrictions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each , ;
(A4) for each , is convex and lower semi-continuous.
Lemma 3.1 [32]
Let F be 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 , , where stands for the solution set of (3.1), and
where is defined by
Corollary 3.2 Let C be a nonempty closed and convex subset of a Hilbert space H. Let be some positive integer, and let be a bifunction satisfying (A1)-(A4). Let and be real number sequences in , and let be a positive real number sequence. Assume that is not empty. Let be a sequence generated in the following manner: and
where . Assume that the following restrictions are satisfied:
-
(a)
;
-
(b)
and ,
where a, b and c are real numbers. Then the sequence converges weakly to .
Corollary 3.3 Let C be a nonempty closed and convex subset of a Hilbert space H. Let be some positive integer, and let be a bifunction satisfying (A1)-(A4). Let and be real number sequences in , and let be a positive real number sequence. Assume that is not empty. Let be a sequence generated in the following manner: and
where x is a fixed element in C and . Assume that the following restrictions are satisfied:
-
(a)
, and ;
-
(b)
, and .
Then the sequence converges strongly to , where is the metric projection from C onto .
References
Browder FE: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 1967, 100: 201–225. 10.1007/BF01109805
Bruck RE: Nonexpansive projections on subsets of Banach spaces. Pac. J. Math. 1973, 47: 341–355. 10.2140/pjm.1973.47.341
Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.
Reich S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 1973, 44: 57–70. 10.1016/0022-247X(73)90024-3
Browder FE: Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. USA 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272
Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 1980, 75: 287–292. 10.1016/0022-247X(80)90323-6
Xu HK: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 2006, 314: 631–643. 10.1016/j.jmaa.2005.04.082
Qin X, Cho SY, Wang L: Iterative algorithms with errors for zero points of m -accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 148
Kim JK: Convergence of Ishikawa iterative sequences for accretive Lipschitzian mappings in Banach spaces. Taiwan. J. Math. 2006, 10: 553–561.
Zegeye H, Shahzad N: Strong convergence theorems for a common zero of a finite family of m -accretive mappings. Nonlinear Anal. 2007, 66: 1161–1169. 10.1016/j.na.2006.01.012
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
Yang S: Zero theorems of accretive operators in reflexive Banach spaces. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 2
Song J, Chen M: A modified Mann iteration for zero points of accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 347
Cho SY, Kang SM: Zero point theorems for m -accretive operators in a Banach space. Fixed Point Theory 2012, 13: 49–58.
Hao Y: Zero theorems of accretive operators. Bull. Malays. Math. Soc. 2011, 34: 103–112.
Yuan Q, Cho SY: A regularization algorithm for zero points of accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 341
Kim JK, Buong N: Regularization inertial proximal point algorithm for monotone hemicontinuous mapping and inverse strongly monotone mappings in Hilbert spaces. J. Inequal. Appl. 2010., 2010: Article ID 451916
Qing Y, Cho SY, Qin X: Convergence of iterative sequences for common zero points of a family of m -accretive mappings in Banach spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 216173
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.
Wu C, Lv S, Zhang Y: Some results on zero points of m -accretive operators in reflexive Banach spaces. Fixed Point Theory Appl. 2014., 2014: Article ID 118
Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199
Martinet B: Régularisation d’inéquations variationelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér. 1970, 4: 154–158.
Martinet B: Détermination approchée d’un point fixe d’une application pseudo-contractante. C. R. Acad. Sci. Paris, Ser. A-B 1972, 274: 163–165.
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056
Rockafellar RT: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1976, 1: 97–116. 10.1287/moor.1.2.97
Lim TC, Xu HK: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. 1994, 22: 1345–1355. 10.1016/0362-546X(94)90116-3
Bruck RE: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 1973, 179: 251–262.
Liu L: Ishikawa-type and Mann-type iterative processes with errors for constructing solutions of nonlinear equations involving m -accretive operators in Banach spaces. Nonlinear Anal. 1998, 34: 307–317. 10.1016/S0362-546X(97)00579-8
Hao Y: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 218573
Gornicki J: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment. Math. Univ. Carol. 1989, 30: 249–252.
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
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
Acknowledgements
This first author thanks the Fundamental Research Funds for the Central Universities (2014ZD44). The authors are grateful to the reviewers’ suggestions which improved the contents of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors designed the algorithms and established weak and strong convergence analysis. Both authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Wang, S., Li, T. Weak and strong convergence theorems for common zeros of accretive operators. J Inequal Appl 2014, 282 (2014). https://doi.org/10.1186/1029-242X-2014-282
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-282