- Research
- Open access
- Published:
Some new iterative algorithms with errors for common solutions of two finite families of accretive mappings in a Banach space
Fixed Point Theory and Applications volume 2014, Article number: 176 (2014)
Abstract
The purpose of this paper is to prove some new theorems of strong convergence to common solutions for two finite families of accretive mappings in a real uniformly smooth and uniformly convex Banach space by means of some new iterative algorithms with errors, which extend the corresponding works by some authors. As applications, the theorems of strong convergence to common fixed points of two finite families of pseudo-contractive mappings are presented.
MSC:47H05, 47H09, 47H10.
1 Introduction
Let E be a real Banach space with norm and let denote the dual space of E. We use → and ⇀ to denote strong and weak convergence, respectively. We denote the value of at by .
We use J to denote the normalized duality mapping from E to , which is defined by
It is well known that J is single-valued if is strictly convex. Moreover, , for and . We call J weakly sequentially continuous if each which converges weakly to x implies that converges in the sense of weak∗ to Jx.
Let C be a nonempty, closed, and convex subset of E and Q be a mapping of E onto C. Then Q is said to be sunny [1] if , for all and .
A mapping Q of E into E is said to be a retraction [1] if . If a mapping Q is a retraction, then for every , where is the range of Q.
A mapping is said to be nonexpansive if , for . We use to denote the fixed point set of T, that is, . A mapping is said to be demiclosed at p if whenever is a sequence in such that and then .
A subset C of E is said to be a sunny nonexpansive retract of E [2] if there exists a sunny nonexpansive retraction of E onto C and it is called a nonexpansive retract of E if there exists a nonexpansive retraction of E onto C. If E is reduced to a Hilbert space H, then the metric projection is a sunny nonexpansive retraction from H to any closed and convex subset C of H. But this is not true in a general Banach space. We note that if E is smooth and Q is a retraction of C onto , then Q is sunny and nonexpansive if and only if for , , [3].
A mapping is called pseudo-contractive [2] if there exists such that holds for all .
Interest in pseudo-contractive mappings stems mainly from their firm connection with the important class of nonlinear accretive mappings. A mapping is said to be accretive if , for , , , and . If A is accretive, then we can define, for each , a nonexpansive single-valued mapping by , which is called the resolvent of A. We also know that for an accretive mapping A, , where . An accretive mapping A is said to be m-accretive if , for .
It is well known that if A is an accretive mapping, then the solutions of the problem correspond to the equilibrium points of some evolution equations. Hence, the problem of finding a solution with has been studied by many researchers (see [4–12] and the references contained therein).
One classical method for studying the problem , where A is an m-accretive mapping, is the following so-called proximal method (cf. [4]), presented in a Hilbert space:
where . It was shown that the sequence generated by (1.1) converges weakly or strongly to a zero point of A under some conditions.
On the other hand, one explicit iterative process was first introduced, in 1967, by Halpern [13] in the frame of Hilbert spaces:
where and is a nonexpansive mapping. It was proved that under some conditions, the sequence produced by (1.2) converges strongly to a point in .
In 2007, Qin and Su [6] presented the following iterative algorithm:
They showed that generated by (1.3) converges strongly to a point in .
Motivated by iterative algorithms (1.1) and (1.2), Zegeye and Shahzad extended their discussion to the case of finite m-accretive mappings. They presented in [14] the following iterative algorithm:
where with and . If , they proved that generated by (1.4) converges strongly to the common point in () under some conditions.
The work in [14] was then extended to the following one presented by Hu and Liu in [15]:
where with and . We have and . If , they proved that converges strongly to the common point in () under some conditions.
In 2009, Yao et al. presented the following iterative algorithm in the frame of Hilbert space in [16]:
Here is a nonexpansive mapping with . Suppose and are two real sequences in satisfying
-
(a)
and ;
-
(b)
.
Then constructed by (1.6) converges strongly to a point in .
The following lemma is commonly used in proving the convergence of the iterative algorithms in a Banach space.
Lemma 1.1 ([17])
Let E be a real uniformly smooth Banach space, then there exists a nondecreasing continuous function with and for , such that for all , the following inequality holds:
Motivated by the work in [14] and [16], and after imposing an additional condition on the function β in Lemma 1.1 that
where is a constant satisfying some conditions, Shehu and Ezeora presented the following result.
Theorem 1.1 ([2])
Let E be a real uniformly smooth and uniformly convex Banach space, and let C be a nonempty, closed, and convex sunny nonexpansive retract of E, where is the sunny nonexpansive retraction of E onto C. Supposed the duality mapping is weakly sequentially continuous. For each , let be an m-accretive mapping such that . Let satisfy (a) and (b). Let be generated iteratively by
Here with , for . , for , and . Then converges strongly to the common point in , where .
How do we show the convergence of the iterative sequence in (1.8) if β loses the additional condition (1.7)? How about the convergence of if different has different coefficient in (1.8)?
To answer these questions, Wei and Tan presented the following iterative scheme in [18]:
where is the error sequence and is a finite family of m-accretive mappings. , , for , , , for . Some strong convergence theorems are obtained.
In this paper, our main purpose is to extend the discussion of (1.9) from one family of m-accretive mappings to that of two families of m-accretive mappings and . We shall first present and study the following three-step iterative algorithm (A) with errors :
where , and . For , . For , . and are real numbers in and , . , for , and , for and .
Later, we introduce and study the following one:
where , are real numbers in , , and and , for and .
More details will be presented in Section 3. Some strong convergence theorems are obtained, which can be regarded as the extension of the work done in [2, 6, 14, 15, 18], etc. As a consequence, some new iterative algorithms are constructed to converge strongly to the common fixed point of two finite families of pseudo-contractive mappings from C to E.
2 Preliminaries
Now, we list some results we need in sequel.
Lemma 2.1 ([19])
Let E be a real uniformly convex Banach space and let C be a nonempty, closed, and convex subset of E and is a nonexpansive mapping such that , then is demiclosed at zero.
Lemma 2.2 ([15])
Let E be a strictly convex Banach space which has a uniformly Gâteaux differential norm, and let C be a nonempty, closed, and convex subset of E. Let be a finite family of accretive mappings with , satisfying the following range conditions:
Let be real numbers in such that and , where and , then is nonexpansive and .
Lemma 2.3 ([12])
In a real Banach space E, the following inequality holds:
where .
Lemma 2.4 ([20])
Let , , and be three sequences of nonnegative real numbers satisfying
where such that (i) and , (ii) either or . Then .
Lemma 2.5 ([21])
Let and be two bounded sequences in a Banach space E such that , for . Suppose satisfying . If , then .
Lemma 2.6 ([22])
Let E be a Banach space and let A be an m-accretive mapping. For , , and , we have
where and .
3 Main results
Lemma 3.1 ([2])
Let E be a real uniformly smooth and uniformly convex Banach space. Let C be a nonempty, closed, and convex sunny nonexpansive retract of E, and be the sunny nonexpansive retraction of E onto C. Let be nonexpansive with . Suppose that the duality mapping is weakly sequentially continuous. If for each , define by
Then is a contraction and has a fixed point , which satisfies , as .
Lemma 3.2 ([2])
Under the assumptions of Lemma 3.1, suppose further that β in Lemma 1.1 satisfies (1.7), where is a sufficiently large constant such that , is in and , then .
Remark 3.1 Lemma 1.1 with additional condition (1.7) is employed as a key tool to prove Lemma 3.2. In the following lemma, we shall show that Lemma 2.3 can be used instead of Lemma 1.1, which simplifies the proof and weakens the assumption.
Lemma 3.3 Only under the assumptions of Lemma 3.1, the result of Lemma 3.2 is true, which ensures that the assumption is weaker than that in Lemma 3.2.
Proof To show that , it suffices to show that for any sequence such that , we have .
In fact, Lemma 3.1 implies that such that , . By using Lemma 2.3, we have for ,
This implies that
In particular,
Since ,
is bounded.
Without loss of generality, we can assume that converges weakly to . Using Lemma 3.1 and Lemma 2.1, we have .
Substituting for p in (3.3), we obtain
Then from (3.4) and the weak convergence of J, we have , as .
Then from , we see that , as .
Suppose there exists another sequence , as and . Then from Lemma 3.1 that and is demi-closed at zero, we have . Moreover, repeating the above proof, we have , as . Next, we want to show that .
Using (3.2), we have
By letting , (3.5) implies that
Interchanging and in (3.6), we obtain
Then (3.6) and (3.7) ensure
which implies that .
Therefore, .
This completes the proof. □
Lemma 3.4 Let E be a strictly convex Banach space and let C be a nonempty, closed, and convex subset of E. Let () be a finite family of m-accretive mappings such that .
Let be real numbers in such that and , where and , for , and , then is nonexpansive and , for .
Proof The proof is from [18]. For later use, we present the proof in the following.
It is easy to check that is nonexpansive and .
On the other hand, for , then .
For , we have
Therefore, , which implies that . Similarly, .
Then , which implies from the strict convexity of E that .
Therefore, , for . We have , which completes the proof. □
Similar to Lemma 3.4, we have the following lemma.
Lemma 3.5 Let E and C be the same as those in Lemma 3.4. Let be a finite family of m-accretive mappings such that .
Let be real numbers in such that and , where and , for , then is nonexpansive and , for .
Lemma 3.6 Let E, C, , and be the same as those in Lemmas 3.4 and 3.5. Suppose . Then are nonexpansive and .
Proof From Lemmas 3.4 and 3.5, we can easily check that are nonexpansive and . So, it suffices to show that since is trivial.
For , then .
For , then . Now,
Then repeating the discussion in Lemma 3.4, we know that . Then , thus , which completes the proof. □
Theorem 3.1 Let E be a real uniformly smooth and uniformly convex Banach space. Let C be a nonempty, closed, and convex sunny nonexpansive retract of E, where is the sunny nonexpansive retraction of E onto C. Let be m-accretive mappings, where , . Suppose that the duality mapping is weakly sequentially continuous and . Let be generated by the iterative algorithm (A), where , and , for , , for , . , where , for , , for , . Suppose , , , and are three sequences in , and satisfy the following conditions:
-
(i)
, , as ;
-
(ii)
;
-
(iii)
;
-
(iv)
and , for and ;
-
(v)
and , for and ;
-
(vi)
, as , and .
Then converges strongly to a point .
Proof We shall split the proof into five steps:
Step 1. , , , , and are all bounded.
We shall first show that ,
where .
By using the induction method, we see that for , ,
Suppose that (3.9) is true for . Then, for ,
Thus (3.9) is true for all . Since , (3.9) ensures that is bounded.
For , from , we see that is bounded.
Since , is bounded. Since both and are bounded, is bounded. Similarly, , , , , and are all bounded, for ; .
Then we set .
Step 2. and .
In fact,
Next, we discuss .
If , then, using Lemma 2.6,
If , then imitating the proof of (3.11), we have
Combining (3.11) and (3.12), we have
Putting (3.13) into (3.10), we have
Similarly, we have
and
Therefore,
Thus . Using Lemma 2.5, we have from (3.17) and then .
Step 3. .
In fact,
Then (3.18) and step 2 imply that , as , since .
Step 4. , where is an element in D.
From Lemma 3.6, we know that is nonexpansive and . Then Lemma 3.1 and Lemma 3.3 imply that there exists such that for . Moreover, , as .
Since , is bounded. Let . Then from step 1, we know that is a positive constant. Using Lemma 2.3, we have
So , which implies that in view of step 3.
Since is bounded and J is uniformly continuous on each bounded subset of E, , as .
Moreover, noticing the fact that
we have .
Since , and J is uniformly continuous on each bounded subset of E,
Step 5. , as , where is the same as in step 4.
Let . By using Lemma 2.3 again, we have
Let , then (3.20) reduces to .
From (3.19), (3.20), and the assumptions, by using Lemma 2.4, we know that , as .
This completes the proof. □
If in Theorem 3.1, , then we have the following theorem.
Theorem 3.2 Let E and D be the same as those in Theorem 3.1. Suppose that the duality mapping is weakly sequentially continuous. Let () and () be two finite families of m-accretive mappings. Let , , and satisfy the some conditions presented in Theorem 3.1.
Let be generated by the following scheme:
Then converges strongly to a point , where and are the same as those in Theorem 3.1.
Lemma 3.7 Let E, C and be the same as those in Lemma 3.5. .
Let be real numbers in such that and , where and , for , and , then is nonexpansive and , for .
Proof It is easy to check that is nonexpansive and .
On the other hand, for , then .
For , then
Therefore, , which implies that . Similarly, .
Then , which implies from the strict convexity of E that .
Therefore, , and then we can easily see that , for . Thus , which completes the proof. □
Lemma 3.8 Let E and C be the same as those in Lemma 3.4. Let and be the same as those in Lemmas 3.4 and 3.7, respectively. Suppose . Then are nonexpansive and .
Proof From Lemmas 3.4 and 3.7, we can easily check that are nonexpansive and . So, it suffices to show that since is trivial.
For , then .
For , then . Now,
Then repeating the discussion in Lemma 3.4, we know that . Then , thus , which completes the proof. □
Theorem 3.3 Let E, C, , , and D be the same as those in Theorem 3.1. Let be m-accretive mappings, for , and . Suppose that the duality mapping is weakly sequentially continuous and . Let be generated by the iterative algorithm (B), where , and , for , , for , and . Suppose , , , and are three sequences in and satisfy the following conditions:
-
(i)
, , as ;
-
(ii)
;
-
(iii)
;
-
(iv)
and , for and ;
-
(v)
and , for and ;
-
(vi)
, as , and .
Then converges strongly to a point .
Proof We shall split the proof into five steps:
Step 1. , , , , and are all bounded.
Similar to the proof of step 1 in Theorem 3.1, we can get the result of step 1.
Then are all bounded.
Step 2. and .
In fact,
Similar to (3.13), we know that
where .
Repeating (3.22), we have
Then (3.22) and (3.23) imply that
By induction, we have
Going back to (3.21), we have
Therefore, similar to (3.17), we have
Thus . Using Lemma 2.5, we have from (3.27) and then .
Similar to Theorem 3.1, we have
Step 3. .
Step 4. , where is an element in D.
From Lemma 3.8, we know that is nonexpansive and . Then Lemma 3.1 and Lemma 3.3 imply that there exists such that for . Moreover, , as . Then copy step 4 in Theorem 3.1, the result follows.
Step 5. , which is the same as that in step 4.
Copy step 5 in Theorem 3.1, the result follows.
This completes the proof. □
If in Theorem 3.3, , then we have the following theorem.
Theorem 3.4 Let E and D be the same as those in Theorem 3.3. Suppose that the duality mapping is weakly sequentially continuous. Let () and () be two finite families of m-accretive mappings. Let , and satisfy the some conditions presented in Theorem 3.3.
Let be generated by the following scheme:
Then converges strongly to a point , where and are the same as those in Theorem 3.3.
Next, we apply Theorems 3.1 and 3.3 to the cases of finite pseudo-contractive mappings.
Theorem 3.5 Let E be a real uniformly smooth and uniformly convex Banach space. Let C be a nonempty, closed, and convex sunny nonexpansive retract of E, where is the sunny nonexpansive retraction of E onto C. Let be pseudo-contractive mappings such that and are m-accretive, where , . Suppose that the duality mapping is weakly sequentially continuous and . Let be generated by the iterative algorithm (A), where , and , for , , for , . , where , for , , for , . Suppose , , , and are three sequences in and satisfying the following conditions:
-
(i)
, , as ;
-
(ii)
;
-
(iii)
;
-
(iv)
and , for and ;
-
(v)
and , for and ;
-
(vi)
, as , and .
Then converges strongly to a point .
Proof Let and , for and . Then the result follows from Theorem 3.1. □
Similarly, from Theorem 3.3, we have the following result.
Theorem 3.6 Let E, C, and D be the same as those in Theorem 3.5. Let be pseudo-contractive mappings such that and are m-accretive mappings, where , . Suppose that the duality mapping is weakly sequentially continuous and . Let be generated by the iterative algorithm (B), where is the same as that in Theorem 3.5 and , where , for , , for , . Suppose , , , and are three sequences in and satisfying the following conditions:
-
(i)
, , as ;
-
(ii)
;
-
(iii)
;
-
(iv)
and , for and ;
-
(v)
and , for and ;
-
(vi)
, as , and .
Then converges strongly to a point .
References
Takahashi W: Proximal point algorithms and four resolvents of nonlinear operators of monotone type in Banach spaces. Taiwan. J. Math. 2008, 12(8):1883–1910.
Shehu Y, Ezeora JN: Path convergence and approximation of common zeroes of a finite family of m -accretive mappings in Banach spaces. Abstr. Appl. Anal. 2010., 2010: Article ID 285376
Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Application. Yokohama Publishers, Yokohama; 2000.
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14(5):877–898.
Chen RD, Liu YJ, Shen XL: Iterative approximation of a zero of accretive operator in Banach space. Nonlinear Anal. 2009, 71: e346-e350.
Qin XL, Su YF: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl. 2007, 329: 415–424.
Mainge PE: Viscosity methods for zeroes of accretive operators. J. Approx. Theory 2006, 140: 127–140.
Qin XL, Cho SY, Wang L: Iterative algorithms with errors for zero points of m -accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 148
Ceng LC, Wu SY, Yao JC: New accuracy criteria for modified approximate proximal point algorithms in Hilbert spaces. Taiwan. J. Math. 2008, 12(7):1691–1705.
Xu HK: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 2006, 314: 631–643.
Cho JC, Kang SM, Zhou HY: Approximate proximal point algorithms for finding zeroes of maximal monotone operator in Hilbert spaces. J. Inequal. Appl. 2008., 2008: Article ID 598191
Ceng LC, Khan AR, Ansari QH, Yao JC: Strong convergence of composite iterative schemes for zeros of m -accretive operators in Banach spaces. Nonlinear Anal. 2009, 70: 1830–1840.
Halpern B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 1967, 73: 957–961.
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.
Hu LG, Liu LW: A new iterative algorithm for common solutions of a finite family of accretive operators. Nonlinear Anal. 2009, 70: 2344–2351.
Yao Y, Liou YC, Marino G: Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 279058
Reich S: Approximating fixed points of nonexpansive mappings. Panam. Math. J. 1994, 4(2):23–28.
Wei L, Tan RL: Iterative scheme with errors for common zeros of finite accretive mappings and nonlinear elliptic systems. Abstr. Appl. Anal. 2014., 2014: Article ID 646843
Browder FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 1965, 54: 1041–1044.
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256.
Suziki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005., 2005: Article ID 685918
Barbu V: Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff, Leyden; 1976.
Acknowledgements
This paper is supported by the National Natural Science Foundation of China (No. 11071053), Natural Science Foundation of Hebei Province (No. A2014207010), Key Project of Science and Research of Hebei Educational Department (ZH2012080) and Key Project of Science and Research of Hebei University of Economics and Business (2013KYZ01).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All 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
Wei, L., Tan, R. Some new iterative algorithms with errors for common solutions of two finite families of accretive mappings in a Banach space. Fixed Point Theory Appl 2014, 176 (2014). https://doi.org/10.1186/1687-1812-2014-176
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-176