Abstract
m-d-Accretive mappings, which are totally different from m-accretive mappings in non-Hilbertian Banach spaces, belong to another type of nonlinear mappings with practical backgrounds. The purpose of this paper is to present some new iterative schemes by means of convex combination methods to approximate the common zeros of finitely many m-d-accretive mappings. Some strong and weak convergence theorems are obtained in a real uniformly smooth and uniformly convex Banach space by using the techniques of the Lyapunov functional and retraction. The restrictions are weaker than in the previous corresponding works. Moreover, an example of m-d-accretive mapping is exemplified, from which we can see the connections between m-d-accretive mappings and the nonlinear elliptic equations.
MSC:47H05, 47H09, 47H10.
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1 Introduction and preliminaries
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 .
The normalized duality mapping J from E to is defined by
We call J weakly sequentially continuous if is sequence in E which converges weakly to x it follows that converges in weak∗ to Jx.
A mapping is said to be demi-continuous [1] on E if , as , for any sequence strongly convergent to x in E. A mapping is said to be hemi-continuous [1] on E if , for any . A mapping is said to be non-expansive if , for .
The Lyapunov functional is defined as follows [2]:
for .
It is obvious from the definition of φ that
for all . We also know that
We use to denote the set of fixed points of a mapping . That is, . A mapping is said to be generalized non-expansive [4] if and , for and .
Let C be a nonempty closed subset of E and let Q be a mapping of E onto C. Then Q is said to be sunny [4] if , for all and . A mapping is said to be a retraction [4] if for every . If E is a smooth and strictly convex Banach space, then a sunny generalized non-expansive retraction of E onto C is uniquely decided, which is denoted by .
Let I denote the identity operator on E. A mapping is said to be accretive if , for and it is called m-accretive if , for .
If A is accretive, we can define, for each , a single-valued mapping by , which is called the resolvent of A. And, is a non-expansive mapping [1]. In the process of constructing iterative schemes to approximate zeros of an accretive mapping A, the non-expansive property of plays an important role.
A mapping is said to be d-accretive [5] if , for . And it is called m-d-accretive if , for . However, the resolvent of an m-d-accretive mapping is not a non-expansive mapping.
An operator is said to be monotone if , for , . A monotone operator B is said to be maximal monotone if , for . An operator is said to be strictly monotone if , for , , .
It is clear that in the frame of Hilbert spaces, (m-)accretive mappings, (m-)d-accretive mappings and (maximal) monotone operators are the same. But in the frame of non-Hilbertian Banach spaces, they belong to different classes of important nonlinear operators, which have practical backgrounds. During the past 50 years or so, a large number of researches have been done on the topics of constructing iterative schemes to approximate the zeros of m-accretive mappings and maximal monotone operators. However, rarely related work on d-accretive mappings can be found.
As we know, in 2000, Alber and Reich [5] presented the following iterative schemes for the d-accretive mapping T in a real uniformly smooth and uniformly convex Banach space:
and
They proved that the iterative sequences generated by (1.3) and (1.4) converge weakly to the zero point of T under the assumptions that T is uniformly bounded and demi-continuous.
In 2007, Guan [6] presented the following projective method for the m-d-accretive mapping A in a real uniformly smooth and uniformly convex Banach space:
where , and is the generalized projection from onto . It was shown that the iterative sequence generated by (1.5) converges strongly to the zero point of A under the assumptions that A is demi-continuous, the normalized duality mapping J is weakly sequentially continuous, and satisfies
for and . The restrictions are extremely strong and it is hardly for us to find such an m-d-accretive mapping which both is demi-continuous and satisfies (1.6).
The so-called block iterative scheme for solving the problem of image recovery proposed by Aharoni and Censor [7] inspired us. In a finite-dimensional space H, the block iterative sequence is generated in the following way: and
where is a non-expansive retraction from H onto , and is a family of nonempty closed and convex subsets of H. , , and , for and .
In [8], Kikkawa and Takahashi applied the block iterative method to approximate the common fixed point of finite non-expansive mappings in Banach spaces in the following way and obtained the weak convergence theorems: , and
where , , and , for and .
In this paper, we shall borrow the idea of block iterative method which highlights the convex combination techniques. Our main work can be divided into three parts. In Section 2, we shall construct iterative schemes by convex combination techniques for approximating common zeros of m-d-accretive mappings. Some weak convergence theorems are obtained in a Banach space. In Section 3, we shall construct iterative schemes by convex combination and retraction techniques for approximating common zeros of m-d-accretive mappings. Some strong convergence theorems are obtained in a Banach space. In Section 4, we shall present a nonlinear elliptic equation from which we can define an m-d-accretive mapping. Our main contributions lie in the following aspects:
-
(i)
The restrictions are weaker. The semi-continuity of the d-accretive mapping A and the inequality of (1.6) are no longer needed.
-
(ii)
The Lyapunov functional is employed in the process of estimating the convergence of the iterative sequence. This is mainly because the resolvent of a d-accretive mapping is not non-expansive.
-
(iii)
The connection between a nonlinear elliptic equation and an m-d-accretive mapping is set up, from which we cannot only find a good example of m-d-accretive mapping but also see the iterative construction of the solution of the nonlinear elliptic equation.
In order to prove our convergence theorems, we also need the following lemmas.
The duality mapping has the following properties:
-
(i)
If E is a real reflexive and smooth Banach space, then is single-valued.
-
(ii)
If E is reflexive, then J is a surjection.
-
(iii)
If E is a real uniformly smooth and uniformly smooth Banach space, then is also a duality mapping. Moreover, J and are uniformly continuous on each bounded subset of E and , respectively.
-
(iv)
E is strictly convex if and only if J is strictly monotone.
Lemma 1.2 [10]
Let E be a real smooth and uniformly convex Banach space, be a maximal monotone operator, then is a closed and convex subset of E and the graph of B, is demi-closed in the following sense: with in E, and with in it follows that and .
Lemma 1.3 [2]
Let E be a real reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed subset of E, and let be a sunny generalized non-expansive retraction. Then, for and , .
Lemma 1.4 [3]
Let E be a real smooth and uniformly convex Banach space, and let and be two sequences in E. If either or is bounded and , as , then , as .
Lemma 1.5 [11]
Let and be two sequences of nonnegative real numbers and , for . If , then exists.
2 Weak convergence theorems
Theorem 2.1 Let E be a real uniformly smooth and uniformly convex Banach space. Let be a finite family of m-d-accretive mappings, , , , for . and . Let . Suppose that the normalized duality mapping is weakly sequentially continuous. Let be generated by the following iterative algorithm:
Suppose the following conditions are satisfied:
-
(i)
, , for ;
-
(ii)
, , for ;
-
(iii)
, , for .
Then converges weakly to a point .
Proof For , let and .
We split the proof into the following six steps.
Step 1. For , and satisfy the following two inequalities, respectively:
In fact, using (1.2), we know that for ,
Since is d-accretive and ,
From (2.4) we know that (2.2) is true. So is (2.3).
Step 2. is bounded.
, using (2.2) and (2.3), we have
Lemma 1.5 implies that exists. Then (1.1) ensures that is bounded.
Step 3. is maximal monotone, for each i, .
Since is d-accretive, then ,
Therefore, is monotone, for each i, .
Since , , then , there exists satisfying , . Using Lemma 1.1(ii), there exists such that . Thus , which implies that , . Thus is maximal monotone, for each i, .
Step 4. , for each i, .
Since , then there exists such that , where . Using Lemma 1.1(ii) again, there exists such that . Thus , for each i, . That is, , for each i, .
Step 5. , where denotes the set of all of the weak limit points of the weakly convergent subsequences of .
Since is bounded, there exists a subsequence of , for simplicity, we still denote it by such that , as .
For , using (2.2) and (2.3), we have
which implies that
Using the assumptions and the result of Step 2, we know that , as , for . Then Lemma 1.4 ensures that , as , for .
Let , since is d-accretive, then
Since both and are bounded, then letting and using Lemma 1.1(iii), we have
. That is, , for . From Step 3 and Lemma 1.2, we know that , which implies that . And then .
Step 6. , as , where is the unique element in D.
From Steps 2 and 5, we know that there exists a subsequence of such that , as . If there exists another subsequence of such that , as , then from Step 2, we know that
Similarly,
From (2.5) and (2.6), we have , which implies that since J is strictly monotone.
This completes the proof. □
Remark 2.1 If E reduces to the Hilbert space H, then (2.1) becomes the iterative scheme for approximating common zeros of m-accretive mappings.
Remark 2.2 The iterative scheme (2.1) can be regarded as two-step block iterative scheme.
Remark 2.3 If , then (2.1) becomes to the following one approximating the zero point of an m-d-accretive mapping A:
If, moreover, , , then (2.7) becomes the so-called double-backward iterative scheme for the m-d-accretive mapping A:
3 Strong convergence theorems
Theorem 3.1 Let E be a real uniformly smooth and uniformly convex Banach space. Let be a finite family of m-d-accretive mappings, , , , for . and . Let . Suppose the normalized duality mapping is weakly sequentially continuous. Let be generated by the iterative scheme:
Suppose the following conditions are satisfied:
-
(i)
, , for ;
-
(ii)
, , for ;
-
(iii)
, , for .
Then converges strongly to , where is the sunny generalized non-expansive retraction from E onto D, as .
Proof We split the proof into six steps.
Step 1. is well defined.
Noticing that
then from Lemma 1.1(iii), we can easily know that () is a closed subset of E.
For , using (2.2), we know that
which implies that . Thus , for .
Since is a nonempty closed subset of E, there exists a unique sunny generalized non-expansive retraction from E onto , which is denoted by . Therefore, is well defined.
Step 2. is bounded.
Using Lemma 1.3, , . Thus is bounded and then (1.1) ensures that is bounded.
Step 3. , where denotes the set of all of the weak limit points of the weakly convergent subsequences of .
Since is bounded, there exists a subsequence of , for simplicity, we still denote it by such that , as .
Since , using Lemma 1.3, we have
which implies that exists. Thus . Lemma 1.4 implies that , as .
Since , then , which implies that , as .
, using (2.2) again, we have
Then
as . Lemma 1.4 implies that , as , where .
The remaining part is similar to that of Step 5 in Theorem 2.1, then we have .
Step 4. is a Cauchy sequence.
If, on the contrary, there exist two subsequences and of such that , .
Since exists, using Lemma 1.3 again,
as . Lemma 1.4 implies that , which makes a contradiction. Thus is a Cauchy sequence.
Step 5. D is a closed subset of E.
Let with , as . Then , for . In view of Lemma 1.1(ii), there exists such that . Using Lemma 1.1(iii), , as . Since , and is maximal monotone, Lemma 1.2 ensures that . Thus, , for . And then . Therefore, D is closed subset of E, which ensures there exists a unique sunny generalized non-expansive retraction from E onto D.
Step 6. , as .
Since is a Cauchy sequence, there exists such that , as . From Step 5, .
Now, we prove that .
Using Lemma 1.3, we have the following two inequalities:
and
Letting , from (3.3), we know that
From (3.2) and (3.4), . Thus . So in view of Lemma 1.4, .
This completes the proof. □
Remark 3.1 Combining the techniques of convex combination and the retraction, the strong convergence of iterative scheme (3.1) is obtained.
Remark 3.2 It is obvious that the restrictions in Theorems 2.1 and 3.1 are weaker.
4 Connection between nonlinear mappings and nonlinear elliptic equations
We have mentioned that in a Hilbert space, m-d-accretive mappings and m-accretive mappings are the same, while in a non-Hilbertian Banach space, they are different. So, there are many examples of m-d-accretive mappings in Hilbert spaces. Can we find one mapping that is (m-)d-accretive but not (m-)accretive?
In Section 4.1, we shall review the work done in [12], where an m-accretive mapping is constructed for discussing the existence of solution of one kind nonlinear elliptic equations.
In Section 4.2, we shall construct an m-d-accretive mapping based on the same nonlinear elliptic equation presented in Section 4.1, from which we can see that it is quite different from the m-accretive mapping defined in Section 4.1.
4.1 m-Accretive mappings and nonlinear elliptic equations
The following nonlinear elliptic boundary value problem is extensively studied in [12, 13]:
In (4.1), Ω is a bounded conical domain of a Euclidean space with its boundary (see [14]). is a given function, ϑ is the exterior normal derivative of Γ, is a given function satisfying Carathéodory’s conditions such that the mapping is defined and there exists a function such that , for and . is the subdifferential of a proper, convex, and semi-lower-continuous function. is a monotone and continuous function, and there exist positive constants , , and such that, for , the following conditions are satisfied:
-
(i)
;
-
(ii)
;
-
(iii)
,
where denotes the inner product in .
In [12], they first present the following definitions.
Definition 4.1 [12]
Define the mapping by
for any .
Definition 4.2 [12]
Define the mapping by , for any .
Definition 4.3 [12]
Define a mapping as follows:
For , .
Definition 4.4 [12]
Define a mapping as follows:
-
(i)
If , then
For , we set .
-
(ii)
If , then define as the -closure of defined in Definition 4.3.
Then they get the following important result in [12].
Proposition 4.1 [12]
Both A and are m-accretive mapping.
Later, by using the perturbations on ranges of m-accretive mappings, the sufficient condition on the existence of solution of (4.1) is discussed.
Theorem 4.1 [12]
If () satisfies the following condition:
then (4.1) has a solution in .
The meaning of , , , and can be found in the following two definitions.
For and , let be the element with least absolute value if and , where or <0, respectively, in the case . Finally, let (in the extended sense) for . Then define measurable functions on Γ.
Define and .
4.2 Examples of m-d-accretive mappings
Now, based on nonlinear elliptic problem (4.1), we are ready to give the example of m-d-accretive mapping in the sequel.
Lemma 4.1 [10]
Let E be a Banach space, if is an everywhere defined, monotone, and hemi-continuous mapping, then B is maximal monotone.
Definition 4.7 Let and .
Define the mapping by
for any .
Proposition 4.2 () is maximal monotone.
Proof We split the proof into four steps.
Step 1. B is everywhere defined.
In fact, for , from the property (i) of α, we have
Thus B is everywhere defined.
Step 2. B is monotone.
Since α is monotone, then, for ,
which implies that B is monotone.
Step 3. B is hemi-continuous.
To show that B is hemi-continuous. It suffices to prove that for and , as .
In fact, since α is continuous,
as .
Step 4. B is maximal monotone.
Lemma 4.1 implies that B is maximal monotone.
This completes the proof. □
Remark 4.1 [10]
There exists a maximal monotone extension of B from to , which is denoted by .
Definition 4.8 For , the normalized duality mapping is defined by
for .
Define the mapping () as follows:
Proposition 4.3 The mapping () is m-d-accretive.
Proof Since is monotone, for ,
Thus A is d-accretive.
In view of Remark 4.1, is maximal monotone, then , for .
For , there exists such that . Using Lemma 1.1 again, there exists such that . Then . Thus and then , for .
Thus A is m-d-accretive.
This completes the proof. □
Proposition 4.4 .
Proof It is obvious that .
On the other hand, if , then . Let be such that . From the property (iii) of α, we have
which implies that .
Thus .
This completes the proof. □
Remark 4.2 From Propositions 4.2 and 4.3, we know that the restriction on the m-d-accretive mapping in Theorem 2.1 or 3.1 that is valid.
Remark 4.3 If (4.1) is reduced to the following:
then it is not difficult to see that is exactly the solution of (4.2), from which we cannot only see the connections between the zeros of an m-d-accretive mapping and the nonlinear equation but also see that the work on designing the iterative schemes to approximate zeros of nonlinear mappings is meaningful.
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Acknowledgements
Li Wei 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).
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The main idea was proposed by LW, and YL and RPA participated in the research. All authors read and approved the final manuscript.
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Wei, L., Liu, Y. & Agarwal, R.P. Convergence theorems of convex combination methods for treating d-accretive mappings in a Banach space and nonlinear equation. J Inequal Appl 2014, 482 (2014). https://doi.org/10.1186/1029-242X-2014-482
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DOI: https://doi.org/10.1186/1029-242X-2014-482