Iterative Scheme with Errors for Common Zeros of Finite Accretive Mappings and Nonlinear Elliptic Systems

Abstract

We present a new iterative scheme with errors to solve the problems of finding common zeros of finite -accretive mappings in a real Banach space. Strong convergence theorems are established, which extend the corresponding works given by some authors. Moreover, the relationship between zeros of -accretive mappings and one kind of nonlinear elliptic systems is investigated, from which we can see that some restrictions imposed on the iterative scheme are valid and the solution of one kind of nonlinear elliptic systems can be approximated by a suitably defined iterative sequence.

1. Introduction and Preliminaries

Let be a real Banach space with norm and let denote the dual space of . We use “” and “” to denote strong and weak convergence, respectively. We denote the value of at by .

Let be a nonempty, closed, and convex subset of and let be a mapping of onto . Then is said to be sunny [1] if , for all and .

A mapping of into is said to be a retraction [1] if . If a mapping is a retraction, then for every , where is the range of .

A mapping is said to be nonexpansive if , for all . We use to denote the fixed point set of ; that is, . A mapping is said to be demiclosed at if whenever is a sequence in such that and , it follows that .

A subset of is said to be a sunny nonexpansive retract of [2] if there exists a sunny nonexpansive retraction of onto and it is called a nonexpansive retract of if there exists a nonexpansive retraction of onto . If is reduced to a Hilbert space , then the metric projection is a sunny nonexpansive retraction from to any closed and convex subset of . But this is not true in a general Banach space. We note that if is smooth and is a retraction of onto , then is sunny and nonexpansive if and only if for all , , [3].

We use to denote the normalized duality mapping from to which is defined by It is well known that is single-valued if is strictly convex. Moreover, , for all and . We call that is weakly sequentially continuous if is a sequence in which converges weakly to it follows that converges in weak* to .

A mapping is called accretive if , for all and it is called -accretive if , for all . Let denote the set of zeros of ; that is, . We denote by (for ) the resolvent of ; that is, . Then is nonexpansive and .

Interest in accretive mappings, which is an important class of nonlinear operators, stems mainly from their firm connection with equations of evolution. It is well known that many physically significant problems can be modelled by initial value problems of the form where is an accretive mapping. Typical examples where such evolution equations occur can be found in the heat, wave, or Schrodinger equations. If is dependent on , then (2) is reduced to whose solutions correspond to the equilibrium of the system (2). Consequently, within the past 40 years or so, considerable research efforts have been devoted to methods for finding approximate solutions of (3). An early fundamental result of accretive operators, due to Browder [4]. One classical method for studying the problem , where is an -accretive mapping, is the following so-called proximal method (c.f. [5]): where . It was shown that the sequence generated by (4) converges weakly or strongly to a zero point of under some conditions.

Recall that the following normal Mann iterative scheme to approximate the fixed point of a nonexpansive mapping was introduced by Mann [6]: It was proved that, under some conditions, the sequence produced by (5) converges weakly to a point in .

Later, many mathematicians try to combine the ideas of proximal method and Mann iterative method to approximate the zeros of -accretive mappings; see, for example, [7–14] and the references therein.

In particular, in 2007, Qin and Su [7] presented the following iterative scheme: And they showed that generated by the above scheme converges strongly to a zero of .

Motivated by iterative schemes (4) and (5), Zegeye and Shahzad extended their discussion to the case of finite -accretive mappings. They presented in [15] the following iterative scheme: where with and . If  , they proved that generated by (7) converges strongly to the common zeros of under some conditions.

The work in [15] was then extended to the following one presented by Hu and Liu in [16]: where with and . , , , and . If , they proved that converges strongly to the common zeros of under some conditions.

In 2009, Yao et al. presented the following iterative scheme in the frame of Hilbert space in [17]: where is a nonexpansive mapping with . Suppose and are two real sequences in satisfying(a) and ;(b).

Then constructed by (9) converges strongly to a fixed point of .

Motivated by the work in [15, 17], Shehu and Ezeora presented the following result in [2].

Theorem 1. Let be a real uniformly smooth and uniformly convex Banach space, and let be a nonempty, closed, and convex sunny nonexpansive retract of , where is the sunny nonexpansive retraction of onto . Suppose the duality mapping is weakly sequentially continuous. For each , let be an -accretive mapping such that . Let , satisfy (a) and (b). Let be generated iteratively by where with , for , , for , and . Then converges strongly to the common zero of , where .

Inspired by the work in [2], we present the following iterative scheme with errors: where is the error sequence and is a finite family of -accretive mappings. , , for ; , , for . More details of iterative scheme will be presented in Section 2. And, some strong convergence theorems are obtained.

Note that there are some differences between our work and Shehu and Ezeora’s in [2] in the following aspects.(i) in iterative scheme is different from in (10) since the former is changing with and the latter is not, which results in in scheme having different coefficient for each different .(ii)The idea of three-step iteration is employed in our paper.(iii)The error sequence is considered in the iterative scheme .(iv)Recall that, in [2], Lemma 8 is a key tool to prove the convergence of generated by (10). In particular, to obtain the main result, they imposed an additional condition on the function in Lemma 8 that , where is a constant satisfying some conditions. One question arises: how to show the convergence of the iterative sequence if does not satisfy this additional condition? To answer the question, we will use Lemma 4 instead of Lemma 8.

In Section 3, we will discuss the relationship between zero point of finite -accretive mappings and the solution of one kind of nonlinear elliptic systems involving -Laplacian operators. The discussion helps us not only to see that the topic of constructing iterative schemes to approximate zeros of -accretive mappings is meaningful but also to see that the solution of -Laplacian elliptic systems can be obtained by an iterative scheme.

Next, we list some results we need in sequel.

Lemma 2 (see [18]). Let be a real uniformly convex Banach space, let be a nonempty, closed, and convex subset of , and is a nonexpansive mapping such that ; then, is demiclosed at zero.

Lemma 3 (see [16]). Let be a strictly convex Banach space which has a uniformly Gâteaux differential norm, and let be a nonempty, closed, and convex subset of . 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 4 (see [13]). In a real Banach space , the following inequality holds: where .

Lemma 5 (see [19]). Let , , and be three sequences of nonnegative real numbers satisfying where such that (i) and and (ii) either or . Then .

Lemma 6 (see [20]). Let and be two bounded sequences in a Banach space such that , for . Suppose satisfying . If , then .

Lemma 7 (see [21]). Let be a Banach space and let be an -accretive mapping. For , , and , one has where and .

Lemma 8 (see [22]). Let 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:

2. Strong Convergence Theorem

Lemma 9 (see [2]). Let be a real uniformly smooth and uniformly convex Banach space. Let be a nonempty, closed, and convex sunny nonexpansive retract of , and let be the sunny nonexpansive retraction of onto . Let be nonexpansive with . Suppose that the duality mapping is weakly sequentially continuous. If for each we define by then has a fixed point , which is convergent strongly to the fixed point of , as . That is, .

Lemma 10. Let be a strictly convex Banach space and let be a nonempty, closed, and convex subset of . Let be a finite family of -accretive mappings such that .
Let be real numbers in such that and , where and , for , and ; then, is nonexpansive and , for .

Proof. The main idea of the proof is essentially from that in [15] or Lemma 3. For the sake of completeness, we present the proof in the following.
It is easy to check that is nonexpansive and .
On the other hand, for all , then .
For all , then Therefore, , which implies that . Similarly, .
Then , which implies from the strictly convexity of that .
Therefore, , for . And then , which completes the proof.

Lemma 11. Let , , and be the same as those in Lemma 10. Suppose . Then is nonexpansive and .

Proof. From Lemma 10, we have . It is easy to check that is nonexpansive. So, it suffices to show that since is trivial.
For all , then .
For all , then . Now, Therefore, , which implies that . Similarly, .
Then repeating the discussion in Lemma 10, we know that , for . And then , which completes the proof.

Theorem 12. Let be a real uniformly smooth and uniformly convex Banach space. Let be a nonempty, closed, and convex sunny nonexpansive retract of , where is the sunny nonexpansive retraction of onto . Let be -accretive mappings, where . Suppose that the duality mapping is weakly sequentially continuous and . Let be generated by the iterative scheme , where , and , for , , for , . Suppose that , , , and are three sequences in and satisfying the following conditions:(i), , as ;(ii);(iii);(iv) and , for and ;(v), as , and .
Then converges strongly to a point .

Proof. We will split the proof into five steps.
Step  1. , , , , and are all bounded.
We will first show that where .
By using the induction method, we see that, for , ,
Suppose that (19) is true for . Then, for ,
Thus (19) is true for all . Since , then (19) ensures that is bounded.
For all , from , we see that is bounded.
Since , then is bounded. Since both and are bounded, then is bounded. Similarly, , , and are all bounded, for .
Then we set .
Step  2. and .
In fact,
Next, we discuss .
If , then, by using Lemma 7, we have
If , then imitating the proof of (23), we have
Combining (23) and (24), we have
Putting (25) into (22), we have
Similarly, we have
Therefore, from (26) and (27), we have
From Step  1, we know that . Using Lemma 6, we have from (28) that and then .
Step  3. and .
In fact,
Noticing the results of Steps  1 and  2 and , we have .
Since , then , as .
Step  4. , where is an element in .
From Lemma 11, we know that is nonexpansive and . Then Lemma 9 implies that there exists such that for . Moreover, , as .
Since , then is bounded. Let . Then from Step  1, we know that is a positive constant. Using Lemma 4, we have
So , which implies that in view of Step  3.
Since is bounded and is uniformly continuous on each bounded subset of , then , as .
Moreover, noticing the fact that we have .
Since + and is uniformly continuous on each bounded subset of , then
Step  5. , as , where is the same as that in Step  4.
Let . By using Lemma 4 again, we have
Let ; then (33) reduces to +.
From (32), (33), and the assumptions, by using Lemma 5, we know that , as .
This completes the proof.

If, in Theorem 12, , then we have the following theorem.

Theorem 13. Let and be the same as those in Theorem 12. Suppose that the duality mapping is weakly sequentially continuous. Let be a finite family of -accretive mappings. Let , , , , and satisfy the conditions presented in Theorem 12.
Let be generated by the following scheme:
Then converges strongly to a point , where is the same as that in Theorem 12.

3. Nonlinear Elliptic Systems with -Laplacian

In what follows in this paper, unless otherwise stated, we will assume that , . We use , , and to denote the norms of spaces , , and , respectively, where .

Now, we will examine the following nonlinear elliptic systems:

In , is a bounded conical domain of a Euclidean space with its boundary , (c.f. [23]). is a given function and , for . denotes the exterior normal derivative of .

Lemma 14 (see [24]). Define the mapping by for any . Then, is everywhere defined, strictly monotone, hemicontinuous, coercive, and maximal monotone, for .

Lemma 15 (see [24]). Define the mapping by for any , . Then, is -accretive, for .

Lemma 16 (see [24]). For , define the mapping in the following way:(i)if , , then for any , ,(ii)if , we define as the -closure of defined in (i) above.
Then, is -accretive, for .

Lemma 17 (see [25]). Let be a bounded conical domain in . If , then ; if and , then , where .

Lemma 18 (see [26]). Let denote the closed subspace of all constant functions in . Let be the quotient space . For , define the mapping by . Then, there is a constant , such that ,

Theorem 19. For , one has .

Proof. (i) .
Let ; then , which implies that . That is, .
On the other hand, suppose . Then , for all . Then . Therefore, the result holds.
(ii) .
Suppose and . Then let , and we can easily see that in view of the definition of .
On the other hand, let . Then there exist and in such that , in and . Now, define the following functions:
Then for , the function is continuous and . Therefore, there exists such that . So, for , we may assume that there exist such that and , for . Let ; then , for .
Now, compute the following:
Using Lemma 18, Then Lemma 17 implies that
From (38), (39), and (40), we know that , as . Then in . Since Nemytskyi mapping is continuous, then in . And then .
This completes the proof.

Remark 20. Theorem 19 helps us to see the assumption that in Theorems 12 and 13 are valid.

Remark 21. If we set , then is also -accretive. And then iterative scheme can be used to approximate the element in , which is just the solution of nonlinear elliptic system .

Remark 22. If , then is reduced to the following nonlinear elliptic systems with -Laplacian:

Moreover, the solution of is the zero of finitely many suitably defined -accretive mappings, which can be approximated by iterative scheme .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (no. 11071053), the Natural Science Foundation of Hebei Province (no. A2014207010), the Key Project of Science and Research of Hebei Educational Department (ZH2012080), and the Key Project of Science and Research of Hebei University of Economics and Business (2013KYZ01).