Abstract
We propose and analyze a new iterative scheme with inertial items to approximate a common zero point of two countable d-accretive mappings in the framework of a real uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems by employing some new techniques compared to the previous corresponding studies. We give some numerical examples to illustrate the effectiveness of the main iterative scheme and present an example of curvature systems to emphasize the importance of the study of d-accretive mappings.
1. Introduction and Preliminaries
In this paper, we assume that is a real Banach space and is the dual space of . Suppose that is a nonempty closed and convex subset of . The symbols “”, “” and “” denote the value of at , the strong convergence, and the weak convergence either in or , respectively.
The normalized duality mapping is defined as follows:
Lemma 1. (see [1]). Assume that is real uniformly convex and uniformly smooth Banach space. Then (1) is single-valued and surjective and, for and , ; (2) is the normalized duality mapping from to ; (3) both and are uniformly continuous on each bounded subset of or , respectively.
Definition 1. (see [2]). The Lyapunov functional is defined as follows: Similarly, the Lyapunov functional defined on can be defined and denoted by .
Lemma 2 (see [3]). Let be a uniformly smooth and uniformly convex Banach space, and let and be two sequences in . If either or is bounded and , as , then , as .
Definition 2. (see [4]). Let be a sequence of nonempty closed and convex subsets of ; then(1), which is called strong lower limit of , is defined as the set of all such that there exists for almost all and it tends to as in the norm(2), which is called weak upper limit of , is defined as the set of all such that there exists a subsequence of and for every and it tends to as in the weak topology(3)If , then the common value is denoted by
Lemma 3. (see [4]). Let be a decreasing sequence of closed and convex subsets of , that is, , if . Then, converges in and .
Definition 3. (see [1, 2]). Suppose that is a real uniformly smooth and uniformly convex Banach space and is a nonempty closed and convex subset of ; then, for each , there exists a unique element such that . Such an element is denoted by and is called the metric projection of onto .
Lemma 4. (see [5]). Suppose that is a real uniformly smooth and uniformly convex Banach space and is a sequence of nonempty closed and convex subsets of . If exists and is not empty, then , for .
Definition 4. (1)A mapping is said to be accretive [6] if , , (2)A mapping is said to be d-accretive [7] if , ,
It is easy to see that accretive mappings and d-accretive mappings are identical in a Hilbert space, while they are different in a non-Hilbert space.
For a nonlinear mapping , we use and to denote the fixed point set and zero point set of , respectively.
Lemma 5. (see [8, 9]). Suppose that is a real uniformly smooth and uniformly convex Banach space. Let be d-accretive mapping such that . Under the assumption that , one has the following: (1), , and ,(2)If , , , and , as , then .
Definition 5. (see [10]). Let be a nonempty closed subset of and let be a mapping of onto . Then is said to be sunny if , for all and . A mapping is said to be a retraction if for every . If is a smooth and strictly convex Banach space, then a sunny generalized nonexpansive retraction of onto is uniquely decided, which is denoted by .
Definition 6. (see [3]). If is a real uniformly smooth and uniformly convex Banach space and is a nonempty closed and convex subset of , then, for each , there exists a unique element satisfying . In this case, , define by , and is called the generalized projection from onto .
Lemma 6 (see [11]). Suppose that is a real uniformly convex Banach space and . Then there exists a continuous and strictly increasing function with satisfyingfor , with and .
Accretive mappings have been extensively studied until now and some works can be seen in [12–16] and the references therein. However, until 2000, some valuable research work has been done on d-accretive mappings. As we know, in 2000, Alber and Reich [17] presented the following iterative schemes for d-accretive mapping in a real uniformly smooth and uniformly convex Banach space:
They proved that the iterative sequences generated by (5) and (6) converge weakly to the zero point of under the assumption that is uniformly bounded and demicontinuous.
In 2006, Guan [18] presented the following projective method for the accretive mapping in a real uniformly smooth and uniformly convex Banach space :
It was shown that the iterative sequences generated by (7) converge strongly to the zero point of under the assumptions that (1) , (2) is demicontinuous, and (3) is weakly sequentially continuous and satisfiesfor and . The restrictions are extremely strong, and it is hard for us to find such a accretive mapping that is demicontinuous and satisfies (8).
In 2014, Wei et al. [7] presented the following block iterative schemes for approximating common zero points of accretive mappings in a Banach space :
Under mild assumptions, generated by (9) is proved to be weakly convergent to an element in , while (10) is proved to be strongly convergent to an element in .
In [19], the study on finite d-accretive mappings is extended to that for infinite d-accretive mappings :
Then, sequence generated by (11) is proved to be strongly convergent to an element in .
A new idea can be seen in (11), where the iterative element can be chosen arbitrarily, which is different from the traditional one, for example, (7) in [18]. However, it is found that, for each iterative step in (11), countable sets should be evaluated for . To simplify it theoretically, the following iterative scheme is designed in [9]:
Then, generated by (12) is proved to be strongly convergent to an element in .
In 2020, Wei et al. [8] extended the discussion on countable d-accretive mappings to that for two groups of countable d-accretive mappings and and construct two key groups of sets and , where the iterative elements and can be chosen arbitrarily in and , respectively.
Then generated by (13) is proved to be strongly convergent to an element in .
Recall that the inertial-type algorithm was first proposed by Polyak [20] as an acceleration process in solving a smooth convex minimization problem. The inertial-type algorithm involves a two-step iterative method where the next iterate is defined by making use of the previous two iterates. For example, in 2015, Lorenz and Pock [21] proposed the following inertial forward-backward algorithm for approximating zero points of , where and are accretive-type mappings in Hilbert space :
In (14), the term is called the inertial term.
In this paper, motivated by the previous work, some new work is done in the construction of new iterative schemes: (i) the inertial term is inserted for the purpose of possible acceleration; (ii) the combination expressions of or are different from those in (11)–(13). Numerical experiments are conducted, and it is very interesting that the rate of convergence is so quick that only eight steps are enough for some special cases and for different choices of iterative elements. To emphasize the importance of the topic, a kind of curvature systems is studied and is taken as an example of d-accretive mappings.
2. Iterative Schemes and Strong Convergence Theorems
2.1. Iterative Schemes
In this section, we suppose that the following conditions are satisfied: (A1) is a real uniformly convex and uniformly smooth Banach space; and are the normalized duality mappings. (A2) are d-accretive mappings such that , for each . (A3) , , and are real number sequences in , for . and are real number sequences in . and are real number sequences in . (A4) , and are real number sequences in such that . (A5) and are the error sequences in . (A6) ; .
We construct the following iterative scheme:
2.2. Strong Convergence Theorems
Theorem 1. Consider , , for , , , , , , , , , and , as . Then, the iterative sequence , as .
Proof. The proof is split into eight steps. Step 1. . Since , there exists such that , . It follows from Lemma 1 that there exists such that . Therefore, . Next, we shall use inductive method to prove that , . . For , it is obvious that . Suppose that the result is true for . Then, if , it follows from the definition of the Lyapunov functional, the convexity of , and Lemma 5 that Thus, . Using Lemma 5 repeatedly, similar to the above discussion, one has Then , which implies that . Step 2. Both and are closed and convex subsets of , for . If , the result is obvious. If , since is a closed and convex subset of , for . Since is a closed and convex subset of , for . Step 3., as . The result follows from the results of Steps 1 and 2 and Lemmas 3 and 4. Step 4. and . Since ; then, for , there exists such that Then . Similarly, . This ensures that is well defined. Step 5. , as . Since , . It follows from Step 3 and that is bounded. Since and is convex, for , . Using Lemma 6, one has Therefore, , as . Therefore, , as . Combining with Step 3, , . Step 6., , , and , as . In fact, since with , , as . Similarly, , as . Since , Since , it follows from Step 5 and Lemma 2 that , as . Therefore, , as . Since , Since , there exists a subsequence of , which is still denoted by such that , and then , which ensures that , as . Step 7.. , and using Lemma 5, we have From iterative scheme (15), we know that which implies that Since , there exists a subsequence of , which is still denoted by such that . Then , as . Using Lemma 5 again, we have , . Therefore, . Similarly, From iterative scheme (15), we have Therefore, Since , there exists a subsequence of , which is still denoted by such that Repeating the above process, Repeating (30), (31), and Lemma 5, one has . Therefore, . Step 8., as . Using Steps 1 and 7, . Since the metric projection is unique, .This completes the proof.
Remark 1. If , then . Two-step inertial iterative scheme (15) reduces to the traditional inertial iterative scheme.
Remark 2. If , then ; and two-step iterative scheme (15) extends the corresponding work of (13) in [8].
Remark 3. If or is chosen as (or ) or (or ), (15) becomes a projection iterative scheme with inertial items.
Corollary 1. In Hilbert space , iterative scheme (15) becomes as follows: where and , . Under the assumptions of Theorem 1, the result of Theorem 1 is still true.
3. Numerical Experiments
Theorem 2. Let , , and , . Let , , , and , . Let ; , . ; , . For initial value , , the iterative sequence generated by (32) converges strongly to by the eighth step for two different choices of and in the corresponding sets and , respectively.
Proof. For the special example, we can easily see that all of the assumptions of Corollary 1 are satisfied; and the iterative scheme (32) can be simplified as follows: Now, compute step by step and choose two different groups of values of and in and , respectively; we can get the two following tables.
Remark 4. From Tables 1 and 2 derived from the numerical experiments done in Theorem 2, we may find that (1) is an interval that permits us to choose intermediate iterative element flexibly; (2) two extreme values of in , the largest and the smallest, are chosen, from which we can see that the convergence of the iterative sequence is not affected.
4. Curvature Systems
To emphasize the importance of d-accretive mappings, the connection among d-accretive mappings, iterative schemes, and nonlinear boundary value problems is set up.
Definition 7. (see [22]). A single-valued mapping is hemicontinuous if , as , .
Definition 8. (see [22, 23]). A multivalued mapping is monotone if , , and . The monotone operator is called maximally monotone if , .
Lemma 7. (see [22]). If is everywhere defined, monotone, and hemicontinuous, then it is maximally monotone.
Example 1. We shall investigate the following curvature systems:where and denote the norm and inner-product in , respectively. is the bounded conical domain of with its boundary , is the normal derivative of , , is a nonnegative constant, and is a given function. For , , , and . If , then suppose that ; if , then suppose that . We use and to denote the norm in and , respectively, .
Lemma 8. For , define as follows: ,Then, is everywhere defined, hemicontinuous, and monotone, .
Proof. The proof is split into three steps. Step 1. is everywhere defined. If , then If , then Thus is everywhere defined. Step 2. is monotone. For , From the fact that , is monotone; we know that is monotone. Step 3. is hemicontinuous. and ; using Lebesgue’s dominated convergence theorem, one hasTherefore, is hemicontinuous.
This completes the proof.
Lemma 9. (see [8]). For , define as follows: ,Then is maximally monotone, for .
Lemma 10. (see [22]). For each , there exist the maximal monotone extension of and the maximal monotone extension of , which are denoted by and , respectively.
Lemma 11. (see [24]). For , if , then the normalized duality mapping is defined by , . Then, is defined by , .
Based on the above lemmas and imitating Theorems 3.10, 3.11, and 3.12 in [8], one has the following results.
Theorem 3. For , define as follows: , . Then is d-accretive and , .
Theorem 4. Define the mapping by , . Then is d-accretive and , .
Define the mapping by , , where is a constant. Then is d-accretive and , .
Theorem 5. If, in (34), , where is a constant, then is the solution of (34). Moreover, .
Proof. It is obvious that is the solution of (34). If , then and . Since , .
This completes the proof.
Remark 5. From Theorem 5, we can see the relationship between the solution of curvature systems (34) and common zero points of two groups of d-accretive mappings. This will help us to approximate the solution of curvature systems by using iterative schemes introduced in Section 2.
Data Availability
All data generated and analyzed during this study are included within this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The first three authors were supported by the Natural Science Foundation of Hebei Province under Grant no. A2019207064, Science and Technology Key Project of Hebei Education Department under Grant no. ZD2019073, and Key Project of Science and Research of Hebei University of Economics and Business under Grant no. 2018ZD06.