Abstract
In this research, the modified subgradient extragradient method and -mapping generated by a finite family of finite Lipschitzian demicontractions are introduced. Then, a strong convergence theorem for finding a common element of the common fixed point set of finite Lipschitzian demicontraction mappings and the common solution set of variational inequality problems is established. Furthermore, numerical examples are given to support the main theorem.
1. Introduction
Let be a real Hilbert space and be a nonempty closed convex subset of with the inner product and norm .
The fixed point problem for the mapping is to find such that
The term is denoted by the set of fixed points of , that is, . Fixed point problem has been widely studied and developed in the various literature studies, see [1].
Definition 1. Let be a real Hilbert space.(i)A mapping is said to be nonexpansive if(ii)A mapping is said to be quasinonexpansive if and(iii)A mapping is called -strictly pseudocontractive if there exists a constant such thatIf , then a nonexpansive mapping is a quasinonexpansive mapping. Also, if , then a strictly pseudocontractive reduces to a nonexpansive mapping.
In a real Hilbert space, the inequality (4) is equivalent to
Definition 2 (see [2]). A mapping is called demicontractive if and there exists a constant such thatThe class of demicontractive mappings covers a variety of nonlinear mappings, including strictly pseudocontractive mappings, quasinonexpansive mappings, and nonexpansive mappings.
By using the same technique as in the proof of (5), we see that (6) is equivalent to the inequality shown below if is a demicontractive mapping.Several mathematicians have taken an interest in studying the common fixed point of the finite family of nonlinear mappings and their characteristics during the past few decades; see [3–6].
In 2009, -mapping for nonlinear mappings is introduced by Kangtunyakarn and Suantai [6] as follows:where for every . This mapping is called K-mapping generated by and . Furthermore, .
Let is a nonlinear mapping. The variational inequality problem (VIP) is to finding an element such thatThe solution set of the problem (9) is denoted by .
Stampacchia [7] introduced and investigated variational inequalities in 1964. In addition to offering a comprehensive, unifying framework for the study of optimization, equilibrium problems, and related problems, it also serves as a helpful computational framework for the resolution of various problems in a wide range of applications. For additional information, see [8–13]. Various approaches are investigated to solve variational inequality problems and the related optimization problems through iterative methods.
Several researchers have presented a variety of iterative algorithms designed for solving the variational inequality problem (VIP). The projected gradient method (GM), which can be defined as follows, is the most fundamental projection technique for solving the VIP.where denotes the metric projection mapping, is the -strongly monotone, and is Lipschitz continuous with .
In 1976, Korpelevich [14] and Antipin [15] proposed the extragradient method (EGM) in a finite-dimensional Euclidean space as follows:where and are monotones and is Lipschitz continuous. If is nonempty, the sequence generated by (11) converges to a solution of VIP.
According to [16–18] and related references, the EGM has undergone modifications and enhancements in the past few years.
Later, in 2012, Censor et al. [19] defined the subgradient extragradient method (SEGM) by modifying the EGM and replacing the second projection with a projection onto a half-space which is presented as follows:Weak convergence theorem is obtained for SEGM (4) under some control conditions.
Recently, in 2021, Kheawborisut and Kangtunyakarn [20] introduced the modified subgradient extragradient method (MSEGM) as follows:where be -inverse strongly monotone mappings, , , with , and is a nonexpansive mapping. Afterwards, under certain control settings, a strong convergence theorem is obtained.
If and , then the modified subgradient extragradient method (MSEGM) reduces to the subgradient extragradient method (SEGM).
Motivated by the recent research, the S-subgradient extragradient method (SSEGM) is introduced as follows:where is a nonexpansive mapping. If , then the S-subgradient extragradient method (SSEGM) reduces to the modified subgradient extragradient method (MSEGM).
In this paper, inspired by [6, 20], the S-subgradient extragradient method and -mapping generated by a finite family of finite Lipschitzian demicontraction mappings are proposed. Under some control conditions, strong convergence theorems are proved. Moreover, numerical examples are given to support the main theorem.
2. Preliminaries
The notations and are denoted weak convergence and strong convergence, respectively. For each , there exists a unique nearest point such that
The mapping is called the metric projection of onto . Also, is a firmly nonexpansive mapping from onto , that is,
Moreover, for any and , if and only if
Definition 3. Let be a mapping. Then,(i) is said to be -Lipschitz continuous if there is a positive real number such that(ii) is called -inverse strongly monotone if there is a positive real number such that
Lemma 4 (see [21]). Let be a sequence of nonnegative real numbers satisfyingwhere is a sequence in (0,1) and is a sequence such that(1);(2) or .
Then, .
Lemma 5. Let be a real Hilbert space. Then, the following properties hold:(i)For all and (ii), for each .
Lemma 6 (see [22]). For , let be an -inverse strongly monotone with and . Therefore, these properties hold:(i);(ii) is a nonexpansive mapping.
Here, , for , and
Definition 7 (see [6]). Let be a nonempty closed convex subset of a real Banach space. Let be a finite family of -demicontractive mapping of into itself and let be real numbers with for every . Define a mapping as follows:This mapping is said to be the -mapping generated by and .
The following lemmas are needed to prove the main result.
Lemma 8. Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of -demicontractive mapping of into itself with , for all , and . Let be real numbers with , for all and . Let be the -mapping generated by and . Then, there hold the following properties:(i);(ii) is a nonexpansive mapping.
Proof. To prove (i), it is clear that .
Next, we prove that . To show this, suppose and .
By the definition of -mapping, we obtainBy (23), it follows thatThen, we haveHence, , that is,By the definition of and (26), we getthat is,From (23) and (28), we havewhich follows that , that is,By the definition of , (23) and (28), this implies thatwhich yields thatUsing the same method, we getNext, we claim that . Sinceand , we havewhich implies thatHence,Therefore,Finally, applying the same proof as in (23), is a quasinonexpansive mapping.
Lemma 9. Let be a nonempty closed convex subset of a real Hilbert space . For , let be a finite family of -demicontractive mappings of into itself and -Lipschitzian mappings with and . For each and , let and be real numbers with and such that as . For each , let and be the K-mappings generated by and and and , respectively. Therefore, for each bounded sequence in , we have
Proof. Let be a bounded sequence in and let and be generated by and and and , respectively. For each , we haveFor , we getFrom (40) and (41), we obtainBy (42) and the condition as for all , hence we obtain .
Lemma 10 (see [23]). Let be a sequence of real numbers that do not decrease at infinity in the sense that there exists a subsequence of such that for all . Also, we consider the sequence of integers defined by
Then, is a nondecreasing sequence verifying
Lemma 11 (see [20]). Let be a real Hilbert space, for every , let be -inverse strongly monotone mappings with . Let and be sequences generated by
Then, the following inequality holds:where and with for every .
3. Strong Convergence Theorem
Theorem 12. Let be a closed convex subset of a real Hilbert space . For , let be -inverse strongly monotone mappings. For , let be -inverse strongly monotone mappings with . For , let be a finite family of -demicontractive mappings and -Lipschitzian mappings with , and . For every and , let be the K-mapping generated by and where , for some and . Let the sequence and be generated by andwhere , with , , with and with satisfying the following conditions:(i) and ;(ii), for some .
Then, and converge strongly to .
Proof. Let . First, we will show that is bounded. Considerwhere . Suppose that . By Lemma 5, we haveBy the definition of , (49), Lemmas 6 and 8, we obtainFrom Lemma 11 and , we haveFrom (50) and (51), we getBy induction, we obtainThis implies that is a bounded sequence. Next, from (50), observe thatThis follows thatTake . Thus, we getNext, the following two possible cases are considered.
Case 13. Put , for all . Assume that there is no such that, for any , . In this case, we haveSince , it yields from (56) that . Hence, we obtainFrom condition (ii), we getSincethen, from (59), we obtainNext, we choose a subsequence of such thatSince is bounded, this follows that as , where . Assume that . This follows by Lemma 6 that , that is, . By nonexpansiveness of , (59), we getThis is a contradiction. Then, we obtainApplying the same proof as in (63), we also haveFor any and , , without loss of generality, we haveLet be the -mapping generated by and . From Lemma 8, we obtain that is nonexpansive and . By Lemma 9, we getSuppose that . Therefore, we obtain . From (61) and (67), we haveThis is a contradiction. Therefore, it follows that . Applying Lemma 8, it implies thatFrom (64), (65), and (69), it follows thatSince and (70), we can conclude thatLet , from Lemma 8, (49) and (51), we haveApplying Lemma 5, the definition of , (72), and , thus we getFrom (71), the conditions (i), (ii), and Lemma 4, we can conclude that converges strongly to . From (59), we also obtain that converges strongly to .
Case 14. Assume that there exists a subsequence such that for all . In this case, we can define by . Then, we obtain as and . This implies by (72) thatUsing the same method as in Case 13, it yields thatSince is a bounded sequence, then there exists a subsequence such thatApplying the same proof of Case 13 for , we getBy Lemma 4, we haveHence, by Lemma 10, we obtainTherefore, we can conclude that converges strongly to . From (59), we also have converging strongly to . The proof is complete.
Remark 15. Since the S-subgradient extragradient method covers various type of iterations such as the modified subgradient extragradient method (MSEGM), the subgradient extragradient method (SEGM), and the extragradient method (EGM), Theorem 12 can be seen as a modification and extension of several research papers, see, for example, [14, 15, 19, 20].
4. Numerical Examples
Numerical examples are provided in this section to back up the main result.
Example 1. Let and be the set of real numbers. Define the mappings byFor , let be defined byPut , , , and . Choose and . Then, and converge strongly to 0.
Solution. Clearly, all sequences satisfy all conditions of Theorem 12. Moreover, is 0-demicontractive mappings and -Lipschitzian mappings, for all . Choose and , thus we get . Since is a -mapping generated by and , thenHence, we obtainBy Theorem 12, the sequences and converge strongly to 0.
Table 1 and Figure 1 show the values of sequences and where and .
Example 2. Let be the two-dimensional space of real numbers with an inner product defined by and a usual norm given by , for every . Suppose . Define the mappings byFor , let be defined byTake . All sequences and other parameters are defined as in Example 1. Let and . Therefore, and converge strongly to (0,0).
Solution. Clearly, all sequences, parameters, and mappings satisfy all conditions of Theorem 12. Hence,Applying Theorem 12, the sequences and converge strongly to (0,0).
Table 2 and Figure 2 show the values of sequences and where and .
5. Conclusion
This study proposes a new subgradient extragradient method for approximating a common fixed point of a finite family of demicontractive mappings and Lipschitzian mappings and a common solution of variational inequality problems. It can also be considered as an extension and modification of several currently used techniques for solving variational inequality problems as well as a fixed point problem with some associated mappings. As special cases of Theorem 12, previous publications such as [14, 15, 19, 20] can be considered. Also, numerical illustrations of the main theorem are given [24, 25].
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This research was supported by the Lampang Rajabhat University.