Abstract
In this research, we introduced the -mapping generated by a finite family of contractive mappings, Lipschitzian mappings and finite real numbers using the results of Kangtunyakarn (2013). Then, we prove the strong convergence theorem for fixed point sets of finite family of contraction and Lipschitzian mapping and solution sets of the modified generalized equilibrium problem introduced by Suwannaut and Kangtunyakarn (2014). Finally, numerical examples are provided to illustrate our main theorem.
1. Introduction
Let be a nonempty closed convex subset of a real Hilbert space . Let be bifunction. The equilibrium problem for is to determine its equilibrium point, i.e., the set
Equilibrium problems were first introduced by Muu and Oettli [1] in 1992. It contains various problems such as variational inequality problem, fixed point problem, optimization problem and Nash equilibrium problem. Iterative methods for the equilibrium problems are widely studied, see, for example, [2–9].
If we take , where is a nonlinear mapping, then the equilibrium problem (1) is equivalent to finding an element such thatwhich is well-known as the variational inequality problem. The solution set of the problem (2) is denoted by .
Variational inequality problem were first defined and studied by Stampacchia [10] in 1964. The variational inequality theory is an important tool based on studying a wide class of problems such as economics, optimization, operations research and engineering sciences. Several iterative algorithms have been used for solving variational inequality problem and related optimization problems (see [11–15] and the references therein).
Let be the family of all nonempty closed bounded subsets of and be the Hausdorff metric on defined aswhere and .
A multivalued mapping is said to be -Lipschitz continuous if there exists a constant such that
Let a multi-valued mapping, be a real-valued function and an equilibrium-like function, that is, for every satisfying the following properties:
is an upper semicontinuous function from , for all fixed , that is, for , whenever and as ,
is a concave function, for all fixed ;
is a convex function, for all fixed .
In 2009, Ceng et al. [16] introduced the generalized equilibrium problem as follows:
Furthermore, denotes the solution set of the generalized equilibrium problem.
In 2012, Kangtunyakarn [7] investigated the strong convergence theorem using CQ method for two solution sets of the generalized equilibrium problem and fixed point problem of nonlinear mappings.
In 2014, by modifying the generalized equilibrium problem (6), Suwannaut and Kangtunyakarn [17] introduced the modified generalized equilibrium problem as follows:where is a self-mapping on . Also, represents the solution set of . If , (7) reduces to (6). They also obtain the strong convergence theorem under some mild conditions.
Definition 1. Let be a self-mapping on . Then is called(i)nonexpansive if(ii)contractive if there exists such that(iii)inverse-strongly monotone if there exists a real number such thatIt is well-known that is demiclosed if is a nonexpansive mapping, see [18]. Moreover, is used to represent the set of fixed points of .
Definition 2. (see [19]). A mapping is called -strictly pseudo-contractive if there exists a constant such thatBrowder and Petryshyn [19] introduced and studied the class of strictly pseudo-contractive mapping as an important generalization of the class of nonexpansive mappings. It is trivial to prove that every nonexpansive mapping is strictly pseudo-contractive.
Definition 3. A mapping is called -Lipschitzian if there exists satisfying the following inequality:Note that if , becomes a contractive mapping. If , is said to be a nonexpansive mapping. In fact, all four classes of mappings mentioned in Definitions 1 and 2 are subclasses of Lipchitzian mapping.
Over the past decades, many mathematicians are interested in studying the fixed point of finite family of nonlinear mappings and their properties, (see [6–8, 17, 20–23]).
In 2009, Kangtunyakarn and Suantai [24] defined -mapping for a finite family of nonexpansive mappings. Let be defined bywhere is a finite family of nonexpansive mappings and . Moreover, under some control conditions, and is a nonexpansive mapping.
Later, Kangtunyakarn and Suantai [6] introduced the -mapping for a finite family of nonexpansive mappings. Let be defined bywhere is a finite family of nonexpansive mappings and , where and for every . Moreover, under some control conditions, and is a nonexpansive mapping.
If we take , then the -mapping reduces to the -mapping.
In 2013, using the concept of -mapping, Kangtunyakarn [25] introduced -mapping for a finite family of nonexpansive mappings and strictly pseudo-contractive mappings as follows. Let be defined bywhere be a finite family of nonexpansive mappings and be a finite family of strictly pseudo-contractive mappings, is the identity mapping and , where and for every . Also, under some control conditions, and is a nonexpansive mapping.
If , for every , then the -mapping becomes the -mapping.
Based on the previous research work, we give our theorem for MGEP and S-mapping for Lipschitzian mappings and some important results as follows:(i)We first establish Lemmas 2 and 3 showing fixed point results and some properties of S-mapping for Lipschitzian mappings under some control conditions.(ii)We prove a strong convergence theorem of the sequences generated by iterative scheme for finding a common solution of generalized equilibrium problems and fixed point problem for a finite family of contractive mappings and Lipschitzian mappings.(iii)We give some illustrative numerical examples supporting our main theorem and our examples show that our main result is not true is some conditions fail. Moreover, the main theorem can be used to approximate the value of pi.
2. Preliminaries
Throughout this work, the notations “” and “” denote weak convergence and strong convergence, respectively.
Lemma 1 (see [26]). Let be a sequence of nonnegative real numbers satisfyingwhere is a sequence in and is a sequence such that(1),(2) or .Then, .
Theorem 1 (see [16]). Let be a lower semicontinuous and convex functional. Let be -Lipschitz continuous with constant , and be an equilibrium-like function satisfying . Let be a constant. For each , take arbitrarily and define a mapping as follows:Then, the following hold:(a) is single-valued;(b) is firmly nonexpansive (that is, for any , ) if for all and all ;(c);(d) is closed and convex.
Definition 4 (see [6]). Let be a nonempty closed convex subset of a real Banach space. For every , let be a finite family of a -contractive mapping and -Lipschitzian mapping of into itself, respectively, with and . For every , let , where and . Define a mapping as follows:This mapping is called the -mapping generated by , and .
Lemma 2. For every , be a finite family of a -contractive mapping and be -Lipschitzian mapping of into itself, respectively, with , and . For every , let , where and . Let be the -mapping generated by , and . Then there hold the following statement:(i);(ii) is a nonexpansive mapping.
Proof. First, it is clear that .
Next, claim that .
Let and . By Definition 4, we haveFrom (20), it yields thatThis implies that , that is,Then, by Definition 4, we obtainAgain, from (20), we getwhich follows thatWe deduce thatthat is, .
From (23) and (26), we haveBy (22) and (27), it yields thatFrom (20) and (26), we derive thatwhich implies thatThen we obtain , that is,By the definition of , (26) and (31), we getBy (20), we have thatwhich follows thatThus, we getBy (32) and (35), we obtainBy (31) and (35), it follows thatBy using the same method described above, we easily obtain that and , for each .
From (20), we obtainwhich implies thatHence, we have , that is,By the definition of and (40), it yields thatwhich follows thatThen, we obtain , that is,Therefore, we can conclude thatFinally, by applying the similar method of (20), is a nonexpansive mapping.
Lemma 3. For each , let be a finite family of -contractive mappings and be a finite family of -Lipschitzian mappings of into itself, respectively, with , and . For each , let , , where , and satisfying the following conditions: as , for and , for . For every , let and be the -mapping generated by , and and generated by , and , , respectively. Then, for any bounded sequences in , there hold the following statement:(i);(ii).
Proof. Let be a bounded sequence in . For fixed and for all , let and be generated by , and , and , , respectively.
First, we will show that (i) holds. For every , we getFor , we haveBy (45) and (46), we getBy (47) and the fact that as , for every and , we can deduce that .
Finally, we shall prove that (ii) holds. For any , we getFor and the similar argument as (46), we haveFrom (48), (49) and the same method as (47), we obtainHence, , for every and , we have .
3. Strong Convergence Theorem
Theorem 2. Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of -contractive mappings and be a finite family of -Lipschitzian mappings of into itself, respectively, with , for all . Assume that . For each , be -Lipschitz continuous with coefficients , be equilibrium-like function satisfying (H1)–(H3). Let be a lower semicontinuous and convex function and be an -inverse strongly monotone mapping. For every , let be the -mapping generated by , and , , where , where , and , for all . For every , let be the sequence generated by and , there exists sequences and , such thatwhere is a contraction mapping with a constant and , , with , . Suppose the following statement are true:(i) and ;(ii);(iii), for each and with ;(iv), for every and ;(v), , , , , , , for each and ;(vi)For each , there exists such thatfor every , and , for , where .
Then , and converge strongly to , for all .
Proof. The proof will be splited into six steps.
Step 1. Claim that is nonexpansive, for each .
From (51), we getfor every and . From (53) and Theorem 1, it yields thatPut for all . From (52), we havefor all and .
From (55), it implies that Theorem 1 holds.
Let . Since is -inverse strongly monotone with , it deduces thatThus is a nonexpansive mapping, for all .
Step 2. Prove that , and , are bounded.
Let . From Theorem 1, observe thatBy nonexpansiveness of , we derive thatBy induction, we obtain . It follows that is bounded so are and , .
Step 3. Show that .
By the definition of , we obtainBy using the same method of proof in Step 3 in [17], we obtainSubstitute (60) into (59), we getUsing the conditions (i), (v), Lemmas 1 and 3 (ii), we obtain
Step 4. Claim that .
By following the same method as in Step 4 of [17], we deduce thatand alsoFrom the definition of and (63), we derive thatwhich implies thatFrom (62), (64) and the conditions (i), (ii), (iii), we getConsiderThen, by (67), this follows thatSincethen, from (62) and (69), we obtainBy the definition of , we obtainFrom (67), (69) and (71) and the conditions (i) and (ii), we can conclude that
Step 5. Prove that and are Cauchy sequences, for each .
Let , by (62), there exists such thatTherefore, for any and , we derive thatSince , we get . From (79), taking , we obtain is a Cauchy sequence in a Hilbert space . Let . Since be -Lipschitz continuous on with coefficients , for every , and (51), we havewhere . From (62), (76) and the condition (vi), we obtainIt is obvious that and are Cauchy sequences in a Hilbert space , for all . So we let , and for every .
Next, claim that , for each .
Because , we obtainSincetaking , we havewhich implies that
Step 6. Finally, show that , and converge strongly to , for every .
Without loss of generality, we can assume that as . By (69), we easily obtain that as .
For each , , without loss of generality, we may assume thatLet be the -mapping generated by , and . By Lemma 2, we have is nonexpansive and .
From Lemma 3 (i), we obtainSinceby (73) and (83), we haveSince as , (85) and is demiclosed, we deduce thatFinally, we show that .
Since as and (67), we getFrom (51), we havefor every and . From (67), (87), the condition and the lower semicontinuity of , we deduce thatfor every and , which follows by (81) thatIt follows thatFrom (86) and (91), we haveTherefore, we obtain the sequence converges strongly to . Moreover, from (67) and (69), we have and converge strongly to , for every . This completes the proof.
The following corollary is a direct consequence of Theorem 2. Therefore, the proof is omitted.
Corollary 1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of -contractive mappings and be a finite family of -Lipschitzian mappings of into itself, respectively, with , for all . Assume that . For each , be -Lipschitz continuous with coefficients , be equilibrium-like function satisfying (H1)–(H3). Let be a lower semicontinuous and convex function. For every , let be the -mapping generated by , and , , where , where , and , for all . For every , let be the sequence generated by and , there exists sequences and , such thatwhere is a contraction mapping with a constant and , , with , . Suppose the following statement are true:(i) and ;(ii);(iii), for each and with ;(iv), for every and ;(v), , , , , , , for each and ;(vi)For each , there exists such that
for every , and , for , where . Then , and converge strongly to , for all .
In 2014, Suwannaut and Kangtunyakarn [17] introduced the viscosity approximation method for the modified generalized equilibrium problem and a finite family of strictly pseudo-contractive mappings in Hilbert spaces. Let be a finite family of -strictly pseudo-contractive mappings with . For , let be the sequence generated by and , there exists sequences and , such thatwhere is a -mapping generated by a finite family of strictly pseudo-contractive mappings and real numbers. Then, under some control conditions, the sequences and converge strongly to , for all .
Remark 1. The iterative method (51) is a modification and extension of the iteration (95) as follows:(1)-mapping can be reduced to -mapping. Therefore, -mapping is a special case of -mapping.(2)In this research, a finite family of Lipschitzian mappings is considered instead of using a finite family of strictly pseudo-contractive mappings.
4. Numerical Examples
In this section, we give numerical examples to support our main theorem.
Example 1. Let the mappings , , be defined byFor , let and be defined byFor , let , be defined byLet , , , and , for each . For every and let , , . Then, the sequences , and converge strongly to 0, for each .
Solution. It is clear that the sequences , and satisfy all the conditions of Theorem 2. It is easy to show that is a contractive mapping with coefficient and is -Lipschitzian mapping. Since , , , then for . Since is -mapping generated by , and , we obtainFrom the definition of and , we deduce thatFor , we have that satisfy all conditions in Theorem 2 andThereforeLet , where is a quadratic function of with coefficients , , and . Determine the discriminant of as follows:From (102), we have , for every . If has most one solution in , thus we have . This deduces thatFor each , we rewrite (51) as follows:From Theorem 2, the sequences , and generated by (105) converge strongly to 0, for every .
The numerical values of all sequences , and , are shown in Table 1 and Figure 1, where and .
Next, we will give an numerical example for the iterative method (95) in the work of Suwannaut and Kangtunyakarn [17].
(a)
(b)
Example 2. For , let the mapping be defined byandLet all parameters and mappings be defined the same as mentioned in Example 1. It is obvious that is -strictly pseudo-contractive mapping, for each and all parameters and mappings satisfy all conditions of Theorem 2 in [17]. Thus, we getThe numerical values of all sequences , and , are shown in Table 2 and Figure 1, where and .
Remark 2. From the above numerical results, we can conclude that(i)Table 1 shows that the sequences , , and converge to 0, where and the convergence of all sequences can be guaranteed by Theorem 2.(ii)Table 2 shows that the sequences , , and converge to 0, where and the convergence of all sequences can be guaranteed by Theorem 2 in [17].(iii)From Tables 1 and 2, we have that the iterative method (51) converges faster than the iterative method (95).Similar to Example 1, we give another example for Theorem 2. Moreover, we use this numerical example to approximate the value of .
Example 3. Let the mappings , , be defined byFor every , let and be defined byFor , let , be defined byLet all parameter sequences be defined as in Example 1. Then, the sequences , and converge strongly to , for each .
Solution. It is clear that the sequences , and satisfy all the conditions of Theorem 2. It is obvious that is a contractive mapping with coefficient and is -Lipschitzian mapping. From the definition of and , we obtainFor each , it is obvious that satisfy all conditions in Theorem 2 and . Then we haveThen we deduce thatLet , where is a quadratic function of with coefficients , , and . Determine the discriminant of as follows:From (114), we have , for every . If has most one solution in , thus we get . This yields thatFor each , (51) becomesFrom Theorem 2, , and generated by (117) converge strongly to , for every .
The numerical values of all sequences , and , are shown in Table 3 and Figure 2, where and .
Remark 3. (i)From Table 3 and Figure 2, the sequences , , and converge to , where .(ii)The convergence of , , and can be guaranteed by Theorem 2.(iii)Using this as an example, Theorem 2 can be used to approximate the value of .
5. Conclusion
In this research, we study and analyze the viscosity iterative method for approximating a common solution of the modified generalized equilibrium problems and a common fixed point of a finite family of Lipchitzian mappings. It can be seen as an improvement and modification of some existing algorithms for solving an equilibrium problem and a fixed point problem of Lipchitzian mappings and some related mappings. Some previous research works, for example, [6, 7, 16, 17, 25] can be considered as special cases of Theorem 2. Moreover, some numerical examples for our main theorem are provided.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The two authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
Acknowledgments
This research is supported by Lampang Rajabhat University and the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang. No funding to declare.