Abstract
The purpose of this paper is to modify the generalized equilibrium problem introduced by Ceng et al. (J. Glob. Optim. 43:487-502, 2012) and to introduce the K-mapping generated by a finite family of strictly pseudo-contractive mappings and finite real numbers modifying the results of Kangtunyakarn and Suantai (Nonlinear Anal. 71:4448-4460, 2009). Then we prove the strong convergence theorem for finding a common element of the set of fixed points of a finite family of strictly pseudo-contractive mappings and a finite family of the set of solutions of the modified generalized equilibrium problem. Moreover, using our main result, we obtain the additional results related to the generalized equilibrium problem.
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1 Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm . A mapping is contractive if there exists a constant such that
We now recall some well-known concepts and results as follows.
Definition 1.1 Let be a mapping. Then B is called
-
(i)
monotone if
-
(ii)
υ-strongly monotone if there exists a positive real number υ such that
-
(iii)
ξ-inverse strongly monotone if there exists a positive real number ξ such that
-
(iv)
μ-Lipschitz continuous if there exists a nonnegative real number such that
Definition 1.2 Let be a mapping. Then:
-
(i)
An element is said to be a fixed point of T if and denotes the set of fixed points of T.
-
(ii)
Mapping T is called nonexpansive if
-
(iii)
T is said to be κ-strictly pseudo-contractive if there exists a constant such that
(1.1)
Note that the class of κ-strictly pseudo-contractions strictly includes the class of nonexpansive mappings, that is, nonexpansive mapping is a 0-strictly pseudo-contraction mapping. In a real Hilbert space H (1.1) is equivalent to
Remark 1.1 is a κ-strictly pseudo-contraction if and only if is -inverse strongly monotone.
In the last decades, many researcher have studied fixed point theorems associated with various types of nonlinear mapping; see, for instance, [1–4]. Fixed point problems arise in many fields such as the vibration of masses attached to strings or nets [5] and a network bandwidth allocation problem [6] which is one of the central issues in modern communication networks. For applications to neural networks, fixed point theorems can be used to design dynamic neural network in order to solve steady state solutions [7]. For general information on neural networks, see for instance, [8, 9].
Let be bifunction. The equilibrium problem for F is to determine its equilibrium point, i.e., the set
Equilibrium problems were introduced by [10] in 1994 where such problems have had a significant impact and influence in the development of several branches of pure and applied sciences. Various problems in physics, optimization, and economics are related to seeking some elements of ; see [10, 11]. Many authors have been investigating iterative algorithms for the equilibrium problems; see, for example, [11–15].
Let be the family of all nonempty closed bounded subsets of H and be the Hausdorff metric on defined as
where , and .
Let C be a nonempty closed convex subset of H. Let be a real-valued function, a multivalued mapping and an equilibrium-like function, that is, for all which satisfies the following conditions with respect to the multivalued mapping .
(H1) For each fixed , is an upper semicontinuous function from , that is, for , whenever and as ,
(H2) For each fixed , is a concave function.
(H3) For each fixed , is a convex function.
In 2009, Ceng et al. [16] introduced the generalized equilibrium problem as follows:
The set of such solutions of is denoted by . In the case of and , then is denoted by .
By using Nadler’s theorem [17], they introduced the following algorithm:
Let and , there exist sequences and such that
They proved the strong convergence theorem of the sequence generated by (1.4) as follows.
Theorem 1.2 ([16])
Let C be a nonempty, bounded, closed and convex subset of a real Hilbert space H and let be a lower semicontinuous and convex functional. Let be ℋ-Lipschitz continuous with constant μ, be an equilibrium-like function satisfying (H1)-(H3) and S be a nonexpansive mapping of C into itself such that . Let f be a contraction of C into itself and let , , and be sequences generated by (1.4), where and satisfy
If there exists a constant such that
for all , and , , where , then for , there exists such that is a solution of and
In 2012, Kangtunyakarn [12] introduced the iterative algorithm as follows.
Algorithm 1.3 ([12])
Let , be -pseudo-contraction mappings of C into itself and and let be the S-mappings generated by and , where , , and for all , , , for all . Let and , , there exist sequences , and such that
where are ℋ-Lipschitz continuous with constants , , respectively, are equilibrium-like functions satisfying (H1)-(H3), is an α-inverse strongly monotone mapping and is a β-inverse strongly monotone mapping.
He proved under some control conditions on , , , and that the sequence generated by (1.5) converges strongly to , where , are defined by , , and is a solution of the following system of variational inequalities:
By modifying the generalized equilibrium problem (1.3), we introduced the modified generalized equilibrium problem as follows:
where is a mapping. The set of such solutions of is denoted by . If , (1.6) reduces to (1.3).
In this paper, motivated by Theorem 1.2, Algorithm 1.3 and (1.6), we modify the generalized equilibrium problem introduced by Ceng et al. [16] and introduce the K-mapping generated by a finite family of strictly pseudo-contractive mappings and finite real numbers modifying the results of Kangtunyakarn and Suantai [13]. Then we prove the strong convergence theorem for finding a common element of the set of fixed points of a finite family of strictly pseudo-contractive mappings and a finite family of the set of solutions of the modified generalized equilibrium problem. Moreover, using our main result, we obtain the additional results related to the generalized equilibrium problem.
2 Preliminaries
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. We denote weak convergence and strong convergence by the notations ‘⇀’ and ‘→’, respectively.
Recall that the (nearest point) projection from H onto C assigns to each the unique point satisfying the property
The following lemmas are needed to prove the main theorem.
Lemma 2.1 ([18])
Let H be a real Hilbert space. Then the following identities hold:
-
(i)
, ;
-
(ii)
, .
Lemma 2.2 ([19])
Let H be a real Hilbert space. Then for all and for such that the following equality holds:
Lemma 2.3 ([18])
For a given and ,
Furthermore, is a firmly nonexpansive mapping of H onto C and satisfies
Lemma 2.4 (Demiclosedness principle [20])
Assume that T is a nonexpansive self-mapping of closed convex subset C of a Hilbert space H. If T has a fixed point, then is demiclosed. That is, whenever is a sequence in C weakly converging to some and the sequence strongly converges to some y it follows that . Here, I is the identity mapping of H.
Lemma 2.5 ([21])
Let C be a nonempty closed convex subset of a real Hilbert space H and be a self-mapping of C. If S is a κ-strict pseudo-contractive mapping, then S satisfies the Lipschitz condition
Lemma 2.6 ([22])
Let be a sequence of nonnegative real numbers satisfying
where is a sequence in and is a sequence such that
-
(1)
;
-
(2)
or .
Then .
Definition 2.1 A multivalued mapping is said to be ℋ-Lipschitz continuous if there exists a constant such that
where is the Hausdorff metric on .
Lemma 2.7 (Nadler’s theorem [17])
Let be a normed vector space and is the Hausdorff metric on . If , then for every and , there exists such that
Theorem 2.8 ([16])
Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H, and let be a lower semicontinuous and convex functional. Let be ℋ-Lipschitz continuous with constant μ, and be an equilibrium-like function satisfying (H1)-(H3). Let be a constant. For each , take arbitrarily and define a mapping as follows:
Then we have the following:
-
(a)
is single-valued;
-
(b)
is firmly nonexpansive (that is, for any , ) if
for all and all , ;
-
(c)
;
-
(d)
is closed and convex.
Definition 2.2 ([13])
Let C be a nonempty closed convex subset of a real Banach space. Let be a finite family of -strictly pseudo-contractive mapping of C into itself and let be real numbers with for every . Define a mapping as follows:
Such a mapping K is called the K-mapping generated by and .
The following lemmas are needed to prove our main result.
Lemma 2.9 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a finite family of -strictly pseudo-contractive mapping of C into itself with , for all , and . Let be real numbers with , for all and . Let K be the K-mapping generated by and . Then the following properties hold:
-
(i)
;
-
(ii)
K is a nonexpansive mapping.
Proof To prove (i), it is easy to see that .
Next, we claim that . To show this, let and .
By the definition of K-mapping, we get
From (2.2), it yields
This implies that
Therefore , that is,
By the definition of and (2.3), we have
that is,
Again by (2.2) and (2.4), we obtain
which implies that , that is,
By the definition of , (2.4), and (2.5), we get
from which it follows that
Using the same argument, we can conclude that
Next, we show that . Since
and , we obtain
from which it follows that
Therefore
Hence
To prove (ii), we claim that K is a nonexpansive mapping.
Let . Then we obtain
which implies that
From (2.9) and , we obtain
that is, K is a nonexpansive mapping. □
Lemma 2.10 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a finite family of -strictly pseudo-contractive mappings of C into itself with and . For every and , let and be real numbers with and such that as and . For every , let K and be the K-mappings generated by and and and , respectively. Then, for every bounded sequence in C, the following properties hold:
-
(i)
;
-
(ii)
.
Proof Let be a bounded sequence in C and let and be generated by and and and , respectively.
First, we shall prove that (i) holds. For each , we obtain
For , we have
By (2.10) and (2.11), we get
By (2.12) and the fact that as for all , we deduce that .
Next, we will claim that (ii) holds. For each , we obtain
For , we have
From (2.13) and (2.14), we obtain
where , for all . Hence, by (2.15) and for all , we have . □
In 2010, Kangtunyakarn and Suantai [23] introduced the S-mapping generated by the finite family of -strictly pseudo-contractions in Hilbert space as in the following definition.
Definition 2.3 ([23])
Let C be a nonempty closed convex subset of real Hilbert space. Let be a finite family of -strictly pseudo-contractions of C into itself. For each , let where and . Define the mappings as follows:
This mapping is called S-mapping generated by and .
Furthermore, they obtained the following important lemma.
Lemma 2.11 ([23])
Let C be a nonempty closed convex subset of real Hilbert space. Let be a finite family of -strictly pseudo-contractions of C into itself with and and let , , where , , for all and , , for all . Let S be the mapping generated by and . Then and S is a nonexpansive mapping.
By putting and , for all , we see that the S-mapping reduces to the K-mapping as defined in Definition 2.2. Moreover, from Lemma 2.11, we have the following result.
Lemma 2.12 Let C be a nonempty closed convex subset of real Hilbert space. Let be a finite family of -strictly pseudo-contractions of C into itself with and and let , for all and . Let K be the mapping generated by and . Then and K is a nonexpansive mapping.
Remark 2.13 For the result of Lemma 2.9 in our work, we obtain some improvement as follows:
-
(i)
We relax the conditions of and in Lemma 2.12 in sense that is not depended on , for all .
-
(ii)
We do not assume the condition .
Example 2.14 Let ℝ be the set of real numbers and let be defined by
and , for all . Let K be the K-mapping generated by and . Then .
Solution. It is easy to see that is -strictly pseudo-contractive mapping with . We obtain and , for all . By the definition of a K-mapping, we have
Observe that . Then, by Lemma 2.12, we obtain
Next, we give an example for Lemma 2.9.
Example 2.15 Let ℝ be the set of real numbers and let be defined by
and , for all . Let K be the K-mapping generated by and . Choose and , from which it follows that . Then, by Lemma 2.9, we obtain .
3 Strong convergence theorem
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. For every , be ℋ-Lipschitz continuous with coefficients , be equilibrium-like function satisfying (H1)-(H3). Let be a lower semicontinuous and convex function and be an α-inverse strongly monotone mapping. Let be a finite family of -strictly pseudo-contractive mappings and with . For every , let be the K-mapping generated by and where , for all and . For every , let be the sequence generated by and , there exist sequences and such that
where be a contraction mapping with a constant ξ and with , . Suppose the following conditions hold:
-
(i)
and ;
-
(ii)
;
-
(iii)
, for all and with ;
-
(iv)
, for all and ;
-
(v)
, , , , , , for all ;
-
(vi)
for each , there exists such that
(3.2)
for all , and , for , where .
Then and converges strongly to , for every .
Proof The proof shall be divided into seven steps.
Step 1. We will prove that is nonexpansive, for all .
From (3.1), we have
for every . From (3.3) and Theorem 2.8, we obtain
Put for all . From (3.2), we have
for all and , .
From (3.4), we find the implication that Theorem 2.8 holds.
It obvious to see that is a nonexpansive mapping, for every .
Indeed, A is α-inverse strongly monotone with , we get
Thus is a nonexpansive mapping, for all .
Step 2. We will show that is bounded.
Let . By nonexpansiveness of , we have
By induction, we have , . It follows that is bounded and so is , .
Step 3. We will show that .
By the definition of , we obtain
From , for all , we have
and
In particular, we obtain
and
Summing up (3.6) with (3.7) and applying (3.4), we get
which implies that
It follows that
From (3.8), we obtain
from which it follows that
From (3.9), we have
From (3.5) and (3.10), we obtain
Applying the conditions (i), (v), Lemma 2.6, and Lemma 2.10(ii), we obtain
Step 4. We will show that , .
Since is a firmly nonexpansive mapping, for every , we obtain
which implies that
From the nonexpansiveness of and , for every , we have
From the definition of and (3.13), we get
from which it follows that
From (3.11), (3.14), and the conditions (i), (ii), (iii), and (iv), we obtain
From the definition of and (3.12), we have
which implies that
From (3.11), (3.15), (3.16), and the conditions (i), (ii), (iii), we get
By the definition of , we obtain
From (3.11), (3.17), and the conditions (i) and (ii), we get
Step 5. We show that , and are Cauchy sequences, for every .
Let , by (3.11), there exists such that
Thus, for any and , we have
Since , we get . From (3.20), taking , we obtain is a Cauchy sequence in a Hilbert space H. Let . Since be ℋ-Lipschitz continuous on H with coefficients , for every , and (3.1), we have
where . From (3.11), (3.21), and the condition (v), we obtain
By continuing the same argument as (3.19) and (3.20), we have is a Cauchy sequence in a Hilbert space H, for all . Let , for every . Using the same method as above and the condition (v), we see that is a Cauchy sequence, for all . Put , for every .
Next, we will prove that , for all .
Since , we obtain
Since
taking , we have
which implies that
Step 6. We will show that , where .
To show this, choose a subsequence of such that
Without loss of generality, we can assume that as .
For every , , for all , without loss of generality, we may assume that
Let K be the K-mapping generated by and . By Lemma 2.9, we see that K is nonexpansive and .
From Lemma 2.10(i), we obtain
Since
by (3.18) and (3.24), we have
Since as , by (3.25) and Lemma 2.4, we have
Next, we show that .
Since as and (3.17), we have
From (3.1), we obtain
for every and . From (3.17), (3.27), the condition (H1), and the lower semicontinuity of φ, we get
for every and , from which it follows by (3.23) that
It implies that
Since and as , then . From (3.26) and (3.28), we have
Indeed, since as , by (3.29) and Lemma 2.3, we obtain
Step 7. Finally, we will prove that and converges strongly to , for every .
By Lemma 2.1(ii), we have
which implies that
Applying the condition (i), (3.30), and Lemma 2.6, we have the sequence converges strongly to . From (3.17), we also obtain converges strongly to , for every . This completes the proof. □
The following corollaries are consequences which are applied by Theorem 3.1. Therefore, we omit the proof.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. For every , be ℋ-Lipschitz continuous with coefficients , be equilibrium-like function satisfying (H1)-(H3). Let be a lower semicontinuous and convex function and be an α-inverse strongly monotone mapping. Let be κ-strictly pseudo-contractive mapping with and . For every , let be a sequence of real numbers where and . For every , let be the sequence generated by and , there exist sequences and such that
where be a contraction mapping with a constant ξ and with , . Suppose the following conditions hold:
-
(i)
and ;
-
(ii)
;
-
(iii)
, for all and with ;
-
(iv)
, for all and ;
-
(v)
, , , , , , for all ;
-
(vi)
for each , there exists such that
(3.32)
for all and , for , where .
Then and converges strongly to , for every .
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. For every , be ℋ-Lipschitz continuous with coefficients , be equilibrium-like function satisfying (H1)-(H3). Let be a lower semicontinuous and convex function. Let be a finite family of -strictly pseudo-contractive mappings and with . For every , let be the K-mapping generated by and where , for all and . For every , let be the sequence generated by and , there exist sequences and such that
where is a contraction mapping with a constant ξ and with , . Suppose the following conditions hold:
-
(i)
and ;
-
(ii)
;
-
(iii)
, for all and with ;
-
(iv)
, for all and ;
-
(v)
, , , , , , for all ;
-
(vi)
for each , there exists such that
(3.34)
for all and , for , where .
Then and converges strongly to , for every .
Remark 3.4 From Corollary 3.3, put , then the iterative scheme (3.33) reduces to
which is a modification of iterative scheme (1.4) in the results of Ceng et al. [16]. By assuming the initial condition , and the following conditions hold:
-
(i)
and ;
-
(ii)
;
-
(iii)
, for all ;
-
(iv)
, , , , ;
-
(v)
there exists such that
for all and , for , where .
Then and converge strongly to .
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Acknowledgements
The authors are greatly thankful to the referees for their useful comments and suggestions which improved the content of this paper. This research is supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.
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Suwannaut, S., Kangtunyakarn, A. Strong convergence theorem for the modified generalized equilibrium problem and fixed point problem of strictly pseudo-contractive mappings. Fixed Point Theory Appl 2014, 86 (2014). https://doi.org/10.1186/1687-1812-2014-86
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DOI: https://doi.org/10.1186/1687-1812-2014-86