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A new iterative method for equilibrium problems and fixed point problems for infinite family of nonself strictly pseudocontractive mappings
Fixed Point Theory and Applications volume 2013, Article number: 286 (2013)
Abstract
The purpose of this paper is to present an iterative method for finding a common element of the set of solutions for an equilibrium problem and the set of common fixed points for a countably infinite family of nonself -strictly pseudocontractive mappings from a closed convex subset C into a Hilbert space H.
MSC:47H17, 47H20.
1 Introduction and preliminaries
Let H be a real Hilbert space with the inner product and the norm , respectively. Let C be a nonempty closed convex subset of H, and let be a bifunction. The equilibrium problem for G is to find such that
The set of solutions of (1.1) is denoted by . The equilibrium problem (1.1) includes as special cases numerous problems in physics, optimization, economics, transportation, and engineering.
Assume that the bifunction G satisfies the following standard properties.
Assumption A Let be a bifunction satisfying conditions (A1)-(A4).
-
(A1)
, ;
-
(A2)
, ;
-
(A3)
For each , is lower semicontinuous and convex;
-
(A4)
, .
Recall that a mapping T in H is said to be λ-strictly pseudocontractive in the terminology of Browder and Petryshyn [1], if there exists a constant such that
for all , the domain of T, where I is the identity operator in H. Clearly, when , T is nonexpansive, i.e.,
It means that the class of λ-strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.
A single-valued mapping A in H is said to be monotone, if
and λ-inverse strongly monotone, if there exists a positive constant λ such that
Therefore, any λ-inverse strongly monotone mapping is monotone and Lipschitz continuous with the Lipschitz constant . It is well known [2] that if T is a nonexpansive mapping, then is -inverse strongly monotone. It is easy to verify that if T is λ-strictly pseudocontractive, then it is -inverse strongly monotone. Hence, is Lipschitz continuous with the Lipschitz constant .
Let be a countably infinite family of nonself -strictly pseudocontractive mappings from C into H with the set of fixed points (i.e., ). Set . Assume that
The problem studied in this paper is to find an element
If , , then problem (1.2) is reduced to (1.1) and shown in [3] and [4] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, the Nash equilibria in noncooperative games, vector equilibrium problems, as well as certain fixed point problems (see also [5]). For finding approximative solutions of (1.1), there exist several approaches: the regularization approach [6–10], the gap-function approach [8, 11–13], and iterative procedure approach [14–30].
In the case that and , , (1.2) is a problem of finding a fixed point for a λ-strictly pseudocontractive mapping in C (see [31]).
In the case that and , , (1.2) is a problem of finding a common fixed point for a finite family of -strictly pseudocontractive mappings in C studied in [32], where the following algorithm is constructed:
Let and , , be three sequences in satisfying for all , and let be a sequence in C. Then the sequence generated by
where , is called the implicit iteration process with mean errors for a finite family of strictly pseudocontractive mappings . The sequence converges weakly to a common fixed point of the maps .
If , , then (1.2) is a problem of finding a fixed point for a λ-strictly pseudocontractive mapping in C, which is an equilibrium point for G (see [33]).
Theorem 1.1 [33]
Let C be a nonempty closed convex subset of a Hilbert space H. Let G be a bifunction from to satisfying (A1)-(A4), and let S be a nonexpansive mapping of C into H such that
Let f be a contraction of H into itself, and let and be sequences generated by and
for all , where and satisfy
Then and converge strongly to , where .
If , , then (1.2) is a problem of finding a common fixed point for a finite family of -strictly pseudocontractive mapping from C into H, which is an equilibrium point for G. We have constructed regularization algorithms (see [10, 19, 29]). First algorithm is defined as follows. The regularization solution is an element being a solution for the single equilibrium problem
where α is the regularization parameter.
Theorem 1.2 [19]
For each , problem (1.4) has a unique solution such that
-
(i)
, .
-
(ii)
, .
-
(iii)
, , where d is a positive constant.
In the second algorithm, the regularization solution is defined on the base of the inertial proximal point algorithm proposed by Alvarez and Attouch [34], where the sequence is defined by an equilibrium problem
and and are the sequences of positive numbers.
Theorem 1.3 [19]
Assume that the parameters , and are chosen such that
-
(i)
, ,
-
(ii)
, ,
-
(iii)
,
-
(iv)
, .
Then the sequence converges strongly to the element , as .
In the case that , Maingé [35] considered the following iteration method:
where is a given contraction with constant , is a starting point, , for all and with the following conditions:
-
C1.
.
-
C2.
For all ,
-
(a)
, or ,
-
(b)
, or ,
-
(c)
, or .
-
(a)
-
C3.
for all .
Then the sequence given by (1.5) converges strongly to the unique fixed point of , where is the metric projection from H onto ℱ.
For solving (1.2), Ceng and Yao [36] proposed the following algorithm:
Let be an arbitrary element and
where , , and are three sequences in such that , and the mapping is defined by
Theorem 1.4 [36]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction satisfying (A1)-(A4), and let be a sequence of nonexpansive self-mappings on C such that
Suppose that , , and are sequences in such that , and is a real sequence. Suppose that the following conditions are satisfied:
-
(i)
and ;
-
(ii)
;
-
(iii)
and ;
-
(iv)
and .
Let f be a contraction of C into itself and given . Then the sequences and generated by (1.6), where is a sequence in for some , converge strongly to , where .
2 Main results
In this section, for solving problem (1.2), we present a new iterative method.
Let be an arbitrary element in a Hilbert space H, the sequence of iterations is defined by finding such that
where is the metric projection of H onto C, and , , and are three sequences of positive numbers such that
Also, we construct a regularization solution for (1.2) by solving the following variational inequality problem: find such that
where , is a sequence of positive numbers satisfying , and α is a small regularization parameter tending to zero.
We need the following important lemmas for the proof of our main results.
Lemma 2.1 [7]
Let C be a nonempty closed convex subset of a Hilbert space H, and let G be a bifunction of into satisfying (A1)-(A4). Let and . Then there exists such that
Lemma 2.2 [7]
Assume that satisfies conditions (A1)-(A4). For and , define a mapping as follows:
Then the following statements hold:
-
(i)
is single-valued;
-
(ii)
is firmly nonexpansive, i.e., for any ,
-
(iii)
;
-
(iv)
is closed and convex.
It is easy to see that is a nonexpansive mapping.
Lemma 2.3 [37]
Let , , and be the sequences of positive numbers satisfying the conditions:
-
(i)
, ,
-
(ii)
, .
Then .
Assume that T is a λ-strictly pseudocontractive mapping of a closed convex subset C of a Hilbert space H. Then is demiclosed at zero; that is, whenever is a sequence in C weakly converging to some , and the sequence strongly converges to zero, it follows that .
Now, we are in a position to introduce and prove the main results.
Theorem 2.5 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction satisfying (A1)-(A4), and let be a sequence of nonself -strictly pseudocontractive mappings from C into H such that
Assume that satisfies condition (2.2). Then for each , problem (2.3) has a unique solution such that
-
(i)
, and , ,
-
(ii)
.
Proof First, we prove that problem (2.3) has a unique solution. Set
Let be a common fixed point of . Since
and , the mapping A is well defined, and converges absolutely for each . It is easy to see that A is Lipschitz continuous with the Lipschitz constant and monotone.
Set
and
Then, problem (2.3) is equivalent to that: find such that
where
and
It is not difficult to verify that for each , is a bifunction. Therefore, also is a bifunction, i.e., satisfies Assumption A. By using Lemma 2.2 with and , we can conclude that problem (2.5) (consequently (2.3)) has a unique solution for each .
(i) Next, we prove that
Since , we have , . Thus,
Since
from (2.7), follows (2.6). Then, there exists a subsequence of that converges weakly to some element . Now, we have to prove that . The -inverse strongly monotone property of implies that
Therefore,
By virtue of Lemma 2.4, we have . Since the closed convex set has only one element with the minimal norm, using (2.6) and the weak convergence of , we can conclude that all the sequence converges strongly to as .
(ii) By virtue of (2.4), (2.5) and the monotone property of , we obtain
or
for each . This completes the proof. □
Remark Obviously, if , where is the solution of (2.4) with , as , then .
Theorem 2.6 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction satisfying (A1)-(A4), and let be a sequence of nonself -strictly pseudocontractive mappings from C into H such that
Assume that satisfies condition (2.2), and , satisfy the following conditions:
Then the sequence given by (2.1) converges strongly to , that is,
Proof Clearly, iteration in (2.1) has the form
where .
Let be the solution of (2.3) when . Then
Set . Obviously,
From the nonexpansive property of , the monotone and Lipschitz continuous properties of , , (2.8), and (2.9), we have
and
Thus,
Set
It is not difficult to check that and satisfy the conditions in Lemma 2.3 for sufficiently large n. Hence, . Since , we have
This completes the proof. □
Remark The sequences , , and with
satisfy all the necessary conditions in Theorem 2.6.
3 Application
In this section, we show that algorithm (2.1) can be applied to find an element
where is the set of solutions of the following variational inequality problem:
involving a monotone hemicontinuous mapping . If is a λ-inverse strongly monotone mapping, and is a countably infinite family of nonexpansive self mappings in C, then problem (3.1) is recently studied in [40] and [41].
Let be a monotone hemicontinuous mapping. Then it is easy to see that is a bifunction, i.e., satisfies Assumption A. Using Lemma 2.2, we construct a nonexpansive mapping such that
Then, . So, as are nonexpansive. Consequently, both mappings and are -strictly pseudocontractive. Thus, the mapping is also Lipschitz continuous with the Lipschitz constant . Then, algorithm (2.1) has the form:
Let be an arbitrary element in a Hilbert space H, the sequence of iteration is defined by
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction satisfying (A1)-(A4), let be a sequence of nonself -strictly pseudocontractive mappings from C into H, and let be a hemicontinuous monotone mapping from C into H such that
Assume that satisfies condition (2.2), and , satisfy the following conditions:
Then
where is defined by (3.3).
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Acknowledgements
This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by Ministry of Education of the Republic of Korea (2013R1A1A2054617).
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The main idea of this paper was proposed by JKK. JKK and NB prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
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Kim, J.K., Buong, N. A new iterative method for equilibrium problems and fixed point problems for infinite family of nonself strictly pseudocontractive mappings. Fixed Point Theory Appl 2013, 286 (2013). https://doi.org/10.1186/1687-1812-2013-286
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DOI: https://doi.org/10.1186/1687-1812-2013-286