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A viscosity hybrid steepest-descent method for a system of equilibrium and fixed point problems for an infinite family of strictly pseudo-contractive mappings
Journal of Inequalities and Applications volume 2012, Article number: 224 (2012)
Abstract
Based on a viscosity hybrid steepest-descent method, in this paper, we introduce an iterative scheme for finding a common element of a system of equilibrium and fixed point problems of an infinite family of strictly pseudo-contractive mappings which solves the variational inequality for . Furthermore, we also prove the strong convergence theorems for the proposed iterative scheme and give a numerical example to support and illustrate our main theorem.
MSC:46C05, 47D03, 47H05,47H09, 47H10, 47H20.
1 Introduction
Throughout this paper, we assume that H is a real Hilbert space with inner product and norm . Let C be a nonempty closed convex subset of H. A self-mapping is said to be a contraction on C if there exists a constant such that , . We denote by the collection of mappings f verifying the above inequality and note that each has a unique fixed point in C.
A mapping is said to be λ-strictly pseudo-contractive if there exists a constant such that
and we denote by the set of fixed points of the mapping T; that is, .
Note that T is the class of λ-strictly pseudo-contractive mappings including the class of nonexpansive mappings T on C (that is, , ) as a subclass. That is, T is nonexpansive if and only if T is 0-strictly pseudo-contractive.
A mapping is called k-Lipschitzian if there exists a positive constant k such that
F is said to be η-strongly monotone if there exists a positive constant η such that
Definition 1.1 A bounded linear operator A is said to be , if there exists a constant such that
In 2006, Marino and Xu [1] introduced the following iterative scheme: for ,
They proved that under appropriate conditions of the sequence , the sequence generated by (1.4) converges strongly to the unique solution of the variational inequality , , which is the optimality condition for the minimization problem
where h is a potential function for γf (i.e., for ).
In 2010, Jung [2] extended the result of Marino and Xu [1] to the class of k-strictly pseudo-contractive mappings with and introduced the following iterative scheme: for ,
where is a mapping defined by . He proved that the sequence generated by (1.5) converges strongly to a fixed point q of T, which is the unique solution of the variational inequality
Later, Tian [3] considered the following iterative method for a nonexpansive mapping with ,
where F is a k-Lipschitzian and η-strongly monotone operator. He proved that the sequence generated by (1.6) converges to a fixed point q in , which is the unique solution of the variational inequality
In 2010, Saeidi [4] introduced the following modified hybrid steepest-descent iterative algorithm for finding a common element of the set of solutions of a system of equilibrium problems for a family and the set of common fixed points for a family of infinitely nonexpansive mappings with respect to W-mappings (see [5]):
where B is a relaxed -cocoercive, k-Lipschitzian mapping such that . Then, under weaker hypotheses on coefficients, he proved the strongly convergence of the proposed iterative algorithm to the unique solution of the variational inequality.
Recently, Wang [6] extended and improved all the above results. He introduced a new iterative scheme: for ,
where is a mapping defined by (2.3), and F is a k-Lipschitzian and η-strongly monotone operator with . He proved that the sequence generated by (1.7) converges strongly to a common fixed point of an infinite family of -strictly pseudo-contractive mappings, which is a unique solution of the variational inequality
Very recently, He, Liu and Cho [7] introduced an explicit scheme which was defined by the following suitable sequence:
They generated -mapping by and where is a family of nonexpansive mappings from H into itself. They found that if , and satisfy appropriate conditions and , then converges strongly to , which satisfies the variational inequality for all .
In this paper, we introduce a new iterative scheme in a Hilbert space H which is a mixed iterative scheme of (1.7) and (1.8). We prove that the sequence converges strongly to a common element of the set of solutions of the system of equilibrium problems and the set of common fixed points of an infinite family of strictly pseudo-contractive mappings by using a viscosity hybrid steepest-descent method. The results obtained in this paper improved and extended the above mentioned results and many others. Finally, we give a simple numerical example to support and illustrate our main theorem in the last part.
2 Preliminaries
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. We have
Recall that the nearest projection from H to C assigns to each the unique point satisfying the property
We recall some lemmas which will be needed in the rest of this paper.
Lemma 2.1 In a Hilbert space H, the following inequality holds:
Lemma 2.2 Let B be a k-Lipschitzian and η-strongly monotone operator on a Hilbert space H with , , and . Then is a contraction with a contractive coefficient and .
Proof From (1.2), (1.3) and (2.1), we have
where , and so, .
Hence, S is a contraction with a contractive coefficient . □
Lemma 2.3 Let H be a Hilbert space. For a given and ,
Lemma 2.4 Let H be a real Hilbert space. For q which solves the variational inequality , , , the following statement is true:
where .
Proof From Lemma (2.3), it follows that
□
Lemma 2.5 [8]
Let C be a closed convex subset of a Hilbert space H and be a nonexpansive mapping with ; if the sequence weakly converges to x and converges strongly to y, then .
Lemma 2.6 [9]
Let and be bounded sequences in a Banach space E and be a sequence in which satisfies the following condition:
Suppose that , and . Then .
Let be a sequence of non-negative real numbers satisfying
where , and satisfy the following conditions:
-
(i)
and ;
-
(ii)
or ;
-
(iii)
(), .
Then .
Lemma 2.8 [12]
Let C be a nonempty closed convex subset of a real Hilbert space H and be a λ-strictly pseudo-contractive mapping. Define a mapping by for all and . Then S is a nonexpansive mapping such that .
In this work, we defined the mapping by
where are real numbers such that , where is a -strictly pseudo-contractive mapping of C into itself and . By Lemma 2.8, we know that is a nonexpansive mapping and . As a result, it can be easily seen that is also a nonexpansive mapping.
Lemma 2.9 [5]
Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let be nonexpansive mappings of C into itself such that and be real numbers such that for each . Then for any and , the limit exists.
By using Lemma 2.8, one can define the mapping W of C into itself as follows:
Such a mapping W is called the modified W-mapping generated by , and .
Lemma 2.10 [5]
Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let be nonexpansive mappings of C into itself such that and be real numbers such that for each . Then .
Combining Lemmas 2.7-2.9, one can get that .
Lemma 2.11 [13]Let C be a nonempty closed convex subset of a Hilbert space H, be a family of infinite nonexpansive mappings with , be a real sequence such that , for each . If K is any bounded subset of C, then
For solving the equilibrium problem, let us give the following assumptions on a bifunction , which were imposed in [14]:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each , ;
(A4) for each , is convex and lower semicontinuous.
Lemma 2.12 [14]
Let C be a nonempty closed convex subset of H, and let F be a bifunction of into ℝ satisfying (A1)-(A4). Then for and , there exists such that
Lemma 2.13 [15]
Let C be a nonempty closed convex subset of H, and let F be a bifunction of into ℝ satisfying (A1)-(A4). For , define a mapping as follows:
for all . Then the following conclusions hold:
-
(1)
is single-valued;
-
(2)
is firmly nonexpansive, i.e., for any ,
-
(3)
;
-
(4)
is closed and convex.
Lemma 2.14 [5]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an infinite family of nonexpanxive mappings with and be a real sequence such that for each . Then:
-
(1)
is nonexpansive and for each ;
-
(2)
for each and for each positive integer k, the limit exists;
-
(3)
the mapping defined by is a nonexpansive mapping satisfying and it is called the W-mapping generated by and ;
-
(4)
if K is any bounded subset of C, then .
3 Main results
In this section, we will introduce an iterative scheme by using a viscosity hybrid steepest-descent method for finding a common element of the set of variational inequalities, fixed points for an infinite family of strictly pseudo-contractive mappings and the set of solutions of a system of equilibrium problems in a real Hilbert space.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, let be a -strictly pseudo-contractive mapping with , be a finite family of bifunctions into ℝ satisfying (A1)-(A4) and be a real sequence such that for each . Let B be a k-Lipschitzian and η-strongly monotone operator on C with and with and . Assume that . Let the mapping be defined by (2.3). Let be the sequence generated by and
where and are sequences in which satisfy the following conditions:
(C1) and ;
(C2) for some constant ;
(C3) , for each .
Then the sequence converges strongly to , where , which is the unique solution of the variational inequality
or equivalently, q is the unique solution of the minimization problem
where h is a potential function for γf.
Proof We will divide the proof of Theorem 3.1 into several steps.
Step 1. We show that is bounded. Let . Since for each , is nonexpansive. Given for and , for each , we have
Consider,
From Lemma 2.2, (3.1) and (3.3), it follows that
By mathematical induction, we have
and we obtain is bounded. So are , and .
Step 2. We claim that if is a bounded sequence in C, then
for every . From Step 2 of the proof in [[16], Theorem 3.1], we have for ,
Note that for every , we have
So, we note that
Now, applying (3.7) to (3.8), we conclude (3.6).
Step 3. We show that .
We define a sequence by , so that . We now observe that
It follows from (3.9) that
We observe that
and compute
Consider,
where is a constant such that for all .
Substituting (3.11) and (3.13) into (3.10), we can obtain
where .
It follows from (3.14) that
Hence, we have
From and the condition and for some , it follows that
By Lemma 2.5, we obtain
From and by (3.16), we get
Hence,
Step 4. We claim that .
It follows that
By the conditions (C1) and (C2), we obtain
Step 5. We show that
for any and . We note that is firmly nonexpansive by Lemma 2.12, then we observe that
and hence
It follows that
where
It follows from the condition (C1) that
So, we obtain
Using the condition (C1), (3.17) and (3.22), we obtain
Step 6. We show that , where .
The Banach contraction principle guarantees that has a unique fixed point q which is the unique solution of (3.1). Let be a subsequence of such that
Since is bounded, then there exists a subsequence which converges weakly to . Without loss of generality, we can assume that . We claim that .
Next, we need to show that . First, by (A2) and given and , we have
Thus,
From (A4), is a lower semicontinuous and convex, and thus weakly semicontinuous. The condition (C3) and (3.23) imply that
in norm. Therefore, letting in (3.24) yields
for all and . Replacing y with with and using (A1) and (A4), we obtain
Hence, , for all and . Letting and using (A3), we conclude that for all and . Therefore,
that is,
Next, we show that . By Lemma 2.6, we have
and . Assume that , then . Therefore, from the Opial property of a Hilbert space, (3.27), (3.28) and Step 4, we have
It is a contradiction. Thus z belongs to . Hence, .
Hence, by Lemma 2.4, we obtain
Step 7. We claim that converges strongly to . We observe that
where . Put and . It follows that
From (C1), (C2) and Step 5, it follows that and . Hence, by Lemma 2.7, the sequence converges strongly to q. □
Using Theorem 3.1, we obtain the following corollaries.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an infinite family of nonexpansive mappings with , be a finite family of bifunctions into ℝ satisfying (A1)-(A4) and be a real sequence such that for each . Let B be a k-Lipschitzian and η-strongly monotone operator on C with and with and . Assume that . Let the mapping be defined by (2.3). Let be the sequence generated by and
where and are the sequences in which satisfy the following conditions:
(C1) and ;
(C2) for some constant ;
(C3) , for each .
Then the sequence converges strongly to where , which is the unique solution of the variational inequality
Remark 3.3 Corollary 3.2 extends and improves Theorem 3.1 from f an infinite family of nonexpansive mappings to a family of strictly pseudo contractive mappings.
If in Theorem 3.1, we obtain the following corollary.
Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an infinite family of nonexpansive mappings with , be a finite family of bifunctions into ℝ satisfying (A1)-(A4) and be a real sequence such that for each . Let B be a k-Lipschitzian and η-strongly monotone operator on C with and with and . Assume that . Let the mapping be defined by (2.3). Let be the sequence generated by and
where and are the sequences in which satisfy the following conditions:
(C1) and ;
(C2) for some constant ;
(C3) , for each .
Then the sequence converges strongly to , where , which is the unique solution of the variational inequality
If , , , and in Theorem 3.1, we obtain the following corollary:
Corollary 3.5 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an infinite family of nonexpansive mappings with , be a finite family of bifunctions into ℝ satisfying (A1)-(A4) and be a real sequence such that for each . Let B be a k-Lipschitzian and η-strongly monotone operator on C with and with and . Assume that . Let the mapping be defined by (2.3). Let be the sequence generated by and
where and are the sequences in which satisfy the following conditions:
(C1) and ;
(C2) for some constant ;
(C3) , for each .
Then the sequence converges strongly to where , which is the unique solution of the variational inequality
4 Numerical example
In this section, we give a real numerical example of Theorem 3.1 as follows.
Example 4.1 Let , , . , , , , , , , for every and . Then is the sequence generated by
and as , where 0 is the unique solution of the minimization problem
where is a constant.
Proof We divide the proof into four steps.
Step 1. Using the idea in [7], we can show that
where
Since , , , with the definition of , in Lemma 2.13, we have
By the equivalent property of the nearest projection from H to C, we can conclude that if we take , . By (3) in Lemma 2.13, we have
Step 2. We show that
Since , where is a -strictly pseudo-contractive mapping and , it can be easily seen that is a nonexpansive mapping. By (2.3), we have
and we compute (2.3) in the same way as above, so we obtain
Since , , , hence,
Step 3. We prove
where 0 is the unique solution of the minimization problem
Since we let , γ is a real number, so we choose . From (4.3), (4.4) and (4.7), we can obtain a special sequence of Theorem 3.1 as follows:
Since , , we have
Combining it with (4.6), we obtain
It is obvious that , 0 is the unique solution of the minimization problem , where is a constant number.
Step 4. In this step, we give the numerical results that support our main theorem as shown by plotting graphs using Matlab 7.11.0. We choose two different initial values as and in Table 1, Figure 1, and Figure 2, respectively. From the example, we can see that converges to 0. □
References
Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028
Jung JS: Strong convergence of iterative methods for k -strictly pseudo-contractive mappings in Hilbert spaces. Appl. Math. Comput. 2010, 215: 3746–3753. 10.1016/j.amc.2009.11.015
Tian M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 2010, 73: 689–694. 10.1016/j.na.2010.03.058
Saeidi S: Modified hybrid steepest-descent methods for variational inequalities and fixed points. Math. Comput. Model. 2010, 52: 134–142. 10.1016/j.mcm.2010.01.023
Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwan. J. Math. 2001, 5(2):387–404.
Wang S: A general iterative method for obtaining an infinite family of strictly pseudo-contractive mappings in Hilbert spaces. Appl. Math. Lett. 2011, 24: 901–907. 10.1016/j.aml.2010.12.048
He H, Liu S, Cho YJ: An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. J. Comput. Appl. Math. 2011. doi:10.1016/j.cam.2011.03.003
Goebel K, Kirk WA: Topics on Metric Fixed-Point Theory. Cambridge University Press, Cambridge; 1990.
Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integral. J. Math. Anal. Appl. 2005, 35: 227–239.
Liu LS: Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332
Zhou Y: Convergence theorems of fixed points for k -strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2008, 69: 456–462. 10.1016/j.na.2007.05.032
Chang S-S: A new method for solving equilibrium problem and variational inequality problem with application to optimization. Nonlinear Anal. 2009, 70: 3307–3319. 10.1016/j.na.2008.04.035
Colao V, Acedob GL, Marinoa G: An implicit method for finding common solution of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. Nonlinear Anal. 2009, 71: 2708–2713. 10.1016/j.na.2009.01.115
Combettes PL, Hirstaoga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.
Colao V, Marino G, Xu HK: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 2008, 344: 340–352. 10.1016/j.jmaa.2008.02.041
Acknowledgements
This research was partially finished during the last visit at Kyungnam University, South Korea. The first author would like to thank the Office of the Higher Education Commission, Thailand, for the financial support of the PhD Program at KMUTT. The third author would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under the project NRU-CSEC No.55000613) for financial support during the preparation of this manuscript. Finally, the authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and improving the original version of this paper.
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Witthayarat, U., Kim, J.K. & Kumam, P. A viscosity hybrid steepest-descent method for a system of equilibrium and fixed point problems for an infinite family of strictly pseudo-contractive mappings. J Inequal Appl 2012, 224 (2012). https://doi.org/10.1186/1029-242X-2012-224
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DOI: https://doi.org/10.1186/1029-242X-2012-224