Abstract
In this paper, equilibrium problems and fixed-point problems based on an iterative method are investigated. It is proved that the sequence generated in the purposed iterative process weakly converges to a common element of the fixed-point set of an asymptotically strict pseudocontraction in the intermediate sense and the solution set of a system of equilibrium problems in the framework of real Hilbert spaces.
MSC:47H05, 47H09, 47J25, 90C33.
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1 Introduction
Approximating solutions of nonlinear operator equations based on iterative methods is now a hot topic of intensive research efforts. Indeed, many well-known problems can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection , where is some positive integer is of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such point. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration, and radiation therapy treatment planning is to find a point in the intersection of common fixed-point sets of a family of nonlinear mappings (see [1–7]). There many classic algorithms, for example, the Picard iterative algorithm, the Mann iterative algorithm, the Ishikawa iterative algorithm, steepest descent iterative algorithms, hybrid projection algorithms, and so on. In this paper, we shall investigate fixed-point and equilibrium problems based on a Mann-like iterative algorithm.
The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, equilibrium problems and fixed-point problems of asymptotically strict pseudocontractions in the intermediate sense are discussed based on a Mann iterative algorithm. Weak convergence theorems are established in Hilbert spaces. And some deduced results are also obtained.
2 Preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with an inner product and norm . Let C be a nonempty, closed, and convex subset of H and F a bifunction of into , where stands for the set of real numbers. In this paper, we consider the following equilibrium problem.
The set of such an is denoted by , i.e.,
Given a mapping , let for all . Then if and only if
that is, z is a solution of the classical variational inequality. In this paper, we use to stand for the set of solutions of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of the equilibrium problem (2.1).
To study the equilibrium problem (2.1), we may assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each ,
(A4) for each , is convex and lower semicontinuous.
Let be a mapping. In this paper, we use to denote the fixed-point set of S. Recall the following definitions.
S is said to be nonexpansive if
If C is a bounded, closed, and convex subset of H, then is nonempty, closed, and convex. Recently, many beautiful convergence theorems for fixed points of nonexpansive mappings (semigroups) have been established in Banach spaces (see [8–10] and the references therein).
S is said to be asymptotically nonexpansive if there exists a sequence with as such that
It is known that if C is a nonempty, bounded, and closed convex subset of a Hilbert space H, then every asymptotically nonexpansive self-mapping has a fixed point. Further, the set of fixed points of S is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence problems of iterative processes for such a class of mappings.
S is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:
Putting
we see that as . Then (2.2) is reduced to the following:
The class of asymptotically nonexpansive mappings in the intermediate sense was considered in [11] and [12] as a generalization of the class of asymptotically nonexpansive mappings. It is known that if C is a nonempty, closed, and convex bounded subset of a real Hilbert space, then every asymptotically nonexpansive self-mapping in the intermediate sense has a fixed point.
S is said to be strictly pseudocontractive if there exists a constant such that
For such a case, S is also said to be a κ-strict pseudocontraction. It is clear that every nonexpansive mapping is a 0-strict pseudocontraction. We also remark that if , then S is said to be pseudocontractive.
S is said to be an asymptotically strict pseudocontraction if there exist a sequence with as and a constant such that
For such a case, S is also said to be an asymptotically κ-strict pseudocontraction. It is clear that every asymptotically nonexpansive mapping is an asymptotically 0-strict pseudocontraction. We also remark here that if , then S is said to be an asymptotically pseudocontractive mapping which was introduced by Schu [13] in 1991.
S is said to be an asymptotically strict pseudocontraction in the intermediate sense if there exist a sequence with as and a constant such that
For such a case, S is also said to be an asymptotically κ-strict pseudocontraction in the intermediate sense. Putting
then we see that as . We know that (2.4) is reduced to the following:
The class of asymptotically strict pseudocontractions in the intermediate sense was introduced by Sahu, Xu, and Yao [14] as a generalization of the class of asymptotically strict pseudocontractions; see [14] for more details.
Recently, many authors considered the weak convergence of iterative sequences for the classical variational inequality, the equilibrium problem (2.1), and fixed-point problems based on iterative methods (see [15–24]).
In 2003, Takahashi and Toyoda [23] considered the classical variational inequality and a nonexpansive mapping. To be more precise, they proved the following theorem.
Theorem 1 Let C be a closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly-monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such that. Letbe a sequence generated by
for every, wherefor someandfor some. Thenconverges weakly to, where.
In 2007, Tada and Takahashi [24] considered the equilibrium problem (2.1) and a nonexpansive mapping. To be more precise, they proved the following result.
Theorem 2 Let C be a nonempty, closed, and convex subset of H. Let F be a bifunction fromtosatisfying (A 1)-(A 4) and let S be a nonexpansive mapping of C into H, such that. Letandbe sequences generated byand let
for each, wherefor someandsatisfies. Thenconverges weakly to, where.
In this paper, motivated by the results announced in [23] and [24], we consider the equilibrium problem (2.1) and an asymptotically strict pseudocontraction in the intermediate sense based on a Mann-like iterative process. We show that the sequence generated in the purposed iterative process converges weakly to a common element of the fixed-point set of an asymptotically strict pseudocontraction in the intermediate sense and the solution set of the equilibrium problem (2.1). The results presented in this paper improve and extend the corresponding results announced by Takahashi and Toyoda [23], and Tada and Takahashi [24].
In order to prove our main results, we also need the following lemmas.
The following lemma can be found in [25] and [26].
Lemma 2.1 Let C be a nonempty, closed, and convex subset of H, and letbe a bifunction satisfying (A 1)-(A 4). Then for anyand, there existssuch that
Further, define
for alland. Then the following statements hold:
-
(a)
is single-valued;
-
(b)
is firmly nonexpansive, i.e., for any ,
-
(c)
;
-
(d)
is closed and convex.
Lemma 2.2 ([14])
Let H be a real Hilbert space, C a nonempty, closed, and convex subset of H, anda uniformly continuous and asymptotically strict pseudocontraction in the intermediate sense. Ifis a sequence in C such thatand, then.
Lemma 2.3 ([27])
Let H be a real Hilbert space, and, for all. Suppose thatandare sequences in H such that
and for some,
Then.
Lemma 2.4 ([28])
Let, , andbe three nonnegative sequences satisfying the following condition:
whereis some nonnegative integer, and. Then the limitexists.
Lemma 2.5 ([29])
Let H be a real Hilbert space. Letbe real numbers insuch that. Then we have the following:
for any given bounded sequencein H.
3 Main results
Theorem 3.1 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Letbe a bifunction fromtowhich satisfies (A 1)-(A 4) for each, whereis some positive integer. Letbe a uniformly continuous and asymptotically κ-strict pseudocontraction in the intermediate sense. Assume thatis nonempty. Let, , , andbe sequences in, anda bounded sequence in C. Letbe a positive sequence such thatanda sequence infor each. Letbe a sequence generated in the following manner:
Assume that the following restrictions are satisfied:
-
(a)
;
-
(b)
and , where is defined in (2.5);
-
(c)
, and ;
-
(d)
and for each ,
where a, b, c, and d are some real constants. Then the sequenceweakly converges to some point in.
Proof Let . Then we see that
Put for each . In view of Lemma 2.5, we see from the restriction (c) that
It follows that
From Lemma 2.4, we obtain the existence of the limit of the sequence . Notice that
This implies that
In view of (3.4) and , where , we see from Lemma 2.5 that
In view of Lemma 2.5, we obtain from restriction (c) that
This implies from (3.5) that
It follows that
In view of the restrictions (b), (c), and (d), we obtain that
Since is bounded, we see that there exists a subsequence of which converges weakly to ω. We can get from (3.7) that converges weakly to ω for each . Note that
From the assumption (A2), we see that
Replacing n by , we arrive at
In view of (3.7) and the assumption (A4), we get that
For any with and , let for each . Since and , we have , and hence for each . It follows that
which yields that
Letting for each , we obtain from the assumption (A3) that
This implies that for each . This proves that .
Next, we show that . Put , we obtain that
From (3.2), we see that
On the other hand, we have
In view of Lemma 2.3, we obtain that
Note that
This implies from (3.7) that
On the other hand, we have
It follows from (3.9) and (3.10) that
Note that
which yields that
This implies from the restriction (c) and (3.11) that
Notice that
This implies from the restriction (c) and (3.9) that
On the other hand, we have
Since S is uniformly continuous, we obtain from (3.12) and (3.13) that
In view of Lemma 2.2, we obtain that . This proves that . Let be another subsequence of converging weakly to , where . In the same way, we can show that . Notice that we have proved that exists for each . Assume that , where Q is a nonnegative number. By virtue of Opial’s condition of H, we have
This is a contradiction. Hence, . This completes the proof. □
If , then Theorem 3.1 is reduced to the following.
Corollary 3.2 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let F be a bifunction fromtowhich satisfies (A 1)-(A 4). Letbe a uniformly continuous and asymptotically κ-strict pseudocontraction in the intermediate sense. Assume thatis nonempty. Let, , , andbe sequences in, anda bounded sequence in C. Letbe a positive sequence such that. Letbe a sequence generated in the following manner:
Assume that the following restrictions are satisfied:
-
(a)
;
-
(b)
and , where is defined in (2.5);
-
(c)
, and ;
where a, b, and c are some real constants. Then the sequenceconverges weakly to some point in.
If for all and for all , then Corollary 3.2 is reduced to the following.
Corollary 3.3 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Letbe a uniformly continuous and asymptotically κ-strict pseudocontraction in the intermediate sense with. Let, , , andbe sequences in, anda bounded sequence in C. Letbe a positive sequence such that. Letbe a sequence generated in the following manner:
Assume that the following restrictions are satisfied:
-
(a)
;
-
(b)
and , where is defined in (2.5);
-
(c)
, and ;
where a, b, and c are some real constants. Then the sequenceconverges weakly to some point in.
For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem 3.1 the following results immediately.
Corollary 3.4 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Letbe a bifunction fromtowhich satisfies (A 1)-(A 4) for each, whereis some positive integer. Letbe a uniformly continuous and asymptotically nonexpansive mapping in the intermediate sense. Assume thatis nonempty. Let, , , andbe sequences in, anda bounded sequence in C. Letbe a positive sequence such thatanda sequence infor each. Letbe a sequence generated in the following manner:
Assume that the following restrictions are satisfied:
-
(a)
;
-
(b)
and , where
-
(c)
, and ;
-
(d)
and for each ,
where a, b, c, and d are some real constants. Then the sequenceconverges weakly to some point in.
References
Chang SS, Yao JC, Kim JK, Yang L: Iterative approximation to convex feasibility problems in Banach space. Fixed Point Theory Appl. 2007., 2007: Article ID 046797
Qin X, Cho SY, Kang SM: Common fixed points of a pair of non-expansive mappings with applications to convex feasibility problems. Glasg. Math. J. 2010, 52: 241–252. 10.1017/S0017089509990309
Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J. Math. Comput. Sci. 2011, 1: 1–18.
Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008
Kim JK, Cho SY, Qin X: Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings. J. Inequal. Appl. 2010., 2010: Article ID 312602
Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal., Real World Appl. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017
Kim JK, Nam YM, Sim JY: Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonexpansive type mappings. Nonlinear Anal. 2009, 71: e2839-e2848. 10.1016/j.na.2009.06.090
Lau ATM, Miyake H, Takahashi W: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces. Nonlinear Anal. 2007, 67: 1211–1225. 10.1016/j.na.2006.07.008
Lau AT, Takahashi W: Invariant submeans and semigroups of nonexpansive mappings on Banach spaces with normal structure. J. Funct. Anal. 1996, 142: 79–88. 10.1006/jfan.1996.0144
Shioji N, Takahashi W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 1997, 125: 3641–3645. 10.1090/S0002-9939-97-04033-1
Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 1993, 65: 169–179.
Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Isr. J. Math. 1974, 17: 339–346. 10.1007/BF02757136
Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 1991, 158: 407–413. 10.1016/0022-247X(91)90245-U
Sahu DR, Xu HK, Yao JC: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal. 2009, 70: 3502–3511. 10.1016/j.na.2008.07.007
Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011
Lv S:Generalized systems of variational inclusions involving -monotone mappings. Adv. Fixed Point Theory 2011, 1: 1–14.
Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi- ϕ -nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 2010, 234: 750–760. 10.1016/j.cam.2010.01.015
Kim JK, Kim KS: A new system of generalized nonlinear mixed quasivariational inequalities and iterative algorithms in Hilbert spaces. J. Korean Math. Soc. 2007, 44: 823–834. 10.4134/JKMS.2007.44.4.823
Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10
Kim JK, Cho SY, Qin X: Some results on generalized equilibrium problems involving strictly pseudocontractive mappings. Acta Math. Sci. 2011, 31: 2041–2057.
Yang S, Li W: Iterative solutions of a system of equilibrium problems in Hilbert spaces. Adv. Fixed Point Theory 2011, 1: 15–26.
Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031
Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003, 118: 417–428. 10.1023/A:1025407607560
Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mappings and an equilibrium problem. J. Optim. Theory Appl. 2007, 133: 359–370. 10.1007/s10957-007-9187-z
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.
Schu J: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884
Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309
Hao Y, Cho SY, Qin X: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 218573
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The authors are grateful to the reviewers and the editor for their useful comments and advice. The work was supported by the Kyungnam University Research Fund, 2012.
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Qing, Y., Kim, J.K. Weak convergence of algorithms for asymptotically strict pseudocontractions in the intermediate sense and equilibrium problems. Fixed Point Theory Appl 2012, 132 (2012). https://doi.org/10.1186/1687-1812-2012-132
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DOI: https://doi.org/10.1186/1687-1812-2012-132