In this paper, the problem of approximating a common element in the common fixed point set of an infinite family of nonexpansive mappings, in the solution set of a variational inequality involving an inverse-strongly monotone mapping and in the solution set of an equilibrium problem is investigated based on a general iterative algorithm. Strong convergence of the iterative algorithm is obtained in the framework of Hilbert spaces. The results obtained in this paper improve the corresponding results announced by many authors.
AMS Subject Classification:47H09, 47J05, 47J25.
Hinweise
Competing interests
The author declares that they have no competing interests.
1 Introduction and preliminaries
Let H be a real Hilbert space, whose inner product and norm are denoted by and respectively. Let C be a nonempty, closed and convex subset of H and be a mapping. In this paper, we use to denote the set of fixed points of T. Recall that T is said to be a κ-contraction iff there exists a constant such that
T is said to be a nonexpansive mapping iff
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Let be a mapping. Recall that B is said to be an α-inverse-strongly monotone iff there exits a positive constant α such that
The classical variational inequality is to find such that
(1.1)
In this paper, we use to denote the solution set of the variational inequality.
Let be the metric projection from H onto C. It is also known that satisfies
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Moreover, is characterized by the properties and for all . One can see that the variational inequality is equivalent to a fixed point problem. The element is a solution of the variational inequality if and only if u is a fixed point of the mapping , where is a constant and I is the identity mapping. This alternative equivalent formulation has played a significant role in the studies of the variational inequality and related optimization problems.
Recall that an operator A is strongly positive on H iff there exists a constant with the property
Recall that a set-valued mapping is said to be monotone if for all , and imply . A monotone mapping is maximal if the graph of of S is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping S is maximal iff for , for every implies . Let B be a monotone map of C into H and let be the normal cone to C at , i.e., and define
Then S is maximal monotone and iff ; see [1] and the references therein.
Let F be a bifunction of into ℝ, where ℝ is the set of real numbers. The equilibrium problem for is to find such that
(1.2)
The set of solutions of the problem (1.2) is denoted by . Numerous problems in physics, optimization and economics reduce to finding a solution of (1.2). Recently, many iterative algorithms have been studied to solve the equilibrium problem (1.2); see, for instance, [2‐19].
For solving the equilibrium problem (1.2), let us 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.
In 2007, Takahashi and Takahashi [17] proved the following result.
Theorem TTLetCbe a nonempty closed convex subset ofH. LetFbe a bifunction fromtoRsatisfying (A1)-(A4) and letTbe a nonexpansive mapping ofCintoHsuch that . Letfbe a contraction ofHinto itself and letandbe sequences generated byand
whereandsatisfy , , , , and . Thenandstrongly converge to some pointz, where .
Recently, Plubtieng and Punpaeng [19] further improved the above results by involving a strongly positive self-adjoint operator. To be more precise, they proved the following results.
Theorem PPLetHbe a real Hilbert space, letFbe a bifunction fromsatisfying (A1)-(A4) and letTbe a nonexpansive mapping onHsuch that . Letfbe a contraction ofHinto itself withand letAbe a strongly positive bounded linear operator onHwith the coefficientand . Letbe a sequence generated byand
whereandsatisfy , , , , and . Thenandstrongly converge to some pointz, where .
In 2008, Su, Shang and Qin [2] considered the variational inequality (1.1), and the equilibrium problem (1.2) based on a composite iterative algorithm and proved the following theorem.
Theorem SSQLetCbe a nonempty closed convex subset ofH. LetFbe a bifunction fromtoRsatisfying (A1)-(A4). LetAbeα-inverse-strongly monotone and letTbe a nonexpansive mapping ofCintoHsuch that . Letfbe a contraction ofHinto itself and letandbe sequences generated byand
where , where , andsatisfy , , , , , and . Thenandstrongly converge to some pointz, where .
The above results only involve a single mapping, we will consider an infinite family of mappings in this paper. To be more precise, we study the mapping defined by
(1.3)
where are real numbers such that , are an infinite family of mappings of C into itself.
Considering , we have the following lemmas which are important in proving our main results.
LetCbe a nonempty closed convex subset of a strictly convex Banach spaceE. Letbe nonexpansive mappings ofCinto itself such thatis nonempty, and letbe real numbers such thatfor any . Then, for everyand , the limitexists.
Using Lemma 1.1, one can define the mapping W of C into itself as follows:
Such a W is called the W-mapping generated by and . Throughout this paper, we will assume that , where b is some constant.
LetCbe a nonempty closed convex subset of a strictly convex Banach spaceE. Letbe nonexpansive mappings ofCinto itself such thatis nonempty, and letbe real numbers such thatfor any . Then .
In this paper, based on a general iterative algorithm, we study the problem of approximating a common element in the common fixed point set of an infinite family of nonexpansive mappings, in the solution set of a variational inequality involving an inverse-strongly monotone mapping and in the solution set of an equilibrium problem. Strong convergence of the iterative algorithm is obtained in the framework of Hilbert spaces.
In order to obtain the strong convergence, we need the following tools.
Lemma 1.3In Hilbert spaces, the following inequality holds:
LetHbe a Hilbert space. LetBbe a strongly positive linear bounded self-adjoint operator with the constantandfbe a contraction with the constantκ. Assume that . LetTbe a nonexpansive mapping with a fixed pointof the contraction . Thenconverges strongly asto a fixed pointofT, which solves the variational inequality
LetKbe a nonempty closed convex subset of a Hilbert spaceH, be a family of infinitely nonexpansive mappings with , be a real sequence such thatfor each . IfCis any bounded subset ofK, then .
2 Main results
Theorem 2.1LetCbe a nonempty closed convex subset of a Hilbert spaceH. LetFbe a bifunction fromto ℝ which satisfies (A1)-(A4). Letbe an infinite family of nonexpansive mappings ofCintoC. Letbe anα-inverse-strongly monotone mapping. LetAbe a strongly positive linear bounded self-adjoint operator onHwith the coefficient . Assume thatand . Letbe aκ-contraction. Letbe a sequence generated in the following iterative process:
whereis generated in (1.3), , are real number sequences in , andare positive real number sequences. Assume that the following restrictions are satisfied:
(a)
;
(b)
, ;
(c)
, ;
(d)
, for somes, with .
Thenconverges strongly to , where , which solves the following variational inequality:
Proof We divide the proof into five steps.
Step 1. Show that the sequence is bounded.
Notice that is nonexpansive. Indeed, we see from the restriction (d) that
which implies the mapping is nonexpansive. Fix . Since , we have
Put
where
It follows that
Since as , we may assume, with no loss of generality, that for all n. It follows that
which yields
This in turn implies that
This completes the proof that the sequence is bounded. This completes the proof of Step 1.
Step 2. Show that .
In view of and , we see that
(2.1)
and
(2.2)
Putting in (2.1) and in (2.2), we find that
and
It follows from (A2) that
That is,
Without loss of generality, let us assume that there exists a real number m such that for all n. It follows that
It follows that
(2.3)
where is some real constant such that .
On the other hand, we have
(2.4)
where . Substituting (2.3) into (2.4) yields
(2.5)
where . Notice that
(2.6)
Since and are nonexpansive, we see from (1.3) that
(2.7)
where is a constant such that . Substituting (2.3), (2.5) and (2.7) into (2.6) yields
where is a constant such that
It follows from the restrictions (b) and (c) that
By virtue of Lemma 1.8, we obtain that
(2.8)
On the other hand, we have
This implies from (2.8) that
(2.9)
This completes the proof of Step 2.
Step 3. Show that .
Notice that . It follows that
This implies from the restriction (b) that
(2.10)
For any , we find that
That is,
This in turn implies that
from which it follows that
It follows from the restriction (b) and (2.9) that
(2.11)
Notice that
(2.12)
On the other hand, we have
(2.13)
Substituting (2.12) into (2.13), we find that
This in turn implies that
It follows from the restrictions (a), (b) and (d) that
(2.14)
On the other hand, we have
which yields
(2.15)
Substituting (2.15) into (2.13) yields
It follows that
In view of the restrictions (a), (b) and (d), we find from (2.9) that
(2.16)
Notice that
In the light of (2.8), (2.10), (2.11) and (2.16), we find that . On the other hand, we have
It follows from Lemma 1.9 that
(2.17)
This completes the proof of Step 3.
Step 4. Show that , where .
To see this, we choose a subsequence of such that
(2.18)
Correspondingly, there exists a subsequence of . Since is bounded, there exists a subsequence of which converges weakly to w. Without loss of generality, we can assume that . Since , we have
It follows from (A2) that
It follows that
In view of the restriction (c), we obtain from (2.11) that
Since , we have from (A4) that for all . For t with and , let . Since and , we have and hence . So, from (A1) and (A4), we have
That is, . It follows from (A3) that for all and hence . On the other hand, we see that . If , then we have the following. Since Hilbert spaces are Opial’s spaces, we find from (2.17) that
which derives a contradiction. Thus, we have . Next, let us first show that . Put
Since B is monotone, we see that S is maximal monotone. Let . Since and , we have
On the other hand, we have from that
That is,
It follows from the above that
which implies from (2.16) that . We have and hence . This completes the proof . On the other hand, we find from (2.18) that
(2.19)
This completes the proof of Step 4.
Step 5. Show .
It follows from Lemma 1.3 that
which implies that
(2.20)
where is a constant such that . On the other hand, we have
(2.21)
Substituting (2.20) into (2.21) yields
(2.22)
Let and
This implies that
(2.23)
In view of the restriction (b), we find from (2.8) and (2.11) that
We can easily draw the desired conclusion with the aid of Lemma 1.4. This completes the proof of Step 5. The proof is completed. □
From Theorem 2.1, we have the following results.
Corollary 2.2LetCbe a nonempty closed convex subset of a Hilbert spaceH. Letbe an infinite family of nonexpansive mappings ofCintoC. Letbe anα-inverse-strongly monotone mapping. LetAbe a strongly positive linear bounded self-adjoint operator onHwith the coefficient . Assume thatand . Letbe aκ-contraction. Letbe a sequence generated in the following iterative process:
whereis generated in (1.3), , are real number sequences in , andare positive real number sequences. Assume that the following restrictions are satisfied:
(a)
;
(b)
, ;
(c)
;
(d)
for somes, with .
Thenconverges strongly to , where , which solves the following variational inequality:
Proof Putting and , we can immediately draw the desired conclusion from Theorem 2.1. □
Corollary 2.3LetCbe a nonempty closed convex subset of a Hilbert spaceH. LetFbe a bifunction fromto ℝ, which satisfies (A1)-(A4). Letbe anα-inverse-strongly monotone mapping. LetAbe a strongly positive linear bounded self-adjoint operator onHwith the coefficient . Assume thatand . Letbe aκ-contraction. Letbe a sequence generated in the following iterative process:
whereuis a fixed element inC, , are real number sequences in , andare positive real number sequences. Assume that the following restrictions are satisfied:
(a)
;
(b)
, ;
(c)
, ;
(d)
, for somes, with .
Then, converges strongly to , where , which solves the following variational inequality:
Proof Putting , where I is the identity mapping and , for all , we can immediately draw the desired conclusion from Theorem 2.1. □
Corollary 2.4LetCbe a nonempty closed convex subset of a Hilbert spaceH. LetFbe a bifunction fromto ℝ which satisfies (A1)-(A4). Letbe an infinite family of nonexpansive mappings ofCintoC. Letbe anα-inverse-strongly monotone mapping. Assume that . Letbe aκ-contraction. Letbe a sequence generated in the following iterative process:
whereis generated in (1.3), , are real number sequences in , andare positive real number sequences. Assume that the following restrictions are satisfied:
(a)
;
(b)
, ;
(c)
, ;
(d)
, for somes, with .
Thenconverges strongly to , where , which solves the following variational inequality:
Proof Putting , where I is the identity mapping and , we can immediately draw the desired conclusion from Theorem 2.1. □
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Competing interests
The author declares that they have no competing interests.