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General iterative algorithms for mixed equilibrium problems, variational inequalities and fixed point problems
Fixed Point Theory and Applications volume 2014, Article number: 80 (2014)
Abstract
In this paper, we introduce and analyze a general iterative algorithm for finding a common solution of a mixed equilibrium problem, a general system of variational inequalities and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some mild conditions, we derive the strong convergence of the sequence generated by the proposed algorithm to a common solution, which also solves some optimization problem. The result presented in this paper improves and extends some corresponding ones in the earlier and recent literature.
MSC:49J30, 47H10, 47H15.
1 Introduction
Let H be a real Hilbert space with the inner product and the norm . Let C be a nonempty, closed and convex subset of H, and let be a nonlinear mapping. Throughout this paper, we use to denote the fixed point set of T. A mapping is said to be nonexpansive if
Let be a real-valued bifunction and be a real-valued function, where R is the set of real numbers. The so-called mixed equilibrium problem (MEP) is to find such that
which was considered and studied in [1, 2]. The set of solutions of MEP (1.2) is denoted by . In particular, whenever , MEP (1.2) reduces to the equilibrium problem (EP) of finding such that
which was considered and studied in [3–7]. The set of solutions of the EP is denoted by . Given a mapping , let for all . Then if and only if for all . Numerous problems in physics, optimization and economics reduce to finding a solution of the EP.
Throughout this paper, assume that is a bifunction satisfying conditions (A1)-(A4) and that is a lower semicontinuous and convex function with restriction (B1) or (B2), where
(A1) for all ;
(A2) F is monotone, i.e., for any ;
(A3) F is upper-hemicontinuous, i.e., for each ,
(A4) is convex and lower semicontinuous for each ;
(B1) for each and , there exists a bounded subset and such that for any ,
(B2) C is a bounded set.
The mappings are said to be an infinite family of nonexpansive self-mappings on C if
and denoted by is the fixed point set of , i.e., . Finding an optimal point in the intersection of fixed point sets of mappings , , is a matter of interest in various branches of sciences.
Recently, many authors considered some iterative methods for finding a common element of the set of solutions of MEP (1.2) and the set of fixed points of nonexpansive mappings; see, e.g., [2, 8, 9] and the references therein.
A mapping is said to be
-
(i)
monotone if
-
(ii)
strongly monotone if there exists a constant such that
In such a case, A is said to be η-strongly monotone;
-
(iii)
inverse-strongly monotone if there exists a constant such that
In such a case, A is said to be ζ-inverse-strongly monotone.
Let be a nonlinear mapping. The classical variational inequality problem (VIP) is to find such that
We use to denote the set of solutions to VIP (1.4). One can easily see that VIP (1.4) is equivalent to a fixed point problem. That is, is a solution of VIP (1.4) if and only if u is a fixed point of the mapping , where is a constant. Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [10]. Not only the existence and uniqueness of solutions are important topics in the study of VIP (1.4), but also how to actually find a solution of VIP (1.4) is important. Up to now, there have been many iterative algorithms in the literature for finding approximate solutions of VIP (1.4) and its extended versions; see e.g., [3, 11–14].
Recently, Plubtieng and Punpaeng [15] and Ceng et al. [16, 17] considered some iterative methods for VIP (1.4) and its extended versions and got some strong convergence theorems. As a generalization of VIP (1.4), the general system of variational inequalities (GSVI) is to find such that
which and are two positive constants. GSVI (1.5) is considered and studied in [17], and the solution set of GSVI (1.5) is denoted by . In particular, whenever and , GSVI (1.5) reduces to VIP (1.4). Ceng et al. [17] transformed GSVI (1.5) into a fixed point problem in the following way.
Lemma 1.1 (see [17])
For given , is a solution of GSVI (1.5) if and only if is a fixed point of the mapping defined by
where and is the projection of H onto C.
In particular, if the mapping is -inverse strongly monotone for , then the mapping G is nonexpansive provided for . We denote by Γ the fixed point set of the mapping G.
On the other hand, Moudafi [1] introduced the viscosity approximation method for nonexpansive mappings (see also [18] for further developments in both Hilbert spaces and Banach spaces).
A mapping is called α-contractive if there exists a constant such that
Let f be a contraction on C. Starting with an arbitrary initial , define a sequence recursively by
where T is a nonexpansive mapping of C into itself and is a sequence in . It is proved in [1, 18] that under appropriate conditions imposed on the sequence generated by (1.6) strongly converges to the unique solution to the VIP
A linear bounded operator A is said to be -strongly positive on H if there exists a constant such that
Recently, Marino and Xu [19] introduced the following general iterative process:
where A is a strongly positive bounded linear operator on H. They proved that under appropriate conditions imposed on the sequence generated by (1.8) converges strongly to the unique solution to the VIP
which is the optimality condition for the minimization problem
where h is a potential function for γf (i.e., for all ).
In 2007, Takahashi and Takahashi [5] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the EP and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let be a nonexpansive mapping. Starting with arbitrary initial , define sequences and recursively by
They proved that under appropriate conditions imposed on and , the sequences and converge strongly to , where .
Subsequently, Plubtieng and Punpaeng [15] introduced a general iterative process for finding a common element of the set of solutions of the EP and the set of fixed points of a nonexpansive mapping in a Hilbert space.
Let be a nonexpansive mapping. Starting with an arbitrary , define sequences and by
They proved that under appropriate conditions imposed on and , the sequence generated by (1.11) converges strongly to the unique solution to the VIP
which is the optimality condition for the minimization problem
where h is a potential function for γf (i.e., for all ).
Let be an infinite family of nonexpansive self-mappings on C and be a sequence of nonnegative numbers in . For any , define a mapping of C into itself as follows:
Such a mapping is called the W-mapping generated by and .
Recently, Yao et al. [6] proved the following strong convergence result.
Theorem 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction satisfying conditions (A1)-(A4). Let be an infinite family of nonexpansive self-mappings on C such that . Suppose that , and are three sequences in such that and . Suppose that the following conditions are satisfied:
-
(i)
and ;
-
(ii)
;
-
(iii)
and .
Let f be a contraction on H, and let be given arbitrarily. Then the sequences and generated iteratively by
converge strongly to , the unique solution of the minimization problem
where h is a potential function for f.
Very recently, Chen [20] proved the following strong convergence theorem.
Theorem 1.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an infinite family of nonexpansive self-mappings on C such that the common fixed point set . Let be an α-contraction, and let be a strongly positive bounded linear operator with a constant . Let γ be a constant such that . For an arbitrary initial point , one defines a sequence iteratively
where is a real sequence in . Assume that the sequence satisfies the following conditions:
(C1) ;
(C2) .
Then the sequence generated by (1.14) converges in norm to the unique solution , which solves the VIP
More recently, Rattanaseeha [7] introduced an iterative algorithm:
and proved the following strong convergence theorem.
Theorem 1.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from to R satisfying (A1)-(A4). Let f be an α-contraction on H with , and let be an infinite family of nonexpansive self-mappings on C such that . Let be a -strongly positive bounded linear operator with . Let be a sequence of real numbers such that , . Let be the W-mapping of C into itself generated by (1.13). Let W be defined by , . Let and be sequences generated by (1.16), where is a sequence in and is a sequence in such that the following conditions hold:
(C1) ;
(C2) ;
(C3) .
Then both and converge strongly to , which is the unique solution to the VIP
Equivalently, .
Let be a real-valued bifunction, be a real-valued function, A be a strongly positive bounded linear operator on H, and f be an l-Lipschitz continuous mapping on H. Motivated by the above facts, in this paper we propose and analyze a general iterative algorithm:
where , , , and is the W-mapping defined by (1.13), for finding a common solution of MEP (1.2), GSVI (1.5) and the fixed point problem of an infinite family of nonexpansive self-mappings on C. It is proven that under mild conditions imposed on , and , the sequences and generated by (1.17) converge strongly to , which is the unique solution of the VIP
or, equivalently, the unique solution of the minimization problem
where h is a potential function for γf (i.e., for all ). Whenever and , our problem of finding reduces to the problem of finding in Theorem 1.3. The results presented in this paper improve and extend the corresponding theorems in [7].
2 Preliminaries
Let H be a real Hilbert space with the inner product and the norm , and let C be a closed convex subset of H. We indicate weak convergence and strong convergence by using the notations → and ⇀, respectively. A mapping is called l-Lipschitz continuous if there exists a constant such that
In particular, if then f is called a nonexpansive mapping; if then f is called a contraction. Recall that a mapping is said to be a firmly nonexpansive mapping if
The metric (or nearest point) projection from H onto C is the mapping which assigns to each point the unique point satisfying the property
Some important properties of projections are gathered in the following proposition.
Proposition 2.1 For given and :
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, .
Consequently, is a firmly nonexpansive mapping of H onto C and hence nonexpansive and monotone.
Let be an α-inverse-strongly monotone mapping. Then it is obvious that A is monotone and -Lipschitz continuous. We also have that for all and ,
So, whenever , is a nonexpansive mapping.
Given a positive number . Let be the solution set of the auxiliary mixed equilibrium problem, that is, for each ,
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction satisfying conditions (A1)-(A4), and let be a lower semicontinuous and convex function with restriction (B1) or (B2). Then the following hold:
-
(a)
for each , ;
-
(b)
is single-valued;
-
(c)
is firmly nonexpansive, i.e., for any ,
-
(d)
for all and ,
-
(e)
;
-
(f)
is closed and convex.
Remark 2.1 It is easy to see from conclusions (c) and (d) in Proposition 2.2 that
and
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma 2.1 Let X be a real inner product space. Then the following inequality holds:
Lemma 2.2 Let H be a real Hilbert space. Then the following hold:
-
(a)
for all ;
-
(b)
for all and with ;
-
(c)
If is a sequence in H such that , it follows that
We have the following crucial lemmas concerning the W-mappings defined by (1.13).
Lemma 2.3 (see [[21], Lemma 3.2])
Let be a sequence of nonexpansive self-mappings on C such that , and let be a sequence in for some . Then, for every and , the limit exists, where is defined by (1.13).
Remark 2.2 (see [[6], Remark 3.1])
It can be known from Lemma 2.3 that if D is a nonempty bounded subset of C, then for there exists such that for all ,
Remark 2.3 (see [[6], Remark 3.2])
Utilizing Lemma 2.3, we define a mapping as follows:
Such a W is called the W-mapping generated by and . Since is nonexpansive, is also nonexpansive. Indeed, observe that for each ,
If is a bounded sequence in C, then we put . Hence, it is clear from Remark 2.2 that for an arbitrary , there exists such that for all ,
This implies that
Lemma 2.4 (see [[21], Lemma 3.3])
Let be a sequence of nonexpansive self-mappings on C such that , and let be a sequence in for some . Then .
Lemma 2.5 (see [[22], demiclosedness principle])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C with . 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 operator of H.
Lemma 2.6 Let be a monotone mapping. In the context of the variational inequality problem, the characterization of the projection (see Proposition 2.1(i)) implies
Lemma 2.7 (see [19])
Let A be a -strongly positive bounded linear operator on H and assume . Then .
Lemma 2.8 (see [23])
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a real sequence such that
-
(i)
;
-
(ii)
or .
Then .
3 Main results
We will introduce and analyze a general iterative algorithm for finding a common solution of MEP (1.2), GSVI (1.5) and the fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences, we will prove strong convergence of the proposed algorithm.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from to R satisfying conditions (A1)-(A4), and let be a lower semicontinuous and convex function with restriction (B1) or (B2). Let the mapping be -inverse strongly monotone for . Let be a sequence of nonexpansive self-mappings on C, and let be a sequence in for some . Let A be a -strongly positive bounded linear operator on H and be an l-Lipschitz continuous mapping with . Let be the W-mapping defined by (1.13). Assume that , where Γ is the fixed point set of the mapping with for . Let and be two sequences in and be a sequence in such that:
-
(i)
, and ;
-
(ii)
and ;
-
(iii)
and .
Given arbitrarily, the sequences and generated iteratively by
converge strongly to , which is the unique solution of the VIP
or, equivalently, the unique solution of the minimization problem
where h is a potential function for γf.
Proof Taking into account that and , we may assume, without loss of generality, that for all . Since A is a -strongly positive bounded linear operator on H, we know that ,
and for with ,
that is, is positive. It follows that
We observe that is a contraction. Indeed, for all , we have
By the Banach contraction principle, we deduce that has a unique fixed point . That is, . In addition, by Proposition 2.2 we have for all .
We divide the rest of the proof into several steps.
Step 1. We show that is bounded. Indeed, take arbitrarily. Since and , is -inverse-strongly monotone for , by Proposition 2.2(c) we deduce from , , that for any ,
(This shows that G is nonexpansive.) Thus, from (3.1), (3.4) and , we get
By induction, we get
Therefore, is bounded and so are the sequences , and .
Step 2. We show that as . Indeed, we write . Then for each . Define for each . Then from the definition of , we obtain
It follows from Lemma 2.7 that
From (1.13), since , and are all nonexpansive, we have
where for some . Furthermore, we estimate . Taking into account that and , we may assume, without loss of generality, that and for some . Utilizing Remark 2.1, we get
where for some .
Note that
Hence it follows from (3.5)-(3.7) that
where for some . Since , , and , from and Lemma 2.8 we conclude that
Step 3. We show that . Indeed, for simplicity, we write , and . Then and
From (3.1), (3.4) and Proposition 2.1(i) and Lemma 2.2(b), we obtain that for ,
which immediately implies that
Since , and , , we deduce from the boundedness of , and that
Also, in terms of the firm nonexpansivity of and the -inverse strong monotonicity of for , we obtain from , and (3.4) that
and
Thus, we have
and
Consequently, from (3.4), (3.9) and (3.11) it follows that
which yields
Since , and , we deduce from the boundedness of , , , and that
Furthermore, from (3.4), (3.9) and (3.12) it follows that
which leads to
Since , and , we deduce from the boundedness of , , , and that
Note that
Hence from (3.13) and (3.14) we get
Step 4. We show that and . Indeed, by Proposition 2.2(c) we have
That is,
From (3.4), (3.9) and (3.16) it follows that
So, we get
Since and , we deduce from the boundedness of , and that
In the meantime, we observe that
From (3.10), (3.15) and (3.17) it follows that
Also, note that
From (3.15), (3.18), Remark 2.3 and the boundedness of we immediately obtain
Step 5. We show that
where uniquely solves the minimization problem (3.3).
Indeed, as previously, we have proven that is the unique fixed point of the mapping (i.e., ). That is, is the unique solution of VIP (3.2). Equivalently, is the unique solution of the minimization problem (3.3).
First, we observe that there exists a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to some w. Without loss of generality, we may assume that . From (3.17), we have that . By (3.15) and (3.19) we have that and as . Utilizing similar arguments to those of (3.4), we know that G is nonexpansive. Hence, by Lemma 2.5 we obtain and (due to Lemma 2.4). Next, we prove that . As a matter of fact, from , we know that
From (A2) it follows that
Replacing n by , we have
Put for all and . Then, from (3.21) we have
So, from (A4), the weak lower semicontinuity of φ, and , we have
From (A1), (A4) and (3.22) we also have
and hence
Letting , we have, for each ,
This implies that . Therefore, . Consequently, from (3.2) and (3.20) we have
Step 6. Finally, we show that as .
Indeed, taking into account that and , we obtain from (3.4) and Proposition 2.1(i) that
which immediately implies that
where and . Note that implies and that (3.23) leads to
Applying Lemma 2.8 to (3.24), we infer that the sequence converges strongly to . This completes the proof. □
Putting the identity mapping, and in Theorem 3.1, we have the following result.
Corollary 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from to R satisfying conditions (A1)-(A4), and let be a lower semicontinuous and convex function with restriction (B1) or (B2). Let the mapping be -inverse strongly monotone for . Let A be a -strongly positive bounded linear operator on H and be an α-contraction with . Assume that , where Γ is the fixed point set of the mapping with for . Let and be two sequences in and be a sequence in such that:
-
(i)
, and ;
-
(ii)
and ;
-
(iii)
and .
Given arbitrarily, the sequences and generated iteratively by
converge strongly to , which is the unique solution of the VIP
or, equivalently, the unique solution of the minimization problem
where h is a potential function for γf.
Putting and in Theorem 3.1, we have the following result.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from to R satisfying conditions (A1)-(A4). Let the mapping be -inverse strongly monotone for . Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let A be a -strongly positive bounded linear operator on H and be an α-contraction with . Let be the W-mapping defined by (1.13). Assume that , where Γ is the fixed point set of the mapping with for . Let and be two sequences in and be a sequence in such that:
-
(i)
, and ;
-
(ii)
and ;
-
(iii)
and .
Given arbitrarily, the sequences and generated iteratively by
converge strongly to , which is the unique solution of the VIP
or, equivalently, the unique solution of the minimization problem
where h is a potential function for γf.
Putting in Corollary 3.1, we have the following result.
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from to R satisfying conditions (A1)-(A4). Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let A be a -strongly positive bounded linear operator on H and be an α-contraction with . Let be the W-mapping defined by (1.13). Assume that . Let and be two sequences in and be a sequence in such that:
-
(i)
, and ;
-
(ii)
and ;
-
(iii)
and .
Given arbitrarily, the sequences and generated iteratively by
converge strongly to , which is the unique solution of the VIP
or, equivalently, the unique solution of the minimization problem
where h is a potential function for γf.
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All the authors were partially supported by the National Science Foundation of China (11071169) and PhD Program Foundation of the Ministry of Education of China (20123127110002).
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Wang, XJ., Ceng, LC., Hu, HY. et al. General iterative algorithms for mixed equilibrium problems, variational inequalities and fixed point problems. Fixed Point Theory Appl 2014, 80 (2014). https://doi.org/10.1186/1687-1812-2014-80
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DOI: https://doi.org/10.1186/1687-1812-2014-80