General iterative algorithms for mixed equilibrium problems, variational inequalities and fixed point problems
© Wang et al.; licensee Springer. 2014
Received: 15 October 2013
Accepted: 11 March 2014
Published: 26 March 2014
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.
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 ;
(A4) is convex and lower semicontinuous for each ;
(B2) C is a bounded set.
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.
- (i)monotone if
- (ii)strongly monotone if there exists a constant such that
- (iii)inverse-strongly monotone if there exists a constant such that
In such a case, A is said to be ζ-inverse-strongly monotone.
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 . 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].
which and are two positive constants. GSVI (1.5) is considered and studied in , 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.  transformed GSVI (1.5) into a fixed point problem in the following way.
Lemma 1.1 (see )
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.
where h is a potential function for γf (i.e., for all ).
They proved that under appropriate conditions imposed on and , the sequences and converge strongly to , where .
Subsequently, Plubtieng and Punpaeng  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.
where h is a potential function for γf (i.e., for all ).
Such a mapping is called the W-mapping generated by and .
Recently, Yao et al.  proved the following strong convergence result.
where h is a potential function for f.
Very recently, Chen  proved the following strong convergence theorem.
where is a real sequence in . Assume that the sequence satisfies the following conditions:
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:
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 .
Some important properties of projections are gathered in the following proposition.
Consequently, is a firmly nonexpansive mapping of H onto C and hence nonexpansive and monotone.
So, whenever , is a nonexpansive mapping.
for each , ;
- (c)is firmly nonexpansive, i.e., for any ,
- (d)for all and ,
is closed and convex.
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
for all ;
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 [, 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 [, Remark 3.1])
Remark 2.3 (see [, Remark 3.2])
Lemma 2.4 (see [, 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 [, 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 2.7 (see )
Let A be a -strongly positive bounded linear operator on H and assume . Then .
Lemma 2.8 (see )
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.
, and ;
where h is a potential function for γf.
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.
Therefore, is bounded and so are the sequences , and .
where for some .
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).