A strong convergence theorem for equilibrium problems and split feasibility problems in Hilbert spaces
© Tang et al.; licensee Springer. 2014
Received: 16 June 2013
Accepted: 31 January 2014
Published: 13 February 2014
The main purpose of this paper is to introduce an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces. Under suitable conditions we prove that the sequence converges strongly to a common element of the set of solutions of equilibrium problems and the set of solutions of split feasibility problems. Our result extends and improves the corresponding results of some others.
MSC:90C25, 90C30, 47J25, 47H09.
The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, the Nash equilibrium problems and others, see, for instance, [1–3]. Some methods have been proposed to solve the EP, see, e.g., [4–6] and [7, 8].
where A is a given real matrix, and C and Q are nonempty, closed and convex subsets in and , respectively.
Due to its extraordinary utility and broad applicability in many areas of applied mathematics (most notably, fully discretized models of problems in image reconstruction from projections, in image processing, and in intensity-modulated radiation therapy), algorithms for solving convex feasibility problems have been received great attention (see, for instance [10–13] and also [14–18]).
where and are the orthogonal projection onto C and Q, respectively, is any positive constant and denotes the adjoint of A.
Motivated and inspired by the research going on in the sections of equilibrium problems and split feasibility problems, the purpose of this article is to introduce an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces. Under suitable conditions we prove the sequence converges strongly to a common element of the set of solutions of equilibrium problems and the set of solutions of split feasibility problems. Our result extends and improves the corresponding results of He et al.  and some others.
2 Preliminaries and lemmas
the weak ω-limit set of .
A mapping is said to be demi-closed at origin, if for any sequence with and , then .
It is easy to prove that if is a firmly nonexpansive mapping, then T is demi-closed at the origin.
∇f is -Lipschitz, i.e., , .
Lemma 2.2 (See, for example, )
T is firmly nonexpansive.
is firmly nonexpansive.
This implies that is firmly nonexpansive.
(iii) ⇒ (i): From (iii) we immediately know that T is firmly nonexpansive.
Throughout this paper, let us assume that a bifunction satisfies the following conditions:
(A1) , ;
(A2) F is monotone, i.e., , ;
(A3) , ;
(A4) for each , is convex and lower semicontinuous.
is firmly nonexpansive;
Ω is closed and convex.
The following results play an important role in this paper.
Lemma 2.4 ()
Lemma 2.5 ()
Lemma 2.6 ()
, or .
3 Main results
We are now in a position to prove the following theorem.
then the sequence converges strongly to .
Proof Since the solution set Ω of EP and the solution set of SPF (1.2) are both closed and convex, Γ (≠∅) is closed and convex. Thus, the metric projection is well defined.
This implies that the sequence is bounded. From (3.2) and (3.6) we know that and both are bounded.
Now, we prove by employing the technique studied by Maingé . For the purpose we consider two cases.
Since T is demi-closed at origin, from (3.14) and (3.15) we have , i.e., .
It follows from (3.19), (3.20), and (3.22) that . In view of and that S is demi-closed at origin, we get .
Applying Lemma 2.6 to (3.25), from the condition (i) we obtain , that is, .
Noting the inequality (3.26), this shows that , that is, . This completes the proof of Theorem 3.1. □
This study was supported by the Scientific Research Fund of Sichuan Provincial Education Department (13ZA0199) and the Scientific Research Fund of Sichuan Provincial Department of Science and Technology (2012JYZ011) and by the National Natural Science Foundation of China (Grant No. 11361070).
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