Viscosity iteration methods for a split feasibility problem and a mixed equilibrium problem in a Hilbert space
© Deng et al.; licensee Springer. 2012
Received: 2 March 2012
Accepted: 27 November 2012
Published: 14 December 2012
In this paper, we consider and analyze two viscosity iteration algorithms (one implicit and one explicit) for finding a common element of the solution set of a mixed equilibrium problem and the set Γ of a split feasibility problem in a real Hilbert space. Furthermore, we derive the strong convergence of a viscosity iteration algorithm to an element of under mild assumptions.
In the sequel, we indicate by the set of solutions of our mixed equilibrium problem. If , we denote by . The mixed equilibrium problem (1.3) has become a rich source of inspiration and motivation for the study of a large number of problems arising in economics, optimization problems, variational inequalities, minimax problem, Nash equilibrium problem in noncooperative games and others (e.g., [9–14]).
It is our purpose in this paper to consider and analyze two viscosity iteration algorithms (one implicit and one explicit) for finding a common element of a solution set Γ of the split feasibility problem (1.1) and a set of the mixed equilibrium problem (1.3) in a real Hilbert space. Furthermore, we prove that the proposed viscosity iteration methods converge strongly to a particular solution of the mixed equilibrium problem (1.3) and the split feasibility problem (1.1).
Proposition 2.1 (Basic properties of projections )
for all and ;
for all ;
for all and .
We also consider some nonlinear operators which are introduced in the following.
- (i)Monotone if
- (ii)Strongly monotone if there exists a constant such that
- (iii)Inverse-strongly monotone (ism) if there exists a constant such that
- (iv)k-Lipschitz continuous if there exists a constant such that
nonexpansive if , ;
- (b)firmly nonexpansive if is nonexpansive. , where is nonexpansive, Alternatively, T is firmly nonexpansive if and only if
average if , where and is nonexpansive. In this case, we also claimed that T is ϵ-averaged. A firmly nonexpansive mapping is -averaged.
Proposition 2.5 ()
T is nonexpansive if and only if the complement is -ism.
If T is v-ism, then for , γT is -ism.
T is averaged if and only if the complement is v-ism for . Indeed, for , T is α-averaged if and only if is -ism.
Proposition 2.6 ()
If for some and if S is averaged and V is nonexpansive, then S is averaged.
T is firmly nonexpansive if and only if the complement is firmly nonexpansive.
If for some , S is firmly nonexpansive and V is nonexpansive, then T is averaged.
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then the composite is α-averaged, where .
- (v)If the mappings are averaged and have a common fixed point, then
(Here the notation denotes the set of fixed points of the mapping T, that is, .)
For solving the mixed equilibrium problem for a bifunction , let us assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, that is, for all ;
(A4) for each , is convex and lower semicontinuous.
Lemma 2.8 ()
Let C be a convex closed subset of a Hilbert space H. Let be a bifunction such that
(f1) , ;
(f2) is monotone and supper hemicontinuous;
(f3) is lower semicontinuous and convex.
Let be a bifunction such that
(h1) , ;
(h2) is monotone and upper semicontinuous;
(h3) is convex.
- (H)for fixed and , there exists a bounded set and such that for all , , for and . Let be a mapping defined by(2.1)
called a resolvent of and .
is a single value;
is firmly nonexpansive;
and it is closed and convex.
- (i)Minimization problem:
- (ii)Tikhonov’s regularization problem:(2.2)
where is the regularization parameter.
Proposition 2.10 ()
If the SFP is consistent, then the strong exists and is the minimum-norm solution of the SFP.
Proposition 2.11 ()
is attained at a point in the set .
Remark 2.12 ()
Remark 2.13 The mapping is nonexpansive.
Hence, it is clear that T is nonexpansive.
Lemma 2.14 ()
for all n;
Then converges in norm to the minimum-norm solution of the SFP (1.1).
Lemma 2.15 ()
Let and be bounded sequences in a Banach space X and let be a sequence in . Suppose that for all and . Then, .
Lemma 2.16 ()
Let K be a nonempty closed convex subset of a real Hilbert space H and be a nonexpansive mapping with . If is a sequence in K weakly converging to x and if converges strongly to y, then ; in particular, if , then .
3 Main results
where B is a k-Lipschitz and η-strongly monotone operator on H with , and , and is a β-contraction mapping, . Let be two bifunctions. In order to find a particular solution of the variational inequality (3.1), we construct the following implicit algorithm.
for all , where is a real sequence in , is defined by Lemma 2.8 and is introduced in Remark 2.12.
We show that the sequence defined by (3.2) converges to a particular solution of the variational inequality (3.1). As a matter of fact, in this paper, we study a general algorithm for solving the variational inequality (3.1).
where , and the sequence of and satisfy the conditions (i)-(iv) in Lemma 2.14.
Therefore, is a contraction mapping when . □
From Lemma 3.2 and using the Banach contraction principle, there exists a unique fixed point of in C, i.e., we obtain the following algorithm.
for all , where and are two real sequences in , is defined by Lemma 2.8 and is introduced in Remark 2.12.
At this point, we would like to point out that Algorithm 3.3 includes Algorithm 3.1 as a special case due to the fact that the contraction g is a possible nonself-mapping.
Proof Next, we divide the remainder of the proof into several steps.
Step 1. We prove that the sequence is bounded.
This indicates that is bounded. It is easy to deduce that and are also bounded.
Step 2. We prove that .
Step 3. We prove .
Since is bounded, without loss of generality, we may assume that converges weakly to a point . Hence, and .
Step 4. We show .
and hence . From (A3), we have for all . Therefore, .
Next, we prove .
Step 5. .
Hence, the weak convergence of implies that strongly.
Therefore, . That is, is the unique fixed point in Ω of the contraction . □
Therefore, is a particular solution of the variational inequality (3.1).
Next, we introduce an explicit algorithm for finding a solution of the variational inequality (3.1). This scheme is obtained by discretizing the implicit scheme (3.3). We show the strong convergence of this algorithm.
Proof First, we prove that the sequence is bounded. Indeed, pick .
Next, we show .
where and . It is clear that and . Hence, all the conditions of Lemma 2.17 are satisfied. Therefore, we immediately deduce that .
Remark 3.7 If we take , by the similar argument as that in Theorem 3.6, we deduce immediately that is a particular solution of the variational inequality (3.1). This completes the proof. □
4 Application in the multiple-set split feasibility problem
where are integers, and are closed convex subsets of Hilbert spaces and , and is a bounded linear operator. The special case where , called the split feasibility problem (1.1), was introduced by Censor and Elfving  for modeling phase retrieval and other image restoration problems.
Proposition 4.1 Given , solves the SFP if and only if solves the fixed point (4.2).
From this proposition, we can easily obtain that MSSFP (4.1) is equivalent to a common fixed point problem of finitely many nonexpansive mappings, as we show in the following.
which is L-Lipschitz continuous with the constant and thus is -ism. It is claimed that if , is nonexpansive. Therefore, fixed point algorithms for nonexpansive mappings can be applied to MSSFP (4.1).
From Algorithm 3.1, Algorithm 3.3 and Proposition 4.1, we consider our results on the optimization method for solving MSSFP (4.1), and obtain the following two algorithms.
for all , is defined by Lemma 2.8 and ∇g is introduced in (4.3).
for all , where are two real sequences in , is defined by Lemma 2.8 and ∇g is introduced in (4.3).
In addition, we would like to point out that Algorithm 4.3 includes Algorithm 4.2 as a special case due to the fact that the contraction f is a possible nonself-mapping. According to Theorem 3.4, we obtain the following theorem.
Theorem 4.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let B be a k-Lipschitz and η-strongly monotone operator on H with , and , and the sequence of and satisfy the conditions (i)-(iv) in Lemma 2.14. Let be two bifunctions which satisfy the conditions (f1)-(f4), (h1)-(h3) and (H) in Lemma 2.8. Let be a β-contraction. Assume , Γ is the solution set of MSSFP (4.1). Then the sequence generated by implicit Algorithm 4.3 converges in norm, as , to the unique solution of the variational inequality (3.1). In particular, if we take , then the sequence defined by Algorithm 4.3 converges in norm, as , to the unique solution of the variational inequality (3.1).
Since and U are nonexpansive, and following the proof of Theorem 3.4, we obtain the sequence converges strongly to a fixed point of U which is also a common fixed point of or a solution of MSSFP (4.1). □
From Theorem 3.6, we introduce an explicit algorithm for finding a common fixed point and for solving the variational inequality (3.1) and multiple set feasibility problem (4.1). This scheme is obtained by discretizing the implicit scheme (4.8).
Since and U are nonexpansive, following the proof of Theorem 3.6, we can easily claim that the sequence converges strongly to the common fixed point of which solves the mixed equilibrium problem (), and U is a solution of MSSFP (4.1). □
According to , we can obtain the following proposition.
Proposition 4.6 is a solution of MSSFP (4.1) if and only if .
From Proposition 4.6 and Algorithm 3.3, we obtain the corresponding algorithm and the convergence theorems for MSSFP (4.1).
for all , where and are two real sequences in , is defined by Lemma 2.8 and ∇f is introduced in (4.9).
Proof From Proposition 4.6, we know that is a nonexpansive mapping. Thus, using the proof of Theorem 3.4, we obtain that the sequence converges strongly to a fixed point of or a solution of MSSFP (4.1), and this fixed point is a solution of the set of mixed equilibrium problem (1.3). □
This work is supported in part by National Natural Science Foundation of China (71272148), the Ph.D. Programs Foundation of Ministry of Education of China (20120032110039) and China Postdoctoral Science Foundation (Grant No. 20100470783).
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