Abstract
The purpose of this paper is to introduce and study the general split equality problem and general split equality fixed point problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequences generated by the proposed new algorithm converges strongly to a solution of the general split equality fixed point problem and the general split equality problem for quasi-nonexpansive mappings in Hilbert spaces. As an application, we shall utilize our results to study the null point problem of maximal monotone operators, the split feasibility problem, and the equality equilibrium problem. The results presented in the paper extend and improve the corresponding results announced by Moudafi et al. (Nonlinear Anal. 79:117-121, 2013; Trans. Math. Program. Appl. 1:1-11, 2013), Eslamian and Latif (Abstr. Appl. Anal. 2013:805104, 2013) and Chen et al. (Fixed Point Theory Appl. 2014:35, 2014), Censor and Elfving (Numer. Algorithms 8:221-239, 1994), Censor and Segal (J. Convex Anal. 16:587-600, 2009) and some others.
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1 Introduction
Let C and Q be nonempty closed convex subsets of real Hilbert spaces and , respectively. The split feasibility problem (SFP) is formulated as
where is a bounded linear operator. In 1994, Censor and Elfving [1] first introduced the SFP in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. It has been found that the SFP can also be used in various disciplines such as image restoration, and computer tomograph and radiation therapy treatment planning [3–5]. The SFP in an infinite-dimensional real Hilbert space can be found in [2, 4, 6–10].
Assuming that the SFP is consistent, it is not hard to see that solves SFP if and only if it solves the fixed point equation
where and are the metric projection from onto C and from onto Q, respectively, is a positive constant and is the adjoint of A.
A popular algorithm to be used to solve SFP (1.1) is due to Byrne’s CQ-algorithm [2]:
where with λ being the spectral radius of the operator .
Recently, Moudafi [11] introduced the following split equality problem (SEP):
where and are two bounded linear operators. Obviously, if (identity mapping on ) and , then (1.2) reduces to (1.1). This kind of split equality problem (1.2) allows asymmetric and partial relations between the variables x and y. The interest is to cover many situations, such as decomposition methods for PDEs, applications in game theory, and intensity-modulated radiation therapy.
In order to solve the split equality problem (1.2), Moudafi [11] introduced the following relaxed alternating CQ-algorithm:
where
and (respectively ) is a convex and subdifferentiable function. Under suitable conditions, he proved that the sequence defined by (1.4) converges weakly to a solution of the split equality problem (1.2).
Each nonempty closed convex subset of a Hilbert space can be regarded as a set of fixed points of a projection. In [12], Moudafi and Al-Shemas introduced the following split equality fixed point problem:
where and are two firmly quasi-nonexpansive mappings, and denote the fixed point sets of S and T, respectively.
To solve the split equality fixed point problem (1.5) for firmly quasi-nonexpansive mappings, Moudafi et al. [11–13] proposed the following iteration algorithm:
Very recently, Eslamian and Latif [14] and Chen et al. [15] introduced and studied some kinds of general split feasibility problem and split equality problem in real Hilbert spaces, and under suitable conditions some strong convergence theorems are proved.
Motivated by the above works, the purpose of this paper is to introduce the following general split equality fixed point problem:
and the general split equality problem:
For solving the GSEFP (1.7) and GSEP (1.8), in Sections 3 and 4, we propose an algorithm for finding the solutions of the general split equality fixed point problem and general split equality problem in a Hilbert space. We establish the strong convergence of the proposed algorithms to a solution of GSEFP and GSEP. As applications, in Section 5 we utilize our results to study the split feasibility problem, the null point problem of maximal monotone operators, and the equality equilibrium problem.
2 Preliminaries
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. In the sequel, denote by the set of fixed points of a mapping T and by and , the strong convergence and weak convergence of a sequence to a point , respectively.
Recall that a mapping is said to be nonexpansive, if , . A typical example of a nonexpansive mapping is the metric projection from H onto defined by . The metric projection is firmly nonexpansive, i.e.,
and it can be characterized by the fact that
Definition 2.1 A mapping is said to be quasi-nonexpansive, if , and
Lemma 2.2 [16]
Let H be a real Hilbert space, and be a sequence in H. Then, for any given sequence of positive numbers with such that for any positive integers i, j with , the following holds:
Lemma 2.3 [17]
Let H be a real Hilbert space. For any , the following inequality holds:
Lemma 2.4 [18]
Let be a sequence of real numbers. If there exists a subsequence of such that for all , then there exists a nondecreasing sequence with such that for all (sufficiently large) positive integer numbers n, the following holds:
In fact,
Definition 2.5 (Demiclosedness principle)
Let C be a nonempty closed convex subset of a real Hilbert space H, and be a mapping with . Then is said to be demi-closed at zero, if for any sequence with and , then .
Remark 2.6 It is well known that if is a nonexpansive mapping, then is demi-closed at zero.
Lemma 2.7 Let , , and be sequences of positive real numbers satisfying for all . If the following conditions are satisfied:
-
(1)
and ,
-
(2)
, or ,
then .
3 Strong convergence theorem for general split equality fixed point problem
Throughout this section we always assume that
-
(1)
, , are real Hilbert spaces;
-
(2)
and are two families of one-to-one and quasi-nonexpansive mappings;
-
(3)
and are two bounded linear operators;
-
(4)
, where , is a k-contractive mapping on with ;
-
(5)
, , Γ is the set of solutions of GSEFP (1.7),
-
(6)
for any given , the iterative sequence is generated by
(3.1)where , , are the sequences of nonnegative numbers with
We are now in a position to give the following main result.
Lemma 3.1 Let be a point in , i.e., and . Then the following statements are equivalent:
-
(i)
is a solution to GSEFP (1.7);
-
(ii)
for each and ;
-
(iii)
for each and for each , solves the fixed point equations:
(3.2)
Proof (i) ⇒ (ii). If is a solution to GSEFP (1.7), then for each , , and . This implies that for each , , and
-
(ii)
⇒ (iii). If , and , it is easy to see that (3.2) holds.
-
(iii)
⇒ (i). From (3.2), for each we have . Since and both are one-to-one, so is . Hence we have , for any . This implies that , and so
i.e., .
This completes the proof of Lemma 3.1. □
Lemma 3.2 If , where , then is a nonexpansive mapping.
Proof In fact for any , we have
This completes the proof. □
Theorem 3.3 Let , , , , , A, B, f, C, Q, Γ, P, G, , satisfy the above conditions (1)-(5). Let be the sequence defined by (3.1). If the solution set Γ of GSEFP (1.7) is nonempty and the following conditions are satisfied:
-
(i)
, for each ;
-
(ii)
, and ;
-
(iii)
for each ;
-
(iv)
for each , where ;
-
(v)
for each , the mapping is demi-closed at zero,
then the sequence converges strongly to which is a solution of GSEFP (1.7).
Proof (I) First we prove that the sequence is bounded.
In fact, for any given , it follows from Lemma 3.1 that
By the assumptions and Lemma 3.2, for each , is nonexpansive, and for each , is quasi-nonexpansive, hence we have
By induction, we can prove that
This shows that is bounded, and so is .
-
(II)
Now we prove that the following inequality holds:
(3.3)
Indeed, it follows from (3.1) and Lemma 2.2 that for each
This implies that for each
Inequality (3.3) is proved.
It is easy to see that the solution set Γ of GSEFP (1.7) is a nonempty closed and convex subset in , hence the metric projection is well defined. In addition, since is a contractive mapping, there exists a such that
-
(III)
Now we prove that .
For this purpose, we consider two cases.
Case I. Suppose that the sequence is monotone. Since is bounded, is convergent. Since , in (3.3) taking and letting , in view of conditions (ii) and (iii), we have
On the other hand, by Lemma 2.3 and (3.1), we have
Simplifying we have
where , , .
By condition (ii), and , and so .
Next we prove that
In fact, since is bounded in , there exists a subsequence with (some point in ), and such that
In view of (3.5)
Again by the assumption that for each , the mapping is demi-closed at zero, hence we have
By Lemma 3.1, this implies that . In addition, since , we have
This shows that (3.7) is true. Taking , , and in Lemma 2.7, all conditions in Lemma 2.7 are satisfied. We have .
Case II. If the sequence is not monotone, by Lemma 2.4, there exists a sequence of positive integers: , (where is large enough) such that
Clearly is nondecreasing, as , and for all
Therefore is a nondecreasing sequence. According to Case I, and . Hence we have
This implies that and is a solution of GSEFP (1.7).
This completes the proof of Theorem 3.3. □
Remark 3.4 Theorem 3.3 extends and improves the main results in Moudafi et al. [11–13] in the following aspects:
-
(a)
For the mappings, we extend the mappings from firmly quasi-nonexpansive mappings to an infinite family of one-to-one quasi-nonexpansive mappings.
-
(b)
For the algorithms, we propose new iterative algorithms which are different from ones given in [11–13].
-
(c)
For the convergence, the iterative sequence proposed by our algorithm converges strongly to a solution of GSEFP (1.7). But the iterative sequences proposed in [11–13] are only of weak convergence to a solution of the split equality problem.
4 Strong convergence theorem for general split equality problem
Throughout this section we always assume that
-
(1)
, , are real Hilbert spaces; and are two families of nonempty closed and convex subsets with and ;
-
(2)
(resp. ) is the metric projection from onto (resp. onto ), and , , and ;
-
(3)
and are two bounded linear operators;
-
(4)
f, G, are the same as in Theorem 3.3.
The so-called general split equality problem (GSEP) is
Lemma 4.1 Let , , , P, , A, B, f, C, Q, G, be the same as above. Then a point is a solution to GSEP (4.1), if and only if for each and for each , solves the following fixed point equations:
Proof In fact, a point is a solution of GSEP (4.1)
This completes the proof of Lemma 4.1. □
The metric projections and are nonexpansive with and , . This implies that the metric projections and all are quasi-nonexpansive. In addition, by Lemma 3.2, for each and each , the mapping is nonexpansive. By Remark 2.6, for each and each , the mapping is demi-closed at zero.
Consequently, we have the following.
Theorem 4.2 Let , , , P, , A, B, f, C, Q, G, be the same as above. Let be the sequence generated by
If the solution set of GSEP (4.1) is nonempty and the following conditions are satisfied:
-
(i)
, for each ;
-
(ii)
, and ;
-
(iii)
for each ;
-
(iv)
for each , where ,
then the sequence defined by (4.3) converges strongly to a solution of GSEP (4.1) and .
Proof Taking , , and , in Theorem 3.3, we know that and both are nonexpansive with and and so they are quasi-nonexpansive mappings, and and . Therefore all conditions in Theorem 3.3 are satisfied. The conclusion of Theorem 4.2 can be obtained from Lemma 4.1 and Theorem 3.3 immediately. □
Remark 4.3 Theorem 4.2 extends and improves the corresponding results in Censor and Elfving [1], Moudafi et al. [11, 12], Eslamian and Latif [14], Chen et al. [15], Censor and Segal [19].
5 Applications
In this section we shall utilize the results presented in the paper to give some applications.
5.1 Application to split feasibility problem
Let and be two nonempty closed convex subsets and be a bounded linear operator. The so-called split feasibility problem (SFP) [1] is to find
Let and be the metric projection from onto C and onto Q, respectively. Thus and . From Theorem 4.2 we have the following.
Theorem 5.1 Let , be two real Hilbert spaces, be a bounded linear operator and I be the identity mapping on . Let and be nonempty closed convex subsets and and are the metric projections from onto C and onto Q, respectively. Let be the sequence generated by :
where f is the mapping as given in Theorem 4.2 and
If the solution set of SFP (5.1) is nonempty and the following conditions are satisfied:
-
(i)
, for each ;
-
(ii)
, and ;
-
(iii)
;
-
(iv)
, where ,
then the sequence defined by (5.2) converges strongly to a solution of SFP (5.1) and .
Proof In Theorem 4.2 taking , , , , and , the conclusions of Theorem 5.1 can be obtained from Theorem 4.2 immediately. □
Remark Theorem 5.1 generalizes and extends the main results of Censor and Elfving [1] and Censor and Segal [19] from weak convergence to strong convergence.
5.2 Application to null point problem of maximal monotone operators
Let , , , A, B, be the same as in Theorem 3.3. Let , and be two strictly maximal monotone operators. It is well known that the associated resolvent mappings and of M and N, respectively, are one-to-one nonexpansive mappings, and
Denote , , , and , then the general split equality fixed point problem (1.7) is reduced to the following null point problem related to strictly maximal monotone operators M and N ():
From Theorem 3.3 we can obtain the following.
Theorem 5.2 Let , , , A, B, f, G, be the same as in Theorem 3.3. Let C, Q, S, and T be the same as above. Let be the sequence generated by
where , . If the solution set of (5.5) is nonempty and the following conditions are satisfied:
-
(i)
, for each ;
-
(ii)
, and ;
-
(iii)
;
-
(iv)
, where ,
then the sequence defined by (5.6) converges strongly to , which is a solution of (5.5).
Proof Since and both are one-to-one nonexpansive with and . Hence they are one-to-one quasi-nonexpansive mappings and is demi-closed at zero. Therefore all conditions in Theorem 3.3 are satisfied. The conclusions of Theorem 5.2 can be obtained from Theorem 3.3 immediately. □
5.3 Application to equality equilibrium problem
Let D be a nonempty closed and convex subset of a real Hilbert H. A bifunction is said to be a equilibrium function, if it satisfies the following conditions:
(A1) , for all ;
(A2) g is monotone, i.e., for all ;
(A3) for all ;
(A4) for each , is convex and lower semi-continuous.
The so-called equilibrium problem with respective to the equilibrium functions g and D is
Its solution set is denoted by .
For given and , the resolvent of the equilibrium function g is the operator defined by
It is well known that the resolvent of the equilibrium function g has the following properties [20]:
-
(1)
is single-valued;
-
(2)
and is a nonempty closed and convex subset of D;
-
(3)
is a nonexpansive mapping, and so it is quasi-nonexpansive.
Definition 5.3 Let be two equilibrium functions and, for given , let and be the resolvents of h and j (defined by (5.8)), respectively. Denote by , , , and . Then the equality equilibrium problem with respective to the equilibrium functions h, j, and D is
where are two linear and bounded operators.
The following theorem can be obtained from Theorem 3.3 immediately.
Theorem 5.4 Let H be a real Hilbert space, D be a nonempty and closed convex subset of H. Let G, f be the same as in Theorem 3.3. For given , let h, j, , , S, T, C, Q be the same as above. Let be the sequence generated by :
where , . If the solution set of (5.9) is nonempty and the following conditions are satisfied:
-
(i)
, for each ;
-
(ii)
, and ;
-
(iii)
;
-
(iv)
, where ,
then the sequence converges strongly to , which is a solution of (5.9).
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Chang, Ss., Agarwal, R.P. Strong convergence theorems of general split equality problems for quasi-nonexpansive mappings. J Inequal Appl 2014, 367 (2014). https://doi.org/10.1186/1029-242X-2014-367
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DOI: https://doi.org/10.1186/1029-242X-2014-367