Abstract
The purpose of this paper is first to introduce and study the general split equality variational inclusion problems and the general split equality optimization problems in the setting of infinite-dimensional Hilbert spaces and then propose a new simultaneous iterative algorithm. Under suitable conditions, some strong convergence theorems for the sequences generated by the proposed algorithm converging strongly to a solution for these two kinds of problems are proved. As special cases, we shall utilize our results to study the split feasibility problems, the split equality equilibrium problems, and the split optimization problems. The results presented in the paper not only extend and improve the corresponding recent results announced by many authors, but they also provide an affirmative answer to an open question raised by Moudafi in his recent work.
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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 solves the (1.1) is due to Byrne’s CQ-algorithm [2]:
where with λ being the spectral radius of the operator .
Recently, Moudafi [11, 12] introduced the following split equality feasibility problem (SEFP):
where and are two bounded linear operators. Obviously, if (identity mapping on ) and , then (1.3) reduces to (1.1). The kind of split equality problems (1.3) allows asymmetric and partial relations between the variables x and y. The interest is to cover many situations, such as decomposition methods for PDEs, and applications in game theory and intensity-modulated radiation therapy.
In order to solve the split equality feasibility problem (1.3), Moudafi [11] introduced the following simultaneous iterative method:
and under suitable conditions he proved the weak convergence of the sequence to a solution of (1.3) in Hilbert spaces.
At the same time, he raised the following open question.
Moudafi’s Open Question 1.1 Is there any strong convergence theorem of an alternating algorithm for the split equality feasibility problem (1.3) in real Hilbert spaces?
More recently, Eslamian and Latif [13], Chen et al. [14], Chuang [15] and Chang and Wang [16] introduced and studied some kinds of general split feasibility problem, general split equality problem, and split variational inclusion problem in real Hilbert spaces. Under suitable conditions some strong convergence theorems are proved. Also a comprehensive survey and update bibliography on split feasibility problems are given in Ansari and Rehan [17].
Motivated by the above works and related literature, in this paper, we continue to consider the problem (1.3). We obtain some strongly convergent theorems to a solution of the problem (1.3) which provide an affirmative answer to Moudafi’s open question.
For the purpose we first introduce and consider the following more general problems.
(I) General split equality variational inclusion problem:
where , and are three real Hilbert spaces, and , are two families of set-valued maximal monotone mappings, and are two linear and bounded operators.
(II) General split equality optimization problem:
where , , and are three real Hilbert spaces, and are two linear and bounded operators, and are two countable families of proper, convex, and lower semicontinuous functions.
The following problems are special cases of Problem I and II.
(III) Split equality feasibility problems.
Let and be two nonempty closed convex subsets and , be two bounded linear operators. As mentioned above the so-called ‘split equality feasibility problem’ (SEFP) is to find
Let and be the indicator functions of C and Q, respectively, i.e.,
Denote by and the normal cones of C and Q at x and y, respectively:
It is easy to know that and both are proper convex and lower semicontinuous functions on and , respectively, and the sub-differentials and both are maximal monotone operators. We define the resolvent operator of by
Here
Hence we have
This implies that for any . Similarly, we also have , and for any . Therefore the (SEFP) (1.3) is equivalent to the following split equality optimization problem, i.e., to find , and such that
(IV) Split equality equilibrium problem.
Let D be a nonempty closed and convex subset of a real Hilbert space 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 semicontinuous.
The so-called equilibrium problem with respect to the equilibrium function g is
Its solution set is denoted by .
For given and , the resolvent of the equilibrium function g is the operator defined by
Proposition 1.2 [18]
The resolvent operator of the equilibrium function g has the following properties:
-
(1)
is single-valued;
-
(2)
and is a nonempty closed and convex subset of D;
-
(3)
is a firmly nonexpansive mapping.
Let be two equilibrium functions. For given , let and be the resolvent of h and g (defined by (1.9)), respectively.
The so-called split equality equilibrium problem with respective to h, g, and D is to find , such that
where are two linear and bounded operators.
By Proposition 1.2, the (1.10) is equivalent to find , such that for each
Letting , , by Proposition 1.2, C and Q both are nonempty closed and convex subset of D. Hence the problem (1.10) is equivalent to the following split equality feasibility problem:
(V) Split optimization problem.
Let and be two real Hilbert spaces, be a linear and bounded operators, and be two proper convex and lower semicontinuous functions. The split optimization problem (SOP) is to find , such that
Denote by and , then the (SOP) (1.12) is equivalent to the following split variational inclusion problem (SVIP): to find such that
For solving (GSEVIP) (1.5) and (GSEOP) (1.6), in Sections 3 and 4, we propose a new simultaneous type iterative algorithm. Under suitable conditions some strong convergence theorems for the sequences generated by the algorithm are proved in the setting of infinite-dimensional Hilbert spaces. As special cases, we shall utilize our results to study the split feasibility problem, split equality equilibrium problem and the split optimization problem. By the way, we obtain a strongly convergent iterative sequence to a solution of the problem (1.3), which provides an affirmative answer to the open question raised by Moudafi [11]. The results presented in the paper extend and improve the corresponding results announced by Moudafi et al. [11, 12, 19], Eslamian and Latif [13], Chen et al. [14], Censor et al. [1, 3–5, 20], Chuang [15], Naraghirad [21], Chang and Wang [16], Ansari and Rehan [17], and some others.
2 Preliminaries
We first recall some definitions, notations, and conclusions.
Throughout this paper, we assume that H is a real Hilbert space and C is a nonempty closed convex subset of H. In the sequel, we 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 nonexpansive mapping is the metric projection from H onto defined by . The metric projection is firmly nonexpansive, if
and it can be characterized by the fact that
A mapping is said to be quasinonexpansive, if , and
It is easy to see that if T is a quasi-nonexpansive mapping, then is a closed and convex subset of C. Besides, T is said to be a firmly nonexpansive, if
Lemma 2.1 [22]
Let H be a real Hilbert space, and be a sequence in H. Then, for any given sequence of positive numbers with for any positive integers i, j with the following holds:
Lemma 2.2 [23]
Let H be a real Hilbert space. For any , the following inequality holds:
Lemma 2.3 [24]
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 number n, the following holds:
In fact,
Definition 2.4 (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 demiclosed at zero, if for any sequence with and , .
Remark 2.5 [25]
It is well known that if is a nonexpansive mapping, then is demiclosed at zero.
Lemma 2.6 Let , and be sequences of positive real numbers satisfying for all . If the following conditions are satisfied:
-
(1)
and ,
-
(2)
, or ,
then .
Lemma 2.7 [15]
Let H be a real Hilbert space, be a set-valued maximal monotone mapping, , and let be the resolvent mapping of B defined by , then
-
(i)
for each , is a single-valued and firmly nonexpansive mapping;
-
(ii)
and ;
-
(iii)
is a firmly nonexpansive mapping for each ;
-
(iv)
suppose that , then for each , each and each
-
(v)
suppose that . Then for each , each , and each .
Lemma 2.8 Let , be two real Hilbert spaces, be a linear bounded operator and be the adjoint of A. Let be a set-valued maximal monotone mapping, , and let be the resolvent mapping of B, then
-
(i)
;
-
(ii)
;
-
(iii)
if , then is a nonexpansive mapping.
Proof By Lemma 2.7(iii), the mapping is firmly nonexpansive, hence the conclusions (i) and (ii) are obvious.
Now we prove the conclusion (iii).
In fact, for any , it follows from the conclusions (i) and (ii) that
This completes the proof of Lemma 2.8. □
3 General split equality variational inclusion problem and strong convergence theorems
Throughout this section we assume that
-
(1)
, , are three real Hilbert spaces;
-
(2)
and are two families of set-valued maximal monotone mappings, and are given positive numbers;
-
(3)
and are two bounded linear operators and , are the adjoint of A and B, respectively;
-
(4)
, where , is a k-contractive mapping on with ;
-
(5)
the set of solutions of (GSEVIP) (1.5) ,
-
(6)
for any given , the iterative sequence is generated by
(3.1)
or its equivalent form:
where , , are the sequences of nonnegative numbers satisfying
We are now in a position to give the following results.
Lemma 3.1 Let , , , A, B, , , , , , G, be the same as above. If (the solution set of (GSEVIP) (1.5)), then is a solution of (GSEVIP) (1.5) if and only if for each , and for any given and
Proof Indeed, if is a solution of (GSEVIP) (1.5), then by Lemma 2.7(ii), for each , and for any and we have
Hence we have , and so
This implies that (3.2) is true.
Conversely, if satisfies (3.2), then we have
We make the assumption that the solution set Ω of (GSEVIP) (1.5) is nonempty. Hence the sets and both are nonempty. By Lemma 2.7(v) and (3.3), we have
and so
Similarly, by Lemma 2.7(v) and (3.3) again, one gets
Adding up (3.4) and (3.5), we have
Simplifying it, we have
Since , taking , for each , we have and and . In (3.6), taking and , we have
Hence from (3.3) and (3.7)
It follows from (3.7) and (3.8) that is a solution of (GSEVIP) (1.5).
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, , , , , , G, , f be the same as above. Let be the sequence defined by (3.1). If the solution set Ω of (GSEVIP) (1.5) is nonempty and the following conditions are satisfied:
-
(i)
, for each ;
-
(ii)
, and ;
-
(iii)
for each ;
-
(iv)
for each , where ,
then the sequence converges strongly to , which is a solution of (GSEVIP) (1.5).
Proof (I) First we prove that the sequence is bounded.
In fact, for any given , it follows from Lemma 3.1, Lemma 3.2, and condition (iv) that
and is a nonexpansive mapping. Also by Lemma 2.7(i), for each , is a firmly nonexpansive mapping. 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:
Indeed, it follows from (3.1) and Lemma 2.1 that for each
This implies that for each
Inequality (3.3) is proved.
It is easy to see that the solution set Ω of (GSEVIP) (1.5) is a closed and convex subset in . By the assumption that Ω is nonempty, so it 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 unique such that
(III) Now we prove that converges strongly to .
For the purpose, we consider two cases.
Case I. Suppose that the sequence is monotone. Since is bounded, is convergent. Since , in (3.9) taking and letting , in view of conditions (ii) and (iii), we have
On the other hand, by Lemma 2.2 and (3.1), we have
Simplifying it we have
where
By condition (ii), and , and so is .
Next we prove that
In fact, since is bounded in , there exists a subsequence with (some point in ), and such that
Since
and is a nonexpansive mapping, by Remark 2.5, is demiclosed at zero, hence we have
By Lemma 3.1, this implies that . In addition, since , we have
This shows that (3.13) is true. Taking , , and in Lemma 2.6, therefore all conditions in Lemma 2.6 are satisfied. We have .
Case II. If the sequence is not monotone, by Lemma 2.3, there exists a sequence of positive integers: , (where large enough) such that
Clearly is a 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 (GSEVIP) (1.5).
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, 12, 19], Eslamian and Latif [13], Chen et al. [14], Chuang [15], Naraghirad [21] and Ansari and Rehan [17].
4 General split equality optimization problem and strong convergence theorems
Let , , and be three real Hilbert spaces. Let and be two linear and bounded operators. The so-called general split equality optimization problem (GSEOP) is to find , and such that for each
where and are two families of proper, lower semicontinuous, and convex functions.
For each denote by and . Then the mappings and , both are set-valued maximal monotone mappings, and
Therefore (GSEOP) (4.1) is equivalent to the following general split equality variational inclusion problem (GSEVIP): to find and such that
Therefore, the following theorem can be obtained from Theorem 3.3 immediately.
Theorem 4.1 Let , , , A, B, , , , be the same as above. Let , G, , f be the same as in Theorem 3.3. Let be the sequence defined by (3.1). If the solution set of (GSEVIP) (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 converges strongly to , which is a solution of (GSEOP) (4.1).
By using Theorem 3.3 and Theorem 4.1, now we give some corollaries for the split equality feasibility problem, the split equality equilibrium problem, and the split optimization problem.
Let , , , C, Q, A, B be the same as in the split equality feasibility problem (1.3). Let and be the indicator function of C and Q, respectively, defined by (1.7). In Theorem 4.1, take , , and , therefore we have the following.
Corollary 4.2 Let , , , A, B, , , be the same as above. Let G, , f be the same as in Theorem 4.1. Let be the sequence generated by
or its equivalent form
If the solution set of (SEFP) (1.3) is nonempty and the following conditions are satisfied:
-
(i)
, for each ;
-
(ii)
, and ;
-
(iii)
;
-
(iv)
for each , where ,
then the sequence converges strongly to , which is a solution of (SEFP) (1.3).
Remark 4.3 Since the simultaneous iterative sequence (4.4) converges strongly to a solution of (SEFP) (1.3). Therefore it provides an affirmative answer to Moudafi’s open question 1.1 [11].
Let be two equilibrium functions. For given , let and be the resolvents of h and g (defined by (1.9)), respectively.
The so-called split equality equilibrium problem with respective to h, g, and D is to find , such that
where are two linear and bounded operators.
By Proposition 1.2, the (4.5) is equivalent to find , such that for each
Letting , , by Proposition 1.2, C and Q both are nonempty closed and convex subset of D. Hence the problem (4.5) (and so the problem (4.6)) is equivalent to the following split equality feasibility problem:
In Corollary 4.2 taking , from Corollary 4.2 we have the following.
Corollary 4.4 Let D, C, Q be the same as above. Let A, B, , , , G, , f be the same as in Corollary 4.2. For any given , let be the sequence generated by
If the solution set of (4.5) is nonempty and the following conditions are satisfied:
-
(i)
, for each ;
-
(ii)
, and ;
-
(iii)
;
-
(iv)
for each , where ,
then the sequence converges strongly to , which is a solution of (4.5).
Let and be two real Hilbert spaces, be a linear and bounded operators, and be two proper convex and lower semicontinuous functions. The split optimization problem (SOP) is to find , such that
Denote and , then the (SOP) (4.9) is equivalent to the following split variational inclusion problem (SVIP): to find such that
In Theorem 4.1 taking , (the identity mapping on ) and
then from Theorem 4.1 we have the following.
Corollary 4.5 Let , , A, I, , , U, K, be the same as above. Let , f be the same as in Theorem 4.1. For any given , let be the sequence defined by
or its equivalent form:
If , the solution set of (SOP) (4.9) 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 (SOP) (4.9).
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The authors would like to express their thanks to the editors and the referees for their helpful suggestion and advices. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
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Chang, Ss., Wang, L., Tang, Y.K. et al. Moudafi’s open question and simultaneous iterative algorithm for general split equality variational inclusion problems and general split equality optimization problems. Fixed Point Theory Appl 2014, 215 (2014). https://doi.org/10.1186/1687-1812-2014-215
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DOI: https://doi.org/10.1186/1687-1812-2014-215