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Weak and Strong Convergence Theorems for the Multiple-Set Split Equality Common Fixed-Point Problems of Demicontractive Mappings
We consider mixed parallel and cyclic iterative algorithms in this paper to solve the multiple-set split equality common fixed-point problem which is a generalization of the split equality problem and the split feasibility problem for the demicontractive mappings without prior knowledge of operator norms in real Hilbert spaces. Some weak and strong convergence results are established. The results obtained in this paper generalize and improve the recent ones announced by many others.
For modeling inverse problems which arise from phase retrieval and in medical image reconstruction, in 1994, Censor and Elfving  firstly introduced the following split feasibility problem (SFP) in finite-dimensional Hilbert spaces.
Let and be nonempty closed convex sets of the Hilbert spaces and , respectively, and let be a bounded linear operator. The split feasibility problem (SFP) is formulated to It has been found that the SFP can be used in many areas such as image restoration, computer tomograph, and radiation therapy treatment planning. Some methods have been proposed to solve split feasibility problems; see, for instance, [2–5].
Assuming that the SFP (1) has a solution, it is not hard to see that where and are the metric projections from onto and from onto , respectively, is a positive constant, and denotes the adjoint of . This implies that SFP can be solved by using fixed-point algorithms.
In 2013, Moudafi and Al-Shemas  introduced the following new split feasibility problem, which is called the split equality fixed-point problem (SEFP). Let be real Hilbert spaces, let and be two bounded linear operators, and let and be two firmly quasi-nonexpansive mappings. The SEFP in  is to where and denote the sets of the fixed points of mappings and , respectively. The goal is to cover many situations, for instance, in decomposition methods for PDFs and applications in game theory and in intensity-modulated radiation therapy (IMRT).
For solving the SEFP (3), Moudafi and Al-Shemas  introduced the following simultaneous iterative method: for firmly quasi-nonexpansive mappings and , where stand for the spectral radii of and , respectively.
In 2014, Zhao  introduced the following simultaneous Mann iterative algorithm: where the step size does not depend on the operator norms and . And she proved the weak convergence of this algorithm (5) to solve SEFP (3) governed by quasi-nonexpansive operators and .
Recently, the multiple-set split equality common fixed-point problem (MSECFP) studied by Zhao and Wang  is towhere are integers. They introduced the following two mixed iterative algorithms for solving the MSECFP (6) of quasi-nonexpansive mappings:and where the step size does not depend on the operator norms and . And they proved the weak convergence of such algorithms.
Very recently, Wang and Kim  introduced the following iterative scheme for solving SEFP (3) of demicontractive mappings, and obtain a strong convergence result with no compactness assumptions on the spaces or the mappings and with no extra conditions on the fixed-point sets.
Inspired and motivated by the works mentioned above, we consider the mixed parallel and cyclic iterative algorithms for MSECFP (6) of demicontractive mappings which are a generalization of quasi-nonexpansive mappings without prior knowledge of operator norms in Hilbert spaces. Under some mild assumptions, we prove weak and strong convergence results of such algorithms for solving MSECFP (6).
Throughout this paper, we always assume that are real Hilbert spaces and let and be the set of positive integers and real numbers, respectively. In what follows, we denote strong and weak convergence in the space by “" and “", respectively, and the set of the fixed points of a mapping by . Also, we use to represent weak -limit set of .
Let be a nonempty closed convex subset of a Hilbert space . The metric (or nearest point) projection from onto is defined as follows: given , the unique point satisfies the property It is well known  that is a nonexpansive mapping and is characterized by the inequality
Definition 1. Let be a real Hilbert space. A mapping is said to be (i)nonexpansive if ;(ii)quasi-nonexpansive if and if ;(iii)firmly nonexpansive if or equivalently (iv) firmly quasi-nonexpansive if and (v) -demicontractive if and there exists a constant such that
Remark 2. Notice that every 0-demicontractive mapping is exactly quasi-nonexpansive. In particular, we say that it is quasi-strictly pseudocontractive  if . Moreover, if , every -demicontractive mapping becomes quasi-nonexpansive. Therefore, it is sufficient to only take in (v) of Definition 1 in Hilbert spaces. However, as seen in (iv) of Definition 1, every firmly quasi-nonexpansive mapping (often called to be a directed operator )) is obviously -demicontractive.
It is worth noting that the class of demicontractive mappings is more general than the class of quasi-nonexpansive mappings and the class of firmly quasi-nonexpansive mappings.
Definition 3. Let be a real Hilbert space. An operator is called demiclosed at origin for any sequence which converges weakly to , and if the sequence converges strongly to 0, then
As a special case of the demicloseness principle on uniformly convex Banach spaces given by , we know that if is a nonempty closed convex subset of a Hilbert space and is a nonexpansive mapping, then the mapping is demiclosed on . Now, the following question naturally arises: If is quasi-nonexpansive, is still demiclosed on ? The answer is negative even at 0 as follows.
Example 4 (see ; Example 2.11). The mapping : is defined by Then, is a quasi-nonexpansive mapping, but is not demiclosed at 0.
Definition 5. Let be a nonempty closed convex subset of a real Hilbert space . A sequence is Fejér-monotone relative to the target set (simply -Fejérian) if
Lemma 6 (see ). Let be a real Hilbert space. Then, for all and ,
Since the square of the norm is convex, we have the following lemma.
Lemma 7. Let be a real Hilbert space. Then, for , and
Lemma 8 (see ). Let be -demicontractive with and set for . Then, is quasi-nonexpansive for and
Lemma 9 (see ; Proposition ). Assume that is a closed convex subset of a Hilbert space . Let be a mapping. If is a -demicontractive mapping (which is also called -quasi-strict pseudocontraction in ), then the fixed-point set is closed and convex.
Lemma 10 (see ). Let and be Banach spaces and be a continuous linear operator from to . Then, is weakly continuous.
Lemma 12 (see ; Lemma ). Let be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence of which satisfies for all Define the sequence of integers as follows: where such that . Then, there hold the following properties: (i) and ;(ii) and
Lemma 13 (see ). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that(a) and ;(b) or Then, .
3. Main Results
In this section, we study two mixed parallel and cyclic iterative algorithms for MSECFP (6) of demicontractive mappings where the step sizes do not depend on the operator norms and , and we prove the weak and strong convergence of such algorithms. Denote the solution set of MSECFP (6) by ; that is,
Given two positive integers and , the -mod function takes values in the set and the -mod function takes values in the set as for for some integers and , and for for some integers and
Put . Define the inner product on as follows: It is easy to see that is a real Hilbert space and Note also that if is a sequence in , there holds the following relation:
Algorithm 14. Let be arbitrary and be integers. Let such that and . Assume that the th iterate has been constructed and , and then we calculate the th iterate via the formulas Assume the step size is chosen in such a way that for all and some small enough , where the index set , otherwise, being any nonnegative value). If , then we take and
Algorithm 15. Let be arbitrary and be integers. Let such that and . Assume that the th iterate has been constructed and , and then we calculate the th iterate via the formulas Assume that are the same as in Algorithm 14 and for the step size is chosen as (30); otherwise, being any nonnegative value). If , then we take andNow, we will see from Lemma 16 that is well defined and bounded. The proof of the following lemma is also added for completing the proof of Lemma in  for the sake of convenience.
Proof. Take , that is, , , and We have By adding the above two equalities and by taking into account, we obtain Consequently, for , that is, , we have or . And since we have Thus, we can choose small enough . This causes to be well defined. From (30), we obtain that and is a fixed positive number, so is bounded.
Assumption 17. Assume that(i)the solution set of (6) is nonempty;(ii) and are -demicontractive and -demicontractive, respectively;(iii) and are demiclosed at origin.Let and . Clearly, is -demicontractive for all and is -demicontractive for all .
Lemma 18. Given two bounded linear operators and , let and be -demicontractive and -demicontractive, respectively. Assume that the solution set of (6) is nonempty. Then, is a nonempty closed convex set.
Proof. By Lemma 9, we have that and are both closed convex subsets, and since and are both linear, it is easy to see that is a closed convex subset in .
Lemma 19 (see ; Lemma ). Let be a bounded sequence of a Hilbert space . Let be a positive integer and . If and , then, for any , there exists a subsequence of depending on such that and .
Theorem 20. Let Assumption 17 be satisfied. Given two bounded linear operators and , if the following conditions are satisfied:(a) and ,(b), then the sequence generated by Algorithm 14 converges weakly to a solution of (6).
Proof. First, we claim that the sequence is -Fejérian. Indeed, taking , that is, , , and , by Algorithm 14, we obtain Similarly, we have It follows from the above inequalities and thatTaking , we have for every and . It follows from (29) that Using Lemma 8 for any , we have Thus, by (43) and (44) and Lemma 7, we obtain By (ii) of Assumption 17 and from (29) and Lemma 6, we also have Then, it follows from (42), (45), and (46) that Hence, is -Fejérian.
Next, we show that, for all , , , , and . In fact, from (50), we see that the sequence is decreasing and bounded below by 0. Consequently, it converges to some finite limit. So, the sequences , , , and are also bounded. Then, from (49) and the assumption on in (30), it follows that which together with (48) implies that Also, from (47) and condition (a), it follows that This, combined with condition (a), implies that, for any , It follows from (47) and condition (b) that Since it follows from Lemma 16 that is bounded, so by (52) we have Similarly, we have It follows from (55) and (58) thatFinally, we claim that . To this end, let . From (28), it follows that , . Since by (57), we use (54) and Assumption 17(iii) to derive that . Given fixed , since and , now we apply Lemma 19 to get a subsequence of , depending on , such that and for all . Moreover, it turns out that by (58) andby virtue of (55). Using Assumption 17(iii) yields for any ; that is, . On the other hand, by Lemma 10, we have , which together with weakly lower semicontinuity of the norm implies that that is, ; hence, . Since is obviously a closed convex set by Lemma 18 and we have shown that in is -Fejérian and , by Lemma 11, we conclude that the sequence generated by Algorithm 14 converges weakly to a point of .
Let be a metric projection from onto We denote the origins of and by and , respectively.
Theorem 21. Let Assumption 17 be satisfied. Given two bounded linear operators and , if the following conditions are satisfied:(1) and ;(2);(3) and , then the sequence generated by Algorithm 15 converges strongly to a solution of (6).
Proof. Set . By (11), we readily seeSince , we have