Multiple-Sets Split Common Fixed-Point Problems for Demicontractive Mappings

In this paper, we are concerned with the multiple-sets split common fixed-point problems whenever the involved mappings are demicontractive. We first study several properties of demicontractive mappings and particularly their connection with directed mappings. By making use of these properties, we propose some new iterative methods for solving multiple-sets split common fixed-point problems, as well as multiple-sets spit feasibility problems. Under mild conditions, we establish their weak convergence of the proposed methods.


Introduction
e split common fixed-point problem (SCFP) requires finding an element in a fixed-point set such that its image under a linear transformation belongs to another fixed-point set. Formally, it consists in finding x ∈ H 1 such that where A: H 1 ⟶ H 2 is a bounded linear mapping from a Hilbert space H 1 into another Hilbert space H 2 , and F(U) and F(T) are respectively the fixed-point sets of nonlinear mappings U: H 1 ⟶ H 1 and T: H 2 ⟶ H 2 . Specially, if U and T are both metric projections, then problem (1) is reduced to the well-known split feasibility problem (SFP) [1]. Actually, the SFP can be formulated as finding x ∈ H 1 such that where C⊆H 1 and Q⊆H 2 are nonempty closed convex sets, and mapping A is as above. ese two problems recently have been extensively investigated since they play an important role in various areas including signal processing and image reconstruction [2][3][4][5][6]. We assume throughout the paper that problem (1) is consistent, which means that its solution set is nonempty. Censor and Segal [7] studied problem (1) when U and T are directed mappings. In this situation, they proposed the following method: where A * is the conjugate of A, I stands for the identity mapping, and τ n is a properly chosen stepsize. It is shown that if τ n is chosen in (0, 2/‖A‖ 2 ), then (7) converges weakly to a solution of (1). Subsequently, this result was extended to more general cases (see, e.g., [8][9][10][11][12][13][14][15][16][17]). Since the choice of the stepsize is related to ‖A‖, thus to implement (7), one has to compute (or at least estimate) the norm ‖A‖, which is generally not easy in practice. A way avoiding this is to adopt variable stepsize which ultimately has no relation with ‖A‖ [9,10,18]. In this connection, Wang and Cui [10] proposed the following stepsize: On the other hand, Wang [19] proposed a new method: It is clear that the selection of stepsizes (8) and (6) does not rely on the norm ‖A‖, which in turn improves the performance of the original algorithm. Assume that U and T are both directed such that I − T and I − U are demiclosed at 0. It is shown that the sequence x n generated by (7) and (8) or (5) and (6) converges weakly to a solution of problem (1). Now, let us consider the multiple-sets split common fixed-point problem (MSCFP) that is more general than the SCFP. Formally, it consists in finding x ∈ H 1 such that where t and s are two positive integers,  (7) is reduced to the wellknown MSFP [20]. Actually, it can be formulated as the problem of finding x ∈ H 1 such that where t and s are two positive integers, A: H 1 ⟶ H 2 is as above, and C i t i�1 ⊂ H 1 and Q j s j�1 ⊂ H 2 are two classes of nonempty convex closed subsets.
Inspired by the works mentioned above, we are aimed to introduce and analyze iterative methods for solving the MSCFP in Hilbert spaces. We first study several properties of demicontractive mappings and especially find its connection with the directed mapping. By making use of these properties, we propose a new iterative algorithm for solving the MSCFP, as well as MSFP. Under mild conditions, we obtain the weak convergence of the proposed algorithm. Our results extend the related works from the case of two-sets to the case of multiple-sets.

Preliminary
roughout the paper, assume that H, H 1 , H 2 are real Hilbert spaces, and F(T) denotes its fixed-point set of a mapping T. e following formula plays an important role in the subsequent analysis.
Lemma 1 (see [21]). Let s, t ∈ R and x, y ∈ H. It then follows that We next recall the definition of several important classes of nonlinear mappings.
Definition 1 (see [21]). Let T be a mapping from H into H.
Definition 2 (see [21]). Let T: H ⟶ H be a mapping with It is clear that a directed mapping is − 1-demicontractive, while a quasinonexpansive mapping is 0-demicontractive. It is also clear that a firmly nonexpansive mapping is − 1-strictly pseudocontractive, while a nonexpansive mapping is 0-strictly pseudocontractive.
It is well known that a mapping T is firmly nonexpansive if and only if 2T − I is nonexpansive (cf. [21]). Analogously, we can easily get the following lemma, which presents a characteristic of directed mappings by using quasinonexpansive mappings.

Lemma 2 A mapping T is directed if and only if 2T − I is quasinonexpansive.
We now study properties of demicontractive mappings.

It then follows that
Since x is chosen arbitrarily, we have ∩ t i�1 F(T i )⊆F(T).

It suffices to show that
To this end, fix any (x, z) ∈ H × F(T). By Lemma 1 and the property of demicontractions that It then follows that us, 2T − I is quasinonexpansive, which implies T is directed.
Let us now prove the second assertion. By Lemma 4, we have F(T) � ∩ t i�1 F(T i ) ≠ ∅. Let x n ⊂ H be such that x n ⇀x and ‖x n − Tx n ‖ ⟶ 0 as n ⟶ ∞. Fix z ∈ F(T).
Since ω i (1 − k i ) > 0, we have lim n ‖x n − T i x n ‖ � 0, which, by our hypothesis, implies lim n ‖x − T i x‖ � 0 for all i � 1, 2 . . . t, that is, x ∈ ∩ t i�1 F(T i ). By Lemma 4, the proof is complete.
□ Finally, we end this section by recalling two weak convergence theorems of iterative methods for approximating a solution of the two-sets SCFP (1).
Theorem 1 (see [10], eorem 3.1). (Assume that U and T are both directed such that I − U and I − T are both demiclosed at 0. en, the sequence x n , generated by (7) and (8), converges weakly to a solution of problem (1).
Theorem 2 (see [19], eorem 3.4). Assume that U and T are both directed such that I − U and I − T are both demiclosed at 0. en, the sequence x n , generated by (5) and (6), converges weakly to a solution of problem (1).

The Case for Demicontractive Mappings
In this section, we are concerned with the multiple-sets split common feasibility problem and we assume that (7) is consistent, which means that its solution set is nonempty. First, motivated by (7) and (8), we propose the first algorithm for solving problem (7).
. Given x n , update the next iteration via Theorem 3. Assume that U i and T j are respectively k i and l j -demicontractive such that I − U i and I − T j are demiclosed at 0 for i � 1, 2, . . . , t and j � 1, 2, . . . , s. en, the sequence x n , generated by Algorithm 1, converges weakly to a solution of (7).
. us, we can rewrite Algorithm 1 as where τ n � 0 if ‖(I − T)Ax n ‖ � 0; otherwise, By Lemma 5, U and T are both directed such as I − T and I − U are demiclosed at 0. It then follows from eorem 1 that x n weakly converges to a point x that satisfies x ∈ F(U) and Ax ∈ F(T). Moreover, by Lemma 4, we conclude that x ∈ ∩ i F(U i ) and Ax ∈ ∩ j F(T j ), that is, x is a solution of problem (7). □ Motivated by (5) and (6), we propose the second algorithm for solving problem (7).

Algorithm 2. Let x 0 be arbitrary and choose
then stop; otherwise, update the next iteration via where Theorem 4. Assume that U i and T j are respectively k i and l j -demicontractive such that I − U i and I − T j are demiclosed at 0 for i � 1, 2, . . . , t and j � 1, 2, . . . , s. en, the sequence x n , generated by Algorithm 2, converges weakly to a solution of (7).
and T � 1/2 s j�1 β j ((1 + l j )I + (1 − l j )T j ). us, we can rewrite Algorithm 2 as x n+1 � x n − τ n [(I − U)x n + A * (I − T)Ax n ], where By Lemma 5, U and T are both directed such as I − T and I − U are demiclosed at 0. It then follows from eorem 2 that x n weakly converges to a point x that satisfies x ∈ F(U) and Ax ∈ F(T). Moreover, by Lemma 4, we conclude that x ∈ ∩ i F(U i ) and Ax ∈ ∩ j F(T j ), that is, x is a solution of problem (7).
e sequence x n , generated by Algorithm 4, converges weakly to a solution of (8).
Proof. It suffices to notice that both P C i and P Q j are − 1-demicontractive, which implies k i � l j � − 1 for all i � 1, . . . , t, j � 1, . . . , s. Applying eorem 4 yields the desired assertion.

Conclusion
In this paper, we consider the MSCFP whenever the involved mappings are demicontractive. We obtained several properties of demicontractive mappings and particularly their connection with directed mappings. ese properties enable us to propose some new iterative methods for solving MSCFP, as well as MSFP. Under mild conditions, we establish their weak convergence of the proposed methods. Our results extend the existing works from the case of twosets to the case of multiple-sets.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares no conflicts of interest.