Iterative Arrangements of the MSCFP for Strictly Pseudocontractive Mappings

In this paper, we consider the multiple-set split common ﬁ xed point problem in Hilbert spaces. We ﬁ rst study a couple of critical properties of strictly pseudocontractive mappings and particularly the property under mix activity. By utilizing these properties, we propose new iterative strategies for settling this problem as well as several connected issues. Under delicate conditions, we state weak convergence of the proposed strategies that expands the current works from the case of two subsets to the instance of multiple subsets. As an application, we give an exhibit of the theoretical results to the multiple-set split equality problem and the elastic net regularization.


Introduction
Let t and s be the two positive integers, and H 1 and H 2 stand for two Hilbert spaces. The well-known split feasibility problem (SFP) [1] is formulated as follows: find a point x ∈ H 1 satisfying the property where C and Q are nonempty closed convex subset of H 1 and H 2 , respectively, and A is a bounded linear mapping from H 1 into H 2 . There are many generalizations of the SFP, one of which is from two groups to multiple groups, that is, multiple-set split feasibility problem (MSFP) [2]. Actually, it can be formulated as the problem of finding x ∈ H 1 such that where A : H 1 ⟶ H 2 is as above and fC i g t i=1 ⊂ H 1 and fQ j g s j=1 ⊂ H 2 are two classes of nonempty convex closed subsets.
The split common fixed point problem (SCFP) [3] is another generalization of the SFP, which requires to find 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 as above and FðUÞ and FðTÞ 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 (3) is reduced to the SFP. As a further extension of the SFP, we recall the multiple-set split common fixed point problem (MSCFP). Indeed, the MSCFP extends the SCFP from two groups to the case of multiple groups. Formally, it consists in finding x ∈ H 1 such that where A : H 1 ⟶ H 2 is as above and FðU i Þ and FðT j Þ are, respectively, the fixed point sets of nonlinear mappings U i : Recently, we [4] considered problem (4) whenever the involved mappings are demicontractive. These issues have been concentrated on broadly in different regions like image reconstruction and signal processing [5][6][7][8][9].
There are many algorithms in the literature that can solve the SCFP problem (see, e.g., [10][11][12][13][14][15][16]). However, in most of these algorithms, the choice of the stepsize is related to kAk: Thus, to implement these algorithms, one has to compute (or at least estimate) the norm kAk, which is generally not easy in practice. A way avoiding this is to adopt variable stepsize which ultimately has no relation with kAk [11,12,17]. In this connection, Wang [18] recently proposed the following method: where A * is the conjugate of A, I stands for the identity mapping, and fτ n g ⊂ ð0,∞Þ is chosen such that It is shown that if mappings U and T are firmly nonexpansive, then the sequence fx n g generated by (5) converges weakly to a solution of problem (3). It is clear that such a choice of the stepsize does not rely on the norm kAk. Kraikaew and Saejung [16] weakened condition (6) as follows: Furthermore, we [19] extended the above results from the class of firmly nonexpansive mappings to the class of strictly pseudocontractive mappings.
Inspired by the above work, we will continue to present and investigate strategies for addressing the MSCFP in Hilbert spaces. We initially explore a few properties of strictly pseudocontractive mappings and track down its soundness under arched combinatorial operation. Exploiting these properties, we propose another iterative algorithm to address the MSCFP, as well as the MSFP. Under gentle conditions, we acquire weak convergence of the proposed algo-rithm. Our outcomes broaden related work from the instance of two groups to the case of multiple groups.

Preliminary
Throughout the paper, assume that H, H 1 , H 2 , and H 3 are real Hilbert spaces, and FðTÞ denotes its fixed point set of a mapping T. For any α, β ∈ ℝ and x, y ∈ H, it is well known that [20] βx + αy k Recall that the mapping T : It is called firmly nonexpansive if It is called k-strictly pseudocontractive ðk < 1Þ if It is clear that the class of strictly pseudocontractive mappings includes the class of nonexpansive mappings, while the latter includes the class of firmly nonexpansive mappings. Indeed, a firmly nonexpansive mapping is −1 -strictly pseudocontractive, while a nonexpansive mapping is 0-strictly pseudocontractive. In general, these inclusion are proper (cf. [20,21]). The following properties of strictly pseudocontractive mappings play an import role in the subsequent analysis. It was shown [21] that if T : H ⟶ H is k -strictly pseudocontractive, then it follows that Moreover, the fixed point set of T is convex and closed. We now collect further properties of strictly pseudocontractive mappings.
Proof. "⇒" Assume T is k-strictly pseudocontractive. Let R = kI + ð1 − kÞT. It is easy to verify that R fulfils (13). It remains to show that R is nonexpansive. To this end, fix 2 Journal of Function Spaces any x, z ∈ H. It then follows from (8) and the property of strictly pseudocontractive mappings that Hence, we have kRx − Rzk ≤ kx − zk; that is, R is nonexpansive.
"⇐" Assume that there is a nonexpansive mapping R such that (13) follows. Choose any x, z ∈ H. It then follows from (8) and the property of nonexpansive mappings that Hence, T is strictly pseudocontractive, and thus, the proof is complete.
Remark 2. Note that a firmly nonexpansive mapping is −1 -strictly pseudocontractive. It is well known that a mapping T is firmly nonexpansive if and only if there is a nonexpansive mapping R such that T = ðI + RÞ/2: The following lemma can be regarded as an extension of this assertion.
Proof. It suffices to show that FðTÞ FðT i Þ and choose any x ∈ FðTÞ. By our hypothesis, there exists k i < 1 such that for every i = 1, 2 ⋯ t. Adding up these inqualities, we have Proof. By our hypothesis, for each i = 1, 2 ⋯ , t, there exists a nonexpansive mapping R i such that where k is defined as in (19). It is readily seen that From Lemma 1, it remains to show that R is nonexpansive. To this end, choose any x, z ∈ H.
Hence, R is nonexpansive, and thus, the proof is complete.

The Case for Strictly Pseudocontractive Mappings
First, let us recall a weak convergence theorem of iterative method (5) for approximating a solution of the two-set split common fixed point problem.
We next consider the MSCFP under the following basic assumption.
(i) MSCFP is consistent; that is, it admits at least one solution Algorithm 1. Let x 0 be arbitrary. Given x n , update the next iteration via where ∑ s j=1 β j = 1, and fτ n g ⊂ ð0,∞Þ are properly chosen stepsizes.

Theorem 6.
Assume that conditions (A1)-(A3) hold and fτ n g is chosen so that where Then, the sequence fx n g, generated by Algorithm 1, converges weakly to a solution of MSCFP.
It seems that the choice of the stepsize above requires the prior information of k i , l j and the norm kAk. However, as shown below, there is a special case in which the selection of stepsizes ultimately has no relation with k i , l j and the norm kAk.

Corollary 7. Assume that conditions (A1)-(A3) hold, and the stepsize is chosen so that
Then, the sequence fx n g generated by Algorithm 1 converges weakly to a solution of MSFP.
Significantly, if the nonlinear mappings in (4) are all metric projections, then the MSCFP is reduced to the MSFP. Consequently, we can apply our outcome to solve the MSFP. As an application of Algorithm 1, we get the following algorithm for solving problem (2).

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Journal of Function Spaces Algorithm 2. Let x 0 be arbitrary. Given x n , update the next iteration via where ∑ s j=1 β j = 1, and fτ n g ⊂ ð0,∞Þ are properly chosen stepsize.

Corollary 8.
Assume that MSFP is consistent. If the stepsize is chosen so that then the sequence fx n g, generated by Algorithm 2, converges weakly to a solution of MSFP.
Proof. Let U = ∑ t i=1 α i P C i and T = ∑ s j=1 β j P Q j : By Lemma 4, we conclude that U and T are both −1-strictly pseudocontractive, that is, firmly nonexpansive. In this situation, we have k = 2/ð1 + kAk 2 Þ: By applying Theorem 6, we at once get the assertion as desired.

Applications
In this part, we first give an application of our theoretical results to the multiple-set split equality problem (MSEP), which is more general than the original split equality problem [22]. Example 1. The multiple-set split equality problem (MSEP) expects to find ðx 1 , x 2 Þ ∈ H 1 × H 2 such that where t and s are two positive integers, A 1 : H 1 ⟶ H 3 and A 2 : H 2 ⟶ H 3 are two bounded linear mappings, and U i : H 1 ⟶ H 1 , i = 1, 2, ⋯, t and T j : H 2 ⟶ H 2 , j = 1, 2, ⋯, s are two classes of nonlinear mappings. We next consider the MSFP under the following basic assumption.
(i) MSEP is consistent; that is, it admits at least one solution Under this situation, we propose a new method for solving problem (32).

Algorithm 3.
For an arbitrary initial guess ðx 0 , y 0 Þ, define ðx n , y n Þ recursively by where fτ n g ⊂ ð0,∞Þ is a sequence of positive numbers.
To proceed the convergence analysis, we consider the product space H ≔ H 1 × H 2 , in which the inner product and the norm are, respectively, defined by where x = ðx 1 , x 2 Þ, y = ðy 1 , y 2 Þ with x 1 , y 1 ∈ H 1 , x 2 , y 2 ∈ H 2 : Define a linear mapping A : H ⟶ H 3 by Let T be the the metric projection onto the set f0g ⊆ H, and define a nonlinear mapping U : H ⟶ H as where α i and β j are as above.
Lemma 11. Let the mapping U be defined as in (36). Then, Moreover, if conditions (B1)-(B3) are met, then U is k-strictly pseudocontractive with

Journal of Function Spaces
Proof. By Lemma 3, it is easy to verify the first assertion. To show the second assertion, fix any x, y ∈ H: By our hypothe- ∑ s j=1 β j T j is l-strictly pseudocontractive with It then follows that From (38), we obtain the result as desired.
Proof. Let z n = ðx n , y n Þ and let A, U, T be defined as above. Thus, problem (32) is equivalently changed into finding z ∈ H such that Moreover, Algorithm 3 can be rewritten as Note that by Lemma 10, U is κ-strictly pseudocontractive and T is −1-strictly pseudocontractive. Hence, by Theorem 5, we conclude that fz n g converges weakly to some z = ðx, yÞ such that Az ∈ 0 f g: ð45Þ By Lemma 11, it is readily seen that x ∈ T i FðU i Þ, y ∈ T j FðT j Þ and A 1 x = A 2 y.
We next give an application of our theoretical results to a problem derived from the real world. In statistics and machine learning, least absolute shrinkage and selection operator (LASSO for short) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the statistical model it produces. It was originally introduced by Tibshirani in [24] who coined out the term and provided further insights into the observed performance.
Subsequently, a number of LASSO variants have been created in order to remedy certain limitations of the original technique and to make the method more useful for particular problems. Among them, elastic net regularization adds an additional ridge regression-like penalty which improves performance when the number of predictors is larger than the sample size, allows the method to select strongly correlated variables together, and improves overall prediction accuracy. More specifically, the LASSO is a regularized regression method with the L 1 penalty, while the elastic net is a regularized regression method that linearly combines the L 1 and L 2 penalties of the LASSO and ridge methods. Here, the L 1 penalty is defined as kxk 1 = ∑ n i=1 jx i j, and the L 2 penalty is defined as kxk 2 = ð∑ n i=1 jx i j 2 Þ 1/2 .
Example 2 (see [25]). The elastic net requires to solve the problem min x∈ℝ n where A ∈ ℝ m×n , y 1 , y 2 ∈ ℝ m , and t 1 , t 2 > 0 are given parameters. This problem is a specific SCFP with T 1 x = y 1 , T 2 x = y 2 , ∀x ∈ ℝ m and where ηðyÞ ∈ ∂ðkyk 1 Þ and U 2 y = y, y k k 2 2 ≤ t 2 , y − y k k 2 2 − t 2 4 y k k 2 , y k k 2 2 > t 2 : Journal of Function Spaces Algorithm 4. Let x 0 be arbitrary. Given x n , update the next iteration via where fα i g 2 i=1 ⊂ ð0, 1Þ with ∑ t i=1 α i = 1 and τ is a properly chosen stepsize.
It is clear that the above mappings are, respectively, firmly nonexpansive and firmly quasi-nonexpansive, which implies that they are, respectively, −1-strictly pseudocontractive and −1-demicontractive mappings. As an application of Theorem 6, we can deduce that the sequence fx n g generated by Algorithm 4 converges to a solution to problem (46) provided that the stepsize is chosen so that

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.