Solving the multiple-set split equality common fixed-point problem of firmly quasi-nonexpansive operators.

In this paper, we propose parallel and cyclic iterative algorithms for solving the multiple-set split equality common fixed-point problem of firmly quasi-nonexpansive operators. We also combine the process of cyclic and parallel iterative methods and propose two mixed iterative algorithms. Our several algorithms do not need any prior information about the operator norms. Under mild assumptions, we prove weak convergence of the proposed iterative sequences in Hilbert spaces. As applications, we obtain several iterative algorithms to solve the multiple-set split equality problem.

Introduction

Let , , and be real Hilbert spaces. The multiple-set split equality common fixed-point problem (MSECFP) is to find with the property where are integers, and are nonlinear operators, and are two bounded linear operators. If and are projection operators, then the MSECFP is reduced to the multiple-set split equality problem (MSEP): where and are nonempty closed convex subsets of real Hilbert spaces and , respectively. When , the MSECFP and MSEP become the split equality common fixed-point problem (SECFP) and split equality problem (SEP), respectively, which were first put forward by Moudafi [1]. These allow asymmetric and partial relations between the variables x and y. They are applied in many situations, for instance, in game theory and in intensity-modulated radiation therapy (see [2] and [3]).

If and , then MSECFP (1.1) reduces to the multiple-set split common fixed-point problem (MSCFP): and MSEP (1.2) reduces to the multiple-set split feasibility problem (MSFP): They play significant roles in dealing with problems in image restoration, signal processing, and intensity-modulated radiation therapy [3-6]. With , MSCFP (1.3) is known as the split common fixed-point problem (SCFP) and MSFP (1.4) is known as the split feasibility problem (SFP). Many iterative algorithms have been developed to solve the MSCFP and the MSFP. See, for example, [7-14] and the references therein.

Note that the SFP can be formulated as a fixed-point equation where and are the (orthogonal) projections onto C and Q, respectively, is any positive constant, and denotes the adjoint of A. This implies that we can use fixed-point algorithms (see [15-21]) to solve SFP. Byrne [22] proposed the so-called CQ algorithm which generates a sequence : where with λ being the spectral radius of the operator . The CQ algorithm is efficient when and are easily calculated. However, if C and Q are complex sets, for example, the fixed-point sets, the efficiency of the CQ algorithm will be affected because the projections onto such convex sets are generally hard to be accurately calculated. To solve the SCFP of nonexpansive operators, Censor and Segal [23] proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm: where with λ being the largest eigenvalue of the matrix .

For solving the constrained MSFP, Censor et al. [6] introduced the following proximity function: where , , and . Then and they proposed the following projection method: where Ω is the constrained set, , and L is the Lipschitz constant of ∇g.

For solving MSCFP (1.3) of directed operators, Censor and Segal [23] introduced a parallel iterative algorithm as follows: where , are nonnegative constants, with and λ being the largest eigenvalue of . They obtained the convergence of iterative algorithm (1.6).

Wang and Xu [24] proposed the following cyclic iterative algorithm for MSCFP (1.3) of directed operators: where , , and . They proved the weak convergence of the sequence generated by (1.7).

For solving MSCFP (1.3), Tang and Liu [25] introduced inner parallel and outer cyclic iterative algorithm: and outer parallel and inner cyclic iterative algorithm: for directed operators and , where , , , , with and . They obtained the weak convergence of the above two mixed iterative sequences to solve MSCFP (1.3) of directed operators.

The SEP proposed by Moudafi [1] is to which can be written as the following minimization problem: Assume that the solution set of the SEP is nonempty. By the optimality conditions, Moudafi [1] obtained the following fixed-point formulation: solves the SEP if and only if where γ, . Therefore, for solving the SECP of firmly quasi-nonexpansive operators, Moudafi [1] introduced the following alternating algorithm: where a nondecreasing sequence , , stand for the spectral radius of and , respectively. In [26], Moudafi and Al-Shemas introduced the following simultaneous iterative method: where , , stand for the spectral radius of and , respectively. Recently, many iterative algorithms have been developed to solve the SEP, SECFP, and MSEP. See, for example, [27-34] and the references therein. Note that in algorithms (1.17) and (1.18), the determination of the step size depends on the operator (matrix) norms and (or the largest eigenvalues of and ). To overcome this shortage, we introduce parallel and cyclic iterative algorithms with self-adaptive step size to solve MSECFP (1.1) governed by firmly quasi-nonexpansive operators. We also propose two mixed iterative algorithms which combine the process of cyclic and parallel iterative methods and do not need the norms of bounded linear operators. As applications, we obtain several iterative algorithms to solve MSEP (1.2).

Preliminaries

Concepts

Throughout this paper, we always assume that H is a real Hilbert space with the inner product and the norm . Let I denote the identity operator on H. Denote the fixed-point set of an operator T by . We denote by → the strong convergence and by ⇀ the weak convergence. We use to stand for the weak ω-limit set of and use Γ to stand for the solution set of MSECFP (1.1).

Definition 2.1

An operator is said to be

nonexpansive if for all ;

firmly nonexpansive if for all ;

firmly quasi-nonexpansive (i.e., directed operator) if and or equivalently for all and .

Definition 2.2

An operator is called demiclosed at the origin if, for any sequence which weakly converges to x, and if the sequence strongly converges to 0, then .

Recall that the metric (nearest point) projection from H onto a nonempty closed convex subset C of H, denoted by , is defined as follows: for each , It is well known that is characterized by the inequality

Remark 2.1

It is easily seen that a firmly nonexpansive operator is nonexpansive. Firmly quasi-nonexpansive operators contain firmly nonexpansive operators with a nonempty fixed-point set. A projection operator is firmly nonexpansive.

Mathematical model

Recall that the SCFP is to find with the property and the SFP is to find with the property: where is a bounded linear operator, and are nonlinear operators, C and Q are closed convex sets of Hilbert spaces and , respectively.

We can formulate SFP (2.2) as an optimization. First, we consider the following proximity function: Then the proximity function is convex and differentiable with gradient where denotes the adjoint of A. Assume that the solution set of the SFP is nonempty, then is a solution of the SFP if and only if , i.e., which is equivalent to for all . For solving the SCFP of directed operators (i.e., firmly quasi-nonexpansive operators), Wang [35] proposed the following algorithm: where the variable size step was chosen: This algorithm can be obtained by the fixed-point Eq. (2.3), where projection operators and are replaced by U and T.

Setting MSEP (1.2) can be written as the following minimization problem: where , , , and . Assume that the solution set of the MSEP is nonempty, by the optimality conditions solves the MSEP if and only if which is equivalent to for γ, . These motivate us to introduce several iterative algorithms with self-adaptive step size for solving MSECFP (1.1) governed by firmly quasi-nonexpansive mappings and MSEP (1.2).

The well-known lemmas

The following lemmas will be helpful for our main results in the next section.

Lemma 2.1

Let H be a real Hilbert space. Then

Lemma 2.2

([36])

Let H be a real Hilbert space. Then, for all and ,

Lemma 2.3

([37])

Let H be a real Hilbert space. Then for any and for , with and .

Lemma 2.4

([38])

Let E be a uniformly convex Banach space, K be a nonempty closed convex subset of E, and be a nonexpansive mapping. Then is demi-closed at origin.

Parallel and cyclic iterative algorithms

In this section, we introduce parallel and cyclic iterative algorithms and prove the weak convergence for solving MSECFP (1.1) of firmly quasi-nonexpansive operators. In our algorithms, the selection of the step size does not need any prior information of the operator norms and .

In what follows, we adopt the following assumptions:

The problem is consistent, namely its solution set Γ is nonempty;

Both and are firmly quasi-nonexpansive operators, and both and are demiclosed at origin (, ).

The sequences , such that and for every , , .

Algorithm 3.1

Let be arbitrary. For , let where the step size is chosen as for small enough , otherwise, (τ being any value in ), the set of indexes .

Remark 3.1

Note that in (3.2) the choice of the step size is independent of the norms and . The value of τ does not influence the considered algorithm, it was introduced just for the sake of clarity.

Lemma 3.1

defined by (3.2) is well defined.

Proof

Taking , i.e., , , and , we have and By adding the two above equalities and by taking into account the fact that , we obtain Consequently, for , that is, , we have or . This leads to the fact that is well defined. □

Theorem 3.1

Assume that and . Then the sequence generated by Algorithm 3.1 weakly converges to a solution of MSECFP (1.1). Moreover, , , and as .

From the condition on , we have is bounded. It follows from Algorithm 3.1 and that Taking , i.e., , , and , we have Similarly, we have By adding the two inequalities (3.5)–(3.6) and taking into account the fact that , we obtain From Algorithm 3.1 we also have By Lemma 2.3 we get and Setting and using (3.7), (3.9)–(3.10), (3.8) can be written as We see that the sequence is decreasing and lower bounded by 0; consequently, it converges to some finite limit which is denoted by . So the sequences and are bounded.

By the conditions on , () and (), from (3.11) we obtain, for all i () and j (), and It follows from (3.3) and (3.13) that Since we get which infers that is asymptotically regular. Similarly, we also have that is asymptotically regular, namely .

Take , i.e., there exists a subsequence of such that as . Combined with the demiclosedness of and at 0, it follows from (3.12) that and for and . So, and . On the other hand, and weakly lower semicontinuity of the norm imply that hence . So .

Next, we will show the uniqueness of the weak cluster point . Indeed, let be another weak cluster point of , then . From the definition of , we have Without loss of generality, we may assume that and . By passing to the limit in relation (3.17), we obtain Reversing the role of and , we also have By adding the two last equalities, we obtain and , which implies that weakly converges to the solution of (1.1). This completes the proof. □

Next, we propose the cyclic iterative algorithm for solving MSECFP (1.1) of firmly quasi-nonexpansive operators.

Algorithm 3.2

Let be arbitrary. For , let where the step size is chosen as in Algorithm 3.1.

Theorem 3.2

The sequence generated by Algorithm 3.2 weakly converges to a solution of MSECFP (1.1). Moreover, , , and as .

Let , we have and By adding the two inequalities (3.19)–(3.20) and taking into account the fact that , we obtain Similar to (3.8), we have We also have and Setting and using (3.21), (3.23)–(3.24), (3.22) can be written as Similar to the proof of Theorem 3.1, we have and Since we get which infers that is asymptotically regular. Similarly, we also have that is asymptotically regular, namely .

Take , i.e., there exists a subsequence of such that as . Noting that the pool of indexes is finite and is asymptotically regular, for any , we can choose a subsequence such that as and for all l. It turns out that By the same reason, for any , we can choose a subsequence such that as and for all m. So, Combined with the demiclosedness of and at 0, it follows from (3.30) and (3.31) that and for and . So, and . Similar to the proof of Theorem 3.1, we can complete the proof. □

Now, we give applications of Theorem 3.1 and Theorem 3.2 to solve MSEP (1.2). Assume that the solution set S of MSEP (1.2) is nonempty. Since the orthogonal projection operator is firmly nonexpansive, by Lemma 2.4 we have the following results for solving MSEP (1.2).

Corollary 3.1

For any given , define a sequence by the following procedure: where the step size is chosen as in Algorithm 3.1. If () and (), then the sequence weakly converges to a solution of MSEP (1.2). Moreover, , , and as .

Corollary 3.2

For any given , define a sequence by the following procedure: where the step size is chosen as in Algorithm 3.1. Then the sequence weakly converges to a solution of MSEP (1.2). Moreover, , , and as .

Mixed cyclic and parallel iterative algorithms

Now, for solving MSECFP (1.1) of firmly quasi-nonexpansive operators, we introduce two mixed iterative algorithms which combine the process of cyclic and simultaneous iterative methods. In our algorithms, the selection of the step size does not need any prior information of the operator norms and , and the weak convergence is proved. We go on making use of assumptions (A1)–(A3).

Algorithm 4.1

Let be arbitrary. For , let where the step size is chosen in the same way as in Algorithm 3.1.

Theorem 4.1

Assume that (). Then the sequence generated by Algorithm 4.1 weakly converges to a solution of MSECFP (1.1). Moreover, , , and as .

Let . We can get (3.5) and (3.20), so

It follows from Algorithm 4.1 that (3.8)–(3.9) and (3.24) are true. Setting , we have By the same reason as in Theorem 3.1, we obtain that, for all i , and So which infers that and are asymptotically regular.

Take , i.e., there exists a subsequence of such that as . Noting that the pool of indexes is finite and is asymptotically regular, for any , we can choose a subsequence such that as and for all l. It turns out that Combined with the demiclosedness of and at 0, it follows from (4.4) and (4.7) that and for and . So, and . Similar to the proof of Theorem 3.1, we can complete the proof. □

Next, we propose another mixed cyclic and parallel iterative algorithm for solving MSECFP (1.1) of firmly quasi-nonexpansive operators.

Algorithm 4.2

Let be arbitrary. For , let where the step size is chosen as in Algorithm 3.1.

Similar to the proof of Theorem 4.1, we can get the following result.

Theorem 4.2

Assume that (). Then the sequence generated by Algorithm 4.2 weakly converges to a solution of MSECFP (1.1) of firmly quasi-nonexpansive operators. Moreover, , and as .

Finally, we obtain two mixed iterative algorithms to solve MSEP (1.2). Assume that the solution set S of MSEP (1.2) is nonempty.

Corollary 4.1

For any given , define a sequence by the following procedure: where the step size is chosen as in Algorithm 3.1. If (), then the sequence weakly converges to a solution of MSEP (1.2). Moreover, , , and as .

Corollary 4.2

For any given , , define a sequence by the following procedure: where the step size is chosen as in Algorithm 3.1. If , then the sequence weakly converges to a solution of MSEP (1.2). Moreover, , , and as .

Results and discussion

To avoid computing the norms of the bounded linear operators, we introduce parallel and cyclic iterative algorithms with self-adaptive step size to solve MSECFP (1.1) governed by firmly quasi-nonexpansive operators. We also propose two mixed iterative algorithms and do not need the norms of bounded linear operators. As applications, we obtain several iterative algorithms to solve MSEP (1.2).

Conclusion

In this paper, we have considered MSECFP (1.1) of firmly quasi-nonexpansive operators. Inspired by the methods for solving SCFP (2.1) and MSCFP (1.3), we introduce parallel and cyclic iterative algorithms for solving MSECFP (1.1). We also present two mixed iterative algorithms which combine the process of parallel and cyclic iterative methods. In our several iterative algorithms, the step size is chosen in a self-adaptive way and the weak convergence is proved.