Abstract
In this paper, we study a kind of extended multiple-set split equality common fixed-point problem with Lipschitz quasi-pseudocontractive operators, which is an extension of multiple-set split equality common fixed-point problem with quasi-nonexpansive operator. We propose two mixed simultaneous iterative algorithms, in which the selecting of the stepsize does not need any priori information about the operator norms. Furthermore, we prove that the sequences generated by the mixed simultaneous iterative algorithms converge weakly to the solution of this problem. Some numerical results are shown to illustrate the feasibility and efficiency of the proposed algorithms.
1. Introduction
Throughout this paper, we always assume that , , and are real Hilbert spaces with inner product and induced norm . Let and be nonempty closed and convex subsets of and , respectively. is a bounded linear operators. Let denote the identity operator. The split feasibility problem (SFP) originally introduced by Censor and Elfving [1] which is to findIt can be used in various disciplines such as image restoration, radiation therapy treatment planning [2, 3]. These applications are in finite-dimensional Hilbert spaces [4–7]. It also can be found in an infinite-dimensional real Hilbert space [8–10] and references therein.
In [11], Censor et al. introduce the multiple-set split feasibility problem (MSFP) which finds application in intensity-modulated radiation therapy. It is formulated as finding a point satisfying the following property:where are integers and and are nonempty closed convex subsets of and , respectively. When , then MSFP (2) is known as SFP (1). About the corresponding results in finite-dimensional spaces, we can see [12–14] and references therein.
Since every closed convex subset of a Hilbert space is the fixed-point set of its associating projection, problems (1) and (2) are all special cases of the so-called multiple-set split common fixed-point problem (MSCFP) which is to findwhere are integers, and are nonlinear operators. In particular, if , then MSCFP (3) reduces to the split common fixed-point problem (SCFP) [15–17] which is to findwhere and are nonlinear operators.
Recently, Moudafi [18] introduced the split equality common fixed-point problem (SECFP) which is to findwhere and are two bounded linear operators. It allows asymmetric and partial relations between the variables and . The interest is to cover many situations, for instance, applications in decomposition methods for , in game theory, and in intensity-modulated radiation therapy (IMRT). At the same time, Moudafi [18] introduced the following iterative methodfor firmly quasi-nonexpansive mappings and . The weak convergence property was proved where was dependent in the spectral radiuses of and Furthermore, Che and Li [19] proposed the following iterative algorithm:for quasi-nonexpansive mappings and and the weak convergence of scheme (7) can also be established. Chang, Wang, and Qin [20] modified the iterative scheme (7) and provided the following framework:for quasi-pseudocontractive mappings and with weak convergence property.
In [21], Zhao et al. study the multiple-set split equality common fixed-point problem (MSECFP) which is to findwhere are integers and and are nonlinear operators. In particular, if , then MSECFP (9) reduces to the split equality common fixed-point problem (SECFP) (5). They proposed the following iterative algorithm:for quasi-nonexpansive mappings, where , ; such that for every and . . Weak convergence results were obtained under certain assumption conditions.
Recently, Li et al. [22] studied the extended split equality common fixed-point problem (ESECFP) which is to findwhere is a nonempty set. For L-Lipschitz and quasi-pseudocontractive mappings and , as well as a kind of special , they proposed the following iterative algorithm:where , , , and They also proved the weak convergence results under mild conditions.
Inspired by the above results, in this paper, we study the following extended multiple-set split equality common fixed-point problem (EMSECFP) which is to findwhere are integers and and are Lipschitzian mappings with and quasi-pseudocontractive mappings with nonempty fixed-point sets which is more general than the class of quasi-nonexpansive mappings. When , the EMSECFP (13) is MSECFP (9). When , the EMSECFP (13) is ESECFP (11). Therefore, EMSECFP (13) is the extension of MSECFP (9) and ESECFP (11). In this paper, we propose two mixed simultaneous iterative algorithms for solving EMSECFP (13), which avoids using the projection and the selecting of stepsize which do not depend on the operator norms. Furthermore, we prove that the sequences generated by the algorithms weakly converge to a solution of the EMSECFP (13). Numerical examples show the feasibility and efficiency of these algorithms.
2. Preliminaries
In this paper, we use and to indicate that the sequence converges weakly to and converges strongly to , respectively. We write to stand for the weak limit set of and use to stand for the solution set of the EMSECFP (13).
Definition 1. A mapping is called
(i) Nonexpansive if for all
(ii) Quasi-nonexpansive, if and for all and .
(iii) Quasi-pseudocontractive, if and for all and
Definition 2. An operator is called demiclosed at the origin, for any sequence which weakly converges to , and if the sequence strongly converges to , then
In the proof of our results, we need the following lemmas.
Lemma 3 (see [23]). Let be a real Hilbert space, then the following conclusions hold:
Lemma 4 (see [20]). Let be a real Hilbert space and be a L-Lipschitzian mapping with . DenoteIf , then the following conclusions hold:
(i) .
(ii) If is demiclosed at 0, then is also demiclosed at 0.
(iii) In addition, if is quasi-pseudocontractive, then the mapping is quasi-nonexpansive; that is,
Lemma 5 (see [21]). Let be a real Hilbert space. Thenfor any and for with and
Lemma 6 (see [16]). Let be a real Hilbert space and be a quasi-nonexpansive mapping. Set for Then the following properties are reached for all
(i) and ;
(ii) ;
(iii)
3. Main Results
In this section, we give the following assumption.
Assumption 7. Assume that
(i) , , and are real Hilbert spaces and is -dimensional Euclidean space. and are two bounded linear operators; and are their adjoint operators, respectively.
(ii) , where is convex function which is subdifferentiable and its subdifferentials are bounded on bounded sets.
(iii) and are all L-Lipschitz, quasi-pseudocontractive with and and and are demiclosed at origin, where
(iv) , , and , , where .
Theorem 8. Let Assumption 7 be true and be small enough. Assume that , ; , such that for every and Suppose that , are arbitrary and , , , , and are generated by the following iteration scheme:whereandAssume , , , and for small enough . Then , , and , where
Proof. It is obvious that for any . Let , that is, and . By Lemma 3, we haveSimilarly,By (23) and (24), we haveCombining (22) and (25), we haveSince , , are quasi-pseudocontractive, from Lemma 4, , , are quasi-nonexpansive. Furthermore, from Lemma 5, we haveSimilarly, since are quasi-pseudocontractive, from Lemma 4, , are quasi-nonexpansive. By Lemma 6 (ii), we haveCombining (27) and (28) with (26), we haveSince , one hasAs a result, we have from (29) and (30) thatWe see that the sequence is decreasing and lower bounded by 0. Consequently it converges to some finite limit. Hence the sequences and are bounded.
According to the assumption of and (29), we haveFurthermore, by the conditions on and , we obtainFrom Lemma 5, (19) and the conference of (27) for , we haveandwhich combining (19) implies thatandFrom (33), (34), and (36), we can conclude thatAccording to (36), (38), andwe havewhich infers that is asymptotically regular. From (33) and (37), we haveAs a result,which implies that is asymptotically regular.
It follows fromthatFurthermore,Taking , from and , we have Let an index be fixed. Noting that the pool of indexes is finite, from (45) and the boundedness of , there exists a subsequence such that as and for all It is easy to see from (33) thatAccording to the demiclosednesses of and at 0, it yields from (33), (34) and (46) that and for and . Hence, and . From Lemma 4, we have and
Furthermore, and the weakly lower semicontinuity of the norm imply thatthat is,
From and , we haveAs follows, we prove .
According to the boundedness of , there exists a constant , such that
Noting that , we obtainAs a result,From the lower semicontinuity of , (49), and (50), we haveThus .
Next, we prove the uniqueness of the weak cluster point of Assume that be other weak cluster point of , then . Define , thenWithout loss of generality, we suppose that . By passing to the limit in relation (52), we haveReserving the role of and , we also haveBy adding the two last equalities, we obtain and , which shows that whole sequence weakly converges to the unique solution of (13). The proof is completed.
Similar to the proof of Theorem 8, we can easily prove the following theorem.
Theorem 9. Let Assumption 7 be true and be small enough. Assume that . and such that for every Suppose that are arbitrary and , , , , and are generated by the following iteration scheme:whereandAssume , for small enough and Then , , and , where
4. Numerical Examples
In this section, we give an example to show some insight into the behavior of the algorithms presented in this paper. The algorithms presented in Theorems 8 and 9 are denoted as Algorithms 1 and 2 separately. The whole codes are written in Matlab 7.0. All the numerical results are carried out on a personal Lenovo Thinkpad computer with Intel(R) Core(TM) i7-6500U CPU 2.50GHz and RAM 8.00GB.
In this section, we need the following lemmas.
Lemma 10 (see [24, 25]). For every , let , . Then for any , we have
Lemma 11 (see [26]). Suppose is a finite convex function, then it is subdifferentiable everywhere and its subdifferentials are uniformly bounded on any bounded subset of .
Example 12. Let , , and . Assume that , , , , , andFind such that .
It is easy to see that and are all Lipschitz quasi-pseudocontractive operators, but not quasi-nonexpansive operators. Therefore, the Example 12 cannot be calculated by the algorithms in [21].
In the experiments, we take , , , , , , and And the stopping criterion is
Tables 1 and 2 show the number of iterations, the cpu time, and the final iteration points for different when the initial point is in Algorithms 1 and 2. We denote by Iter., Sec. and Fina. the number of iterations, the cpu time in seconds, and the final iteration points, respectively.
Figures 1 and 2 present the behaviors of for and when the initial point is , in Algorithms 1 and 2.
It is easy to see that the presentation reveals that .
Furthermore, for testing the stationary property of iterative numbers, we carry out 500 experiments for different initial points which are presented randomly, such asFigures 3 and 4 illustrate the behaviors of iterative numbers for and , which reveals the stationary property of iterative numbers of the Algorithms 1 and 2.
Especially, Figure 5 illustrates the behaviors of and for in Algorithm 1. Figure 6 shows the behaviors of iterative numbers for different in Algorithm 1.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This project is supported by the Natural Science Foundation of China (Grants nos. 11401438 and 11571120), Shandong Provincial Natural Science Foundation (Grant no. ZR2017LA002), and the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J14LI52).