Abstract
We study a kind of split equality fixed point problem which is an extension of split equality problem. We propose a kind of simultaneous iterative algorithm with a way of selecting the step length which does not need any a priori information about the operator norms and prove that the sequences generated by the iterative method converge weakly to the solution of this problem. Some numerical results are shown to confirm the feasibility and efficiency of the proposed methods.
1. Introduction
Let and be nonempty closed and convex subsets of the real Hilbert spaces and , respectively. The split feasibility problem is to findIt can be used in various disciplines such as image restoration and radiation therapy treatment planning [1, 2]. These applications are in finite-dimensional Hilbert spaces [3, 4]. It also can be found in an infinite-dimensional real Hilbert space [5, 6].
Recently, Moudafi [7] introduced a new split equality feasibility problem. Let , , and be real Hilbert spaces. Let and be two bounded linear operators. The split equality feasibility problem is to findwhich 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 (for short, ).
Moudafi [8] introduced the simultaneous iterative method to solve the split equality feasibility problem. Furthermore, Moudafi studied the fixed point formulation to avoid using the projection. Assume and are the sets of fixed points of and , respectively, where and are nonlinear operators such that and . So the split equality fixed point problem is to findwhere and are firmly quasi-nonexpansive mappings. In order to find the solution of the split equality problem, Che and Li [9] proposed the following iterative algorithm:and the weak convergence of the scheme (4) can also be established. Furthermore, Chang et al. [10] modified the iterative scheme (4) and provided a unified framework for solving this problem without using the projection. The framework is as follows.They got the following conclusion that the sequence generated by the above modification converges weakly to a solution of problem (3). Furthermore, some authors [11–14] studied the problems (1)–(3) in Banach space. They proposed effective algorithms and proved their convergence under some conditions.
Recently, He and Sun [15] studied the problem of split convex feasibility and established a strongly convergent alternating algorithm. They proposed the following iterative algorithm to findvia the formulawhere , , , and In some conditions such as , they proved that the sequences , , and , where .
In this article, we study the following problem of extended split equality fixed point, which is to findWhen , problem (8) is problem (3). Therefore, problem (8) is the extension of the split equality fixed point problem. We propose the simultaneous iterative algorithm for solving this problem, which avoids using the projection and the step length sequences do not depend on the operator norms and . Furthermore, we prove the sequences generated by the algorithm weakly converge to a solution of the extended split equality fixed point problem. Numerical examples show the feasibility and efficiency of this algorithm.
2. Preliminaries
In this paper, we recall some concepts, definitions, and conclusions, which are prepared for proving our main results. We write to indicate that the sequence converges weakly to . implies that converges strongly to . We denote by a real Hilbert space with inner product and induced norm .
A mapping is called(i)quasi-nonexpansive, if ,(ii)quasi-pseudo-contractive, if ,
A mapping is said to be metric projection of onto if, for every point , there exists a unique nearest point in denoted by such that
It is well known that is a nonexpansive mapping and is characterized by the following properties:
In the proof of our results, we need the following lemmas.
Lemma 1 (see [16]). Let be a real Hilbert space; then the following conclusions hold.
Lemma 2 (see [16]). Let be a real Hilbert space and be a -Lipschitzian mapping with . DenoteIf , then the following conclusions hold.(i)(ii)If is demiclosed at , then is also demiclosed at .(iii)In addition, if is quasi-pseudo-contractive, then the mapping is quasi-nonexpansive; that is,
3. Main Results
In this section, we assume that(i) and are two -Lipschitzian and quasi-pseudo-contractive mappings with , , and .(ii), , and be real Hilbert spaces. and are two bounded linear operators; and are their adjoint operators, respectively. Let be nonempty closed convex set. We consider the extended split equality fixed point problem (8).
Theorem 3. Let , , and be real Hilbert spaces and be a closed convex level setwhere is convex function which is subdifferentiable on and its subdifferentials are bounded on bounded sets. and are two bounded linear operators with their adjoint operators and , respectively, is a parameter controlling step length, , and . Let , , , , and be sequences generated bywhere where and are demiclosed at . Assume . Then , , and , where .
Proof. It is obvious that for any . Let ; namely, , , and By Lemma 1, we haveBy (20), we haveSimilarly,By (21), (22), (17), and (19), we haveNotice thatThanks to we obtainLetWe obtainThis implies that is a nonincreasing sequence; hence exists. As a result, and are bounded sequences. Rewrite (23) asLetting and taking the limit in (29), we haveThen,which imply that and are asymptotically regular. Furthermore, we getSince and are bounded sequences, there exist weakly convergent subsequences, say such that ; also such that . The Opial property guarantees that the weakly subsequential limit of is unique. So we have , . Therefore , Since and are demiclosed at , and from Lemma 2, by (33), we have , , which imply that , Hence, Furthermore, since , by using the weakly lower semicontinuity of squared norm, we have that is, By (31), we have Now, we prove . We know that is bounded. There exists , such that , where is a constant. Note that , and we haveHence,By the lower semicontinuity of , (38), and (31), we obtainThus . The proof is completed.
4. Consequent Results
In this section, we give some corollaries, which are easily obtained from Theorem 3.
If , we have the following corollary.
Corollary 4. Let , , and be real Hilbert spaces and be a closed convex level set where is convex function which is subdifferentiable on and its subdifferentials are bounded on bounded sets. and are two bounded linear operators with their adjoint operators and , respectively, is a parameter controlling step length, , and . Let , , , , and be sequences generated bywhere where is demiclosed at . Assume . Then and , where .
If , we have the following corollary.
Corollary 5. Let and be real Hilbert spaces and be a closed convex level set where is convex function which is subdifferentiable on and its subdifferentials are bounded on bounded sets. is a bounded linear operator with its adjoint operator , is a parameter controlling step length, , and . Let , , , , and be sequences generated bywhere where and are demiclosed at . Assume . Then , , and , where .
If and , we have the following corollary.
Corollary 6. Let and be real Hilbert spaces and be a closed convex level set where is convex function which is subdifferentiable on and its subdifferentials are bounded on bounded sets. is a bounded linear operator with its adjoint operator , and is a parameter controlling step length, , and . Let , , , , and be sequences generated bywhere where is demiclosed at . Assume . Then , , and , where .
If is an identity operator, we have the following corollary.
Corollary 7. Let and be real Hilbert spaces and be a closed convex level set where is convex function which is subdifferentiable on and its subdifferentials are bounded on bounded sets. is a parameter controlling step length, , and Let , , , , and be sequences generated bywhere , . Assume . Then , , and , where .
5. Numerical Examples
In this section, we give an example to show some insight into the behavior of the algorithm presented in this paper. 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 2.50 GHz and RAM 8.00 GB.
Let , . and are as follows.
Let , where For convenience, we take , , and We choose as the stopping criterion.
Figure 1 presents the behaviors of in different initial points such asIt is easy to see that the presentation reveals that . Table 1 shows the number of iterations and the CPU time for the above four initial points. We denote Iter. and Sec. as the number of iterations and the CPU time in seconds, respectively.
Furthermore, for testing the stationary property of iterative numbers, we carry out 500 experiments for different initial points which are presented randomly, such asseparately. In these cases, we take , and choose as the stopping criterion. Figure 2 illustrates the behaviors of iterative numbers for different initial points, which reveals the stationary property of iterative numbers of the algorithm.
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 (Grant nos. 11401438 and 11571120) and Shandong Provincial Natural Science Foundation (Grant no. ZR2013FL032) and the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J14LI52).