Abstract
We introduce an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups. Moreover, numerical results demonstrate the performance and convergence of our result, which may be viewed as a refinement and improvement of the previously known results announced by many other researchers.
1. Introduction
Recently, Moudafi [1] proposed the following split monotone variational inclusion problem (SMVIP): find a point such that where and are two real Hilbert spaces with inner product and induced norm and and are multivalued maximal monotone mappings.
Moudafi [1] shows that SMVIP (1) includes, as special cases, the split variational inequality problem, the split common fixed point problem, split zero problem, and split feasibility problem [1–7] which have already been studied and used in practice as a model in intensity-modulated radiation therapy treatment planning (see [5, 6]). This formalism is also at the core of modeling of many inverse problems arising for phase retrieval and other real-world problems, for instance, in sensor networks in computerized tomography and data compression [8, 9].
If and , then SMVIP (1) can reduce to the following split variational inclusion problem (SVIP): find a point such that
We know that (2) is the variational inclusion problem and denote its solution set by SOLVIP(). SVIP ((2)-(3)) contains a pair of variational inclusion problems which need to be solved so that the image under a given bounded linear operator , the solution of SVIP (2) in , is the solution of another SVIP (3) in another space ; we denote the solution set of SVIP (3) by SOLVIP(). The solution set of SVIP ((2)-(3)) is denoted by and .
Many works were devoted to the split variational inclusion problem ((2)-(3)). In 2012, Byrne et al. [4] proposed the weak and strong convergence of the following iterative method for SVIP ((2)-(3)): for , compute iterative sequence generated by the following scheme:where is the adjoint of , is the spectral radius of the operator , and and .
In 2014, Kazmi and Rizvi [10] considered the strong convergence of the following iterative method: where , is the adjoint of , is the spectral radius of the operator , and . They proved the sequence generated by (5) strongly converges to the fixed point of nonexpansive mapping and the solution set of SVIP ((2)-(3)).
In 2015, Sitthithakerngkiet et al. [11] proposed the hybrid steepest descent method:where is a sequence of nonexpansive mappings, is a strongly positive bounded linear operator, , is the adjoint of , is the spectral radius of the operator , and and . They revealed that the sequence converges strongly to a point , where is a unique solution of the variational inequalities:
Note that, in algorithms (4), (5), and (6) mentioned above, the determination of the stepsize depends on the operator (matrix) norms (or the largest eigenvalues of ). This means that, in order to implement algorithms (4), (5), and (6), one has first to compute (or, at least, estimate) operator norms of , which is not an easy work in practice.
To overcome this difficulty, López et al. [12] and Zhao and Yang [13] presented useful method for choosing the stepsizes which do not need prior knowledge of the operator norms for solving the split feasibility problems and multiple-set split feasibility problems, respectively.
Motivated by the above results, we introduce a new choice of the stepsize sequence which depends onwhere . The advantage of our choice (8) of the stepsizes lies in the fact that no prior information about the operator norms of is required, and still convergence is guaranteed.
Following the work of Moudafi [1], Kazmi and Rizvi [10], and Byrne et al. [4], we introduce and study an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces. We also prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups. Numerical results are proposed to show that our algorithm is more suitable for SVIP ((2)-(3)) than the proposed algorithms (4) and (6).
2. Preliminaries
Throughout this paper, we denote to be a nonempty closed convex subset of . Let be a mapping. A point is said to be a fixed point of provided . We use to denote the fixed point set of . We write to indicate that the sequence converges weakly to , and implies that converges strongly to . We use standing for the weak -limit set of . For any , there exists a unique nearest point in , denoted by , such that
Before proceeding further, we need to introduce a few concepts.
A mapping is called contraction, if there exists a constant such that
If , then is called nonexpansive.
A mapping is said to be firmly nonexpansive, if
One-parameter family mapping from into itself is said to be a nonexpansive semigroup if it satisfies the following conditions:(i), for all .(ii), for all .(iii)For each , the mapping is continuous.(iv), for all and .
We denote by the set of all common fixed points of ; that is, . It is well known that is closed and convex [14].
Now, we give an example of a nonexpansive semigroup.
Example 1. Let and , where . Then, is said to be a nonexpansive semigroup. In fact,(i), for all ,(ii), for all ,(iii)for each , the mapping is continuous,(iv), for all and .
A set-valued mapping is called monotone if, for all , and imply .
A monotone mapping is called maximal if the graph of is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping is maximal if and only if for , for every implies .
Let be a multivalued maximal monotone mapping. Then, the resolvent mapping , associated with , is defined by for some , where stands for identity operator on . We note that, for all , the resolvent operator is single-valued, nonexpansive, and firmly nonexpansive.
The following principles play an important role in our argument.
A mapping is called demiclosed at the origin if for each sequence which weakly converges to , and the sequence strongly converges to 0, then .
is said to be semicompact, if, for any bounded sequence , ; then, there exists a subsequence such that converges strongly to some point
To establish our results, we need the following technical lemmas.
Lemma 2 (see [15]). If , then(a);(b)for any ;(c)for with ,
Lemma 3 (see [1]). SVIP ((2)-(3)) is equivalent to finding such that ,
Lemma 4 (see [16]). Let be a nonempty bounded closed and convex subset of a real Hilbert space . Let from be a nonexpansive semigroup on C; then, for all ,
Lemma 5 (see [16]). Let be a nonempty bounded closed and convex subset of a real Hilbert space , let be a sequence, and let from be a nonexpansive semigroup on ; if the following conditions are satisfied,(i),(ii),then, .
Lemma 6 (see [17]). Let be a Hilbert space and let be a sequence in such that there exists a nonempty set satisfying the following:(i)For every , exists.(ii)Each weak cluster point of the sequence is in W.Then, there exists such that weakly converges to .
Lemma 7. Let from into itself be a nonexpansive semigroup; then, for , , and ,
Proof. For , , and , it follows from the definition of nonexpansive semigroup that which implies thatFurthermore, Then, Thus, This completes the proof of Lemma 7.
3. Main Results
In this section, we first describe our algorithm and then reveal the convergence analysis of the algorithm.
Now, we propose our algorithm.
Algorithm 8. Let be arbitrary. Assume that has been constructed and ; then, calculate via the rule th iterate via the following formula: where the stepsize is chosen by (8). If , then is the solution of ((2)-(3)) and the iterative process stops. Otherwise, we set and go to (23).
Remark 9. Notice that in (8) the choice of the stepsize is independent of the norms .
Remark 10. The stepsize is bounded. Indeed, it follows from the condition on thatOn the other hand, sincewe obtainThus, and is bounded.
Next, we will discuss the convergence analysis of algorithm (23) for approximating a common solution of SVIP ((2)-(3)) and fixed point problem for nonexpansive semigroups.
Theorem 11. Let and be two real Hilbert spaces and let be a bounded linear operator. Assume that and are maximal monotone mappings and be a one-parameter nonexpansive semigroup on such that . Let the sequences be generated by (23). If and , then converges strongly to .
Proof. Taking , we have , , and . From (23) and Lemma 3, one has Notice that It follows from (28), (27), and (8) that Furthermore, (23), (29), and Lemma 2 lead toHence, is bounded and we also obtain that is bounded.
On the other hand, from Lemma 7, we obtain which means that From (30), we have that is nonincreasing, and then is convergent. Obviously,This together with (32) and the condition on deduce that Observe that It follows from (34) and Lemma 4 that From (29) and (30), we deduce It yields that Thus, Sinceit is easy to show that Consequently, we obtainFurthermore,From (23), we have Hence, which implies that Moreover, from (30), we have Therefore, (43) yields that We compute Equation (34) implies that Consequently,Furthermore, Since and are bounded, we consider a weak cluster point of . Without loss of generality, we may assume that subsequence of converges weakly to . From (48), we have of , which converges weakly to . Furthermore, can be rewritten asTaking limit in (53) and taking into account (43) and (48), together with the fact that the graph of a maximal monotone operator is weakly strongly closed, we have ; that is, . Furthermore, from the asymptotical behavior of and , we deduce that weakly converges to . Applying (43) and the fact that the resolvent is nonexpansive, together with Lemma 3, we have ; that is, . Thus, .
Next, we will prove that . For the sake of contradiction, suppose that . It follows from Opial condition and (36) that This is a contradiction, which shows that . Thus, . Furthermore, from Lemma 6, we deduce that , , and . Due to (36), we obtain that is semicompact, and then and converge strongly to . This completes the proof of Theorem 11.
Theorem 12. Let and be two real Hilbert spaces and let be a bounded linear operator. Assume that and are maximal monotone mappings and let be a nonexpansive mapping such that . Let the sequences be generated by (23). If , then, converges strongly to .
Proof. Clearly, Theorem 12 is valid for a nonexpansive mapping. Therefore, the desired conclusion follows immediately from Theorem 11. This completes the proof.
4. Numerical Examples
We now pay our attention to show a numerical example to demonstrate the performance and convergence of our result. In the experiment, the stopping criterion is . In [4, 11], the stopping criterion is . ST denotes the initial point, IT denotes the iterative number, and SOL denotes a solution of the test problem.
Example 13 (see [11]). Let , and let two operators of matrix multiplication and be defined by and , where and Observe that and are positive linear operators; then, they are maximal monotone. So, we can define the resolvent mappings and on associated with and , where . Let be a nonsingular matrix operator in which elements are random and let be an adjoint of . Assume that , , , and , where , for .
Example 14. Let two operators of matrix multiplication and be defined by and , which are generated by the codes as Observe that and are positive linear operators; then, they are maximal monotone. So, we can define the resolvent mappings on and on associated with and , where . Let be a matrix operator which is generated by the codes asWe choose the initial point .
Figure 1 reveals that the more the iteration steps are, the more slowly the sequence converges to . From Table 1, we see that our method is as completive as the methods of [4, 11], and the sequence is more closed to the same point . From Table 2, as the size increases, our method performs better than the algorithms of [4, 11]. Furthermore, it indicates that our method is promising for solving large scale problems. Thus, we could observe that our algorithm is more suitable for SVIP ((2)-(3)) than the proposed algorithms of [4, 11].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank Professor Yiju Wang for his careful reading of the paper. This project is supported by the Natural Science Foundation of China (Grant nos. 11401438, 11171180, 11171193, and 11126233), the Natural Science Foundation of Shandong Province (Grant no. ZR2013FL032), and Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J14LI52).