- Open Access
A simultaneous iterative method for split equality problems of two finitefamilies of strictly pseudononspreading mappings without prior knowledge ofoperator norms
© Che and Li; licensee Springer. 2015
- Received: 1 July 2014
- Accepted: 8 October 2014
- Published: 5 January 2015
In this article, we first introduce the concept of T-mapping of a finitefamily of strictly pseudononspreading mapping , and we show that the fixed point set of theT-mapping is the set of common fixed points of and T is a quasi-nonexpansive mapping.Based on the concept of a T-mapping, we propose a simultaneousiterative algorithm to solve the split equality problem with a way of selectingthe stepsizes which does not need any prior information about the operatornorms. The sequences generated by the algorithm weakly converge to a solution ofthe split equality problem of two finite families of strictly pseudononspreadingmappings. Furthermore, we apply our iterative algorithms to some convex andnonlinear problems.
MSC: 47H09, 47H05, 47H06, 47J25.
- fixed point
- feasibility problem
- simultaneous iterative method
- weak convergence
- strictly pseudononspreading mapping
which allows asymmetric and partial relations between the variables x andy. The interest is to cover many situations, for instance indecomposition methods for PDEs, applications in game theory and inintensity-modulated radiation therapy (IMRT). In decision sciences, this allows oneto consider agents who interplay only via some components of their decisionvariables, for further details, the interested reader is referred to . In IMRT, this amounts to envisage a weak coupling between the vector ofdoses absorbed in all voxels and that of the radiation intensity, for furtherdetails, the interested reader is referred to [13, 14].
Observe that in the algorithms (1.3) and (1.5) mentioned above, the determination ofthe stepsize depends on the operator (matrix) norms and (or the largest eigenvalues of and ). To implement the alternating algorithm (1.3) andthe simultaneous algorithm (1.5), one has first to compute (or, at least, estimate)operator norms of A and B, which is in general not easy inpractice.
To overcome this difficulty, Lopez et al. and Zhao et al. presented useful method for choosing the stepsizes which do not needprior knowledge of the operator norms for solving the split feasibility problems andmultiple-set split feasibility problems, respectively.
The advantage of our choice (1.6) of the stepsizes lies in the fact that no priorinformation about the operator norms of A and B is required, andstill convergence is guaranteed.
Before proceeding, we need to introduce a few concepts.
Remark 2.1 It is easy to see that and . Furthermore, is well known to contain resolvents and projectionoperators, and includes subgradient projection operators . T is a nonspreading mapping if and only if T is a0-strictly pseudononspreading mapping.
The so-called demiclosedness principle plays an important role in our argument.
To establish our results, we need the following technical lemmas.
Lemma 2.1 ()
The following definition will be useful for our results.
In 2009, Kangtunyakarn and Suantai  introduced T-mapping generated by and as follows.
Using the above definition, we have the following important lemma.
Lemma 2.2LetCbe a nonempty convex subset of real Banach space. Let be a finite family of -strictly pseudononspreading mappings ofCinto itself with , and let be real numbers such that for every . IfTis theT-mapping generated by and , then andTis a quasi-nonexpansive mapping.
Now, we are in a position to prove our convergence results in this section.
for all . We can divide the sequence into two sequences: one satisfies , which is denoted by and the other sequence satisfies , which is denoted by . Following the process of Case 1, we show that theresults hold for the subsequences with and . Thus, we obtain .
The following conclusions can be obtained from Theorem 3.1 immediately.
We now pay attention to applying our simultaneous iterative algorithms to some convexand nonlinear analysis notions; see, for example, .
4.1 Split feasibility problem
4.2 Variational problems via resolvent mappings
and the algorithm is applied to the following form.
We thank Prof. Yiju Wang for his careful reading of the manuscript and thank theanonymous referees and the editor for their constructive comments andsuggestions, which greatly improved this article. This project is supported bythe Natural Science Foundation of China (Grant Nos. 11401438, 11171180,11171193, 11126233), and Project of Shandong Province Higher Educational Scienceand Technology Program (Grant No. J14LI52).
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