Abstract
The purpose of this paper is to study the split feasibility problem and fixed point problem involved in the pseudocontractive mappings. We construct an iterative algorithm and prove its strong convergence.
MSC:47J25, 47H09, 65J15, 90C25.
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1 Introduction
Let and be two real Hilbert spaces and let and be two nonempty, closed, and convex sets. Let be a bounded linear operator with its adjoint . Let and be two nonlinear mappings.
The purpose of this paper is to study the following split feasibility problem and fixed point problem:
Special cases:
(i) Finding a point which satisfies
This problem, referred to as the split feasibility problem, was introduced by Censor and Elfving [1], modeling phase retrieval and other image restoration problems, and further studied by many researchers; see, for instance, [2–7].
(ii) Find a point with the property
This problem, referred to as the split common fixed point problem, was first introduced by Censor and Segal [8].
Next, we recall some existing algorithms for solving (1.1)-(1.3) in the literature.
In order to solve (1.2), Censor and Elfving [1] introduced the following algorithm:
where C and Q are closed and convex sets in , A is a full rank matrix and .
Now (1.4) is not popular because it involves the computation of the inverse .
A more popular algorithm that solves (1.4) seems to be the CQ algorithm presented by Byrne [5, 7]:
where , with L being the largest eigenvalue of the matrix .
Note that solves (1.2) if and only if solves the fixed point equation
The above equivalence relation (1.6) reminds us to use fixed point method to solve (1.2). Many authors have given a continuation of the study on the CQ algorithm and its variant form. For related work, please refer to [9–16]. Especially, the following regularized method was presented by Xu [6]:
It should be pointed out that (1.7) can be used to find the minimum norm solution of (1.2).
For solving (1.3), Censor and Segal [8] invented an algorithm which generates a sequence according to the iterative procedure:
Note that (1.8) is more general than (1.5). Some further generations of this algorithm were studied by Moudafi [17] and Wang and Xu [18] and others; see, for example, [19–22].
Motivated by the results in this direction, the purpose of this paper is to study the split feasibility problem and the fixed point problem involved in the pseudocontractive mappings. We construct an iterative algorithm and prove its strong convergence.
2 Preliminaries
Let H be a real Hilbert space with inner product and norm , respectively. Let C be a nonempty, closed, and convex subset of H.
Recall that a mapping is called pseudocontractive if
for all . It is well known that T is pseudocontractive if and only if
for all . A mapping is called L-Lipschitzian if there exists such that
for all . If , we call T nonexpansive.
We will use to denote the set of fixed points of T, that is,
We know that the metric projection satisfies
It is well known that the metric projection is firmly nonexpansive, that is,
for all .
For all , the following conclusions hold:
and
Lemma 2.1 ([23])
Let H be a real Hilbert space, C a closed convex subset of H. Let be a continuous pseudocontractive mapping. Then
-
(i)
is a closed convex subset of C,
-
(ii)
is demiclosed at zero.
Lemma 2.2 ([24])
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(1)
;
-
(2)
or .
Then .
Lemma 2.3 ([25])
Let be a sequence of real numbers. Assume does not decrease at infinity, that is, there exists at least a subsequence of such that for all . For every , define an integer sequence as
Then as and for all
3 Main results
Let and be two real Hilbert spaces and let and be two nonempty closed convex sets. Let be a bounded linear operator with its adjoint . Let nonexpansive mapping and let be an L-Lipschitzian pseudocontractive mapping with .
We use Γ to denote the set of solutions of (1.1), that is,
In the sequel, we assume .
Now, we present our algorithm for finding .
Algorithm 3.1 For fixed and arbitrarily, let be a sequence defined by
where , , and are three real number sequences in and δ is a constant in .
Theorem 3.2 Assume the following conditions are satisfied:
(C1) ;
(C2) ;
(C3) .
Then the sequence generated by algorithm (3.1) converges strongly to the point , given by .
Proof Set . Then we have and . Set and for all . Thus for all .
Since and are nonexpansive, we have
and
By (2.2), we get
From (2.1), we have
and
Applying equality (2.3), we have
Since T is L-Lipschitzian and , by (3.7), we get
By (2.3) and (3.5), we have
From (3.6), (3.8), and (3.9), we deduce
Since , we derive that
This together with (3.10) implies that
By (2.3), (3.1), (3.11), and (C3), we have
By the convexity of the norm and by using (2.4), we get
Since A is a linear operator with its adjoint , we have
Again using (2.4), we obtain
From (3.4), (3.14), and (3.15), we get
Substituting (3.16) into (3.13) we deduce
From (3.3), (3.12), and (3.17), we get
The boundedness of the sequence yields our result.
Using the firmly nonexpansivenessity of (2.2), we have
Thus
It follows that
Next, we consider two possible cases.
Case 1. Assume there exists some integer such that is decreasing for all . In this case, we know that exists. From (3.19), we deduce
Returning to (3.17), we have
Hence,
which implies that
So,
Note that
It follows from (3.21) that
From (3.3), (3.12), and (3.17), we deduce
It follows that
Therefore,
Observe that
Thus,
This together with (3.24) implies that
Now, we show that
Choose a subsequence of such that
Since the sequence is bounded, we can choose a subsequence of such that . For the sake of convenience, we assume (without loss of generality) that . Consequently, we derive from the above conclusions that
Applying Lemma 2.1, we deduce
Note that and . From (3.27), we deduce
To this end, we deduce
That is to say, .
Therefore,
Using (2.5), we have
Applying Lemma 2.2 and (3.28) to (3.29), we deduce .
Case 2. Assume there exists an integer such that
Set . Then we have
Define an integer sequence for all as follows:
It is clear that is a non-decreasing sequence satisfying
and
for all .
By a similar argument to that of Case 1, we can obtain
and
This implies that
Thus, we obtain
Since , we have from (3.29) that
It follows that
Combining (3.30) and (3.32), we have
and hence
By (3.31), we obtain
This together with (3.33) implies that
Applying Lemma 2.3 to get
Therefore, . That is, . This completes the proof. □
Algorithm 3.3 For arbitrarily, let be a sequence defined by
where , , and are three real number sequences in and δ is a constant in .
Corollary 3.4 Assume the following conditions are satisfied:
(C1) ;
(C2) ;
(C3) .
Then the sequence generated by algorithm (3.34) converges strongly to , which is the minimum norm in Γ.
Algorithm 3.5 For fixed and arbitrarily, let be a sequence defined by
where is a real number sequence in and δ is a constant in .
Corollary 3.6 Suppose , the set of the solutions of (1.2), is nonempty. Assume the following conditions are satisfied:
(C1) ;
(C2) .
Then the sequence generated by algorithm (3.35) converges strongly to .
Algorithm 3.7 For arbitrarily, let be a sequence defined by
where is a real number sequence in and δ is a constant in .
Corollary 3.8 Suppose , the set of the solutions of (1.2), is nonempty. Assume the following conditions are satisfied:
(C1) ;
(C2) .
Then the sequence generated by algorithm (3.36) converges strongly to , which is the minimum norm in .
Algorithm 3.9 For fixed and arbitrarily, let be a sequence defined by
where , , and are three real number sequences in and δ is a constant in .
Corollary 3.10 Suppose , the set of the solutions of (1.3), is nonempty. Assume the following conditions are satisfied:
(C1) ;
(C2) ;
(C3) .
Then the sequence generated by algorithm (3.37) converges strongly to .
Algorithm 3.11 For and arbitrarily, let be a sequence defined by
where , , and are three real number sequences in and δ is a constant in .
Corollary 3.12 Suppose , the set of the solutions of (1.3), is nonempty. Assume the following conditions are satisfied:
(C1) ;
(C2) ;
(C3) .
Then the sequence generated by algorithm (3.38) converges strongly to which is the minimum norm in .
Example 3.13 Let with the inner product defined by for all and the standard norm . Let and . Let for all and let for all . Let for all . Then A is a bounded linear operator with its adjoint . Observe that and . It is easy to see that
and
for all .
But
Hence, T is a Lipschitzian pseudocontractive mapping but not a nonexpansive one.
Note that . Let and . Then we have
Let and for all n. It is not hard to compute that
which shows .
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Acknowledgements
YY was supported in part by NSFC 71161001-G0105. Y-CL was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
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Yao, Y., Agarwal, R.P., Postolache, M. et al. Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theory Appl 2014, 183 (2014). https://doi.org/10.1186/1687-1812-2014-183
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DOI: https://doi.org/10.1186/1687-1812-2014-183