Abstract
An extraordinary split problem, which can be regarded as a superimposition of the split feasibility problem and the split fixed point problem, is considered. A superimposed algorithm is presented. The analysis technique of the suggested algorithm and the corresponding convergence results are demonstrated.
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1 Introduction
1.1 Background
The split feasibility problem (SFP) is formulated as finding \(u^{\ddagger}\) such that
where \(\mathcal{C}\) (≠∅) and \(\mathcal{Q}\) (≠∅) are closed convex subsets of real Hilbert spaces \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\), respectively, and \(\mathcal{A}\) is a bounded linear operator from \(\mathcal{H}_{1}\) to \(\mathcal{H}_{2}\). The mathematical model of the SFP was refined from phase retrievals and the medical image reconstruction by Censor and Elfving [1] in 1994. One effective approach to solve the SFP is algorithmic iteration. There are several effective iterations which are listed as follows.
Existing iterations for the SFP
1. Simultaneous multiprojections (Censor and Elfving [1]):
where \(\mathcal{C}\subset\mathbb{R}^{n}\) and \(\mathcal{Q}\subset\mathbb {R}^{n}\) are closed convex sets, and \(\mathcal{A}\) is an \(n\times n\) matrix.
2. Gradient projections (CQ iteration) [2–6]:
where ϖ is a constant and \(\mathcal{A}^{T}\) denotes the transposition of \(\mathcal{A}\).
3. Averaged CQ iteration [2, 7]:
where \(\alpha_{k}\in\, ]0,1[\), ϖ is a constant and \(\mathcal{A}^{*}\) is the adjoint of \(\mathcal{A}\).
4. Relaxed CQ iteration [3, 8, 9]: Let \(f:\mathcal{H}_{1}\to\mathbb {R}\) and \(g:\mathcal{H}_{2}\to\mathbb{R}\) be two convex functions. Define two level sets and the related subdifferentials
and
Define the relaxed CQ iteration as follows:
where
where \(\xi_{k}\in\partial f(x_{k})\), and
where \(\eta_{k}\in\partial g(\mathcal{A}x_{k})\).
5. Regularized iteration [2, 10]:
where \(\{\alpha_{k}\}\subset\, ]0,1[\) and \(\{\varpi_{k}\}\in\, ]0,\frac{\alpha _{k}}{\|\mathcal{A}\|^{2}+\alpha_{k}}[\).
6. Self-adaptive iteration [11–13]:
where the step-size \(\varpi_{k}=\frac{\tau_{k}\|(\mathcal{I}-\operatorname{proj}_{\mathcal {Q}})\mathcal{A}x_{k}\|^{2}}{2\|\mathcal{A}^{*}(\mathcal{I}-\operatorname{proj}_{\mathcal {Q}})\mathcal{A}x_{k}\|^{2}}\) in which \(\tau_{k}\in\, ]0,2[\).
7. Halpern-type iteration [2]:
where \(u\in\mathcal{C}\) is a fixed point, \(\{\alpha_{k}\}\subset\, ]0,1[\) and \(\varpi_{k}=\frac{\tau_{k}\|(\mathcal{I}-\operatorname{proj}_{\mathcal{Q}})\mathcal {A}x_{k}\|^{2}}{2\|\mathcal{A}^{*}(\mathcal{I}-\operatorname{proj}_{\mathcal{Q}})\mathcal {A}x_{k}\|^{2}}\) in which \(\tau_{k}\in\, ]0,2[\).
The (two-set) split common fixed point problem (SCFP) can be formulated as finding \(u^{\dagger}\) such that
where \(\operatorname{Fix}(\mathcal{T})\) and \(\operatorname{Fix}(\mathcal{S})\) stand for the fixed point sets of the operators \(\mathcal{T}:\mathcal{H}_{1}\to\mathcal {H}_{1}\) and \(\mathcal{S}:\mathcal{H}_{2}\to\mathcal{H}_{2}\).
The SCFP is a natural extension of the SFP and of the convex feasibility problem. The SCFP was firstly considered by Censor and Segal in [14] where \(\mathcal{S}\) and \(\mathcal{T}\) are directed operators which include the orthogonal projections and the sub-gradient projectors.
Existing iterations for the SCFP
1. Censor and Segal’s iteration [14]:
2. Averaged iteration [15, 16]:
3. Halpern-type iteration [17]:
4. Self-adaptive iteration [18]:
where the step-size \(\varpi_{k}=\frac{(1-\tau)\|(\mathcal{I}-\mathcal {S})\mathcal{A}x_{k}\|^{2}}{2\|\mathcal{A}^{*}(\mathcal{I}-\mathcal {S})\mathcal{A}x_{k}\|^{2}}\).
5. Composite iteration [19]:
where \(\{\alpha_{k}\}_{k\in\mathbb{N}}\), \(\{\beta_{k}\}_{k\in\mathbb{N}}\), \(\{\gamma_{k}\}_{k\in\mathbb{N}}\), \(\{\zeta_{k}\}_{k\in\mathbb{N}}\) and \(\{\eta_{k}\}_{k\in\mathbb{N}}\) are five real number sequences in \(]0,1[\), \(\delta\in\, ]0,1[\) is a constant, \(h:\mathcal{H}_{1}\to\mathcal {H}_{1}\) is a contraction and \(\mathcal{B}:\mathcal{H}_{1}\to\mathcal {H}_{1}\) is a strong positive linear bounded operator.
1.2 Problem statement
The purpose of this paper is to study the following split feasibility problem and fixed point problem:
It is obvious that (1.15) includes SFP (1.1) and SCFP (1.9) as special cases.
Motivated by iterations (1.3), (1.11) and (1.14), we will construct a new iteration to approach the solution of (1.15). Strong convergence results are given in the third section.
2 Several notions and lemmas
Assume that \(\mathcal{H}\) is a real Hilbert space. \(\langle\cdot,\cdot \rangle\) and \(\|\cdot\|\) stand for its inner product and norm, respectively. Let (∅≠) \(\mathcal{C}\subset\mathcal{H}\) be a closed convex set.
Definition 2.1
An operator \(\mathcal{P}:\mathcal{C}\to\mathcal{C}\) is said to be \(\mathcal{L}\)-Lipschitzian if
for some constant \(\mathcal{L}>0\).
If \(\mathcal{L}\in[0,1[\), then \(\mathcal{P}\) is called \(\mathcal {L}\)-contraction. If \(\mathcal{L}=1\), then \(\mathcal{P}\) is called nonexpansive.
Definition 2.2
An operator \(\mathcal{P}:\mathcal{C}\to\mathcal{C}\) is said to be firmly nonexpansive if
for all \(u,u^{\dagger}\in\mathcal{C}\).
Definition 2.3
An operator \(\mathcal{P}:\mathcal{C}\to\mathcal{C}\) is said to be pseudo-contractive if
for all \(u,u^{\dagger}\in\mathcal{C}\).
Definition 2.4
An operator \(\mathcal{P}:\mathcal{C}\to\mathcal{C}\) is said to be quasi-pseudo-contractive if
for all \(u\in\mathcal{C}\) and \(u^{\dagger}\in \operatorname{Fix}(\mathcal{P})\).
Definition 2.5
An operator \(\mathcal{P}\) is said to be demiclosed if \(\forall u_{n}\to u^{\ddagger}\) weakly and \(\mathcal{P}(x_{n})\to u\) strongly imply that \(\mathcal{P}(u^{\ddagger})=u\).
Lemma 2.6
([20])
Let \(\{\varrho_{n}\}\subset [0,+\infty[\), \(\{\vartheta_{n}\}\subset\, ]0,1[\) and \(\{\eta_{n}\}\) be three real number sequences. Suppose that \(\{\varrho_{n}\}\), \(\{\vartheta_{n}\}\) and \(\{\eta_{n}\}\) satisfy the following three conditions:
-
(i)
\(\varrho_{n+1}\leq(1-\vartheta_{n})\varrho_{n}+\eta_{n}\vartheta_{n}\),
-
(ii)
\(\sum_{n=1}^{\infty}\vartheta_{n}=\infty\),
-
(iii)
\(\limsup_{n\to\infty}\eta_{n}\leq0\) or \(\sum_{n=1}^{\infty}|\eta_{n}\vartheta_{n}|<\infty\).
Then \(\lim_{n\to\infty}\varrho_{n}=0\).
Lemma 2.7
([21])
Let \(\{\rho_{n}\}\) be a sequence of real numbers. Assume that there exists a subsequence \(\{\rho _{n_{k}}\}\) of \(\{\rho_{n}\}\) such that \(\rho_{n_{k}}\le\rho_{n_{k}+1}\) for all \(k\ge0\). For every \(n\ge N_{0}\), define an integer sequence \(\{\tau (n)\}\) as
Then \(\tau(n)\to\infty\) as \(n\to\infty\) and, for all \(n\ge N_{0}\),
3 Algorithms and convergence
In this section, we first construct an iterative algorithm for solving problem (1.15) and subsequently to prove its convergence. Now we give the assumptions on the underlying spaces, involved operators and additional parameters, throughout.
-
I.
Conditions on the underlying spaces:
-
(UC1):
\(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) are two real Hilbert spaces,
-
(UC2):
\(\mathcal{C}\subset\mathcal{H}_{1}\) and \(\mathcal{Q}\subset \mathcal{H}_{2}\) are two nonempty closed convex sets.
-
(UC1):
-
II.
Conditions on the involved operators:
-
(IO1):
\(\mathcal{A}: \mathcal{H}_{1} \to\mathcal{H}_{2}\) is a bounded linear operator with its adjoint \(\mathcal{A}^{*}\),
-
(IO2):
\(\mathcal{B}\) is a strongly positive bounded linear operator on \(\mathcal{H}_{1}\) with coefficient σ (>0),
-
(IO3):
\(f:\mathcal{C}\to\mathcal{H}_{1}\) is a ρ-contraction,
-
(IO4):
\(\mathcal{S}:\mathcal{Q}\to\mathcal{Q}\) is an \(\mathcal {L}_{1}\)-Lipschitzian quasi-pseudo-contractive operator with \(\mathcal {L}_{1}\) (>1) and \(\mathcal{T}:\mathcal{C}\to\mathcal{C}\) is an \(\mathcal{L}_{2}\)-Lipschitzian quasi-pseudo-contractive operator with \(\mathcal{L}_{2}\) (>1).
-
(IO1):
-
III.
Conditions on the parameters:
-
(AP1):
δ and γ are two positive constants,
-
(AP2):
\(\{\alpha_{n}\}_{n\in\mathbb{N}}\), \(\{\beta_{n}\}_{n\in\mathbb {N}}\), \(\{\gamma_{n}\}_{n\in\mathbb{N}}\), \(\{\zeta_{n}\}_{n\in\mathbb {N}}\) and \(\{\eta_{n}\}_{n\in\mathbb{N}}\) are real number sequences in \(]0,1[\).
-
(AP1):
We use Γ to denote the set of solutions of problem (1.15), that is,
In the sequel, we assume \(\Gamma\ne\emptyset\).
Next, we construct the following iterative algorithm to solve problem (1.15).
Algorithm 3.1
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by
for all \(n\in\mathbb{N}\).
Theorem 3.2
Suppose that \(\mathcal{T}-\mathcal{I}\) and \(\mathcal{S}-\mathcal{I}\) are demiclosed at 0. Assume that the following conditions are satisfied:
- (C1)::
-
\(\lim_{n\to\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha _{n}=\infty\),
- (C2)::
-
\(0< a_{1}<\zeta_{n}<b_{1}<\eta_{n}<c_{1}<\frac{1}{\sqrt{1+\mathcal{L}_{1}^{2}}+1}\),
- (C3)::
-
\(0< a_{2}<\beta_{n}<b_{2}<\gamma_{n}<c_{2}<\frac{1}{\sqrt{1+\mathcal{L}_{2}^{2}}+1}\),
- (C4)::
-
\(0<\delta<\frac{1}{\|\mathcal{A}\|^{2}}\) and \(\sigma>\gamma\rho\).
Then the sequence \(\{x_{n}\}\) generated by algorithm (3.1) converges strongly to the unique fixed point of the contractive mapping \(\operatorname{proj}_{\Gamma}(\gamma f+\mathcal{I}-\mathcal{B})\).
Remark 3.3
In the sequel, we denote the unique fixed point of the mapping \(\operatorname{proj}_{\Gamma}(\gamma f+\mathcal{I}-\mathcal{B})\) by \(z^{\dagger}\), i.e., \(z^{\dagger}=\operatorname{proj}_{\Gamma}(\gamma f+\mathcal{I}-\mathcal{B})z^{\dagger }\). It is clear that \(z^{\dagger}\) solves the variational inequality \(\langle(\gamma f-\mathcal{B})z^{\dagger}, z-z^{\dagger}\rangle\le0\), \(\forall z\in\Gamma\).
In order to prove Theorem 3.2, we need several helpful propositions.
Proposition 3.4
([19])
Let \(\mathcal{H}\) be a real Hilbert space. Let \(\mathcal{U}:\mathcal{H} \to\mathcal{H} \) be an \(\mathcal {L}\)-Lipschitzian operator with \(\mathcal{L}>1\). Then
for all \(\zeta\in(0,\frac{1}{\mathcal{L}})\).
Proposition 3.5
([19])
Let \(\mathcal{H}\) be a real Hilbert space. Let \(\mathcal{U}:\mathcal{H} \to\mathcal{H} \) be an \(\mathcal {L}\)-Lipschitzian quasi-pseudo-contractive operator. Then we have
and the operator \((1-\xi)\mathcal{I}+\xi\mathcal{U}((1-\eta)\mathcal {I}+\eta\mathcal{U})\) is quasi-nonexpansive when \(0<\xi<\eta<\frac {1}{\sqrt{1+\mathcal{L}^{2}}+1}\), that is,
for all \(x\in\mathcal{H}\) and \(u^{\dagger}\in \operatorname{Fix}(\mathcal{U})\).
Proposition 3.6
In any real Hilbert space \(\mathcal{H}\), the following two equalities hold:
and
for all \(u,u^{\dagger}\in\mathcal{H}\).
Proposition 3.7
([19])
Let \(\mathcal{H}\) be a real Hilbert space. Let \(\mathcal{U}:\mathcal{H}\to\mathcal{H}\) be an \(\mathcal {L}\)-Lipschitzian operator with \(\mathcal{L}>1\). If \(\mathcal {I}-\mathcal{U}\) is demiclosed at 0, then \(\mathcal{I}-\mathcal {U}((1-\zeta)\mathcal{I}+\zeta\mathcal{U})\) is also demiclosed at 0 when \(\zeta\in(0, \frac{1}{\mathcal{L}})\).
Next, we prove Theorem 3.2.
Proof
Let \(z^{\dagger}=\operatorname{proj}_{\Gamma}(\gamma f+\mathcal{I}-\mathcal {B})z^{\dagger}\). Subsequently, we obtain \(z^{\dagger}\in\mathcal{C}\cap \operatorname{Fix}(\mathcal{T})\) and \(\mathcal{A}z^{\dagger}\in\mathcal{Q}\cap \operatorname{Fix}(\mathcal{S})\). Note that \(\operatorname{proj}_{\mathcal{Q}}\) is firmly nonexpansive. From (2.1), we deduce
Applying Proposition 3.4 and noting conditions (C2) and (C3), we have
and
for all \(n\in\mathbb{N}\).
By condition (C2) and Proposition 3.5, we derive
This together with (3.4) implies that
By condition (C3) and Proposition 3.5, we derive
Noting that \(\operatorname{proj}_{\mathcal{C}}\) is nonexpansive, we obtain
From (3.1), we get
Observe that
Using (3.3), we obtain
From (3.5), (3.9) and (3.10), we get
According to equality (3.3), we get
Combining the above equality and (3.11), we deduce
In view of condition (C4), we know that \(\delta^{2}\|\mathcal{A}\| ^{2}-\delta<0\). From (3.12), we have
Therefore,
Substituting (3.13) into (3.8) we deduce
From (3.6), (3.7) and (3.14), we get
By induction, we get
Hence, the sequence \(\{x_{n}\}\) is bounded.
Using the firm nonexpansiveness of \(\operatorname{proj}_{\mathcal{C}}\), we have
From (3.6), (3.14) and (3.15), we deduce
It follows that
Next, we consider two possible cases: the sequence \(\{\|x_{n}-z^{\dagger}\|\} \) is either monotone decreasing at infinity (Case 1) or not (Case 2).
-
Case 1.
There exists \(n_{0}\) such that the sequence \(\{\|x_{n}-z^{\dagger}\|\} _{n\ge n_{0}}\) is decreasing.
-
Case 2.
For any \(n_{0}\), there exists an integer \(m\ge n_{0}\) such that \(\| x_{m}-z^{\dagger}\|\le\|x_{m+1}-z^{\dagger}\|\).
In Case 1, we assume that there exists some integer \(m>0\) such that \(\{ \|x_{n}-z^{\dagger}\|\}\) is decreasing for all \(n\ge m\). Then \(\lim_{n\to\infty}\|x_{n}-z^{\dagger}\|\) exists. From (3.16), we deduce
From (3.8), we have
Since \(\{x_{n}\}\) is bounded, there exists a constant M> such that
By (3.18), we deduce
Combining (3.12) and (3.19), we obtain
Hence,
which implies that
Therefore,
Note that \(v_{n}-z_{n}=\zeta_{n}[\mathcal{S}((1-\eta_{n})\mathcal{I}+\eta _{n}\mathcal{S})z_{n}-z_{n}]\). Thus,
Since
it follows that
This together with (3.22) implies that
According to (3.1), we have
It follows from (3.20) and (C1) that
Applying Proposition 3.5, we get
From (3.19), (3.25) and (3.26), we deduce
It follows that
Therefore,
Observe that
Thus,
This together with (3.27) implies that
Next, we show that
Choose a subsequence \(\{y_{n_{i}}\}\) of \(\{y_{n}\}\) such that
Since the sequence \(\{y_{n_{i}}\}\) is bounded, we can choose a subsequence \(\{y_{n_{i_{j}}}\}\) of \(\{y_{n_{i}}\}\) such that \(y_{n_{i_{j}}}\rightharpoonup z\). For the sake of convenience, we assume (without loss of generality) that \(y_{n_{i}}\rightharpoonup z\). Subsequently, we derive from the above conclusions that
and
Note that \(u_{n_{i}}=\operatorname{proj}_{\mathcal{C}}y_{n_{i}}\in\mathcal{C}\) and \(z_{n_{i}}=\operatorname{proj}_{\mathcal{Q}}\mathcal{A}x_{n_{i}}\in\mathcal{Q}\). From (3.30), we deduce \(z\in\mathcal{C}\) and \(\mathcal{A}z\in\mathcal {Q}\) by (3.31). By the demiclosedness of \(\mathcal{T}-\mathcal {I}\) and \(\mathcal{S}-\mathcal{I}\), we deduce \(z\in \operatorname{Fix}(\mathcal{T})\) (by (3.28)) and \(\mathcal{A}z\in \operatorname{Fix}(\mathcal{S})\) (by (3.23)). To this end, we deduce \(z\in\mathcal{C}\cap \operatorname{Fix}(\mathcal{T})\) and \(\mathcal{A}z\in\mathcal{Q}\cap \operatorname{Fix}(\mathcal{S})\). That is to say, \(z\in\Gamma\).
Therefore,
From (3.1), we have
It follows that
Therefore,
Applying Lemma 2.6 and (3.32) to (3.33), we deduce \(x_{n}\to z^{\dagger}\).
Case 2. Assume that there exists an integer \(n_{0}\) such that
Set \(\omega_{n}=\{\|x_{n}-z^{\dagger}\|\}\). Then we have
Define an integer sequence \(\{\tau_{n}\}\) for all \(n\ge n_{0}\) as follows:
It is clear that \(\tau(n)\) is a nondecreasing sequence satisfying
and
for all \(n\ge n_{0}\).
By a similar argument as that of Case 1, we can obtain
and
This implies that
Thus, we obtain
Since \(\omega_{\tau(n)}\le\omega_{\tau(n)+1}\), we have from (3.33) that
It follows that
Combining (3.34) and (3.36), we have
and hence
By (3.35), we obtain
This together with (3.37) implies that
Applying Lemma 2.7 we get
Therefore, \(\omega_{n}\to0\). That is, \(x_{n}\to z^{\dagger}\). This completes the proof. □
4 Applications
The following results can be deduced directly from Algorithm 3.1 and Theorem 3.2.
Algorithm 4.1
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by
for all \(n\in\mathbb{N}\).
Corollary 4.2
Suppose that \(\mathcal{T}-\mathcal{I}\) and \(\mathcal{S}-\mathcal{I}\) are demiclosed at 0. Assume that the following conditions are satisfied:
- (C1)::
-
\(\lim_{n\to\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha _{n}=\infty\),
- (C2)::
-
\(0< a_{1}<\zeta_{n}<b_{1}<\eta_{n}<c_{1}<\frac{1}{\sqrt{1+\mathcal{L}_{1}^{2}}+1}\),
- (C3)::
-
\(0< a_{2}<\beta_{n}<b_{2}<\gamma_{n}<c_{2}<\frac{1}{\sqrt{1+\mathcal{L}_{2}^{2}}+1}\),
- (C4)′::
-
\(0<\delta<\frac{1}{\|\mathcal{A}\|^{2}}\).
Then the sequence \(\{x_{n}\}\) generated by algorithm (4.1) converges strongly to the minimum norm solution \(u^{\clubsuit}\in\Gamma\).
Algorithm 4.3
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by
for all \(n\in\mathbb{N}\).
Corollary 4.4
Assume that the following conditions are satisfied:
- (C1)::
-
\(\lim_{n\to\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha _{n}=\infty\),
- (C4)::
-
\(0<\delta<\frac{1}{\|\mathcal{A}\|^{2}}\) and \(\sigma>\gamma\rho\).
Then the sequence \(\{x_{n}\}\) generated by algorithm (4.2) converges strongly to \(u\in\Gamma_{1}\) (the set of the solutions of (1.1)) provided \(\Gamma_{1}\ne\emptyset\).
Algorithm 4.5
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by
for all \(n\in\mathbb{N}\).
Corollary 4.6
Assume that the following conditions are satisfied:
- (C1)::
-
\(\lim_{n\to\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha _{n}=\infty\),
- (C4)′::
-
\(0<\delta<\frac{1}{\|\mathcal{A}\|^{2}}\).
Then the sequence \(\{x_{n}\}\) generated by algorithm (4.3) converges strongly to the minimum norm solution \(u^{\clubsuit}\in\Gamma _{1}\) provided \(\Gamma_{1}\ne\emptyset\).
Algorithm 4.7
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by
for all \(n\in\mathbb{N}\).
Corollary 4.8
Suppose that \(\mathcal{T}-\mathcal{I}\) and \(\mathcal{S}-\mathcal{I}\) are demiclosed at 0. Assume that the following conditions are satisfied:
- (C1)::
-
\(\lim_{n\to\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha _{n}=\infty\),
- (C2)::
-
\(0< a_{1}<\zeta_{n}<b_{1}<\eta_{n}<c_{1}<\frac{1}{\sqrt{1+\mathcal{L}_{1}^{2}}+1}\),
- (C3)::
-
\(0< a_{2}<\beta_{n}<b_{2}<\gamma_{n}<c_{2}<\frac{1}{\sqrt{1+\mathcal{L}_{2}^{2}}+1}\),
- (C4)::
-
\(0<\delta<\frac{1}{\|\mathcal{A}\|^{2}}\) and \(\sigma>\gamma\rho\).
Then the sequence \(\{x_{n}\}\) generated by algorithm (4.4) converges strongly to \(u\in\Gamma_{2}\) (the set of the solutions of (1.9)) provided \(\Gamma_{2}\ne\emptyset\).
Algorithm 4.9
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by
for all \(n\in\mathbb{N}\).
Corollary 4.10
Suppose that \(\mathcal{T}-\mathcal{I}\) and \(\mathcal{S}-\mathcal{I}\) are demiclosed at 0. Assume that the following conditions are satisfied:
- (C1)::
-
\(\lim_{n\to\infty}\alpha_{n}=0\) and \(\sum_{n=1}^{\infty}\alpha _{n}=\infty\),
- (C2)::
-
\(0< a_{1}<\zeta_{n}<b_{1}<\eta_{n}<c_{1}<\frac{1}{\sqrt{1+\mathcal{L}_{1}^{2}}+1}\),
- (C3)::
-
\(0< a_{2}<\beta_{n}<b_{2}<\gamma_{n}<c_{2}<\frac{1}{\sqrt{1+\mathcal{L}_{2}^{2}}+1}\),
- (C4)′::
-
\(0<\delta<\frac{1}{\|\mathcal{A}\|^{2}}\).
Then the sequence \(\{x_{n}\}\) generated by algorithm (4.5) converges strongly to the minimum norm solution \(u^{\clubsuit}\in\Gamma _{2}\) provided \(\Gamma_{2}\ne\emptyset\).
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Acknowledgements
Yeong-Cheng Liou was supported in part by MOST 101-2628-E-230-001-MY3 and MOST 101-2622-E-230-005-CC3. Abdelouahed Hamdi would like to thank Qatar University for providing excellent research facilities under Grant: QUUG-CAS-DMSP-14/15-4.
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Hamdi, A., Liou, YC., Yao, Y. et al. The common solutions of the split feasibility problems and fixed point problems. J Inequal Appl 2015, 385 (2015). https://doi.org/10.1186/s13660-015-0870-6
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DOI: https://doi.org/10.1186/s13660-015-0870-6