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.
$$ u^{\ddagger}\in\mathcal{C}\quad \mbox{and}\quad \mathcal{A}u^{\ddagger}\in\mathcal{Q}\quad \bigl(\mbox{or }u^{\ddagger}\in \mathcal{C}\cap\mathcal{A}^{-1}\mathcal{Q}\mbox{ when } \mathcal{A}^{-1}\mbox{ exists}\bigr), $$
(1.1)
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.
$$ x_{k+1}=\mathcal{A}^{-1}\operatorname{proj}_{\mathcal{Q}} \bigl(\operatorname{proj}_{\mathcal{A}(\mathcal {C})}(\mathcal{A}x_{k})\bigr),\quad k\in\mathbb{N}, $$
(1.2)
2. Gradient projections (CQ iteration) [2‐6]: where ϖ is a constant and \(\mathcal{A}^{T}\) denotes the transposition of \(\mathcal{A}\).
$$ x_{k+1}=\operatorname{proj}_{\mathcal{C}} \biggl(x_{k}-\frac{\varpi}{\|\mathcal{A}\|^{2}} \mathcal {A}^{T}(\mathcal{I}- \operatorname{proj}_{\mathcal{Q}})\mathcal{A}x_{k}\biggr),\quad k\in \mathbb{N}, $$
(1.3)
3. Averaged CQ iteration [2, 7]: where \(\alpha_{k}\in\, ]0,1[\), ϖ is a constant and \(\mathcal{A}^{*}\) is the adjoint of \(\mathcal{A}\).
$$ x_{k+1}=(1-\alpha_{k})x_{k}+ \alpha_{k} \operatorname{proj}_{\mathcal{C}}\biggl(x_{k}- \frac{\varpi}{\| \mathcal{A}\|^{2}} \mathcal{A}^{*}(\mathcal{I}-\operatorname{proj}_{\mathcal{Q}}) \mathcal {A}x_{k}\biggr),\quad k\in\mathbb{N}, $$
(1.4)
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 \(\eta_{k}\in\partial g(\mathcal{A}x_{k})\).
$$\begin{aligned}& \mathcal{C}:=\bigl\{ x\in\mathcal{H}_{1}|f(x)\le0\bigr\} \quad \text{and}\quad \mathcal {Q}:=\bigl\{ y\in\mathcal{H}_{2}|g(y)\le0\bigr\} , \\& \partial{f(x)}=\bigl\{ z\in\mathcal{H}_{1}|f(u)\ge f(x)+\langle u-x, z \rangle, u\in\mathcal{H}_{1}\bigr\} ,\quad \forall x\in\mathcal{C} \end{aligned}$$
$$ \partial{g(x)}=\bigl\{ w\in\mathcal{H}_{2}|g(v)\ge g(y)+\langle v-y, w \rangle, v\in\mathcal{H}_{2}\bigr\} ,\quad \forall y\in\mathcal{Q}. $$
$$ x_{k+1}=\operatorname{proj}_{\mathcal{C}_{k}} \biggl(x_{k}-\frac{\varpi}{\|\mathcal{A}\|^{2}} \mathcal{A}^{T}(\mathcal{I}- \operatorname{proj}_{\mathcal{Q}_{k}})\mathcal{A}x_{k}\biggr),\quad k\in \mathbb{N}, $$
(1.5)
$$ \mathcal{C}_{k}=\bigl\{ x\in\mathcal{H}_{1}|f(x_{k})+ \langle\xi_{k},x-x_{k}\rangle\le 0\bigr\} , $$
$$ \mathcal{Q}_{k}=\bigl\{ y\in\mathcal{H}_{2}|g( \mathcal{A}x_{k})+\langle\eta _{k},y-\mathcal{A}x_{k} \rangle\le0\bigr\} , $$
5. Regularized iteration [2, 10]: where \(\{\alpha_{k}\}\subset\, ]0,1[\) and \(\{\varpi_{k}\}\in\, ]0,\frac{\alpha _{k}}{\|\mathcal{A}\|^{2}+\alpha_{k}}[\).
$$ x_{k+1}=\operatorname{proj}_{\mathcal{C}}\bigl((1- \alpha_{k}\varpi_{k})x_{k}-\varpi_{k} \mathcal {A}^{*}(\mathcal{I}-\operatorname{proj}_{\mathcal{Q}}) \mathcal{A}x_{k}\bigr),\quad k\in\mathbb{N}, $$
(1.6)
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 \(\tau_{k}\in\, ]0,2[\).
$$ x_{k+1}=\operatorname{proj}_{\mathcal{C}} \bigl(x_{k}-\varpi_{k} \mathcal{A}^{*}(\mathcal {I}- \operatorname{proj}_{\mathcal{Q}})\mathcal{A}x_{k}\bigr),\quad k\in \mathbb{N}, $$
(1.7)
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 \(\tau_{k}\in\, ]0,2[\).
$$ x_{k+1}=\alpha_{k}u+(1-\alpha_{k}) \operatorname{proj}_{\mathcal{C}}\bigl(x_{k}-\varpi_{k} \mathcal {A}^{*}(\mathcal{I}-\operatorname{proj}_{\mathcal{Q}}) \mathcal{A}x_{k}\bigr),\quad k\in\mathbb{N}, $$
(1.8)
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 \(\mathcal{S}:\mathcal{H}_{2}\to\mathcal{H}_{2}\).
$$ u^{\dagger}\in \operatorname{Fix}(\mathcal{T}) \quad \text{and}\quad \mathcal{A}u^{\dagger}\in \operatorname{Fix}(\mathcal{S}), $$
(1.9)
Anzeige
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]:
$$ x_{k+1}=\mathcal{T}\biggl(x_{k}- \frac{\varpi}{\|\mathcal{A}\|^{2}}\mathcal {A}^{*}(\mathcal{I}-\mathcal{S})\mathcal{A}x_{k} \biggr), \quad k\in\mathbb{N}. $$
(1.10)
2. Averaged iteration [15, 16]:
$$ \textstyle\begin{cases} y_{k}=x_{k}-\frac{\varpi}{\|\mathcal{A}\|^{2}} \mathcal{A}^{*}(\mathcal {I}-\mathcal{S})\mathcal{A}x_{k}, \\ x_{k+1}=(1-\alpha_{k})y_{k}+\alpha_{k}\mathcal{T}y_{k},\quad k\in\mathbb{N}. \end{cases} $$
(1.11)
3. Halpern-type iteration [17]:
$$ x_{k+1}=\alpha_{k}u+(1-\alpha_{k}) \mathcal{T}\biggl(x_{k}-\frac{\varpi}{\|\mathcal {A}\|^{2}} \mathcal{A}^{*}(\mathcal{I}- \mathcal{S})\mathcal{A}x_{k}\biggr),\quad k\in \mathbb{N}. $$
(1.12)
4. Self-adaptive iteration [18]: where the step-size \(\varpi_{k}=\frac{(1-\tau)\|(\mathcal{I}-\mathcal {S})\mathcal{A}x_{k}\|^{2}}{2\|\mathcal{A}^{*}(\mathcal{I}-\mathcal {S})\mathcal{A}x_{k}\|^{2}}\).
$$ \textstyle\begin{cases} y_{k}=x_{k}-\varpi_{k} \mathcal{A}^{*}(\mathcal{I}-\mathcal{S})\mathcal{A}x_{k}, \\ x_{k+1}=(1-\lambda)y_{k}+\lambda_{k}\mathcal{T}y_{k}, \quad k\in\mathbb{N}, \end{cases} $$
(1.13)
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.
$$ \textstyle\begin{cases} v_{k}=x_{k}+\frac{\delta}{\|\mathcal{A}\|^{2}} \mathcal{A}^{*}[(1-\zeta _{k})\mathcal{I}+\zeta_{k}\mathcal{S}((1-\eta_{k})\mathcal{I}+\eta_{k}\mathcal {S})-\mathcal{I}]\mathcal{A}x_{k}, \\ u_{k}=\alpha_{n}h(x_{k})+(\mathcal{I}-\alpha_{k}\mathcal{B})v_{k}, \\ x_{k+1}=(1-\beta_{k})u_{k}+\beta_{k}\mathcal{T}((1-\gamma_{k})u_{n}+\gamma _{k}\mathcal{T}u_{k}),\quad k\in\mathbb{N}, \end{cases} $$
(1.14)
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.
$$ \mbox{Find }u^{\dagger}\in\mathcal{C}\cap \operatorname{Fix}( \mathcal{T})\mbox{ such that } \mathcal{A}u^{\dagger}\in\mathcal{Q}\cap \operatorname{Fix}(\mathcal{S}). $$
(1.15)
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.
Anzeige
Definition 2.1
An operator \(\mathcal{P}:\mathcal{C}\to\mathcal{C}\) is said to be \(\mathcal{L}\)-Lipschitzian if for some constant \(\mathcal{L}>0\).
$$\bigl\Vert \mathcal{P}u-\mathcal{P}u^{\dagger}\bigr\Vert \le\mathcal{L} \bigl\Vert u-u^{\dagger}\bigr\Vert , \quad \forall u,u^{\dagger}\in \mathcal{C} $$
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 \(u,u^{\dagger}\in\mathcal{C}\).
$$ \bigl\Vert \mathcal{P}u-\mathcal{P}u^{\dagger}\bigr\Vert ^{2}\le\bigl\Vert u-u^{\dagger}\bigr\Vert ^{2}-\bigl\Vert (\mathcal {I}-\mathcal{P})u-(\mathcal{I}-\mathcal{P})u^{\dagger}\bigr\Vert ^{2} $$
(2.1)
Definition 2.3
An operator \(\mathcal{P}:\mathcal{C}\to\mathcal{C}\) is said to be pseudo-contractive if for all \(u,u^{\dagger}\in\mathcal{C}\).
$$\bigl\langle \mathcal{P}u-\mathcal{P}u^{\dagger},u-u^{\dagger}\bigr\rangle \leq\bigl\Vert u-u^{\dagger}\bigr\Vert ^{2} $$
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 \(u^{\dagger}\in \operatorname{Fix}(\mathcal{P})\).
$$ \bigl\Vert \mathcal{P}u-u^{\dagger}\bigr\Vert ^{2}\leq\bigl\Vert u-u^{\dagger}\bigr\Vert ^{2}+ \Vert \mathcal{P}u-u\Vert ^{2} $$
(2.2)
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:
Then
\(\lim_{n\to\infty}\varrho_{n}=0\).
(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
\(\sum_{n=1}^{\infty}|\eta_{n}\vartheta_{n}|<\infty\).
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
\(n\ge N_{0}\),
$$\tau(n)=\max\{i\le n: \rho_{n_{i}}< \rho_{n_{i}+1}\}. $$
$$\max\{\rho_{\tau(n)}, \rho_{n}\}\le\rho_{\tau(n)+1}. $$
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.
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).
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 \(]0,1[\).
We use Γ to denote the set of solutions of problem (1.15), that is, In the sequel, we assume \(\Gamma\ne\emptyset\).
$$\Gamma=\bigl\{ z^{\dagger}|z^{\dagger}\in\mathcal{C}\cap \operatorname{Fix}(\mathcal{T}), \mathcal {A}z^{\dagger}\in\mathcal{Q}\cap \operatorname{Fix}(\mathcal{S})\bigr\} . $$
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 \(n\in\mathbb{N}\).
$$ \textstyle\begin{cases} z_{n}=\operatorname{proj}_{\mathcal{Q}}\mathcal{A}x_{n}, \\ v_{n}=(1-\zeta_{n})z_{n}+\zeta_{n}\mathcal{S}((1-\eta_{n})z_{n}+\eta_{n}\mathcal {S}z_{n}), \\ y_{n}=\alpha_{n}\gamma f(x_{n})+(\mathcal{I}-\alpha_{n}\mathcal{B})(x_{n}-\delta \mathcal{A}^{*}(\mathcal{A}x_{n}-v_{n})), \\ u_{n}=\operatorname{proj}_{\mathcal{C}}y_{n}, \\ x_{n+1}=(1-\beta_{n})u_{n}+\beta_{n}\mathcal{T}((1-\gamma_{n})u_{n}+\gamma _{n}\mathcal{T}u_{n}) \end{cases} $$
(3.1)
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:
Then the sequence
\(\{x_{n}\}\)
generated by algorithm (3.1) converges strongly to the unique fixed point of the contractive mapping
\(\operatorname{proj}_{\Gamma}(\gamma f+\mathcal{I}-\mathcal{B})\).
(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
\(\sigma>\gamma\rho\).
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
\(\zeta\in(0,\frac{1}{\mathcal{L}})\).
$$\operatorname{Fix}\bigl(\bigl((1-\zeta)\mathcal{I}+\zeta\mathcal{U}\bigr) \mathcal{U}\bigr)=\operatorname{Fix}\bigl(\mathcal {U}\bigl((1-\zeta)\mathcal{I}+ \zeta\mathcal{U}\bigr)\bigr)=\operatorname{Fix}(\mathcal{U}) $$
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
\(u^{\dagger}\in \operatorname{Fix}(\mathcal{U})\).
$$ \bigl\Vert \mathcal{U}\bigl((1-\eta)x+\eta\mathcal{U}x\bigr)-u^{\dagger} \bigr\Vert ^{2}\le\bigl\Vert x-u^{\dagger}\bigr\Vert ^{2}+(1-\eta)\bigl\Vert x-\mathcal{U}\bigl((1-\eta)x+\eta\mathcal{U}x \bigr)\bigr\Vert ^{2}, $$
$$ \bigl\Vert (1-\xi)x+\xi\mathcal{U}\bigl((1-\eta)x+\eta\mathcal{U}x \bigr)-u^{\dagger}\bigr\Vert \le\bigl\Vert x-u^{\dagger}\bigr\Vert $$
Proposition 3.6
In any real Hilbert space
\(\mathcal{H}\), the following two equalities hold:
and
for all
\(u,u^{\dagger}\in\mathcal{H}\).
$$ \bigl\Vert \zeta u+(1-\zeta)u^{\dagger}\bigr\Vert ^{2}=\zeta \Vert u\Vert ^{2}+(1-\zeta)\bigl\Vert u^{\dagger}\bigr\Vert ^{2}-\zeta(1-\zeta)\bigl\Vert u-u^{\dagger}\bigr\Vert ^{2}, \quad \zeta\in[0,1] $$
(3.2)
$$ \bigl\Vert u+u^{\dagger}\bigr\Vert ^{2}=\Vert u\Vert ^{2}+2\bigl\langle u,u^{\dagger}\bigr\rangle +\bigl\Vert u^{\dagger}\bigr\Vert ^{2} $$
(3.3)
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 \(n\in\mathbb{N}\).
$$ \bigl\Vert z_{n}-\mathcal{A}z^{\dagger}\bigr\Vert ^{2}=\bigl\Vert \operatorname{proj}_{\mathcal{Q}}\mathcal {A}x_{n}-\operatorname{proj}_{\mathcal{Q}}\mathcal{A}z^{\dagger} \bigr\Vert ^{2}\le\bigl\Vert \mathcal {A}x_{n}- \mathcal{A}z^{\dagger}\bigr\Vert ^{2}-\Vert \mathcal{A}x_{n}-z_{n}\Vert ^{2}. $$
(3.4)
$$\operatorname{Fix}\bigl(\mathcal{S}\bigl((1-\eta_{n})\mathcal{I}+ \eta_{n}\mathcal{S}\bigr)\bigr)=\operatorname{Fix}(\mathcal{S}) $$
$$\operatorname{Fix}\bigl(\mathcal{T}\bigl((1-\gamma_{n})\mathcal{I}+ \gamma_{n}\mathcal {T}\bigr)\bigr)=\operatorname{Fix}(\mathcal{T}) $$
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.
$$\begin{aligned} \bigl\Vert v_{n}-\mathcal{A}z^{\dagger}\bigr\Vert =&\bigl\Vert \bigl[(1-\zeta_{n})\mathcal{I}+\zeta_{n}\mathcal {S} \bigl((1-\eta_{n})\mathcal{I}+\eta_{n}\mathcal{S}\bigr) \bigr]z_{n}-\mathcal{A}z^{\dagger}\bigr\Vert \\ =&\bigl\Vert \bigl[(1-\zeta_{n})\mathcal{I}+\zeta_{n} \mathcal{S}\bigl((1-\eta_{n})\mathcal {I}+\eta_{n}\mathcal{S} \bigr)\bigr]z_{n} \\ &{} -\bigl[(1-\zeta_{n})\mathcal{I}+\zeta_{n}\mathcal{S} \bigl((1-\eta_{n})\mathcal {I}+\eta_{n}\mathcal{S}\bigr)\bigr] \mathcal{A}z^{\dagger}\bigr\Vert \\ \le&\bigl\Vert z_{n}-\mathcal{A}z^{\dagger}\bigr\Vert . \end{aligned}$$
$$ \bigl\Vert v_{n}-\mathcal{A}z^{\dagger}\bigr\Vert ^{2}\le\bigl\Vert \mathcal{A}x_{n}- \mathcal{A}z^{\dagger}\bigr\Vert ^{2}-\Vert \mathcal{A}x_{n}-z_{n}\Vert ^{2}. $$
(3.5)
$$\begin{aligned} \bigl\Vert x_{n+1}-z^{\dagger}\bigr\Vert =&\bigl\Vert \bigl[(1-\beta_{n})\mathcal{I}+\beta_{n}\mathcal {T} \bigl((1-\gamma_{n})\mathcal{I}+\gamma_{n}\mathcal{T}\bigr) \bigr]u_{n}-z^{\dagger}\bigr\Vert \\ =&\bigl\Vert \bigl[(1-\beta_{n})\mathcal{I}+\beta_{n} \mathcal{T}\bigl((1-\gamma_{n})\mathcal {I}+\gamma_{n} \mathcal{T}\bigr)\bigr]u_{n} \\ &{} -\bigl[(1-\beta_{n})\mathcal{I}+\beta_{n}\mathcal{T} \bigl((1-\gamma_{n})\mathcal {I}+\gamma_{n}\mathcal{T}\bigr) \bigr]z^{\dagger}\bigr\Vert \\ \le&\bigl\Vert u_{n}-z^{\dagger}\bigr\Vert . \end{aligned}$$
(3.6)
$$ \bigl\Vert u_{n}-z^{\dagger}\bigr\Vert =\bigl\Vert \operatorname{proj}_{\mathcal{C}} y_{n}-\operatorname{proj}_{\mathcal{C}}z^{\dagger} \bigr\Vert \le\bigl\Vert y_{n}-z^{\dagger}\bigr\Vert . $$
(3.7)
$$\begin{aligned} \bigl\Vert y_{n}-z^{\dagger}\bigr\Vert =&\bigl\Vert \alpha_{n}\gamma f(x_{n})+(\mathcal{I}- \alpha_{n}\mathcal {B}) \bigl(x_{n}-\delta\mathcal{A}^{*}( \mathcal{A}x_{n}-v_{n})\bigr)-z^{\dagger}\bigr\Vert \\ =&\bigl\Vert \alpha_{n}\gamma\bigl(f(x_{n})-f \bigl(z^{\dagger}\bigr)\bigr)+\alpha_{n}\bigl(\gamma f \bigl(z^{\dagger }\bigr)-\mathcal{B}z^{\dagger}\bigr) \\ &{} +(\mathcal{I}-\alpha_{n}\mathcal{B}) \bigl(x_{n}-z^{\dagger}- \delta\mathcal {A}^{*}(\mathcal{A}x_{n}-v_{n})\bigr)\bigr\Vert \\ \le&\alpha_{n}\gamma\bigl\Vert f(x_{n})-f \bigl(z^{\dagger}\bigr)\bigr\Vert +\alpha_{n}\bigl\Vert \gamma f \bigl(z^{\dagger }\bigr)-\mathcal{B}z^{\dagger}\bigr\Vert \\ &{} +\Vert \mathcal{I}-\alpha_{n}\mathcal{B}\Vert \bigl\Vert x_{n}-z^{\dagger}-\delta\mathcal {A}^{*}(\mathcal{A}x_{n}-v_{n}) \bigr\Vert \\ \le&\alpha_{n}\gamma\rho\bigl\Vert x_{n}-z^{\dagger} \bigr\Vert +\alpha_{n}\bigl\Vert \gamma f\bigl(z^{\dagger } \bigr)-\mathcal{B}z^{\dagger}\bigr\Vert \\ &{} +(1-\alpha_{n}\sigma)\bigl\Vert x_{n}-z^{\dagger}+ \delta\mathcal {A}^{*}(v_{n}-\mathcal{A}x_{n})\bigr\Vert . \end{aligned}$$
(3.8)
$$\begin{aligned} \bigl\langle x_{n}-z^{\dagger}, \mathcal{A}^{*}(v_{n}-\mathcal{A}x_{n})\bigr\rangle =&\bigl\langle \mathcal{A}\bigl(x_{n}-z^{\dagger}\bigr), v_{n}- \mathcal{A}x_{n}\bigr\rangle \\ =&\bigl\langle \mathcal{A}x_{n}-\mathcal{A}z^{\dagger}+v_{n}- \mathcal {A}x_{n}-(v_{n}-\mathcal{A}x_{n}), v_{n}-\mathcal{A}x_{n}\bigr\rangle \\ =&\bigl\langle v_{n}-\mathcal{A}z^{\dagger}, v_{n}- \mathcal{A}x_{n}\bigr\rangle -\| v_{n}-\mathcal{A}x_{n} \|^{2}. \end{aligned}$$
(3.9)
$$ \bigl\langle v_{n}-\mathcal{A}z^{\dagger}, v_{n}-\mathcal{A}x_{n}\bigr\rangle =\frac{1}{2} \bigl( \bigl\Vert v_{n}-\mathcal{A}z^{\dagger}\bigr\Vert ^{2}+\Vert v_{n}-\mathcal{A}x_{n}\Vert ^{2}-\bigl\Vert \mathcal{A}x_{n}-\mathcal{A}z^{\dagger} \bigr\Vert ^{2} \bigr). $$
(3.10)
$$\begin{aligned} \bigl\langle x_{n}-z^{\dagger}, \mathcal{A}^{*}(v_{n}-\mathcal{A}x_{n})\bigr\rangle =& \frac{1}{2} \bigl(\bigl\Vert v_{n}-\mathcal{A}z^{\dagger} \bigr\Vert ^{2}+\Vert v_{n}-\mathcal{A}x_{n} \Vert ^{2}-\bigl\Vert \mathcal{A}x_{n}- \mathcal{A}z^{\dagger}\bigr\Vert ^{2} \bigr) \\ &{} -\Vert v_{n}-\mathcal{A}x_{n}\Vert ^{2} \\ \le&\frac{1}{2} \bigl(\bigl\Vert \mathcal{A}x_{n}- \mathcal{A}z^{\dagger}\bigr\Vert ^{2}-\Vert z_{n}- \mathcal{A}x_{n}\Vert ^{2}+\Vert v_{n}- \mathcal{A}x_{n}\Vert ^{2} \\ &{} -\bigl\Vert \mathcal{A}x_{n}-\mathcal{A}z^{\dagger}\bigr\Vert ^{2} \bigr)-\Vert v_{n}-\mathcal {A}x_{n} \Vert ^{2} \\ =&-\frac{1}{2}\Vert z_{n}-\mathcal{A}x_{n}\Vert ^{2}-\frac{1}{2}\Vert v_{n}-\mathcal{A}x_{n} \Vert ^{2}. \end{aligned}$$
(3.11)
$$\begin{aligned} \bigl\Vert x_{n}-z^{\dagger}+\delta\mathcal{A}^{*}(v_{n}- \mathcal{A}x_{n})\bigr\Vert ^{2} =&\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}+\delta^{2} \bigl\Vert \mathcal{A}^{*}(v_{n}-\mathcal{A}x_{n})\bigr\Vert ^{2} \\ &{} +2\delta\bigl\langle x_{n}-z^{\dagger}, \mathcal{A}^{*}(v_{n}- \mathcal {A}x_{n})\bigr\rangle . \end{aligned}$$
$$\begin{aligned} \bigl\Vert x_{n}-z^{\dagger}+\delta \mathcal{A}^{*}(v_{n}-\mathcal{A}x_{n})\bigr\Vert ^{2} \le&\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}+\delta^{2}\Vert \mathcal{A}\Vert ^{2} \Vert v_{n}-\mathcal{A}x_{n}\Vert ^{2} \\ &{} -\delta \Vert z_{n}-\mathcal{A}x_{n}\Vert ^{2}-\delta \Vert v_{n}-\mathcal{A}x_{n}\Vert ^{2} \\ =&\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}+ \bigl(\delta^{2}\Vert \mathcal{A}\Vert ^{2}-\delta\bigr) \Vert v_{n}-\mathcal {A}x_{n}\Vert ^{2} \\ &{} -\delta \Vert z_{n}-\mathcal{A}x_{n}\Vert ^{2}. \end{aligned}$$
(3.12)
$$ \bigl\Vert x_{n}-z^{\dagger}+\delta\mathcal{A}^{*}(v_{n}- \mathcal{A}x_{n})\bigr\Vert ^{2}\le\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}. $$
$$ \bigl\Vert x_{n}-z^{\dagger}+\delta \mathcal{A}^{*}(v_{n}-\mathcal{A}x_{n})\bigr\Vert \le\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert . $$
(3.13)
$$\begin{aligned} \begin{aligned}[b] \bigl\Vert y_{n}-z^{\dagger}\bigr\Vert &\le \alpha_{n}\gamma\rho\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert +\alpha_{n}\bigl\Vert \gamma f\bigl(z^{\dagger}\bigr)- \mathcal{B}z^{\dagger}\bigr\Vert +(1-\alpha_{n}\sigma)\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert \\ &=\bigl[1-(\sigma-\gamma\rho)\alpha_{n}\bigr]\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert +\alpha_{n}\bigl\Vert \gamma f\bigl(z^{\dagger}\bigr)-\mathcal{B}z^{\dagger}\bigr\Vert . \end{aligned} \end{aligned}$$
(3.14)
$$\begin{aligned} \bigl\Vert x_{n+1}-z^{\dagger}\bigr\Vert \le&\bigl[1-(\sigma- \gamma\rho)\alpha_{n}\bigr]\bigl\Vert x_{n}-z^{\dagger} \bigr\Vert +\alpha_{n}\bigl\Vert \gamma f\bigl(z^{\dagger} \bigr)-\mathcal{B}z^{\dagger}\bigr\Vert \\ =&\bigl[1-(\sigma-\gamma\rho)\alpha_{n}\bigr]\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert +(\sigma-\gamma\rho ) \alpha_{n}\frac{\Vert \gamma f(z^{\dagger})-\mathcal{B}z^{\dagger} \Vert }{\sigma-\gamma \rho}. \end{aligned}$$
$$ \bigl\Vert x_{n}-z^{\dagger}\bigr\Vert \le\max\biggl\{ \bigl\| x_{0}-z^{\dagger}\bigr\| , \frac{\|\gamma f(z^{\dagger })-\mathcal{B}z^{\dagger}\|}{\sigma-\gamma\rho}\biggr\} . $$
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).
$$\begin{aligned} \bigl\Vert u_{n}-z^{\dagger}\bigr\Vert ^{2} =&\bigl\Vert \operatorname{proj}_{\mathcal{C}}y_{n}-z^{\dagger} \bigr\Vert ^{2} \\ \le&\bigl\Vert y_{n}-z^{\dagger}\bigr\Vert ^{2}- \Vert \operatorname{proj}_{\mathcal{C}} y_{n}-y_{n}\Vert ^{2} \\ =&\bigl\Vert y_{n}-z^{\dagger}\bigr\Vert ^{2}- \Vert u_{n}-y_{n}\Vert ^{2}. \end{aligned}$$
(3.15)
$$\begin{aligned} \bigl\Vert x_{n+1}-z^{\dagger}\bigr\Vert ^{2} \le& \bigl\Vert u_{n}-z^{\dagger}\bigr\Vert ^{2} \\ \le&\bigl\Vert y_{n}-z^{\dagger}\bigr\Vert ^{2}- \Vert u_{n}-y_{n}\Vert ^{2} \\ \le&\bigl[1-(\sigma-\gamma\rho)\alpha_{n}\bigr]\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}+\frac{\alpha _{n}}{\sigma-\gamma\rho} \bigl\Vert \gamma f\bigl(z^{\dagger}\bigr)-\mathcal{B}z^{\dagger}\bigr\Vert ^{2}-\Vert u_{n}-y_{n}\Vert ^{2}. \end{aligned}$$
$$ \Vert u_{n}-y_{n}\Vert ^{2}\le \bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}-\bigl\Vert x_{n+1}-z^{\dagger}\bigr\Vert ^{2}+ \frac{\alpha _{n}}{\sigma-\gamma\rho}\bigl\Vert \gamma f\bigl(z^{\dagger}\bigr)- \mathcal{B}z^{\dagger}\bigr\Vert ^{2}. $$
(3.16)
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 From (3.1) and (3.2), we have 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, \(z\in\Gamma\).
$$ \lim_{n\to\infty}\|u_{n}-y_{n}\|=0. $$
(3.17)
$$\begin{aligned} \bigl\Vert y_{n}-z^{\dagger}\bigr\Vert \le& \alpha_{n}\gamma\rho\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert +\alpha_{n}\bigl\Vert \gamma f\bigl(z^{\dagger}\bigr)- \mathcal{B}z^{\dagger}\bigr\Vert \\ &{} +(1-\alpha_{n}\sigma)\bigl\Vert x_{n}-z^{\dagger}+ \delta\mathcal {A}^{*}(v_{n}-\mathcal{A}x_{n})\bigr\Vert \\ =&\alpha_{n}\sigma\frac{\gamma\rho \Vert x_{n}-z^{\dagger} \Vert +\Vert \gamma f(z^{\dagger })-\mathcal{B}z^{\dagger} \Vert }{\sigma} \\ &{} +(1-\alpha_{n}\sigma)\bigl\Vert x_{n}-z^{\dagger}+ \delta\mathcal {A}^{*}(v_{n}-\mathcal{A}x_{n})\bigr\Vert . \end{aligned}$$
(3.18)
$$\sup_{n}\biggl\{ \frac{\gamma\rho\|x_{n}-z^{\dagger}\|+\|\gamma f(z^{\dagger })-\mathcal{B}z^{\dagger}\|}{\sigma}\biggr\} < M. $$
$$ \bigl\Vert y_{n}-z^{\dagger}\bigr\Vert ^{2}\le\alpha_{n}\sigma M^{2}+(1- \alpha_{n}\sigma)\bigl\Vert x_{n}-z^{\dagger }+\delta \mathcal{A}^{*}(v_{n}-\mathcal{A}x_{n})\bigr\Vert ^{2}. $$
(3.19)
$$\begin{aligned} \bigl\Vert x_{n+1}-z^{\dagger}\bigr\Vert ^{2} \le& \bigl\Vert y_{n}-z^{\dagger}\bigr\Vert ^{2} \\ \le&(1-\sigma\alpha_{n})\bigl\Vert x_{n}-z^{\dagger} \bigr\Vert ^{2}+(1-\sigma\alpha_{n}) \bigl( \delta^{2}\Vert \mathcal{A}\Vert ^{2}-\delta\bigr)\Vert v_{n}-\mathcal{A}x_{n}\Vert ^{2} \\ &{} -(1-\sigma\alpha_{n})\delta \Vert z_{n}- \mathcal{A}x_{n}\Vert ^{2}+\alpha_{n}\sigma M^{2}. \end{aligned}$$
$$\begin{aligned} 0 \le&(1-\sigma\alpha_{n}) \bigl(\delta-\delta^{2}\Vert \mathcal{A}\Vert ^{2}\bigr)\Vert v_{n}-\mathcal {A}x_{n}\Vert ^{2}+(1-\sigma\alpha_{n})\delta \Vert z_{n}-\mathcal{A}x_{n}\Vert ^{2} \\ \le&(1-\sigma\alpha_{n})\bigl\Vert x_{n}-z^{\dagger} \bigr\Vert ^{2}-\bigl\Vert x_{n+1}-z^{\dagger}\bigr\Vert ^{2}+\alpha _{n}\sigma M^{2}, \end{aligned}$$
$$ \lim_{n\to\infty}\|v_{n}- \mathcal{A}x_{n}\|=\lim_{n\to\infty}\| z_{n}- \mathcal{A}x_{n}\|=0. $$
(3.20)
$$ \lim_{n\to\infty}\|v_{n}-z_{n}\|=0. $$
(3.21)
$$ \lim_{n\to\infty}\bigl\Vert z_{n}- \mathcal{S}\bigl((1-\eta_{n})\mathcal{I}+\eta_{n}\mathcal {S} \bigr)z_{n}\bigr\Vert =\lim_{n\to\infty}\bigl\Vert \mathcal{A}x_{n}-\mathcal{S}\bigl((1-\eta _{n})\mathcal{I}+ \eta_{n}\mathcal{S}\bigr)\mathcal{A}x_{n}\bigr\Vert =0. $$
(3.22)
$$\begin{aligned} \Vert \mathcal{A}x_{n}-\mathcal{S}\mathcal{A}x_{n}\Vert \le&\bigl\Vert \mathcal {A}x_{n}-S\bigl((1-\eta_{n}) \mathcal{I}+\eta_{n}\mathcal{S}\bigr)\mathcal{A}x_{n}\bigr\Vert \\ &{} +\bigl\Vert \mathcal{S}\bigl((1-\eta_{n})\mathcal{I}+ \eta_{n}\mathcal{S}\bigr)\mathcal {A}x_{n}-\mathcal{S} \mathcal{A}x_{n}\bigr\Vert \\ \le&\bigl\Vert \mathcal{A}x_{n}-\mathcal{S}\bigl((1- \eta_{n})\mathcal{I}+\eta_{n}\mathcal {S}\bigr) \mathcal{A}x_{n}\bigr\Vert +\mathcal{L}_{1} \eta_{n}\Vert \mathcal{A}x_{n}-\mathcal {S} \mathcal{A}x_{n}\Vert , \end{aligned}$$
$$ \Vert \mathcal{A}x_{n}-\mathcal{S}\mathcal{A}x_{n}\Vert \le\frac{1}{1-\mathcal {L}_{1}\eta_{n}}\bigl\Vert \mathcal{A}x_{n}-\mathcal{S}\bigl((1- \eta_{n})\mathcal{I}+\eta _{n}\mathcal{S}\bigr) \mathcal{A}x_{n}\bigr\Vert . $$
$$ \lim_{n\to\infty}\|\mathcal{A}x_{n}- \mathcal{S}\mathcal{A}x_{n}\|=0. $$
(3.23)
$$\begin{aligned} \Vert y_{n}-x_{n}\Vert =&\bigl\Vert \alpha_{n}\gamma f(x_{n})-\delta\mathcal{A}^{*}(\mathcal {A}x_{n}-v_{n})-\alpha_{n}\mathcal{B} \bigl(x_{n}-\delta\mathcal{A}^{*}(\mathcal {A}x_{n}-v_{n}) \bigr)\bigr\Vert \\ \le&\delta \Vert \mathcal{A}\Vert \Vert v_{n}- \mathcal{A}x_{n}\Vert +\alpha_{n}\bigl\Vert \gamma f(x_{n})-\mathcal{B}\bigl(x_{n}-\delta\mathcal{A}^{*}( \mathcal{A}x_{n}-v_{n})\bigr)\bigr\Vert . \end{aligned}$$
$$ \lim_{n\to\infty}\|x_{n}-y_{n}\|=0. $$
(3.24)
$$\begin{aligned} \bigl\Vert x_{n+1}-z^{\dagger}\bigr\Vert ^{2} =&\bigl\Vert (1-\beta_{n}) \bigl(u_{n}-z^{\dagger} \bigr)+\beta_{n}\bigl[\mathcal {T}\bigl((1-\gamma_{n})u_{n}+ \gamma_{n}\mathcal{T}u_{n}\bigr)-z^{\dagger}\bigr]\bigr\Vert ^{2} \\ =&(1-\beta_{n})\bigl\Vert u_{n}-z^{\dagger}\bigr\Vert ^{2}+\beta_{n}\bigl\Vert \mathcal{T}\bigl((1-\gamma _{n})u_{n}+\gamma_{n}\mathcal{T}u_{n} \bigr)-z^{\dagger}\bigr\Vert ^{2} \\ &{} -\beta_{n}(1-\beta_{n})\bigl\Vert \mathcal{T} \bigl((1-\gamma_{n})u_{n}+\gamma _{n} \mathcal{T}u_{n}\bigr)-u_{n}\bigr\Vert ^{2}. \end{aligned}$$
(3.25)
$$\begin{aligned}& \bigl\Vert \mathcal{T}\bigl((1-\gamma_{n})u_{n}+ \gamma_{n} \mathcal{T}u_{n}\bigr)-z^{\dagger}\bigr\Vert ^{2} \\& \quad \le\bigl\Vert u_{n}-z^{\dagger}\bigr\Vert ^{2}+(1-\gamma_{n})\bigl\Vert u_{n}-\mathcal{T} \bigl((1-\gamma _{n})u_{n}+\gamma_{n} \mathcal{T}u_{n}\bigr)\bigr\Vert ^{2}. \end{aligned}$$
(3.26)
$$\begin{aligned} \bigl\Vert x_{n+1}-z^{\dagger}\bigr\Vert ^{2} \le& \bigl\Vert u_{n}-z^{\dagger}\bigr\Vert ^{2}- \beta_{n}(\gamma_{n}-\beta_{n})\bigl\Vert u_{n}-\mathcal{T}\bigl((1-\gamma_{n})u_{n}+ \gamma_{n}\mathcal{T}u_{n}\bigr)\bigr\Vert ^{2} \\ \le&\alpha_{n}\sigma M^{2}+(1-\alpha_{n}\sigma) \bigl\Vert x_{n}-z^{\dagger}+\delta\mathcal {A}^{*}(v_{n}- \mathcal{A}x_{n})\bigr\Vert ^{2} \\ &{} -\beta_{n}(\gamma_{n}-\beta_{n})\bigl\Vert u_{n}-\mathcal{T}\bigl((1-\gamma_{n})u_{n}+\gamma _{n}\mathcal{T}u_{n}\bigr)\bigr\Vert ^{2} \\ \le&\alpha_{n}\sigma M^{2}+\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}-\beta_{n}( \gamma_{n}-\beta_{n})\bigl\Vert u_{n}-\mathcal{T} \bigl((1-\gamma_{n})u_{n}+\gamma_{n} \mathcal{T}u_{n}\bigr)\bigr\Vert ^{2}. \end{aligned}$$
$$\begin{aligned}& \beta_{n}(\gamma_{n}-\beta_{n})\bigl\Vert u_{n}-\mathcal{T}\bigl((1-\gamma_{n})u_{n}+\gamma _{n}\mathcal{T}u_{n}\bigr)\bigr\Vert ^{2} \\& \quad \le\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}-\bigl\Vert x_{n+1}-z^{\dagger}\bigr\Vert ^{2}+\alpha_{n}\sigma M^{2}. \end{aligned}$$
$$ \lim_{n\to\infty}\bigl\Vert u_{n}- \mathcal{T}\bigl((1-\gamma_{n})u_{n}+\gamma_{n} \mathcal {T}u_{n}\bigr)\bigr\Vert =0. $$
(3.27)
$$\begin{aligned} \Vert u_{n}-\mathcal{T}u_{n}\Vert \le&\bigl\Vert u_{n}-\mathcal{T}\bigl((1-\gamma_{n})u_{n}+\gamma _{n}\mathcal{T}u_{n}\bigr)\bigr\Vert +\bigl\Vert \mathcal{T}\bigl((1-\gamma_{n})u_{n}+\gamma_{n} \mathcal {T}u_{n}\bigr)-\mathcal{T}u_{n}\bigr\Vert \\ \le&\bigl\Vert u_{n}-\mathcal{T}\bigl((1-\gamma_{n})u_{n}+ \gamma_{n}\mathcal{T}u_{n}\bigr)\bigr\Vert +\mathcal {L}_{2}\gamma_{n}\Vert u_{n}- \mathcal{T}u_{n}\Vert . \end{aligned}$$
$$ \|u_{n}-\mathcal{T}u_{n}\|\le\frac{1}{1-\mathcal{L}_{2}\gamma_{n}}\bigl\Vert u_{n}-\mathcal{T}\bigl((1-\gamma_{n})u_{n}+ \gamma_{n}\mathcal{T}u_{n}\bigr)\bigr\Vert . $$
$$ \lim_{n\to\infty}\|u_{n}- \mathcal{T}u_{n}\|=0. $$
(3.28)
$$\limsup_{n\to\infty}\bigl\langle \gamma f\bigl(z^{\dagger}\bigr)- \mathcal{B}z^{\dagger}, y_{n}-z^{\dagger}\bigr\rangle \le0. $$
$$ \limsup_{n\to\infty}\bigl\langle \gamma f \bigl(z^{\dagger}\bigr)-\mathcal{B}z^{\dagger}, y_{n}-z^{\dagger} \bigr\rangle =\lim_{i\to\infty}\bigl\langle \gamma f\bigl(z^{\dagger } \bigr)-\mathcal{B}z^{\dagger}, y_{n_{i}}-z^{\dagger}\bigr\rangle . $$
(3.29)
$$ \textstyle\begin{cases} x_{n_{i}}\rightharpoonup z, \\ y_{n_{i}}\rightharpoonup z, \\ u_{n_{i}}\rightharpoonup z \end{cases} $$
(3.30)
$$ \textstyle\begin{cases} \mathcal{A}x_{n_{i}}\rightharpoonup\mathcal{A}z, \\ \mathcal{A}y_{n_{i}}\rightharpoonup\mathcal{A}z, \\ \mathcal{A}u_{n_{i}}\rightharpoonup\mathcal{A}z. \end{cases} $$
(3.31)
Therefore, From (3.1), we have It follows that Therefore, Applying Lemma 2.6 and (3.32) to (3.33), we deduce \(x_{n}\to z^{\dagger}\).
$$\begin{aligned} \limsup_{n\to\infty}\bigl\langle \gamma f \bigl(z^{\dagger}\bigr)-\mathcal{B}z^{\dagger}, y_{n}-z^{\dagger} \bigr\rangle =&\lim_{i\to\infty}\bigl\langle \gamma f \bigl(z^{\dagger }\bigr)-\mathcal{B}z^{\dagger}, y_{n_{i}}-z^{\dagger} \bigr\rangle \\ =&\lim_{i\to\infty}\bigl\langle \gamma f\bigl(z^{\dagger}\bigr)- \mathcal{B}z^{\dagger}, z-z^{\dagger}\bigr\rangle \\ \le&0. \end{aligned}$$
(3.32)
$$\begin{aligned} \bigl\Vert y_{n}-z^{\dagger}\bigr\Vert ^{2} =&\bigl\Vert \alpha_{n}\gamma\bigl(f(x_{n})-f\bigl(z^{\dagger} \bigr)\bigr)+\alpha_{n}\bigl(\gamma f\bigl(z^{\dagger }\bigr)- \mathcal{B}z^{\dagger}\bigr) \\ &{} +(\mathcal{I}-\alpha_{n}\mathcal{B}) \bigl(x_{n}-z^{\dagger}- \delta\mathcal {A}^{*}(\mathcal{A}x_{n}-v_{n})\bigr)\bigr\Vert ^{2} \\ \le&\Vert \mathcal{I}-\alpha_{n}\mathcal{B}\Vert ^{2} \bigl\Vert x_{n}-z^{\dagger}-\delta\mathcal {A}^{*}( \mathcal{A}x_{n}-v_{n})\bigr\Vert ^{2} \\ &{} +2\alpha_{n}\gamma\bigl\langle f(x_{n})-f \bigl(z^{\dagger}\bigr), y_{n}-z^{\dagger}\bigr\rangle +2 \alpha_{n}\bigl\langle \gamma f\bigl(z^{\dagger}\bigr)- \mathcal{B}z^{\dagger}, y_{n}-z^{\dagger }\bigr\rangle \\ \le&(1-\alpha_{n}\sigma)^{2}\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}+2\alpha_{n} \gamma\bigl\Vert f(x_{n})-f\bigl(z^{\dagger}\bigr)\bigr\Vert \bigl\Vert y_{n}-z^{\dagger}\bigr\Vert \\ &{} +2\alpha_{n}\bigl\langle \gamma f\bigl(z^{\dagger}\bigr)- \mathcal{B}z^{\dagger}, y_{n}-z^{\dagger}\bigr\rangle \\ \le&(1-\alpha_{n}\sigma)^{2}\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}+\alpha_{n} \gamma\rho\bigl\Vert x_{n}-z^{\dagger}\bigr\Vert ^{2}+ \alpha_{n}\gamma\rho\bigl\Vert y_{n}-z^{\dagger}\bigr\Vert ^{2} \\ &{} +2\alpha_{n}\bigl\langle \gamma f\bigl(z^{\dagger}\bigr)- \mathcal{B}z^{\dagger}, y_{n}-z^{\dagger}\bigr\rangle . \end{aligned}$$
$$\begin{aligned} \bigl\Vert y_{n}-z^{\dagger}\bigr\Vert ^{2} \le& \biggl[1-\frac{2(\sigma-\gamma\rho)\alpha_{n}}{1-\gamma\rho\alpha _{n}} \biggr]\bigl\Vert x_{n}-z^{\dagger} \bigr\Vert ^{2}+\frac{\sigma^{2}\alpha_{n}^{2}}{1-\gamma\rho\alpha _{n}}\bigl\Vert x_{n}-z^{\dagger} \bigr\Vert ^{2} \\ &{} +\frac{2\alpha_{n}}{1-\gamma\rho\alpha_{n}}\bigl\langle \gamma f\bigl(z^{\dagger }\bigr)- \mathcal{B}z^{\dagger}, y_{n}-z^{\dagger}\bigr\rangle . \end{aligned}$$
$$\begin{aligned} \bigl\Vert x_{n+1}-z^{\dagger}\bigr\Vert ^{2} \le&\bigl\Vert y_{n}-z^{\dagger}\bigr\Vert ^{2} \\ \le& \biggl[1-\frac{2(\sigma-\gamma\rho)\alpha_{n}}{1-\gamma\rho\alpha _{n}} \biggr]\bigl\Vert x_{n}-z^{\dagger} \bigr\Vert ^{2}+\frac{\sigma^{2}\alpha_{n}^{2}}{1-\gamma\rho\alpha _{n}}\bigl\Vert x_{n}-z^{\dagger} \bigr\Vert ^{2} \\ &{}+\frac{2\alpha_{n}}{1-\gamma\rho\alpha_{n}}\bigl\langle \gamma f\bigl(z^{\dagger }\bigr)- \mathcal{B}z^{\dagger}, y_{n}-z^{\dagger}\bigr\rangle . \end{aligned}$$
(3.33)
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 \(n\ge n_{0}\).
$$\bigl\Vert x_{n_{0}}-z^{\dagger}\bigr\Vert \le\bigl\Vert x_{n_{0}+1}-z^{\dagger}\bigr\Vert . $$
$$\omega_{n_{0}}\le\omega_{n_{0}+1}. $$
$$\tau(n)=\max\{l\in\mathbb{N}| n_{0}\le l\le n, \omega_{l} \le\omega_{l+1}\}. $$
$$\lim_{n\to\infty}\tau(n)=\infty $$
$$\omega_{\tau(n)}\le\omega_{\tau(n)+1} $$
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. □
$$\begin{aligned} \begin{aligned} &\lim_{n\to\infty} \Vert u_{\tau(n)}-y_{\tau(n)}\Vert = \lim_{n\to\infty} \Vert x_{\tau(n)}-y_{\tau(n)}\Vert =0, \\ &\lim_{n\to\infty} \Vert \mathcal{S}x_{\tau(n)}- \mathcal{A}x_{\tau(n)}\Vert =0 \end{aligned} \end{aligned}$$
$$ \lim_{n\to\infty} \Vert u_{\tau(n)}-\mathcal{T}u_{\tau(n)} \Vert =0. $$
$$\omega_{w}(y_{\tau(n)})\subset\Gamma. $$
$$ \limsup_{n\to\infty}\bigl\langle \gamma f \bigl(z^{\dagger}\bigr)-\mathcal{B}z^{\dagger },y_{\tau(n)}-z^{\dagger} \bigr\rangle \le0. $$
(3.34)
$$\begin{aligned} \omega_{\tau(n)}^{2} \le&\omega_{\tau(n)+1}^{2} \\ \le& \biggl[1-\frac{2(\sigma-\gamma\rho)\alpha_{\tau(n)}}{1-\gamma\rho \alpha_{\tau(n)}} \biggr]\omega_{\tau(n)}^{2}+ \frac{\sigma^{2}\alpha_{\tau (n)}^{2}}{1-\gamma\rho\alpha_{\tau(n)}}\omega_{\tau(n)}^{2} \\ &{}+\frac{2\alpha_{\tau(n)}}{1-\gamma\rho\alpha_{\tau(n)}}\bigl\langle \gamma f\bigl(z^{\dagger}\bigr)- \mathcal{B}z^{\dagger}, y_{\tau(n)}-z^{\dagger}\bigr\rangle . \end{aligned}$$
(3.35)
$$ \omega_{\tau(n)}^{2}\le\frac{2}{2(\sigma-\gamma\rho)-\sigma^{2}\alpha_{\tau (n)}}\bigl\langle \gamma f\bigl(z^{\dagger}\bigr)-\mathcal{B}z^{\dagger}, y_{\tau(n)}-z^{\dagger }\bigr\rangle . $$
(3.36)
$$\limsup_{n\to\infty}\omega_{\tau(n)}\le0, $$
$$ \lim_{n\to\infty}\omega_{\tau(n)}=0. $$
(3.37)
$$\limsup_{n\to\infty}\omega_{\tau(n)+1}^{2}\le\limsup _{n\to\infty}\omega _{\tau(n)}^{2}. $$
$$\lim_{n\to\infty}\omega_{\tau(n)+1}=0. $$
$$0\le\omega_{n}\le\max\{\omega_{\tau(n)}, \omega_{\tau(n)+1}\}. $$
4 Applications
Algorithm 4.1
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by for all \(n\in\mathbb{N}\).
$$ \textstyle\begin{cases} z_{n}=\operatorname{proj}_{\mathcal{Q}}\mathcal{A}x_{n}, \\ v_{n}=(1-\zeta_{n})z_{n}+\zeta_{n}\mathcal{S}((1-\eta_{n})z_{n}+\eta_{n}\mathcal {S}z_{n}), \\ y_{n}=(1-\alpha_{n})(x_{n}-\delta\mathcal{A}^{*}(\mathcal{A}x_{n}-v_{n})), \\ u_{n}=\operatorname{proj}_{\mathcal{C}}y_{n}, \\ x_{n+1}=(1-\beta_{n})u_{n}+\beta_{n}\mathcal{T}((1-\gamma_{n})u_{n}+\gamma _{n}\mathcal{T}u_{n}) \end{cases} $$
(4.1)
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:
Then the sequence
\(\{x_{n}\}\)
generated by algorithm (4.1) converges strongly to the minimum norm solution
\(u^{\clubsuit}\in\Gamma\).
(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)′:
\(0<\delta<\frac{1}{\|\mathcal{A}\|^{2}}\).
Algorithm 4.3
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by for all \(n\in\mathbb{N}\).
$$ x_{n+1}=\operatorname{proj}_{\mathcal{C}} \bigl[ \alpha_{n}\gamma f(x_{n})+(\mathcal {I}-\alpha_{n} \mathcal{B}) \bigl(x_{n}-\delta\mathcal{A}^{*}(\mathcal {A}x_{n}-\operatorname{proj}_{\mathcal{Q}}\mathcal{A}x_{n}) \bigr) \bigr] $$
(4.2)
Corollary 4.4
Assume that the following conditions are satisfied:
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
\(\Gamma_{1}\ne\emptyset\).
(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
\(\sigma>\gamma\rho\).
Algorithm 4.5
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by for all \(n\in\mathbb{N}\).
$$ x_{n+1}=\operatorname{proj}_{\mathcal{C}} \bigl[(1- \alpha_{n}) \bigl(x_{n}-\delta\mathcal {A}^{*}( \mathcal{A}x_{n}-\operatorname{proj}_{\mathcal{Q}}\mathcal{A}x_{n}) \bigr) \bigr] $$
(4.3)
Corollary 4.6
Assume that the following conditions are satisfied:
Then the sequence
\(\{x_{n}\}\)
generated by algorithm (4.3) converges strongly to the minimum norm solution
\(u^{\clubsuit}\in\Gamma _{1}\)
provided
\(\Gamma_{1}\ne\emptyset\).
(C1):
\(\lim_{n\to\infty}\alpha_{n}=0\)
and
\(\sum_{n=1}^{\infty}\alpha _{n}=\infty\),
(C4)′:
\(0<\delta<\frac{1}{\|\mathcal{A}\|^{2}}\).
Algorithm 4.7
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by for all \(n\in\mathbb{N}\).
$$ \textstyle\begin{cases} v_{n}=(1-\zeta_{n})\mathcal{A}x_{n}+\zeta_{n}\mathcal{S}((1-\eta_{n})\mathcal {A}x_{n}+\eta_{n}\mathcal{S}\mathcal{A}x_{n}), \\ y_{n}=\alpha_{n}\gamma f(x_{n})+(\mathcal{I}-\alpha_{n}\mathcal{B})(x_{n}-\delta \mathcal{A}^{*}(\mathcal{A}x_{n}-v_{n})), \\ x_{n+1}=(1-\beta_{n})y_{n}+\beta_{n}\mathcal{T}((1-\gamma_{n})y_{n}+\gamma _{n}\mathcal{T}y_{n}) \end{cases} $$
(4.4)
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:
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
\(\Gamma_{2}\ne\emptyset\).
(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
\(\sigma>\gamma\rho\).
Algorithm 4.9
For given \(x_{0}\in\mathcal{H}_{1}\) arbitrarily, define a sequence \(\{ x_{n}\}\) iteratively by for all \(n\in\mathbb{N}\).
$$ \textstyle\begin{cases} v_{n}=(1-\zeta_{n})\mathcal{A}x_{n}+\zeta_{n}\mathcal{S}((1-\eta_{n})\mathcal {A}x_{n}+\eta_{n}\mathcal{S}\mathcal{A}x_{n}), \\ y_{n}=(1-\alpha_{n})(x_{n}-\delta\mathcal{A}^{*}(\mathcal{A}x_{n}-v_{n})), \\ x_{n+1}=(1-\beta_{n})y_{n}+\beta_{n}\mathcal{T}((1-\gamma_{n})y_{n}+\gamma _{n}\mathcal{T}y_{n}) \end{cases} $$
(4.5)
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:
Then the sequence
\(\{x_{n}\}\)
generated by algorithm (4.5) converges strongly to the minimum norm solution
\(u^{\clubsuit}\in\Gamma _{2}\)
provided
\(\Gamma_{2}\ne\emptyset\).
(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)′:
\(0<\delta<\frac{1}{\|\mathcal{A}\|^{2}}\).
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.