1 Introduction-preliminaries

Let \(H_{1}\) and \(H_{2}\) be Hilbert spaces, and let C and Q be nonempty convex closed sets in \(H_{1}\) and \(H_{2}\), respectively. Let \(A:H_{1}\rightarrow H_{2}\) be a bounded linear mapping.

In 1994, Censor and Elfving [10] introduced the well-known split feasibility problem for modeling inverse problems formulated as follows:

$$ \text{Find} \quad x^{*} \in C\quad \text{such that } Ax^{*}\in Q. $$
(1.1)

It can be formulated as the following convex feasibility problem:

$$ \text{Find}\quad x^{*} \in C\cap A^{-1}(Q). $$

Both split feasibility and convex feasibility problems are much related to a number of real-world applications, for example, signal processing, intensity-modulated radiation therapy, and image reconstruction; see [9, 11, 35] and the references therein. Recently, a number of regularized iterative methods have been introduced and investigated for solutions of the feasibility problems in either Banach or Hilbert spaces by many authors; see [1,2,3,4,5, 16, 17, 19, 28, 31] and the references therein.

Let H be a real Hilbert space endowed with inner product \(\langle \cdot ,\cdot \rangle \) and induced norm \(\|\cdot \|\). Let S be a mapping on H. \(\operatorname{Fix}(S)\) stands for a fixed point set of S. Recall that S is said to be nonexpansive if

$$ \Vert Sx-Sy \Vert \leq \Vert x-y \Vert , \quad \forall x,y\in H. $$

It is well known that every nonexpansive mapping satisfies the following property:

$$ 2\bigl\langle Sx-Sy, (y-Sy)-(x-Sx) \bigr\rangle \leq \bigl\Vert ( x-Sx )-(y-Sy) \bigr\Vert ^{2}, \quad \forall x,y\in H. $$

The mapping S is said to be quasinonexpansive if

$$ \Vert x-Sy \Vert \leq \Vert x-y \Vert , \quad \forall x\in \operatorname{Fix}(S)\neq \emptyset ,y\in H. $$

It is obvious that quasinonexpansive mappings may not be continuous beyond their fixed-point sets. Every quasinonexpansive mapping S satisfies the following property:

$$ 2\bigl\langle x-Sy, (y-Sy) \bigr\rangle \leq \Vert y-Sy \Vert ^{2}, \quad \forall x\in \operatorname{Fix}(S)\neq \emptyset ,y\in H. $$
(1.2)

It is said to be firmly nonexpansive if

$$ \Vert Sx-Sy \Vert ^{2} \leq \langle Sx-Sy,x-y\rangle , \quad \forall x,y\in H. $$

It is is said to be firmly quasinonexpansive if

$$ \Vert x-Sy \Vert ^{2} \leq \langle x-Sy,x-y\rangle ,\quad \forall x \in \operatorname{Fix}(S)\neq \emptyset ,y\in H. $$

It is is said to be contractive if there exists a constant \(\kappa \in (0,1)\) such that

$$ \Vert Sx-Sy \Vert ^{2} \leq \kappa \Vert x-y \Vert ,\quad \forall x,y\in H. $$

Contractive mappings and their extensions are important classes of nonlinear mappings since they are connected with differential equations and nonsmooth optimization; see [7, 8, 14, 21] and the references therein. Recently, they have been extensively analyzed via projection-based iterative methods. It deserves mentioning that the methods based on nearest-point projections are not efficient from the viewpoint of numerical computation. Let \(\operatorname{Proj}_{C}^{H}\) be the nearest-point (metric) projection from H onto C, that is,

$$ \operatorname{Proj}_{C}^{H}y:=\bigl\{ x\in C: \Vert x-y \Vert =\operatorname{dist} _{C}(y)\bigr\} , $$

where \(\operatorname{dist}_{C}(y):=\inf_{x\in C}\|x-y\|\) for \(y\in H\).

To avoid using nearest projections, Yamada [33] recently studied a descent method, which is known as the Yamada descent algorithm. This algorithm is as follows:

$$ u_{n+1}= (I-\alpha _{n+1}\mu F) Tu_{n},\quad \forall n\in \mathbb{N}, $$

where \(\{\alpha _{n}\}\) is a real sequence in \((0,1)\), μ is some positive real number, T is a nonexpansive mapping on H, and F is η-strongly monotone and \(\mathcal{L}\)-Lipschitz continuous on H. Recently, many authors studied the Yamada descent methods for nonexpansive nonlinear operators in Banach or Hilbert spaces; see [13, 22, 23, 26] and the references therein.

Now we recall some useful notions. Let \(F: C\rightarrow H\) be a nonself single-valued operator. It is called

  1. (i)

    monotone if

    $$ \bigl\langle x^{*}-x , Fx^{*}-Fx\bigr\rangle \geq 0,\quad \forall x^{*}, x \in C; $$
  2. (ii)

    strongly monotone if there exists a positive constant \(\eta > 0\) such that

    $$ \eta \bigl\Vert x^{*}-x \bigr\Vert ^{2}\leq \bigl\langle x^{*}-x , Fx^{*}-Fx\bigr\rangle ,\quad \forall x^{*}, x\in C. $$
  3. (iii)

    \(\mathcal{L}\)-Lipschitz if there exists \(\mathcal{L}> 0\) such that

    $$ \bigl\Vert Fx-Fx^{*} \bigr\Vert \leq \mathcal{L} \bigl\Vert x-x^{*} \bigr\Vert ,\quad \forall x^{*},x \in C. $$

Let \(M: H\rightarrow 2^{H}\) be a set-valued monotone mapping. The zero-point set of M is denoted by \(M^{-1}(0)\). Recall that M is said to be monotone if, for all \(x, y\in H\), \(u\in Mx\), and \(v\in My\)

$$ \langle x-y, u-v\rangle \geq 0. $$

It is said to be maximal if its graph \(\operatorname{Graph}(M)\) is not properly contained in the graph of any other monotone mapping. If M is maximally monotone, then \(\operatorname{Graph}(M)\) is weakly strongly closed; see [24] and the references therein. A well-known fact is that for \((x, u)\in H\times H\), \(\langle x-y, u-v \rangle \geq 0\) for all \((y, v)\in \operatorname{Graph}(M)\) implies that \(u\in M(x)\) iff M is maximal. Let N be a maximal monotone operator with domain \(\operatorname{Dom}(N)\) and range H. Define the mapping \(\operatorname{Res}_{\lambda }^{N}: H\rightarrow \operatorname{Dom}(M)\) associated with index λ by

$$ \operatorname{Res}_{\lambda }^{N}x= (\lambda N+\mathrm{Id})^{-1}(x),\quad \forall x \in H, $$

where Id is the identity operator on H. If N is the subdifferential of proper convex lower semicontinuous functions, then the resolvent operator is the known proximity operator. The resolve operator plays a significant role in nonsmooth optimization problems. A variety of nonlinear problems, including variational inequalities and equilibrium problems, can be formulated as finding a zero of a maximal monotone operator. It is known that \(\operatorname{Fix}( \operatorname{Res}_{\lambda }^{N})=N^{-1}(0)\); see [15, 18, 20, 27, 34] and the references therein.

Let N be a set-valued maximal monotone operator on \(H_{1}\), and let M be a set-valued maximal monotone operator on \(H_{2}\). We consider the following split inclusion problem: find \(x^{*}\in H_{1}\) such that

$$ 0\in N\bigl(x^{*}\bigr), y^{*}=Ax^{*} \in H_{2} \quad \text{solves}\quad 0\in M\bigl(y ^{*}\bigr), $$
(1.3)

where A is a linear bounded mapping from \(H_{1}\) to \(H_{2}\). We denote by \(\operatorname{SIP}(M,N)\) the solution set of problem (1.3).

In this paper, we analyze iterative solutions of a split feasibility problem with common fixed-point constraints of a family of nonexpansive mappings. We present solution theorems of the feasibility problem under some weak assumptions imposed on different mappings. For our main result, we also need the following tools.

Let \(S_{i}\) be a nonexpansive mapping on C, and let \(\eta _{i}\) be real numbers with \(0<\eta _{i}<1\) for each \(i\geq 1\). Let \(W_{n}\) be a mapping on C defined for each \(n\geq 1\) by

$$ \begin{aligned} &U_{n,n+1}=I, \\ &U_{n,n}=(1-\eta _{n})I+\eta _{n}S_{n}U_{n,n+1}, \\ &U _{n,n-1}=(1-\eta _{n-1})I+\eta _{n-1}S_{n-1}U_{n,n}, \\ &\vdots \\ &U _{n,k}=(1-\eta _{k})I+\eta _{k}S_{k}U_{n,k+1}, \\ &U_{n,k-1}=(1-\eta _{k-1})I+ \eta _{k-1}S_{k-1}U_{n,k}, \\ &\vdots \\ &U_{n,2}=(1-\eta _{2})I+\eta _{2}S_{2}U_{n,3}, \\ &U_{n,1}=(1-\eta _{1})I+\eta _{1}S_{1}U_{n,2}, \\ &W _{n}=U_{n,1}.\end{aligned} $$
(1.4)

It is clear that \(W_{n}:C\rightarrow C\), governed by \(S_{1},S_{2}, \ldots ,S_{n}\) and \(\eta _{1},\eta _{2},\ldots ,\eta _{n}\), is a nonexpansive mapping; see [29] and the references therein. We further assume that \(0<\eta _{i} \leq \eta < 1\) for \(i\geq 1\), where η is a constant in \((0,1)\).

Lemma 1.1

([29])

Let C be a convex and closed set in a Hilbert space H, and let \(S_{i}\) be nonexpansive mappings on C with fixed points. If \(\bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})\neq \emptyset \), then

  1. (1)

    \(\lim_{n\rightarrow \infty }U_{n,k}\) exists for each positive integer k and each \(x\in C\);

  2. (2)

    the mapping \(W:C\rightarrow C\) defined by

    $$ Wx:=\lim_{n\rightarrow \infty }W_{n}x=\lim _{n\rightarrow \infty }U _{n,1}x,\quad x\in C, $$
    (1.5)

    is a nonexpansive mapping with \(\operatorname{Fix}(W)=\bigcap_{i=1}^{ \infty }\operatorname{Fix}(S_{i})=\operatorname{Fix}(W_{n})\).

Lemma 1.2

([12])

Let C be a convex and closed set in a Hilbert space H, and let \(S_{i}\) be a nonexpansive mappings on C with fixed points. Assume that \(\bigcap_{i=1}^{\infty }F(S_{i})\neq \emptyset \). Then \(\lim_{n\rightarrow \infty }\sup_{x\in K}\|W_{n}x-Wx\|=0 \) for any bounded set \(K\subset C\).

Lemma 1.3

([33])

Let H be a Hilbert space. Let F be an \(\mathcal{L}\)-Lipschitz continuous and η-strongly monotone mapping on the space H. Let \(T^{\alpha }\) be a mapping on the space H defined by \(T^{\alpha }x=x-\mu \alpha Fx\) for \(x\in H\), where α is a real number in \((0,1)\). If \(0< \mathcal{L}^{2}\mu \in (0,2\eta )\) and \(\tau =1-\sqrt{1-\mu (2\eta -\mu \mathcal{L} ^{2})}\in (0,1]\), then

$$ \bigl\Vert T^{\alpha }x-T^{\alpha }y \bigr\Vert \leq (1- \tau \alpha ) \Vert x-y \Vert , \quad \forall x,y\in H. $$

Lemma 1.4

([32])

Let \(\{\alpha _{n}\}\), \(\{\beta _{n}\} \), and \(\{\gamma _{n} \} \) be sequences of real numbers such that \(\alpha _{n}\in [0,1]\), \(\sum_{n=1}^{\infty } \alpha _{n}= \infty \), \(\limsup_{n\rightarrow \infty } \beta _{n}\leq 0\), and \(\sum_{n=1}^{ \infty } \gamma _{n}<\infty \) Let \(\{\lambda _{n} \}\) be a sequence of nonnegative real numbers such that

$$ \lambda _{n+1}\leq (1-\alpha _{n} )\lambda _{n}+\alpha _{n}\beta _{n}+ \gamma _{n}. $$

Then \(\lim_{n\rightarrow \infty }\lambda _{n} =0 \).

Lemma 1.5

([25])

Let \(\{x_{n}\}\) be a sequence in a real Hilbert space H. If \(x_{n}\rightharpoonup x\), then

$$ \liminf_{n\rightarrow \infty } \Vert x_{n}-x \Vert < \liminf _{n\rightarrow \infty } \Vert x_{n}-y \Vert $$

for any \(y\in X\) with \(y\neq x\). This is also equivalent to

$$ \limsup_{n\rightarrow \infty } \Vert x_{n}-x \Vert < \limsup _{n\rightarrow \infty } \Vert x_{n}-y \Vert . $$

Lemma 1.6

([6, resolvent equality])

Let H be a Hilbert space. Let N be a set-valued maximal operator on H. For parameters \(\lambda >0\) and \(\mu >0\), we have

$$ \operatorname{Res}_{\mu }^{N} \biggl( \biggl(1-\frac{\mu }{\lambda } \biggr) \operatorname{Res}_{\lambda }^{N}x+ \frac{\mu }{\lambda }x \biggr)= \operatorname{Res}_{\lambda }^{N}x,\quad \forall x\in H. $$
(1.6)

Lemma 1.7

([30])

Let H be a Hilbert space. Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be bounded sequences in H with \(x_{n+1}=(1-\beta _{n})y _{n}+\beta _{n}x_{n}\) for all \(n\geq 0\) and

$$ \limsup_{n\rightarrow \infty } \bigl( \Vert y_{n+1}-y_{n} \Vert - \Vert x_{n+1}-x_{n} \Vert \bigr)\leq 0, $$

where \(\{\beta _{n}\}\) is a sequence in \((0,1)\) such that \(\liminf_{n\rightarrow \infty }\beta _{n}>0\) and \(\limsup_{n\rightarrow \infty }\beta _{n}<1\). Then \(\lim_{n\rightarrow \infty }\|y_{n}-x_{n}\|=0\).

2 Main results

Theorem 2.1

Let \(H_{1}\) and \(H_{2}\) be Hilbert spaces, and let N and M be set-valued maximal monotone mappings on \(H_{1}\) and \(H_{2}\), respectively. Let \(S_{i}\) be nonexpansive mappings on \(H_{1}\) for all integers \(i\geq 1\). Let \(F:H_{1}\rightarrow H_{1}\) be an \(\mathcal{L}\)-Lipschitz continuous and τ-strongly monotone mapping. Let A be a linear bounded operator from \(H_{1}\) to \(H_{2}\), and let \(A^{*}\) be its adjoint operator. Assume that \(\bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})\cap \operatorname{SIP}(M,N) \neq \emptyset \). Let \(\{ x_{n}\}\) be a vector sequence in \(H_{1}\) generated by the iterative process

$$ \textstyle\begin{cases} x_{1}\in H_{1}, \\ y_{n}=\gamma _{n}\operatorname{Res}_{s_{n}}^{N}(x_{n}+\gamma A^{*}( \operatorname{Res}_{r_{n}}^{M}-I)Ax_{n})+(1-\gamma _{n})x_{n}, \\ x_{n+1}=\beta _{n}(I-\mu \alpha _{n}F)W_{n}y_{n}+(1-\beta _{n})x_{n},\quad n\geq 1, \end{cases} $$

where γ and μ are two positive real numbers, \(\{s_{n}\}\) and \(\{r_{n}\}\) are two positive real number sequences, \(\{\alpha _{n} \}\), \(\{\beta _{n}\}\), and \(\{\gamma _{n}\}\) are real number sequences in \((0,1)\). Suppose that \(\gamma \in (0,\frac{1}{\|A\|^{2}})\), \(\mu \in (0, \frac{2 \tau }{\mathcal{L}^{2}})\), \(\liminf_{n\rightarrow \infty } s_{n}> 0\), \(\lim_{n\rightarrow \infty }|s_{n}-s_{n+1}|<\infty \), \(\liminf_{n\rightarrow \infty }r_{n}> 0\), \(\lim_{n\rightarrow \infty }|r_{n}-r_{n+1}|<\infty \), \(\sum_{n=1}^{ \infty }\alpha _{n}=\infty \), \(\{\beta _{n}\}\) is number sequence in \([\bar{\beta },\bar{\beta }']\), where β̄ and \(\bar{\beta }'\) are two real numbers in \((0,1)\), such that \(\lim_{n\rightarrow \infty }|\beta _{n+1}- \beta _{n}|=0\), and \(\{\gamma _{n}\}\) is a sequence in \([\bar{\gamma },1]\), where \(\bar{\gamma }\in (0,1]\), such that \(\lim_{n\rightarrow \infty }| \gamma _{n+1}- \gamma _{n}|=0\). Then the sequence \(\{ x_{n}\}\) converges strongly to \(\widetilde{x}\in H_{1}\), which is a unique solution of the variational inequality

$$ \langle \widetilde{x} -y, F\widetilde{x} \rangle \leq 0,\quad \forall y\in \bigcap _{i=1}^{\infty }\operatorname{Fix}(S_{i}) \cap \operatorname{SIP}(M,N) . $$

Proof

The proof is split into four steps.

Step 1. We prove that \(\{x_{n}\}\) is a bounded vector sequence in \(H_{1}\).

For any fixed \(p\in \bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i}) \cap \operatorname{SIP}(M,N)\), we conclude \(Ap=\operatorname{Res}_{r _{n}}^{M}Ap\), \(p=\operatorname{Res}_{s_{n}}^{N}p\), and \(p=S_{i}p\) for each \(i\geq 1\). Since Ap is a fixed point of \(\operatorname{Res} _{r_{n}}^{M}\) and \(\operatorname{Res}_{r_{n}}^{M}\) is a (firmly) nonexpansive mapping, we have

$$ \bigl\langle \bigl(\operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax_{n}, \operatorname{Res} _{r_{n}}^{M}Ax_{n}-Ap \bigr\rangle \leq \frac{ \Vert \operatorname{Res}_{r_{n}} ^{M}Ax_{n}-Ax_{n} \Vert ^{2}}{2}. $$
(2.1)

Putting

$$ z_{n}=\operatorname{Res}_{s_{n}}^{N} \bigl(x_{n}+\gamma A^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax_{n}\bigr), $$

(2.1) sends us to

$$ \begin{aligned}[b] & \Vert z_{n}-p \Vert ^{2} \\ &\quad \leq \bigl\Vert \gamma A^{*}\bigl(\operatorname{Res}_{r_{n}} ^{M}-I\bigr)Ax_{n}+(x_{n}-p) \bigr\Vert ^{2} \\ &\quad \leq \gamma ^{2} \Vert A \Vert ^{2} \bigl\Vert \bigl( \operatorname{Res}_{r_{n}}^{M}-I\bigr)Ax_{n} \bigr\Vert ^{2}+2\gamma \bigl\langle A^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I\bigr)Ax_{n},x_{n}-p \bigr\rangle + \Vert x_{n}-p \Vert ^{2} \\ &\quad =\gamma \bigl(\gamma \Vert A \Vert ^{2}-2\bigr) \bigl\Vert \bigl(\operatorname{Res}_{r_{n}} ^{M}-I\bigr)Ax_{n} \bigr\Vert ^{2} \\ &\qquad {}+2\gamma \bigl\langle \bigl(\operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax _{n}, \operatorname{Res}_{r_{n}}^{M}Ax_{n}-Ap \bigr\rangle + \Vert x_{n}-p \Vert ^{2} \\ &\quad \leq \gamma \bigl(\gamma \Vert A \Vert ^{2}-1\bigr) \bigl\Vert \bigl(\operatorname{Res}_{r_{n}} ^{M}-I \bigr)Ax_{n} \bigr\Vert ^{2}+ \Vert x_{n}-p \Vert ^{2}, \end{aligned} $$
(2.2)

which leads to

$$ \begin{aligned}[b] \Vert y_{n}-p \Vert ^{2} &\leq \gamma _{n} \Vert z_{n}-p \Vert ^{2}+(1-\gamma _{n}) \Vert x_{n}-p \Vert ^{2} \\ &\leq \Vert x_{n}-p \Vert ^{2}-\gamma _{n}\gamma \bigl(1-\gamma \Vert A \Vert ^{2}\bigr) \bigl\Vert \bigl(\operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax_{n} \bigr\Vert ^{2}).\end{aligned} $$
(2.3)

The restriction imposed on parameter γ tells us that \(\|y_{n}-p\|\leq \|x_{n}-p\|\). Since \(W_{n}\) is a nonexpansive mapping for each n, we find from Lemma 1.3 that

$$ \begin{aligned} \Vert x_{n+1}-p \Vert &\leq \beta _{n} \bigl\Vert (I-\mu \alpha _{n}F)W_{n}y_{n}-(I- \mu \alpha _{n}F)p-\mu \alpha _{n}Fp \bigr\Vert +(1- \beta _{n}) \Vert x_{n}-p \Vert \\ & \leq \beta _{n} \bigl\Vert (I-\mu \alpha _{n}F)W_{n}y_{n}-(I- \mu \alpha _{n}F)p \bigr\Vert + \mu \beta _{n} \alpha _{n} \Vert Fp \Vert +(1-\beta _{n}) \Vert x_{n}-p \Vert \\ &\leq \beta _{n}(1-\tau \alpha _{n}) \Vert W_{n}y_{n}-W_{n}p \Vert +\mu \beta _{n} \alpha _{n} \Vert Fp \Vert +(1-\beta _{n}) \Vert x_{n}-p \Vert \\ &\leq \beta _{n}(1-\tau \alpha _{n}) \Vert y_{n}-p \Vert +\mu \beta _{n} \alpha _{n} \Vert Fp \Vert +(1-\beta _{n}) \Vert x_{n}-p \Vert \\ &\leq \tau \alpha _{n}\beta _{n} \frac{ \Vert Fp \Vert \mu }{\tau }+(1- \tau \alpha _{n}\beta _{n}) \Vert x_{n}-p \Vert \\ &\leq \max \biggl\{ \frac{ \Vert Fp \Vert \mu }{ \tau }, \Vert x_{n}-p \Vert \biggr\} ,\end{aligned} $$

from which we conclude that \(\{x_{n}\}\) is a bounded vector sequence in \(H_{1}\).

Step 2. We prove that \(\lim_{n\rightarrow \infty }\|x_{n+1}-x_{n}\|=0\). From resolvent equality (1.6) in Lemma 1.6 we see that

$$\begin{aligned} \Vert z_{n}-z_{n+1} \Vert \leq& \bigl\Vert \operatorname{Res}_{s_{n}}^{N} \rho _{n}- \operatorname{Res}_{s_{n+1}}^{N}\rho _{n} \bigr\Vert + \bigl\Vert \operatorname{Res}_{s _{n+1}}^{N} \rho _{n}-\operatorname{Res}_{s_{n+1}}^{N}\rho _{n+1} \bigr\Vert \\ \leq& \bigl\Vert \operatorname{Res}_{s_{n}}^{N}\rho _{n}-\operatorname{Res}_{s _{n+1}}^{N}\rho _{n} \bigr\Vert + \Vert \rho _{n}-\rho _{n+1} \Vert \\ = &\biggl\Vert \operatorname{Res}_{s_{n+1}}^{N} \biggl( \frac{s_{n+1}}{s_{n}}\rho _{n}+\biggl(1-\frac{s _{n+1}}{s_{n}}\biggr) \operatorname{Res}_{s_{n}}^{N}\rho _{n} \biggr)- \operatorname{Res}_{s_{n+1}}^{N}\rho _{n} \biggr\Vert + \Vert \rho _{n}-\rho _{n+1} \Vert \\ =& \biggl\Vert \biggl(\frac{s_{n+1}}{s_{n}}\rho _{n}+\biggl(1- \frac{s_{n+1}}{s_{n}}\biggr) \operatorname{Res}_{s_{n}}^{N} \rho _{n} \biggr)-\rho _{n} \biggr\Vert + \Vert \rho _{n}- \rho _{n+1} \Vert \\ \leq& \biggl\vert 1-\frac{s_{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}- \operatorname{Res}_{s_{n}}^{N}\rho _{n} \bigr\Vert + \Vert \rho _{n+1}-\rho _{n} \Vert , \end{aligned}$$
(2.4)

where

$$ \rho _{n}=x_{n}+\gamma A^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I\bigr)Ax_{n}. $$

It is easy to see that

$$ \begin{aligned} & \bigl\Vert (x_{n+1}-x_{n})- \gamma A^{*} (Ax_{n+1}-Ax_{n}) \bigr\Vert \\ &\quad = \sqrt{ \Vert x _{n+1}-x_{n} \Vert ^{2}-2\gamma \bigl\langle x_{n+1}-x_{n}, A^{*} (Ax_{n+1}-Ax _{n})\bigr\rangle + \bigl\Vert \gamma A^{*} (Ax_{n+1}-Ax_{n}) \bigr\Vert ^{2}} \\ &\quad = \bigl(1-\gamma \Vert A \Vert ^{2}\bigr) \Vert x_{n+1}-x_{n} \Vert ,\end{aligned} $$

which sends us to

$$ \begin{aligned}[b] &\Vert \rho _{n+1}-\rho _{n} \Vert \\ &\quad \leq \gamma \bigl\Vert A^{*} \bigl( \operatorname{Res} _{r_{n+1}}^{M}Ax_{n+1}- \operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr) \bigr\Vert + \bigl\Vert (x _{n+1}-x_{n})-\gamma A^{*} (Ax_{n+1}-Ax_{n}) \bigr\Vert \\ &\quad \leq \Vert x_{n+1}-x _{n} \Vert +\gamma \Vert A \Vert \biggl\vert 1-\frac{r_{n+1}}{r_{n}} \biggr\vert \Big| \bigl\Vert Ax_{n}- \operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert .\end{aligned} $$
(2.5)

Inequalities (2.4) and (2.5) yield

$$ \begin{aligned} \Vert z_{n}-z_{n+1} \Vert \leq{}& \biggl\vert 1-\frac{s_{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}- \operatorname{Res}_{s_{n}}^{N}\rho _{n} \bigr\Vert + \Vert x_{n+1}-x_{n} \Vert \\ &{}+\gamma \Vert A \Vert \bigg|1-\frac{r_{n+1}}{r_{n}}\bigg| \Big| \bigl\Vert Ax_{n}-\operatorname{Res}_{r_{n}}^{M}Ax _{n} \bigr\Vert , \end{aligned} $$

which further leads us to

$$\begin{aligned} \Vert y_{n}-y_{n+1} \Vert \leq& \gamma _{n} \Vert z_{n}-z_{n+1} \Vert +(1-\gamma _{n}) \Vert x _{n}-x_{n+1} \Vert + \vert \gamma _{n}-\gamma _{n+1} \vert \Vert z_{n+1}-x_{z+1} \Vert \\ \leq& \gamma _{n} \biggl\vert 1-\frac{s_{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}-\operatorname{Res}_{s _{n}}^{N} \rho _{n} \bigr\Vert + \Vert x_{n+1}-x_{n} \Vert \\ &{} +\gamma _{n}\gamma \Vert A \Vert \biggl\vert 1- \frac{r _{n+1}}{r_{n}} \biggr\vert \bigl\vert \bigl\Vert Ax_{n}- \operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert \\ &{} + | \gamma _{n}-\gamma _{n+1}| \Vert z_{n+1}-x_{z+1} \Vert . \end{aligned}$$

From Lemma 1.1 we arrive at

$$\begin{aligned}& \Vert W_{n+1}y_{n+1}-W_{n}y_{n} \Vert \\& \quad \leq \Vert W_{n+1}y_{n+1}-W_{n}y_{n+1} \Vert + \Vert W _{n}y_{n+1}-W_{n}y_{n} \Vert \\& \quad \leq { \sup_{x\in \varPsi }}\bigl[ \Vert W_{n+1}x-Wx \Vert + \Vert Wx-W_{n}x \Vert \bigr]+ \Vert y_{n+1}-y_{n} \Vert \\& \quad \leq { \sup_{x\in \varPsi }}\bigl[ \Vert W_{n+1}x-Wx \Vert + \Vert Wx-W_{n}x \Vert \bigr]+\gamma _{n} \biggl\vert 1-\frac{s _{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}-\operatorname{Res}_{s_{n}}^{N}\rho _{n} \bigr\Vert \\& \qquad {} + \Vert x_{n+1}-x_{n} \Vert +\gamma _{n}\gamma \Vert A \Vert \biggl\vert 1-\frac{r_{n+1}}{r _{n}} \biggr\vert \bigl\vert \bigl\Vert Ax_{n}-\operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert \\& \qquad {}+ | \gamma _{n}-\gamma _{n+1}| \Vert z_{n+1}-x_{z+1} \Vert , \end{aligned}$$
(2.6)

where Ψ is a bounded set containing \(\{y_{n}\}\). Inequality (2.6) ensures that

$$ \begin{aligned} & \bigl\Vert (I-\mu \alpha _{n+1}F)W_{n+1}y_{n+1} -(I-\mu \alpha _{n}F)W_{n}y_{n} \bigr\Vert \\ &\quad \leq \bigl\Vert (I-\mu \alpha _{n+1}F)W_{n+1}y_{n+1}-(I- \mu \alpha _{n+1}F)W _{n}y_{n} \bigr\Vert \\ &\qquad {} + \bigl\Vert (I-\mu \alpha _{n+1}F)W_{n}y_{n}-(I- \mu \alpha _{n}F)W _{n}y_{n} \bigr\Vert \\ &\quad \leq (1-\tau \alpha _{n+1}) \Vert W_{n+1}y_{n+1}- W_{n}y _{n} \Vert + \vert \alpha _{n+1}- \alpha _{n} \vert \Vert \mu FW_{n}y_{n} \Vert \\ &\quad \leq (1-\tau \alpha _{n+1}) \Vert W_{n+1}y_{n+1}- W_{n}y_{n} \Vert + \vert \alpha _{n+1}- \alpha _{n} \vert \Vert \mu FW_{n}y_{n} \Vert \\ &\quad \leq (1-\tau \alpha _{n+1}){ \sup_{x\in \varPsi }}\bigl[ \Vert W_{n+1}x-Wx \Vert + \Vert Wx-W_{n}x \Vert \bigr] \\ &\qquad {}+ (1-\tau \alpha _{n+1}) \gamma _{n} \biggl\vert 1-\frac{s_{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}- \operatorname{Res}_{s _{n}}^{N}\rho _{n} \bigr\Vert \\ & \qquad {}+ (1-\tau \alpha _{n+1}) \Vert x_{n+1}-x_{n} \Vert +(1-\tau \alpha _{n+1})\gamma _{n}\gamma \Vert A \Vert \biggl\vert 1- \frac{r_{n+1}}{r_{n}} \biggr\vert \bigl\vert \bigl\Vert Ax_{n}-\operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert \\ &\qquad {} +(1-\tau \alpha _{n+1}) | \gamma _{n}- \gamma _{n+1} | \Vert z_{n+1}-x _{z+1} \Vert + | \alpha _{n+1}-\alpha _{n}| \Vert \mu FW_{n}y_{n} \Vert . \end{aligned} $$

This further leads to

$$ \begin{aligned} & \bigl\Vert (I-\mu \alpha _{n+1}F)W_{n+1}y_{n+1} -(I-\mu \alpha _{n}F)W_{n}y_{n} \bigr\Vert - \Vert x_{n+1}-x_{n} \Vert \\ &\quad \leq { \sup_{x\in \varPsi }}\bigl[ \Vert W_{n+1}x-Wx \Vert + \Vert Wx-W_{n}x \Vert \bigr]+ \gamma _{n} \biggl\vert 1-\frac{s _{n+1}}{s_{n}} \biggr\vert \bigl\Vert \rho _{n}-\operatorname{Res}_{s_{n}}^{N}\rho _{n} \bigr\Vert \\ & \qquad {}+ \gamma _{n}\gamma \Vert A \Vert \biggl\vert 1- \frac{r_{n+1}}{r_{n}} \biggr\vert \bigl\vert \bigl\Vert Ax_{n}- \operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert \\ & \qquad {}+ | \gamma _{n}-\gamma _{n+1}| \Vert z_{n+1}-x_{z+1} \Vert +\bigl( \vert \alpha _{n+1} \vert + \vert \alpha _{n} \vert \bigr) \Vert \mu FW_{n}y _{n} \Vert .\end{aligned} $$

Using Lemma 1.2, the boundedness of operator A, and the restrictions on the parameter sequences \(\{\alpha _{n}\}\), \(\{\gamma _{n}\}\), \(\{s_{n}\}\), and \(\{r_{n}\}\), we obtain that

$$ \limsup_{n\rightarrow \infty } \bigl( \bigl\Vert (I-\mu \alpha _{n+1}F)W_{n+1}y _{n+1} -(I-\mu \alpha _{n}F)W_{n}y_{n} \bigr\Vert - \Vert x_{n+1}-x_{n} \Vert \bigr)\leq 0. $$

With the aid of Lemma 1.7, we conclude that

$$ \lim_{n\rightarrow \infty } \bigl\Vert (I-\mu \alpha _{n}F)W_{n}y_{n}-x_{n} \bigr\Vert =0. $$
(2.7)

Since \(\alpha _{n}\rightarrow 0\) as \(n\rightarrow \infty \), we also have

$$ \lim_{n\rightarrow \infty } \Vert W_{n}y_{n}-x_{n} \Vert =0. $$
(2.8)

From (2.7) we see that

$$ \lim_{n\rightarrow \infty } \Vert x_{n+1}-x_{n} \Vert =0. $$
(2.9)

Since \(\{x_{n}\}\) is a bounded vector sequence in \(H_{1}\), we find that there is a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) that converges weakly to x̄.

Step 3. We prove that \(x\in \bigcap_{i=1}^{\infty }\operatorname{Fix}(S _{i})\cap \operatorname{SIP} (M,N)\).

Put

$$ \varphi _{n}=(I-\mu \alpha _{n}F)W_{n}y_{n}. $$

For any \(p\in \bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})\cap \operatorname{SIP} (M,N)\), we conclude from (2.3) that

$$\begin{aligned} \Vert \varphi _{n}-p \Vert ^{2} \leq &\bigl\Vert (I-\mu \alpha _{n}F)W_{n}y_{n}-(I- \mu \alpha _{n}F)W_{n}p \bigr\Vert ^{2}-2 \mu \alpha _{n}\langle \varphi _{n}-p,Fp \rangle \\ \leq& (1-\tau \alpha _{n})^{2} \Vert W_{n}y_{n}-W_{n}p \Vert ^{2}-2 \mu \alpha _{n}\langle \varphi _{n}-p,Fp\rangle \\ \leq& (1-\tau \alpha _{n})^{2} \Vert y_{n}-p \Vert ^{2}-2\mu \alpha _{n} \langle \varphi _{n}-p,Fp \rangle \\ \leq& (1-\tau \alpha _{n})^{2} \Vert x_{n}-p \Vert ^{2}-\gamma \bigl(1- \gamma \Vert A \Vert ^{2}\bigr) (1-\tau \alpha _{n})^{2} \bigl\Vert \operatorname{Res}_{r_{n}} ^{M}Ax_{n}-Ax_{n} \bigr\Vert ^{2} \\ &{} +2\mu \alpha _{n} \Vert Fp \Vert \Vert \varphi _{n}-p \Vert . \end{aligned}$$

This shows us that

$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} \leq &\beta _{n} \Vert \varphi _{n}-p \Vert ^{2}+(1-\beta _{n}) \Vert x_{n}-p \Vert ^{2} \\ \leq &\Vert x_{n}-p \Vert ^{2}-\beta _{n}\gamma \bigl(1-\gamma \Vert A \Vert ^{2}\bigr) (1-\tau \alpha _{n})^{2} \bigl\Vert \operatorname{Res}_{r_{n}}^{M}Ax_{n}-Ax _{n} \bigr\Vert ^{2} \\ &{} +2\mu \alpha _{n}\beta _{n} \Vert Fp \Vert \Vert \varphi _{n}-p \Vert . \end{aligned}$$

It follows that

$$ \begin{aligned} &\gamma \bigl(1-\gamma \Vert A \Vert ^{2}\bigr) (1-\tau \alpha _{n})^{2}\beta _{n} \bigl\Vert Ax _{n}-\operatorname{Res}_{r_{n}}^{M}Ax_{n} \bigr\Vert ^{2} \\ &\quad \leq \Vert x_{n}-x_{n+1} \Vert \bigl( \Vert x_{n}-p \Vert + \Vert x_{n+1}-p \Vert \bigr) +2\mu \alpha _{n}\beta _{n} \Vert Fp \Vert \Vert \varphi _{n}-p \Vert .\end{aligned} $$

Limit (2.9) and the fact that \(\alpha _{n}\rightarrow 0\) as \(n\rightarrow \infty \) lead us to

$$ \lim_{n\rightarrow \infty } \bigl\Vert Ax_{n}-\operatorname{Res}_{r_{n}}^{M}Ax _{n} \bigr\Vert =0. $$
(2.10)

Next, we have

$$\begin{aligned}& 2 \Vert z_{n}-p \Vert ^{2} \\& \quad \leq 2\bigl\langle \gamma A^{*}\bigl(\operatorname{Res} _{r_{n}}^{M}-I\bigr)Ax_{n}+x_{n}-p, z_{n}-p\bigr\rangle \\& \quad = \gamma ^{2} \bigl\Vert A ^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I\bigr)Ax_{n} \bigr\Vert ^{2} +2\gamma \bigl\langle A ^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax_{n},x_{n}-p \bigr\rangle + \Vert x_{n}-p \Vert ^{2} \\& \qquad {} - \bigl\Vert x_{n}+\gamma A^{*}\bigl( \operatorname{Res}_{r_{n}}^{M}-I\bigr)Ax _{n}-y_{n} \bigr\Vert ^{2}+ \Vert z_{n}-p \Vert ^{2} \\& \quad \leq \gamma ^{2} \Vert A \Vert ^{2} \bigl\Vert \operatorname{Res}_{r_{n}}^{M}Ax_{n}-Ax_{n} \bigr\Vert ^{2} \\& \qquad {}+2\gamma \bigl(\bigl\langle \operatorname{Res}_{r_{n}}^{M} Ax_{n}-Ap, \operatorname{Res}_{r_{n}} ^{M}Ax_{n}-Ax_{n} \bigr\rangle - \bigl\Vert \operatorname{Res}_{r_{n}}^{M}Ax_{n}-Ax _{n} \bigr\Vert ^{2}\bigr) \\& \qquad {} + \Vert z_{n}-p \Vert ^{2}+ \Vert x_{n}-p \Vert ^{2} - \Vert z_{n}-x _{n} \Vert ^{2} -2\gamma \bigl\langle A^{*}\bigl(\operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax _{n},x_{n}-z_{n} \bigr\rangle \\& \qquad {} - \bigl\Vert \gamma A^{*}\bigl(\operatorname{Res} _{r_{n}}^{M}-I \bigr)Ax_{n} \bigr\Vert ^{2}) \\& \quad \leq \Vert x_{n}-p \Vert ^{2}+ \Vert z_{n}-p \Vert ^{2}+2 \Vert A \Vert \gamma \Vert x_{n}-z_{n} \Vert \bigl\Vert \operatorname{Res}_{r_{n}}^{M}Ax _{n}-Ax_{n} \bigr\Vert - \Vert x_{n}-z_{n} \Vert ^{2}, \end{aligned}$$

that is,

$$ \begin{aligned} \Vert z_{n}-p \Vert ^{2} \leq \Vert x_{n}-p \Vert ^{2}+2 \Vert A \Vert \gamma \Vert z_{n}-x_{n} \Vert \bigl\Vert \operatorname{Res}_{s_{n}}^{N}Ax_{n}-Ax_{n} \bigr\Vert - \Vert x_{n}-z_{n} \Vert ^{2}. \end{aligned} $$

This sends us to

$$\begin{aligned} \Vert \varphi _{n}-p \Vert ^{2} \leq& (1-\tau \alpha _{n})^{2} \Vert W_{n}y_{n}-W_{n}p \Vert ^{2}-2\mu \alpha _{n}\langle \varphi _{n}-p,Fp\rangle \\ \leq& (1- \tau \alpha _{n})^{2} \gamma _{n} \Vert z_{n}-p \Vert ^{2}+(1-\tau \alpha _{n})^{2}(1- \gamma _{n}) \Vert x_{n}-p \Vert ^{2} +2\mu \alpha _{n} \Vert \varphi _{n}-p \Vert \Vert Fp \Vert \\ \leq& (1-\tau \alpha _{n})^{2} \Vert x_{n}-p \Vert ^{2}+2(1-\tau \alpha _{n})^{2} \gamma _{n} \Vert A \Vert \gamma \Vert z_{n}-x_{n} \Vert \bigl\Vert \operatorname{Res}_{s_{n}}^{N}Ax _{n}-Ax_{n} \bigr\Vert \\ &{} -(1-\tau \alpha _{n})^{2}\gamma _{n} \Vert x_{n}-z _{n} \Vert ^{2} +2\mu \alpha _{n} \Vert \varphi _{n}-p \Vert \Vert Fp \Vert . \end{aligned}$$

It follows that

$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} \leq &\beta _{n} \Vert \varphi _{n}-p \Vert ^{2}+(1-\beta _{n}) \Vert x_{n}-p \Vert ^{2} \\ \leq& \Vert x_{n}-p \Vert ^{2}+2\beta _{n}(1-\tau \alpha _{n})^{2} \gamma _{n} \Vert A \Vert \gamma \Vert z_{n}-x_{n} \Vert \bigl\Vert \operatorname{Res}_{s_{n}}^{N}Ax _{n}-Ax_{n} \bigr\Vert \\ &{} -\beta _{n}(1-\tau \alpha _{n})^{2} \gamma _{n} \Vert x _{n}-z_{n} \Vert ^{2} +2\mu \alpha _{n}\beta _{n} \Vert \varphi _{n}-p \Vert \Vert Fp \Vert . \end{aligned}$$

Hence

$$\begin{aligned}& \beta _{n}(1-\tau \alpha _{n})^{2}\gamma _{n} \Vert x_{n}-z_{n} \Vert ^{2} \\& \quad \leq \Vert x_{n}-x_{n+1} \Vert \bigl( \Vert x_{n}-p \Vert + \Vert x_{n+1}-p \Vert \bigr) +2 \Vert A \Vert \gamma \Vert z _{n}-x_{n} \Vert \bigl\Vert \operatorname{Res}_{s_{n}}^{N}Ax_{n}-Ax_{n} \bigr\Vert \\& \qquad {} +2\mu \alpha _{n} \Vert \varphi _{n}-p \Vert \Vert Fp \Vert . \end{aligned}$$

Using (2.9) and (2.10), we have that \(x_{n}-z_{n}\rightarrow 0\) as \(n\rightarrow \infty \), that is,

$$ \lim_{n\rightarrow \infty } \bigl\Vert x_{n}- \operatorname{Res}_{s_{n}}^{N}\bigl(x _{n}+ \gamma A^{*}\bigl(\operatorname{Res}_{r_{n}}^{M}-I \bigr)Ax_{n}\bigr) \bigr\Vert =0. $$
(2.11)

Since \(x_{n}-z_{n}\rightarrow 0\) as \(n\rightarrow \infty \), we have that \(\{z_{n}\}\) converges weakly to x̄. Further, \(\{z_{n_{i}}\}\) converges weakly to x̄ as \(i\rightarrow \infty \). The graphs of maximal monotone mappings are weakly-strongly closed. Observe that

$$ \frac{x_{n_{i}}-z_{n_{i}}}{s_{n_{i}}}+\gamma A^{*} \frac{ \operatorname{Res}_{r_{n_{i}}}^{M}Ax_{n_{i}}-Ax_{n_{i}}}{s_{n_{i}}} \in Nz_{n_{i}}. $$

So \(0\in N(\bar{x})\). Fixing a positive real number p, Lemma 1.6 yields that \(\|Ax_{n_{i}}-\operatorname{Res}_{p}^{M}Ax _{n_{i}}\|\rightarrow \) as \(i\rightarrow \infty \), which implies \(0\in M(A\bar{x})\).

We are now in a position to show that \(\bar{x}\in \bigcap_{i=1}^{\infty }\operatorname{Fix}(S_{i})=\operatorname{Fix}(W)\). We have

$$\begin{aligned} \Vert y_{n_{i}}-W_{n_{i}}y_{n_{i}} \Vert \leq& \Vert y_{n_{i}}-W_{n_{i}}y_{n_{i}} \Vert + \Vert W_{n_{i}}y_{n_{i}}-Wy_{n_{i}} \Vert \\ \leq& \Vert y_{n_{i}}-W_{n_{i}}y_{n_{i}} \Vert +\sup _{x\in \varPsi } \Vert W_{n_{i}}x-Wx \Vert . \end{aligned}$$

Relations (2.8) and (2.11) yield that \(\lim_{i\to \infty }\|y_{n_{i}}-W_{n_{i}}y_{n_{i}}\|=0\). If \(\bar{x}\neq W\bar{x}\), then the Opial condition, Lemma 1.5, sends us to

$$\begin{aligned} \limsup_{i\to \infty } \Vert \bar{x}-y_{n_{i}} \Vert < & \limsup_{i\to \infty } \Vert W \bar{x}-y_{n_{i}} \Vert \\ \leq& \limsup_{i\to \infty }\bigl\{ \Vert Wy_{n_{i}}-y _{n_{i}} \Vert + \Vert W\bar{x}-Wy_{n_{i}} \Vert \bigr\} \\ \leq& \limsup_{i\to \infty } \Vert \bar{x}-y_{n_{i}} \Vert , \end{aligned}$$

a contradiction. Thus \(\bar{x}\in \operatorname{Fix}(W)\), that is, \(\bar{x}\in \bigcap^{\infty }_{i=1} \operatorname{Fix}(S_{i})\).

Step 4. We prove that the sequence \(\{x_{n}\}\) is strongly convergent.

Since F is strongly monotone and Lipschitz continuous, we get that the following variational inequality has a unique solution:

$$ \langle \widetilde{x} -y, F\widetilde{x}\rangle \leq 0,\quad \forall y\in \bigcap _{i=1}^{\infty }\operatorname{Fix}(S_{i}) \cap \operatorname{SIP}(M,N) . $$

Thus

$$ \limsup_{n\rightarrow \infty }\langle \widetilde{x}-\varphi _{n},F \widetilde{x}\rangle \leq 0. $$
(2.12)

Lemma 1.1 and Lemma 1.3 send us to

$$\begin{aligned}& \Vert x_{n+1}-\widetilde{x} \Vert ^{2} \\& \quad \leq \beta _{n} \Vert \varphi _{n}- \widetilde{x} \Vert ^{2} + (1-\beta _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2} \\& \quad \leq \beta _{n} \bigl( \bigl\Vert (I-\mu \alpha _{n} F)W_{n}y_{n}-(I-\mu \alpha _{n} F)\widetilde{x} \bigr\Vert ^{2} -2\mu \alpha _{n}\bigl\langle \varphi _{n}- \widetilde{x}, F( \widetilde{x})\bigr\rangle \bigr)+ (1-\beta _{n}) \Vert x_{n}- \widetilde{x} \Vert ^{2} \\& \quad \leq \beta _{n} \bigl( (1-\tau \alpha _{n})^{2} \Vert y_{n}- \bar{x} \Vert ^{2}-2 \mu \alpha _{n} \bigl\langle F (\widetilde{x}), \varphi _{n}- \widetilde{x} \bigr\rangle \bigr) + (1-\beta _{n}) \Vert x_{n}- \widetilde{x} \Vert ^{2} \\& \quad \leq (1-2\tau \beta _{n}\alpha _{n}) \Vert x_{n}- \widetilde{x} \Vert ^{2}+2\tau \beta _{n}\alpha _{n}\varPi _{n}, \end{aligned}$$

where \(\varPi =\frac{\mu }{\tau } \langle F \bar{x}, \widetilde{x}- \varphi _{n} \rangle +\frac{\tau \alpha _{n}}{2}\|x_{n}-\widetilde{x}\| ^{2}\). In light of Lemma 1.4, we find that \(\| x_{n}- \widetilde{x}\| \rightarrow 0\) as \(n\rightarrow \infty \). This completes the proof. □

From Theorem 2.1 we have the following subresult on split inclusion problem (1.3).

Corollary 2.1

Let \(H_{1}\) and \(H_{2}\) be Hilbert spaces, and let N and M be set-valued maximal monotone mappings on \(H_{1}\) and \(H_{2}\), respectively. Let \(F:H_{1}\rightarrow H_{1}\) be an \(\mathcal{L}\)-Lipschitz continuous and τ-strongly monotone mapping. Let A be a linear bounded operator from \(H_{1}\) to \(H_{2}\), and let \(A^{*}\) be its the adjoint operator. Assume that \(\operatorname{SIP}(M,N) \neq \emptyset \). Let \(\{ x_{n}\}\) be the vector sequence in \(H_{1}\) generated by the iterative process

$$ x_{1}\in H_{1},\quad x_{n+1}=\beta _{n}(I-\mu \alpha _{n}F) \operatorname{Res}_{s_{n}}^{N} \bigl(x_{n}+\gamma A^{*}\bigl(\operatorname{Res} _{r_{n}}^{M}-I\bigr)Ax_{n}\bigr)+(1-\beta _{n})x_{n},\quad n\geq 1, $$

where γ and μ are two positive real numbers, \(\{s_{n}\}\) and \(\{r_{n}\}\) are two positive real number sequences, and \(\{\alpha _{n}\}\) and \(\{\beta _{n}\}\) are real number sequences in \((0,1)\). Suppose that \(\gamma \in (0,\frac{1}{\|A\|^{2}})\), \(\mu \in (0, \frac{2 \tau }{\mathcal{L}^{2}})\), \(\liminf_{n\rightarrow \infty } s_{n}> 0\), \(\lim_{n\rightarrow \infty }|s_{n}-s_{n+1}|<\infty \), \(\liminf_{n\rightarrow \infty }r_{n}> 0\), \(\lim_{n\rightarrow \infty }|r_{n}-r_{n+1}|<\infty \), \(\sum_{n=1}^{ \infty }\alpha _{n}=\infty \), \(\{\beta _{n}\}\) is a number sequence in \([\bar{\beta },\bar{\beta }']\), where β̄ and \(\bar{\beta }'\) are two real numbers in \((0,1)\), such that \(\lim_{n\rightarrow \infty }|\beta _{n+1}- \beta _{n}|=0\). Then the sequence \(\{ x_{n}\}\) converges strongly to \(\widetilde{x}\in H_{1}\), which is a unique solution of the variational inequality \(\langle \widetilde{x} -y, F\widetilde{x} \rangle \leq 0\), \(\forall y\in \operatorname{SIP}(M,N)\).

Remark 2.1

In this paper, we investigated the descent iterative methods for split inclusion problem with a common fixed point constraint of an infinite family of nonexpansive mappings. It deserves mentioning that our method does not involve projections. A solution theorem of the problem was established in the framework of Hilbert spaces under some weak assumptions imposed on different mappings and control sequences.