On iterative solutions of a split feasibility problem with nonexpansive mappings

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 and control sequences.

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:

It can be formulated as the following convex feasibility problem:

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

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

The mapping S is said to be quasinonexpansive if

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

It is said to be firmly nonexpansive if

It is is said to be firmly quasinonexpansive if

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

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,

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:

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 \(v\in My\)

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

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

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

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

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

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

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

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

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

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 \(\lim_{n\rightarrow \infty }\|y_{n}-x_{n}\|=0\).

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

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

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

Putting

(2.1) sends us to

which leads to

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

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

where

It is easy to see that

which sends us to

Inequalities (2.4) and (2.5) yield

which further leads us to

From Lemma 1.1 we arrive at

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

This further leads to

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

With the aid of Lemma 1.7, we conclude that

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

From (2.7) we see that

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

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

This shows us that

It follows that

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

Next, we have

that is,

This sends us to

It follows that

Hence

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

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

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

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

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:

Thus

Lemma 1.1 and Lemma 1.3 send us to

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

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.

Acknowledgements

The authors thank the learned referees for their comments and appreciation.

Availability of data and materials

All data generated or analyzed during this study are included in this paper.

Funding

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. KEP-2-130-39. Therefore the authors acknowledge with thanks DSR for technical and financial support.

Affiliations

1. Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

   • M. A. Kutbi
   •  & A. Latif

2. General Education Center, National Yunlin University of Science and Technology, Douliou, Taiwan

   • X. Qin

Contributions

All authors contributed equally to the manuscript and approved the final manuscript.

Corresponding author

Correspondence to A. Latif.

Competing interests

The authors declare that they have no competing interests.

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