An Iterative Method for Split Variational Inclusion Problem and Split Fixed Point Problem for Averaged Mappings ()
1. Introduction
Throughout the paper, unless otherwise stated, let
and
be real Hilbert spaces with their inner product
and norm
. Let C and Q be nonempty closed convex subsets of
and
, respectively. Let
be a bounded linear operator and
is the corresponding adjoint operator of A. A mapping
is called contractive, if there exists a constant
such that
If
, then S is called nonexpansive. In addition, let’s first review the split feasibility problem (SFP): find
such that
. The split feasibility problem (SFP) originated from phase recovery and medical image reconstruction [1] [2] [3] , and it has been widely studied, as shown in [4] [5] [6] . When C and Q in the split feasibility problem (SFP) are fixed point sets of nonlinear operators, the split feasibility problem (SFP) is called the split fixed point problem (SFPP) [7] [8] . More precisely, find
such that
(1.1)
where
and
denote the fixed point sets of two nonlinear
and
. The solution set of the SFPP is denoted by F, that is,
A mapping
is said to be
1) monotone, if
2)
-strongly monotone, if there exists a constant
such that
3)
-inverse strong monotone (
-ism), if there exists a constant
such that
4) firmly nonexpansive, if
A multivalued mapping
is called monotone if for all
,
and
such that
. And
is maximal if the
is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for
,
for
implies that
. Then, the resolvent mapping
associated with M, is defined by
for
, where I stands identity operator on
. Noting that
is single valued and firmly nonexpansive.
Recently, Moudafi [9] introduced the following split monotone variational inclusion problem (SMVIP): Find
such that
(1.2)
and such that
solves
(1.3)
where
and
are multivalued maximal monotone mappings.
Moudafi [9] introduced an iterative algorithm for solving SMVIP (1.2)-(1.3), which is an important extension of the iterative method for split variational inequality given by Censor et al. [10] for split variational inequality problem. As Moudafi pointed out in [9] , SMVIP (1.2)-(1.3) includes as special, the split common fixed point problem, splitting variational inclusion problem, splitting zero point problem and splitting feasibility problem [1] [8] - [25] . These problems have been widely studied and used in practice as a model for intensity modulated radiation planning (IMRT), see [1] [25] . This is the core of many inverse modeling problems caused by phase retrieval and other real-world problems. For example, computer tomography and data compression in sensor networks are shown in [2] [26] .
If
and
, then SMVIP (1.2)-(1.3) can be reduced to the following split variational inclusion problem (SVIP): Find
such that
, (1.4)
and such that
solves
. (1.5)
When looked separately (1.4) is the variational inclusion problem and we denoted its solution set by
. The SVIP (1.4)-(1.5) constitutes a pair of variational inclusion problems which have to be solved so that the image
under a given bounded linear operator A, of the solution
of SVIP (1.4) in
is the solution of another SVIP (1.5) in another space
, we denoted the solution set of SVIP (1.5) by
. And the solution set of SVIP (1.4)-(1.5) is denoted by
.
In 2011, Byrne et al. [24] studied the weak and strong convergence of iterative algorithms for SVIP (1.4)-(1.5): For given
, calculate the iterative sequence
generated by the following method:
On the other hand, Censor and Segal [7] studied iterative algorithms for solving split fixed point problems (SFPP): For given
, calculate the sequence
generated by the following method:
where
and
are two directed operators.
Inspired by Moudafi [9] and Fyrne, Kazmi and Rizvi [27] proposed the following iterative algorithm for SVIP (1.4)-(1.5) and fixed point problems of nonexpansive mappings:
where
,
, L is the spectral radius of the operator
.
Motivated and inspired by the above results and the ongoing research in this direction, we suggest and analyze an iterative algorithm, which is proposed to solve the split variational inclusion problem SVIP (1.4)-(1.5) and split fixed point problem SFPP (1.1) under appropriate conditions. We also prove that the iterative sequence generated by the iterative algorithm converges strongly to the common solution of SVIP (1.4)-(1.5) and SFPP (1.1). The results presented here improve and extend some known results.
2. Preliminaries
We denote the weak and the strong convergence of a sequence
to a point x by
and
, respectively. Let us recall some concepts and results which are needed in sequel. For
, there exists a unique closest point in C denoted by
such that
is called the metric projection of
onto C. As we all know,
is firmly nonexpansive mapping, that is,
(2.1)
In addition,
is characterized by the fact
and
(2.2)
and
(2.3)
In a real Hilbert space, for
and
, the following holds:
(2.4)
Noting that every nonexpansive operator
satisfies the inequality
(2.5)
As a result, we have,
, (2.6)
for details, see e.g., ( [28] , Theorem 3.1) and ( [29] , Theorem 2.1).
A mapping
is called averaged if and only if it can be written as the average of the identity mapping and a nonexpansive mapping, i.e.,
, where
and
is nonexpansive and I is the identity operator on
.
It is easy to see that every averaged mapping is nonexpansive. In addition, the firmly nonexpansive mapping (especially the projection on the nonempty closed convex set and the resolvent operators of the maximal monotone operators) is averaged.
The following are some key properties of averaged operators, see for instance [3] [9] [30] .
Proposition 2.1. (i) If
, where
is averaged,
is nonexpansive and
, then T is averaged.
(ii) The composite of finitely many averaged mappings is averaged.
(iii) If the mapping
is averaged and have a nonempty common fixed point, then
.
(iv) If T is
-inverse strong monotone (
-ism), then for
,
is
-inverse strong monotone (
-ism).
(v) T is averaged if and only if its complement
is
-inverse strong monotone (
-ism) for some
.
Lemma 2.1. [31] Assume that T is nonexpansive self-mapping of a closed convex subset C of a Hilbert space
. If T has a fixed point, then
is demiclosed, i.e., whenever
is a sequence in C converging weakly to some
and the sequence
converges strongly to some y, it follows that
. Here I is the identity mapping on
.
Lemma 2.2. [32] Let
is a sequence of non-negative real numbers such that
,
where
is a sequence in
and
is the sequence in
such that (i)
; (ii)
or
. Then
.
3. Main Results
In this section, we will prove a strong convergence theorem based on the proposed iterative algorithm to calculate the common approximate solutions of SVIP (1.4)-(1.5) and SFPP (1.1).
Theorem 3.1. Let
and
be two real Hilbert spaces, let
be a bounded linear operator with its adjoint operator
. Let
be a contraction mapping with
. Assume that
,
are maximal monotone mappings,
,
are two average mappings and
. For a given
, let the iterative sequence
,
and
be generated by
(3.1)
where
,
, L is the spectral radius of the operator
and
is a sequences in
such that
,
and
. Then the sequence
,
and
all converge strongly to
, where
.
Proof. We divide the proof into the following steps.
Step 1 Let
, then
,
,
,
. By (3.1) we have
(3.2)
Denoting
and from (2.6), we can obtain
(3.3)
It follows from (3.2) and (3.3) that
(3.4)
Since
, we have
. Next we prove
.
By (3.1), we have again
(3.5)
Denoting
, since U is averaged mapping, it follows from (2.6) that
(3.6)
It follows from (3.5) and (3.6) that
(3.7)
Noting
that
, thus we have
(3.8)
Since f is
-contractive, then from (3.1) and (3.8) that
(3.9)
Hence
is bounded and so are
and
.
Step 2. Next, we show that
is asymptotically regular, i.e.,
. for
, since S and U are both averaged mappings, and hence the mapping
is nonexpansive (see [9] ). Hence, we obtain
(3.10)
It follows from (3.1) and (3.10) that
(3.11)
where
. Since, for
, the mapping
is averaged and hence nonexpanding (see [27] ), then we obtain
It follows from (3.10) that
(3.12)
Then, from (3.11) and (3.12), we have
By applying Lemma 2.2 with
and
, we have
(3.13)
Next, since
Then, we have
It follows from (3.13) and
, we obtain
(3.14)
Next, we show that
. From (3.18) and (3.4), we have
(3.18)
Therefore,
Since
and
and (3.13), we obtain
(3.16)
From (3.7) and (3.8), we have
Therefore
Since
and
and (3.13), we obtain
(3.17)
In addition, using (3.2), (3.8) and
, we observe that
Therefore
(3.18)
It follows from (3.8), (3.15) and (3.18) that
Implying that
Since
and from (3.13) and (3.16), we obtain
(3.19)
Next, we show that
. Now, we can write
From (3.14) and (3.19), we get
(3.20)
Next, we show that
. Note that from (3.13) and (3.19), we have
(3.21)
And from (3.13) and (3.14) that
(3.22)
Finally, it follows form (3.1) that
From (3.17), we have
(3.23)
Then, from (3.21)-(3.23), we have
(3.24)
Step 3. We show that
. Since
is bounded, we consider weak cluster point w of
. Hence, there exists a subsequence
of
, which converges weakly to w. Since S and U both are both average mappings, then S and U are also both nonexpansive mappings. According to (3.17) and (3.24) and Lemma 2.1, we have
,
. Thus
.
On the other hand,
can be written as
(3.25)
By pass to limit when
in (3.25) and by taking into (3.16) and (3.19) and the fact that graphs of a maximal monotone operators is weakly-strongly closed, we obtain
, i.e.,
. In addition, since
and
have the same asymptotic behavior,
weakly convergence to Aw. Again, by (3.16) and the fact that the resolvent
is nonexpansive and Lemma 2.1, we obtain that
, i.e.,
. Thus
. Therefore
.
Step 4. We show that
. First, we claim that
.
Since
is bounded, there exist a subsequence
of
satisfy
as
and
. Since
, we have
as
. From step 3, we obtain
. Indeed, we have
(3.26)
where
. Next, we show that
.
which implies that
Therefore, according to (3.26) and Lemma 2.2, we obtain
. Further it follows from
,
and
that
. This completes the proof.
Remark 3.1. Theorem 3.1 improves and extends the corresponding results in [7] [24] .
Remark 3.2. The algorithm is more general than the existing algorithm. The disadvantage is that the spectral radius of the operator is calculated, but the adaptive step size can be used to overcome the difficulties caused by calculating the spectral radius.
Remark 3.3. Numerical experiments are the direstion of our future efforts.
At last, we give two examples to illustrate the validity of our considered common solution problem for SVIP (1.4)-(1.5) and SFPP (1.1) and our convergence result of proposed algorithm (3.1).
Example 3.1. Let
and
be defined by
Then, B is a maximal monotone mapping. We define the mappings
By
, respectively.
It is easy to cheek that A is a bounded linear operator, f is a
-contractive mapping, Sand U are averaged mappings. Let
. Then
are maximal mappings. Let
be the resolvent operators. It is easy to see that
is the common solution to SVIP (1.4)-(1.5) and SFPP (1.1).
Example 3.2. Let
with the normal inner product and norm. We define the operators
by
and
.
Clearly,
and
are maximal monotone operators and their resolvents are given by
and
.
For some
, we also defined the mappings
by
,
,
and
.
Clearly, A is a bounded linear mapping, f is a
-contractive mapping, S and U are two averaged mappings. It is easy to know that
is the common solution to SVIP (1.4)-(1.5) and SFPP (1.1).
Acknowledgements
The authors would like to thank the reviewers for their valuable comments, which have helped to improve the quality of this paper.
Funding
This research was supported by Liaoning Provincial Department of Education under project No. LJKMZ20221491.
Authors’ Contributions
The authors carried out the results and read and approved the current version of the manuscript.