1 Introduction and preliminaries
Throughout this paper, let
and
bereal Hilbert spaces whose inner product and norm are denoted by
and
, respectively; let C andQ be nonempty closed convex subsets of
and
,respectively. A mapping
is said to benonexpansive if
for any
.A mapping
is said to bequasi-nonexpansive if
for any
and
, where
is the set of fixed pointsof T. A mapping
is calledasymptotically nonexpansive if there exists a sequence
satisfying
such that
for any
.A mapping
is semi-compact if, forany bounded sequence
with
, there exists asubsequence
such that
converges strongly to somepoint
.
and
bereal Hilbert spaces whose inner product and norm are denoted by
and
, respectively; let C andQ be nonempty closed convex subsets of
and
,respectively. A mapping
is said to benonexpansive if
for any
.A mapping
is said to bequasi-nonexpansive if
for any
and
, where
is the set of fixed pointsof T. A mapping
is calledasymptotically nonexpansive if there exists a sequence
satisfying
such that
for any
.A mapping
is semi-compact if, forany bounded sequence
with
, there exists asubsequence
such that
converges strongly to somepoint
.The split feasibility problem (SFP) is to find a point
with the property
with the property
(1.1)
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where
is a bounded linear operator.
is a bounded linear operator.Assuming that SFP (1.1) is consistent (i.e., (1.1) has a solution), itis not hard to see that
solves(1.1) if and only if it solves the following fixed point equation:
solves(1.1) if and only if it solves the following fixed point equation:
(1.2)
where
and
arethe (orthogonal) projections onto C and Q, respectively,
isany positive constant, and
denotes the adjoint of A.
and
arethe (orthogonal) projections onto C and Q, respectively,
isany positive constant, and
denotes the adjoint of A.The SFP in finite-dimensional Hilbert spaces was first introduced by Censor andElfving [1] for modeling inverse problems whicharise from phase retrievals and in medical image reconstruction [2]. Recently, it has been found that the SFP can also beused in various disciplines such as image restoration, computer tomograph, and radiationtherapy treatment planning [2‐7].
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Let
and
be two mappings satisfying
and
,respectively; let
be a bounded linear operator. The split common fixed point problem (SCFP) formappings S and T is to find a point
with the property
and
be two mappings satisfying
and
,respectively; let
be a bounded linear operator. The split common fixed point problem (SCFP) formappings S and T is to find a point
with the property
(1.3)
We use Γ to denote the set of solutions of SCFP (1.3), that is,
.
.Since each closed and convex subset may be considered as a fixed point set of aprojection on the subset, hence the split common fixed point problem (SCFP) isa generalization of the split feasibility problem (SFP) and the convexfeasibility problem (CFP) [5].
Split feasibility problems and split common fixed point problems have been studied bysome authors [8‐15]. In 2010,Moudafi [10] proposed the following iterationmethod to approximate a split common fixed point of demi-contractive mappings: forarbitrarily chosen
,
,
and he proved that
converges weakly to a split common fixedpoint
, where
and
are two demi-contractive mappings,
is a bounded linear operator.
converges weakly to a split common fixedpoint
, where
and
are two demi-contractive mappings,
is a bounded linear operator.Using the iterative algorithm above, in 2011, Moudafi [9] also obtained a weak convergence theorem for the split commonfixed point problem of quasi-nonexpansive mappings in Hilbert spaces. After that, someauthors also proposed some iterative algorithms to approximate a split common fixedpoint of other nonlinear mappings, such as nonspreading type mappings [16], asymptotically quasi-nonexpansive mappings[12], κ-asymptoticallystrictly pseudononspreading mappings [17],asymptotically strictly pseudocontraction mappings [18]etc., but they just obtained weak convergence theoremswhen those mappings do not have semi-compactness. This naturally brings us to thefollowing question.
Can we construct an iterative scheme which can guarantee the strong convergence forsplit common fixed point problems without assumption ofsemi-compactness?
In this paper, we introduce the following iterative scheme. Let
,
,the sequence
is defined as follows:
,
,the sequence
is defined as follows:
(1.4)
where
and
are two asymptotically nonexpansive mappings,
is a bounded linear operator,
denotes the adjoint of A. Under some suitable conditions on parameters, theiterative scheme
is shown to converge strongly to a splitcommon fixed point of asymptotically nonexpansive mappings
and
without the assumption of semi-compactness on
and
.
and
are two asymptotically nonexpansive mappings,
is a bounded linear operator,
denotes the adjoint of A. Under some suitable conditions on parameters, theiterative scheme
is shown to converge strongly to a splitcommon fixed point of asymptotically nonexpansive mappings
and
without the assumption of semi-compactness on
and
.The following lemma and results are useful for our proofs.
Lemma 1.1[19]
LetEbe a real uniformly convex Banach space, Kbea nonempty closed subset ofE, and let
be an asymptoticallynonexpansive mapping. Then
isdemiclosed at zero, that is, if
convergesweakly to a point
and
,then
.
be an asymptoticallynonexpansive mapping. Then
isdemiclosed at zero, that is, if
convergesweakly to a point
and
,then
.Let C be a closed convex subset of a real Hilbert space H.
denotes the metric projection of H onto C. It is well known that
ischaracterized by the properties: for
and
,
denotes the metric projection of H onto C. It is well known that
ischaracterized by the properties: for
and
,
(1.5)
and
(1.6)
In a real Hilbert space H, it is also well known that
(1.7)
and
(1.8)
2 Main results
Theorem 2.1Let
and
be twoHilbert spaces,
bea bounded linear operator,
bean asymptotically nonexpansive mapping with the sequence
satisfying
,and
bean asymptotically nonexpansive mapping with the sequence
satisfying
,
and
, respectively.Let
,
,and let the sequence
be defined as follows:
and
be twoHilbert spaces,
bea bounded linear operator,
bean asymptotically nonexpansive mapping with the sequence
satisfying
,and
bean asymptotically nonexpansive mapping with the sequence
satisfying
,
and
, respectively.Let
,
,and let the sequence
be defined as follows:
(2.1)
where
denotesthe adjoint ofA,
and
satisfies
,
,
.If
, then
converges stronglyto
.
denotesthe adjoint ofA,
and
satisfies
,
,
.If
, then
converges stronglyto
.Proof We will divide the proof into five steps.
Step 1. We first show that
is closedand convex for any
.
is closedand convex for any
.Since
,so
isclosed and convex. Assume that
is closedand convex. For any
,since
,so
isclosed and convex. Assume that
is closedand convex. For any
,since
we know that
is closed and convex. Therefore
is closedand convex for any
.
is closed and convex. Therefore
is closedand convex for any
.Step 2. We prove
for any
.
for any
.Let
, then from (2.1) we have
, then from (2.1) we have
(2.2)
where
(2.3)
Substituting (2.3) into (2.2), we can obtain that
(2.4)
In addition, it follows from (2.1) that
(2.5)
Therefore, from (2.4) and (2.5), we know that
and
for any
.
and
for any
.Step 3. We will show that
is a Cauchy sequence.
is a Cauchy sequence.Since
and
,then
and
,then
(2.6)
It means that
is bounded. For any
, by using(1.6), we have
is bounded. For any
, by using(1.6), we have
which implies that
. Thus
is nondecreasing. Therefore, by the boundednessof
,
exists. For somepositive integers m, n with
, from
and (1.6), we have
. Thus
is nondecreasing. Therefore, by the boundednessof
,
exists. For somepositive integers m, n with
, from
and (1.6), we have
(2.7)
Since
exists, it followsfrom (2.7) that
. Therefore
is a Cauchy sequence.
exists, it followsfrom (2.7) that
. Therefore
is a Cauchy sequence.Step 4. We will show that
.
.Since
,we have
,we have
(2.8)
(2.9)
(2.10)
Notice that
, it follows from (2.4)that
, it follows from (2.4)that
thus, since
is bounded and
,from (2.8) we have
is bounded and
,from (2.8) we have
(2.11)
On the other hand, since
we have
Since
and
,we know that
and
,we know that
(2.12)
In addition, since
, we know that
. So from
, we know that
. So from
we can obtain that
(2.13)
Similarly, we have
(2.14)
Step 5. We will show that
converges strongly to an element ofΓ.
converges strongly to an element ofΓ.Since
is a Cauchy sequence, we may assume that
,from (2.8) we have
,which implies that
.So it follows from (2.13) and Lemma 1.1 that
.
is a Cauchy sequence, we may assume that
,from (2.8) we have
,which implies that
.So it follows from (2.13) and Lemma 1.1 that
.In addition, since A is a bounded linear operator, we have that
. Hence, itfollows from (2.14) and Lemma 1.1 that
. This means that
and
converges strongly to
. The proof iscompleted. □
. Hence, itfollows from (2.14) and Lemma 1.1 that
. This means that
and
converges strongly to
. The proof iscompleted. □In Theorem 2.1, as
and
,we have the following result.
and
,we have the following result.Corollary 2.2Let
be aHilbert space,
bean asymptotically nonexpansive mapping with a sequence
satisfying
.The sequence
is defined as follows:
,
be aHilbert space,
bean asymptotically nonexpansive mapping with a sequence
satisfying
.The sequence
is defined as follows:
,
(2.15)
where
and
satisfies
.If
, then
converges strongly to a fixedpoint
ofT.
and
satisfies
.If
, then
converges strongly to a fixedpoint
ofT.In Theorem 2.1, when
and
aretwo nonexpansive mappings, the following result holds.
and
aretwo nonexpansive mappings, the following result holds.Corollary 2.3Let
and
be twoHilbert spaces,
bea bounded linear operator,
and
betwo nonexpansive mappings such that
and
, respectively.Let
,
,and let the sequence
be defined as follows:
and
be twoHilbert spaces,
bea bounded linear operator,
and
betwo nonexpansive mappings such that
and
, respectively.Let
,
,and let the sequence
be defined as follows:
(2.16)
where
denotesthe adjoint ofA,
and
satisfies
.If
, then
converges stronglyto
.
denotesthe adjoint ofA,
and
satisfies
.If
, then
converges stronglyto
.Remark 2.4 When
and
aretwo quasi-nonexpansive mappings and
and
are demiclosed at zero, Corollary 2.3 also holds.
and
aretwo quasi-nonexpansive mappings and
and
are demiclosed at zero, Corollary 2.3 also holds.Example 2.5 Let C be a unit ball in a real Hilbert space
,and let
be amapping defined by
,and let
be amapping defined by
It is proved in Goebel and Kirk [20] that
(i)
,
;
,
;(ii)
,
,
.
,
,
.Taking
,
, itis easy to see that
.So we can take
,and
,
,then
,
, itis easy to see that
.So we can take
,and
,
,then
Therefore
is anasymptotically nonexpansive mapping from C into itself with
.
is anasymptotically nonexpansive mapping from C into itself with
.Let D be an orthogonal subspace of
with thenorm
and the inner product
for
and
. For each
, we define amapping
by
with thenorm
and the inner product
for
and
. For each
, we define amapping
by
It is easy to show that
or
for any
.Therefore
is anasymptotically nonexpansive mapping from D into itself with
since
for anysequence
with
.
or
for any
.Therefore
is anasymptotically nonexpansive mapping from D into itself with
since
for anysequence
with
.Obviously, C and D are closed convex subsets of
and
,respectively. Let
be defined by
for
. ThenA is a bounded linear operator with adjoint operator
for
. Clearly,
,
.
and
,respectively. Let
be defined by
for
. ThenA is a bounded linear operator with adjoint operator
for
. Clearly,
,
.Taking
,
,
,
,
and
,
. Itfollows from Theorem 2.1 that
converges strongly to
.
,
,
,
,
and
,
. Itfollows from Theorem 2.1 that
converges strongly to
.3 Applications and examples
Application to the equilibrium problem
Let H be a real Hilbert space, C be a nonempty closed and convexsubset of H, and let the bifunction
satisfy the following conditions:
satisfy the following conditions:(A1)
,
;
,
;(A2)
,
;
,
;(A3) For all
,
;
,
;(A4) For each
, thefunction
is convex and lower semi-continuous.
, thefunction
is convex and lower semi-continuous.The so-called equilibrium problem for F is to find a point
such that
for all
.The set of its solutions is denoted by
.
such that
for all
.The set of its solutions is denoted by
.Lemma 3.1[21]
LetCbe a nonempty closed convex subset of a HilbertspaceH, and let
be a bifunctionsatisfying (A1)-(A4). Let
and
.Then there exists
suchthat
be a bifunctionsatisfying (A1)-(A4). Let
and
.Then there exists
suchthat
Lemma 3.2[21]
Assume that
satisfies(A1)-(A4). For
and
,define a mapping
as follows:
satisfies(A1)-(A4). For
and
,define a mapping
as follows:
Then
(1)
is single-valued;
is single-valued;(2)
is firmly nonexpansive, that is, for all
,
is firmly nonexpansive, that is, for all
,
(3)
;
;(4)
is nonempty, closed and convex.
is nonempty, closed and convex.Theorem 3.3Let
and
be twoHilbert spaces,
bea bounded linear operator,
bea nonexpansive mapping,
be a bifunctionsatisfying (A1)-(A4). Assume that
and
.Taking
,for arbitrarily chosen
,the sequence
is defined asfollows:
and
be twoHilbert spaces,
bea bounded linear operator,
bea nonexpansive mapping,
be a bifunctionsatisfying (A1)-(A4). Assume that
and
.Taking
,for arbitrarily chosen
,the sequence
is defined asfollows:
(3.1)
where
denotesthe adjoint ofA,
,
and
satisfies
.If
, then thesequence
converges strongly toa point
.
denotesthe adjoint ofA,
,
and
satisfies
.If
, then thesequence
converges strongly toa point
.Proof It follows from Lemma 3.2 that
,
is nonempty, closed and convex and
is a firmly nonexpansive mapping. Hence all conditions in Corollary 2.3 aresatisfied. The conclusion of Theorem 3.3 can be directly obtained fromCorollary 2.3. □
,
is nonempty, closed and convex and
is a firmly nonexpansive mapping. Hence all conditions in Corollary 2.3 aresatisfied. The conclusion of Theorem 3.3 can be directly obtained fromCorollary 2.3. □Let
and
be two real Hilbert spaces. Let C be a closed convex subset of
,K be a closed convex subset of
,
be a bounded linear operator. Assume that F is a bi-function from
into R and G is a bi-function from
intoR. The split equilibrium problem (SEP) is to
and
be two real Hilbert spaces. Let C be a closed convex subset of
,K be a closed convex subset of
,
be a bounded linear operator. Assume that F is a bi-function from
into R and G is a bi-function from
intoR. The split equilibrium problem (SEP) is to
(3.2)
and
(3.3)
Let
denote the solution setof the split equilibrium problem SEP.
denote the solution setof the split equilibrium problem SEP.Example 3.4[22]
Let
,
and
. Let
for all R, then A is a bounded linear operator. Let
and
bedefined by
and
, respectively. Clearly,
and
. So
.
,
and
. Let
for all R, then A is a bounded linear operator. Let
and
bedefined by
and
, respectively. Clearly,
and
. So
.Example 3.5[22]
Let
with the standard norm
and
with the norm
for some
.
and
. Define a bi-function
,where
,
, thenF is a bi-function from
intoR with
. For each
,let
,then A is a bounded linear operator from
into
.In fact, it is also easy to verify that
and
for some
and
.Now define another bi-function G as follows:
for all
.Then G is a bi-function from
intoR with
.
with the standard norm
and
with the norm
for some
.
and
. Define a bi-function
,where
,
, thenF is a bi-function from
intoR with
. For each
,let
,then A is a bounded linear operator from
into
.In fact, it is also easy to verify that
and
for some
and
.Now define another bi-function G as follows:
for all
.Then G is a bi-function from
intoR with
.Clearly, when
, we have
. So
.
, we have
. So
.Corollary 3.6Let
and
be twoHilbert spaces,
bea bounded linear operator,
be a bifunctionsatisfying
and
be a bifunctionsatisfying
.Taking
,for arbitrarily chosen
,the sequence
is defined asfollows:
and
be twoHilbert spaces,
bea bounded linear operator,
be a bifunctionsatisfying
and
be a bifunctionsatisfying
.Taking
,for arbitrarily chosen
,the sequence
is defined asfollows:
(3.4)
where
denotesthe adjoint ofA,
,
and
satisfies
.If
, then thesequence
converges strongly toa point
.
denotesthe adjoint ofA,
,
and
satisfies
.If
, then thesequence
converges strongly toa point
.Remark 3.7 Since Example 3.4 and Example 3.5 satisfy the conditionsof Corollary 2.3, the split equilibrium problems in Example 3.4 andExample 3.5 can be solved by algorithm (3.4).
Application to the hierarchial variational inequality problem
Let H be a real Hilbert space,
and
be two nonexpansive mappings from H to H such that
and
.
and
be two nonexpansive mappings from H to H such that
and
.The so-called hierarchical variational inequality problem for nonexpansive mapping
with respect to a nonexpansive mapping
isto find a point
such that
with respect to a nonexpansive mapping
isto find a point
such that
(3.5)
It is easy to see that (3.5) is equivalent to the following fixed point problem:
(3.6)
where
is the metric projection from H onto
. Letting
and
(the fixed point set ofthe mapping
)and
(theidentity mapping on H), then problem (3.6) is equivalent to the followingsplit feasibility problem:
is the metric projection from H onto
. Letting
and
(the fixed point set ofthe mapping
)and
(theidentity mapping on H), then problem (3.6) is equivalent to the followingsplit feasibility problem:
(3.7)
Hence from Theorem 2.1 we have the following theorem.
Theorem 3.8LetH,
,
,CandQbe the same as above.Let
and
,and let the sequence
be defined asfollows:
,
,CandQbe the same as above.Let
and
,and let the sequence
be defined asfollows:
(3.8)
where
and
satisfies
.If
, then thesequence
converges strongly toa solution of the hierarchical variational inequality problem (3.5).
and
satisfies
.If
, then thesequence
converges strongly toa solution of the hierarchical variational inequality problem (3.5).Acknowledgements
The authors would like to express their thanks to the reviewers and editors for theirhelpful suggestions and advice. This work was supported by the National NaturalScience Foundation of China (Grant No. 11361070) and the Scientific ResearchFoundation of Postgraduate of Yunnan University of Finance and Economics.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as 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 contributed equally to this work. All authors read and approved the finalmanuscript.