R E S E A R C H
Open Access
Weak convergence of a Mann-like
algorithm for nonexpansive and accretive
operators
Xiaolong Qin
1,2and Jen-Chih Yao
2**Correspondence:
2Center for General Education,
China Medical University, Taichung, Taiwan
Full list of author information is available at the end of the article
Abstract
Zero point problems of two accretive operators and fixed point problems of a nonexpansive mappings are investigated based on a Mann-like iterative algorithm. Weak convergence theorems are established in a Banach space.
MSC: 47H06; 47H09; 90C33
Keywords: accretive operator; fixed point; nonexpansive mapping; resolvent; zero point
1 Introduction and preliminaries
LetEbe a real Banach space and letE∗be the dual space ofE. LetR+be the set of
nonneg-ative real numbers. Given a continuous strictly increasing function:h:R+→R+, whereR+
denotes the set of nonnegative real numbers, such thatlimr→∞h(r) =∞andh() = , we associate with it a (possibly multivalued) generalized duality mapJϕ:E→E
∗
, defined as
Jϕ(x) :=
x∗∈E∗:x∗(x) =hxx,hx=x∗, ∀x∈E.
In this paper, we use the generalized duality map associated with the gauge functionh(t) = tq–forq> ,
Jq(x) :=
x∗∈E∗:x∗,x=xq,x∗=xq–, ∀x∈E.
The modulus of convexity ofEis the functionδE() : (, ]→[, ] defined byδE() =
inf{ –x+t :x=y= ,x–y ≥}. Recall thatE is said to be uniformly convex if
δE() > for any∈(, ]. Letp> . We say thatEisp-uniformly convex if there exists a constantcq> such thatδE()≥cppfor any∈(, ].
LetBE={x∈E:x= }. The norm ofEis said to be Gâteaux differentiable if the limit
limt→(x+ty–x)/texists for eachx,y∈BE. In this case,Eis said to be smooth. The norm ofE is said to be uniformly Gâteaux differentiable if for eachy∈BE, the limit is attained uniformly for allx∈BE. The norm ofEis said to be Fréchet differentiable if for eachx∈BE, the limit is attained uniformly for ally∈BE. The norm of Eis said to be uniformly Fréchet differentiable if the limit is attained uniformly for allx,y∈BE.
LetρE: [,∞)→[,∞) be the modulus of smoothness ofEby
ρE(t) =sup
y+x–y–x
– :y ≤t,x∈BE
.
A Banach spaceEis said to be uniformly smooth if ρE(t)
t → ast→. Letq> .Eis said to beq-uniformly smooth if there exists a fixed constantc> such thatρE(t)≤ctq. It is well known thatEis uniformly smooth if and only if the norm ofEis uniformly Fréchet differentiable. IfEisq-uniformly smooth, thenq≤ andEis uniformly smooth [], and hence the norm ofEis uniformly Fréchet differentiable, in particular, the norm ofEis Fréchet differentiable.
Typical examples of both uniformly convex and uniformly smooth Banach spaces are Lp, wherep> . To be more precise,Lpis mini{p, }-uniformly smooth for everyp> . It is well known thatEisp-uniformly convex if and only ifE∗isq-uniformly smooth, where pandqsatisfy the relation
p+
q= .
LetTbe a mapping onE. The fixed point set ofTis denoted byFix(T). Recall thatTis said to be nonexpansive iff
Tx–Ty ≤ x,y, ∀x,y∈E.
LetIdenote the identity operator onE. An operatorA⊂E×Ewith domainD(A) = {z∈E:Az=∅}and rangeR(A) =∪{Az:z∈D(A)}is said to be accretive if, fort> and x,y∈D(A),
x–y ≤t(u–v) + (x–y), ∀u∈Ax,v∈Ay.
It follows from Kato [] thatAis accretive if and only if, forx,y∈D(A), there existsjq(x– y)∈Jq(x–y) such that
u–v,jq(x–y)
≥.
An accretive operatorAis said to bem-accretive ifR(I+rA) =Efor allr> . In Hilbert spaces, an operatorAism-accretive if and only ifAis maximal monotone. In this paper, we useA–() to denote the set of zero points ofA.
Recall that a single valued operatorAonE is said to beα-strongly accretive if there exists a constantα> and somejq(x–y)∈Jq(x–y) such that
Ax–Ay,jq(x–y)
≥αx–yq, ∀x,y∈E.
Ais said to beα-inverse strongly accretive if there exists a constantα> and somejq(x– y)∈Jq(x–y) such that
Ax–Ay,jq(x–y)
≥αAx–Ayq, ∀x,y∈E.
The convex feasibility problem asks to find a point in the intersection of convex sets. This is an important problem in mathematics and engineering; see,e.g., [–] and the ref-erences therein. Oftentimes, the convex sets are given as fixed point sets of projections or (more generally) averaged nonexpansive operators. In this paper, we will focus our at-tention on the problem of finding a common element inFix(T)∩(A+B)–(), whereT
is a nonexpansive mapping,Ais anα-inverse strongly accretive operator andBis an m-accretive operator, in the framework of uniformly convex andq-uniformly smooth Ba-nach spaces. The problem is quite general in the sense that it includes: split feasibility problems, convexly constrained linear inverse problems, fixed point problems, variational inequalities, convexly constrained minimization problems, and Nash equilibrium prob-lems in noncooperative games, as special cases; see, for instance, [–] and the refer-ences therein. Recently, mean valued iterative algorithms have been introduced by many authors to investigate this problem; see, for instance, [–] and the references therein. Related work can also be found,e.g., in [–]. However, there is little work in the exist-ing literature in the settexist-ing of Banach spaces. The aim of this paper is to establish a weak convergence theorem in the framework of Banach spaces based on a Mann-like iterative algorithm. Applications are also provided to support the main results of this article.
In order to obtain our main results, we also need the following lemmas.
Lemma . Let E be a real Banach space.Let A:E→E be a single valued operator and
let B:E→Ebe an m-accretive operator.Then
(A+B)–() =FixJrB(I–rA),
where JB
r(I–rA)is the resolvent of B for a> .
Proof
p∈FixJrB(I–rA) ⇐⇒ p=JrB(I–rA)p
⇐⇒ p+rBp=p–rAp
⇐⇒ p∈(A+B)–().
Lemma .[] Let E be a real q-uniformly smooth Banach space.Then the following
in-equality holds:
x+yq≤ xq+qy,Jq(x+y)
, ∀x,y∈E,
and
x+yq≤ xq+qy,Jq(x)
+Kqyq, ∀x,y∈E,
where Kqis some fixed positive constant.
Lemma .[] Let r> and q> be two fixed real numbers.Then a Banach space E is
ϕ: [,∞)→[,∞)withϕ() = such that
ax+ ( –a)yq≤( –a)yq+axq–w(a)ϕy–x,
where w(a) =aq( –a) + ( –a)qa,for all x,y∈B
r() :={x∈E:x ≤r}and a∈[, ].
Lemma .[] Let E be a real uniformly convex Banach space and let C be a nonempty
closed convex and bounded subset of E.Then there is a strictly increasing and continu-ous convex functionψ: [,∞)→[,∞)withϕ() = such that,for every nonexpansive mapping T:C→C and,for all x,y∈C and t∈[, ],the following inequality holds:
tTx+ ( –t)Ty–Ttx+ ( –t)y≤ψ–y–x–Ty–Tx.
Lemma .[] Let E be a real uniformly convex Banach space,and let T be a
nonexpan-sive mapping on E.Then I–T is demiclosed at zero.
Lemma .[] Let E be a real uniformly convex Banach space.Let E∗ the dual space
of E such that it has the Kadec-Klee property.Suppose that{xn} is a bounded sequence such thatlimn→∞( –a)p–p+axnexists for all a∈[, ]and p,p∈ωw(xn),where
ωw(xn) :{x:∃xni x}denotes the weakω-limit set of{xn}Thenωw(xn)is a singleton.
2 Main results
Theorem . Let E be a real uniformly convex and q-uniformly smooth Banach space with
constant Kq.Let B:D(B)⊂E→E be an m-accretive operator,A:E→E anα-inverse strongly accretive operator and T :E→E a nonexpansive mapping such thatFix(T)∩ (A+B)–()=∅. Let{r
n} be a positive number sequence and let{αn}be a real number sequence in(, )such that {αn} ⊂[α,α],where <α<α< and{rn} ⊂[r,r],where <r<r< (Kqα
q)
q–.Let{x
n}be a sequence generated in the following manner:x∈E and
xn+=αnTxn+ ( –αn)(I+rnB)–(xn–rnAxn),∀n≥,Then{xn}converges weakly to some point inFix(T)∩(A+B)–().
Proof First, we show that{xn}is bounded. From Lemma ., we have
(I–rnA)x– (I–rnA)yq ≤ x–yq–qrn
Ax–Ay,Jq(x–y)
+KqrnqAx–Ayq
≤ x–yq–qrnαAx–Ayq+KqrqnAx–Ayq
=x–yq–αq–Kqrqn–
rnAx–Ayq. (.)
In view of the restriction imposed on{rn}, one sees thatI–rnAis nonexpansive. PutJrBn= (I+rnB)–and fixp∈(A+B)–()∩Fix(T). By using Lemma ., we find from (.) that
It follows thatlimn→∞xn–pexists and, in particular, that{xn}is bounded. Putyn= JB
rn(xn–rnAxn). SinceBism-accretive, we find from Lemma . and (.) that
yn–pq≤ rn
xn–rnAxn–yn rn
–(I–rnA)p–p rn
+ (yn–p) q
=
(I–rnA)xn– (I–rnA)p
+ (yn–p)
q
≤
yn–p q+
(I–rnA)xn– (I–rnA)p q
–
qϕ(yn–p) –
(I–rnA)xn– (I–rnA)p
≤(I–rnA)xn– (I–rnA)p q
–
qϕ(yn–p) –
(I–rnA)xnn– (I–rnA)p
≤ xn–pq–
αq–Kqrnq–
rnAxn–Apq
–
qϕ(yn–p) –
(I–rnA)xnn– (I–rnA)p. (.)
Since · qis convex, we find from (.) and (.) that
xn+–pq≤αnTxn–pq+ ( –αn)yn–pq ≤ xn–pq–
αq–Kqrqn–
rn( –αn)Axn–Apq
– ( –αn)
qϕ(yn–p) –
(I–rnA)xn– (I–rnA)p.
It follows from the restrictions imposed on{αn}and{rn}that
lim
n→∞(yn–xn) – (rnAp–rnAxn)= (.)
and
lim
n→∞Axn–Ap= . (.)
Sinceyn–xn ≤ (yn–xn) – (rnAp–rnAxn)+rnAp–Axn, we find from (.) and (.) that
lim
n→∞xn–J B
rn(xn–rnAxn)= . (.)
SinceBis anm-accretive operator, we have
xn–JrB(I–rA)xn
r –
xn–JrBn(I–rnA)xn rn
,Jq
JrB(I–rA)xn–JrBn(I–rnA)xn
It follows that
xn–JrBn(I–rnA)xnJ B
r(I–rA)xn–JrBn(I–rnA)xn q–
≥rn–r rn
xn–JrBn(I–rnA)xn,Jq
JrB(I–rA)xn–JrBn(I–rnA)xn
≥JrB(I–rA)xn–JrBn(I–rnA)xn q
,
which implies
xn–JrBn(I–rnA)xn≥J B
r(I–rA)xn–JrBn(I–rnA)yn. (.)
From (.), one sees that
lim
n→∞J
B
r(xn–rAxn) –xn= . (.)
In view of Lemma ., one has
xn+–pq≤αnTxn–pq+ ( –αn)yn–pq –αnq( –αn) + ( –αn)qαn
ϕTxn–yn
≤ xn–pq–
αnq( –αn) + ( –αn)qαn
ϕTxn–yn
,
that is,
αqn( –αn) + ( –αn)qαn
ϕTxn–yn
≤ xn–pq–xn+–pq.
This shows thatlimn→∞Txn–yn= . From (.), we findlimn→∞Txn–xn= . In view of Lemma ., we see thatωw(xn)⊂Fix(JrB(I+rA))∩Fix(T) = (A+B)–()∩Fix(T).
Next, we show thatωw(xn) is a singleton set. Define mappingsSn:E→Eby
Snx:=αnTx+ ( –αn)JrBn(I–rnA)x, ∀x∈C.
Set
Sn,m=Sn+m–Sn+m–· · ·Sn, ∀n,m≥.
SinceSnis nonexpansive, we find thatSn,mis also nonexpansive andSn,mxn=xn+m. For all t∈[, ] andn,m≥, put
bn(t) =txn+ ( –t)p–p
and
cn,m=Sn,m
txn+ ( –t)p
–txn+m+ ( –t)p,
cn,m≤ψ–
xn–p–Sn,mxn–Sn,mp
=ψ–xn–p–xn+m–p+p–Sn,mp
≤ψ–xn–p–
xn+m–p–p–Sn,mp
.
This implies that{cn,m}converges uniformly to zero asn→ ∞for allm≥. On the other hand, we have
bn+m(t)≤cn,m+Sn,m
txn+ ( –t)p
–p
≤cn,m+Sn,m
txn+ ( –t)p
–Sn,mp+Sn,mp–p
≤cn,m+bn(t) +Sn,mp–p.
Takinglim supasm→ ∞and then thelim infasn→ ∞, we find thatlim supn→∞bn(t)≤
lim infn→∞bn(t). This proves thatlimn→∞bn(t) exists for anyt∈[, ]. This implies from Lemma . thatωw(xn) is a singleton set. This proves the proof.
Note that, in the framework of Hilbert spaces, the concept of monotonicity coincides with the concept of accretivity. Next, we apply our main results to solve variational in-equality problems and minimizer problems of convex functions in the framework of Hilbert spaces.
LetHbe a Hilbert space with inner product·,·and its induced norm · . LetCbe a nonempty closed convex subset ofHand letProjHCbe the metric projection fromHontoC. Recall the following classical variational inequality: findx∈Csuch thaty–x,Ax ≥, ∀y∈C. The solution set of the variational inequality is denoted byVI(C,A). Projection-gradient methods have been recently investigated for solving the variational inequality. It is well known thatxis a solution to the variational inequality iffxis a fixed point of
ProjHC(I–rA), whereI denotes the identity onH andris a positive real number. IfAis inverse strongly monotone, thenProjHC(I–rA) is a nonexpansive mapping. Moreover. IfC is also bounded, then the existence of solutions of the variational inequality is guaranteed by the nonexpansivity of mappingProjHC(I–rA). LetiCbe a function defined by
iC(x) = ⎧ ⎨ ⎩
, x∈C, ∞, x∈/C.
It is easy to see thatiCis a proper lower and semicontinuous convex function onH, and the subdifferential∂iC ofiCis maximal monotone. Define the resolventJr:= (I+r∂iC)– of subdifferential operator∂iC. Lettingx=Jry, we find that
y∈x+r∂iCx ⇐⇒ y∈x+rNCHx
⇐⇒ y–x,v–x ≤, ∀v∈C ⇐⇒ x=ProjHCy,
whereNH
Corollary . Let H be a real Hilbert space.Let C be a nonempty closed and convex subset of E and letProjHCbe the metric projection from H onto C.Let A anα-inverse strongly mono-tone operator on H and T a nonexpansive mapping on C such thatFix(T)∩VI(C,A)=∅. Let{rn}be a positive number sequence and let{αn}be a real number sequence in(, )such that{αn} ⊂[α,α],where <α<α< and{rn} ⊂[r,r],where <r<r< α.Let{xn}be a sequence generated in the following manner:x∈C and xn+=αnTxn+ ( –αn)ProjHC(xn– rnAxn),∀n≥.Then{xn}converges weakly to some point inFix(T)∩VI(C,A).
Now, we are in a position to consider the problem of finding minimizers of proper lower semicontinuous convex functions. For a proper lower semicontinuous convex function g:H→(–∞,∞], the subdifferential mapping∂gofgis defined by∂g(x) ={x∗∈H:g(x) + y–x,x∗ ≤g(y),∀y∈H},∀x∈H. Rockafellar [] proved that∂gis a maximal monotone operator. It is easy to verify that ∈∂g(v) if and only ifg(v) =minx∈Hg(x).
Corollary . Let H be a real Hilbert space.Let g:H→(–∞,∞]be a proper convex and
lower semicontinuous function and let T:H→H be a nonexpansive mapping such that
Fix(T)∩(∂g)–()=∅.Let{r
n}be a positive number sequence and let{αn}be a real number sequence in(, )such that {αn} ⊂[α,α],where <α<α< and{rn} ⊂[r,r],where <r<r< (qKα
q)
q–.Let{x
n}be a sequence generated in the following manner:x∈H and
xn+=αnTxn+ ( –αn)yn,∀n≥,where yn=minz∈H{g(z) +z–xn+en
rn }.Then{xn}converges weakly to some point inFix(T)∩(A+B)–().
Proof Sinceg:H→(–∞,∞] is a proper convex and lower semicontinuous function, we see that subdifferential ∂g ofg is maximal monotone. PuttingA= , we haveyn=
arg minz∈H{g(z) + z–xn
rn } is equivalent to ∈∂g(yn) +
rn(yn–xn). Hence, we havexn∈ yn+rn∂g(yn). By use of Theorem ., we find the desired conclusion immediately.
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Author details
1Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Sichuan,
China.2Center for General Education, China Medical University, Taichung, Taiwan.
Acknowledgements
The first author was supported by the National Natural Science Foundation of China under Grant No. 11401152. The second author was partially supported by the Grant MOST 103-2923-E-039-001-MY3.
Received: 12 August 2016 Accepted: 4 September 2016 References
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