DSpace at VNU: Some results on zero points of m-accretive operators in reflexive Banach spaces

R E S E A R C H Open AccessSome results on zero points of m-accretive operators in reflexive Banach spaces Chang Qun Wu1, Songtao Lv2*and Yunpeng Zhang3 * Correspondence: sqlvst@yeah.net

R E S E A R C H Open Access

Some results on zero points of m-accretive

operators in reflexive Banach spaces

Chang Qun Wu1, Songtao Lv2*and Yunpeng Zhang3

* Correspondence: sqlvst@yeah.net

2 School of Mathematics and

Information Science, Shangqiu

Normal University, Shangqiu,

Henan, China

Full list of author information is

available at the end of the article

Abstract

A modified proximal point algorithm is proposed for treating common zero points of

a finite family of m-accretive operators A strong convergence theorem is established

in a reflexive, strictly convex Banach space with the uniformly Gâteaux differentiable norm

Keywords: accretive operator; nonexpansive mapping; resolvent; fixed point; zero

point

1 Introduction and preliminaries

Let E be a Banach space and let E∗be the dual of E Let·, · denote the pairing between

E and E∗ The normalized duality mapping J : E→ E∗

is defined by

J (x) =

f ∈ E∗:x, f  = x=f 

, ∀x ∈ E.

A Banach space E is said to strictly convex if and only if x = y = ( – λ)x + λy for

x , y ∈ E and  < λ <  implies that x = y Let U E={x ∈ E : x = } The norm of E is said

to be Gâteaux differentiable if the limit limt→x+ty–x t exists for each x, y ∈ U E In this

case, E is said to be smooth The norm of E is said to be uniformly Gâteaux differentiable

if for each y ∈ U E , the limit is attained uniformly for all x ∈ U E The norm of E is said to be Fréchet differentiable if for each x ∈ U E , the limit is attained uniformly for all y ∈ U E The

norm of E is said to be uniformly Fréchet differentiable if the limit is attained uniformly for all x, y ∈ U E It is well known that (uniform) Fréchet differentiability of the norm of E implies (uniform) Gâteaux differentiability of the norm of E.

Let ρ E: [,∞) → [, ∞) be the modulus of smoothness of E by

ρ E (t) = sup

x + y – x – y

 –  : x ∈ U E,y ≤ t



A Banach space E is said to be uniformly smooth if ρ E (t)

t →  as t →  It is well known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is

single valued and uniformly norm to weak∗continuous on each bounded subset of E Recall that a closed convex subset C of a Banach space E is said to have a normal struc-ture if for each bounded closed convex subset K of C which contains at least two points, there exists an element x of K which is not a diametral point of K , i.e., sup{x – y : y ∈

K } < d(K), where d(K) is the diameter of K.

© 2014 Wu et al.; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribuAttribu-tion, and reproducAttribu-tion in any

Let D be a nonempty subset of a set C Let Proj D : C → D Q is said to be () sunny if for each x ∈ C and t ∈ (, ), we have Proj D (tx + ( – t)Proj D x ) = Proj D x;

() a contraction if ProjD = Proj D;

() a sunny nonexpansive retraction if Proj Dis sunny, nonexpansive, and a contraction

D is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from

C onto D The following result, which was established in [–], describes a

characteriza-tion of sunny nonexpansive retraccharacteriza-tions on a smooth Banach space

Let E be a smooth Banach space and let C be a nonempty subset of E Let Proj C : E → C

be a retraction and J ϕ be the duality mapping on E Then the following are equivalent:

() Proj Cis sunny and nonexpansive;

() x – Proj C x , J ϕ (y – Proj C x) ≤ , ∀x ∈ E, y ∈ C;

() Proj C x – Proj C y≤ x – y, J ϕ (Proj C x – Proj C y), ∀x, y ∈ E.

It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction Proj Cis

coincident with the metric projection from E onto C Let C be a nonempty closed convex

subset of a smooth Banach space E, let x ∈ E, and let x∈ C Then we have from the above

that x= Proj C xif and only ifx – x, J ϕ (y – x) ≤  for all y ∈ C, where Proj Cis a sunny

nonexpansive retraction from E onto C For more additional information on nonexpansive

retracts, see [] and the references therein

Let C be a nonempty closed convex subset of E Let T : C → C be a mapping In this paper, we use F(T) to denote the set of fixed points of T Recall that T is said to be an

α -contractive mapping iff there exists a constant α ∈ [, ) such that Tx – Ty ≤ αx –

y , ∀x, y ∈ C The Picard iterative process is an efficient method to study fixed points of

α -contractive mappings It is well known that α-contractive mappings have a unique fixed

point T is said to be nonexpansive iff Tx – Ty ≤ x – y, ∀x, y ∈ C It is well known that

nonexpansive mappings have fixed points if the set C is closed and convex, and the space

Eis uniformly convex The Krasnoselski-Mann iterative process is an efficient method for

studying fixed points of nonexpansive mappings The Krasnoselski-Mann iterative process

generates a sequence{x n} in the following manner:

x∈ C, x n+= α n Tx n + ( – α n )x n, ∀n ≥ .

It is well known that the Krasnoselski-Mann iterative process only has weak convergence

for nonexpansive mappings in infinite-dimensional Hilbert spaces; see [–] for more

de-tails and the references therein In many disciplines, including economics, image

recov-ery, quantum physics, and control theory, problems arise in infinite-dimensional spaces

In such problems, strong convergence (norm convergence) is often much more desirable

than weak convergence, for it translates the physically tangible property that the energy

x n – x of the error between the iterate x n and the solution x eventually becomes

arbi-trarily small To improve the weak convergence of a Krasnoselski-Mann iterative process,

so-called hybrid projections have been considered; see [–] for more details and the

references therein The Halpern iterative process was initially introduced in []; see []

for more details and the references therein The Halpern iterative process generates a

se-quence{x n} in the following manner:

x ∈ C, x = α u + ( – α )Tx , ∀n ≥ ,

where xis an initial and u is a fixed element in C Strong convergence of Halpern iterative

process does not depend on metric projections The Halpern iterative process has recently

been extensively studied for treating accretive operators; see [–] and the references

therein

Let I denote the identity operator on E An operator A ⊂ E × E with domain D(A) = {z ∈

{Az : z ∈ D(A)} is said to be accretive if for each x i ∈ D(A) and y i ∈ Ax i , i = , , there exists j(x– x)∈ J(x– x) such thaty– y, j(x– x) ≥  An

accretive operator A is said to be m-accretive if R(I + rA) = E for all r >  In this paper,

we use A–() to denote the set of zero points of A For an accretive operator A, we can

define a nonexpansive single valued mapping J r : R(I + rA) → D(A) by J r = (I + rA)–for

each r > , which is called the resolvent of A.

Now, we are in a position to give the lemmas to prove main results

Lemma .[] Let {a n }, {b n }, {c n }, and {d n } be four nonnegative real sequences

satis-fying a n+≤ ( – b n )a n + b n c n + d n, ∀n ≥ n, where n is some positive integer, {b n } is a

number sequence in (, ) such that∞

n =nb n=∞, {c n } is a number sequence such that

lim supn→∞c n ≤ , and {d n } is a positive number sequence such that∞

n =nd n<∞ Then

limn→∞a n= 

Lemma . [] Let C be a closed convex subset of a strictly convex Banach space E.

Let N ≥  be some positive integer and let T i : C → C be a nonexpansive mapping for

each i ∈ {, , , N} Let {δ i } be a real number sequence in (, ) withN

i=δ i =  Suppose

thatN

i=F (T i ) is nonempty Then the mappingN

i=T i is defined to be nonexpansive with

F(N

i=T i) =N

i=F (T i)

Lemma .[] Let {x n } and {y n } be bounded sequences in a Banach space E and let β n

be a sequence in [, ] with  < lim inf n→∞β n≤ lim supn→∞β n <  Suppose that x n+=

( – β n )y n + β n x n for all n ≥  and

lim sup

y n+– y n  – x n+– x n ≤ 

Then lim n→∞y n – x n = 

Lemma .[] Let E be a real reflexive Banach space with the uniformly Gâteaux

differ-entiable norm and the normal structure , and let C be a nonempty closed convex subset of E.

Let f : C → C be α-contractive mapping and let T : C → C be a nonexpansive mapping

with a fixed point Let {x t } be a sequence generated by the following: x t = tf (x t ) + ( – t)Tx t,

where t ∈ (, ) Then {x t } converges strongly as t →  to a fixed point x∗of T , which is the

unique solution in F (T) to the following variational inequality: f (x∗) – x∗, j(x∗– p) ≥ ,

∀p ∈ F(T).

2 Main results

Theorem . Let E be a real reflexive , strictly convex Banach space with the uniformly

Gâteaux differentiable norm Let N ≥  be some positive integer Let A m be an m-accretive

operator in E for each m ∈ {, , , N} Assume that C :=N

m=D (A m ) is convex and has

the normal structure Let f : C → C be an α-contractive mapping Let {α n }, {β n }, and {γ n}

be real number sequences in (, ) with the restriction α n + β n + γ n =  Let {δ n ,i } be a real

number sequence in (, ) with the restriction δ + δ +· · · + δ =  Let {r } be a positive

real numbers sequence and {e n ,i } a sequence in E for each i ∈ {, , , N} Assume that

N

i=A–i () is not empty Let {x n } be a sequence generated in the following manner:

x ∈ C, x n+= α n f (x n ) + β n x n + γ n

N

i=

δ n ,iJ ri (x n + e n ,i), ∀n ≥ ,

where J ri = (I + r i A i)– Assume that the control sequences {α n }, {β n }, {γ n }, and {δ n ,i } satisfy

the following restrictions:

(a) limn→∞α n= ,∞

n=α n=∞;

(b)  < lim infn→∞β n≤ lim supn→∞β n< ;

(c) ∞

n=e n ,m < ∞;

(d) limn→∞δ n ,i = δ i∈ (, )

Then the sequence {x n } converges strongly to ¯x, which is the unique solution to the following

variational inequality:f (¯x) – ¯x, J(p – ¯x) ≤ , ∀p ∈N

i=A–i ()

Proof Put y n=N

i=δ n ,iJ ri (x n + e n ,i ) Fixing p∈N

i=A–

i (), we have

y n – p ≤

N

i=

δ n ,i J ri (x n + e n ,i ) – p

≤

N

i=

δ n ,i (x n + e n ,i ) – p

≤ x n – p +

N

i=

e n ,i

Hence, we have

x n+– p  ≤ α n f (x n ) – p + β n x n – p  + γ n y n – p

≤ α n α x n – p + α n f (p) – p + β n x n – p + γ n x n – p + γ n N

i=

e n ,i

≤  – α n ( – α) x n – p  + α n ( – α) f (p) – p

 – α +

N

i=

e n ,i

≤ maxx n – p, f (p) – p +N

i=

e n ,i

≤ maxx– p, f (p) – p +∞

j=

N

i=

e j ,i

This proves that the sequence{x n } is bounded, and so is {y n} Since

y n – y n–=

N

i=

δ n ,i

J rm (x n + e n ,i ) – J ri (x n–+ e n –,i)

+

N

(δ n ,i – δ n –,i )J ri (x n–+ e n –,i),

we have

y n – y n– ≤

N

i=

δ n ,i J ri (x n + e n ,i ) – J ri (x n–+ e n –,i)

+

N

i=

|δ n ,i – δ n –,i| J r

i (x n–+ e n –,i)

≤ x n – x n– +

N

i=

e n ,i +

N

i=

e n –,i +

N

i=

|δ n ,i – δ n –,i| J ri (x n–+ e n –,i)

≤ x n – x n– +

N

i=

e n ,i +

N

i=

e n –,i  + M

N

i=

|δ n ,i – δ n –,i|,

where Mis an appropriate constant such that

M= max sup

n≥ J r(x n + e n,) , sup

n≥ J r(x n + e n,) , , sup

n≥ J rN (x n + e n ,N) .

Define a sequence{z n } by z n:=xn+–β nxn

–β n , that is, x n+= β n x n + ( – β n )z n It follows that

yz n – z n– ≤ α n

 – β n

f (x n ) – y n + α n–

 – β n–

f (x n–) – y n– +y n – y n–

≤ α n

 – β n f (x n ) – y n + α n–

 – β n– f (x n–) – y n– +x n – x n– +

N

i=

|δ n ,i – δ n –,i |J ri x n–

≤ α n

 – β n

f (x n ) – y n + α n–

 – β n–

f (x n–) – y n– +x n – x n–

+ M

 N

i=

|δ n ,i – δ i| +

N

i=

|δ i – δ n –,i|

 ,

where Mis an appropriate constant such that

M= max sup

n≥J rx n, sup

n≥J rx n, , sup

n≥J rN x n

This implies that

z n – z n– – x n – x n–

≤ α n

 – β n f (x n ) – y n + α n–

 – β n– f (x n–) – y n–

+ M

 N

|δ n ,i – δ i| +

N

|δ i – δ n –,i|



From the restrictions (a), (b), (c), and (d), we find that

lim sup

z n – z n– – x n – x n– ≤ 

Using Lemma ., we find that limn→∞z n – x n =  This further shows that

lim supn→∞x n+– x n  =  Put T =N

i=δ i J ri It follows from Lemma . that T is nonex-pansive with F(T) =N

i=F (J ri) =N

i=A–i () Note that

x n – Tx n

≤ x n – x n+ + x n+– Tx n

≤ x n – x n+ + α n f (x n ) – Tx n + β n x n – Tx n  + γ n y n – Tx n

≤ x n – x n+ + α n f (x n ) – Tx n + β n x n – Tx n  + MN

i=

|δ n ,i – δ i|

This implies that

( – β n)xn – Tx n  ≤ x n – x n+ + α n f (x n ) – Tx n + MN

i=

|δ n ,i – δ i|

It follows from the restrictions (a), (b), and (d) that

lim

n→∞Tx n – x n = 

Now, we are in a position to prove that lim supn→∞f (¯x) – ¯x, J(x n–¯x) ≤ , where ¯x =

limt→x t , and x tsolves the fixed point equation

x t = tf (x t ) + ( – t)Tx t, ∀t ∈ (, ).

It follows that

x t – x n = t

f (x t ) – x n , J(x t – x n)

+ ( – t)

Tx t – x n , j(x t – x n)

= t

f (x t ) – x t , J(x t – x n)

+ t

x t – x n , J(x t – x n)

+ ( – t)

Tx t – Tx n , J(x t – x n)

+ ( – t)

Tx n – x n , J(x t – x n)

≤ tf (x t ) – x t , J(x t – x n)

+x t – x n+Tx n – x n x t – x n , ∀t ∈ (, ).

This implies that



x t – f (x t ), J(x t – x n)

≤

t Tx n – x n x t – x n , ∀t ∈ (, ).

Since limn→∞Tx n – x n = , we find that lim supn→∞x t – f (x t ), J(x t – x n) ≤  Since J is

strong to weak∗uniformly continuous on bounded subsets of E, we find that

f(¯x) – ¯x, J(xn–¯x)

–

x t – f (x t ), J(x t – x n)

≤f(¯x) – ¯x, J(x –¯x)

–

f(¯x) – ¯x, J(x – x)

+f(¯x) – ¯x, J(xn – x t)

–

x t – f (x t ), J(x t – x n)

≤f(¯x) – ¯x, J(xn–¯x) – J(x n – x t)+f(¯x) – ¯x + xt – f (x t ), J(x n – x t)

≤ f (x t) –¯x J (x n–¯x) – J(x n – x t) + ( + α) ¯x – x t x n – x t.

Since x t → ¯x, as t → , we have

lim

t→f(¯x) – ¯x, J(xn–¯x)

–

f (x t ) – x t , J(x n – x t)= .

For > , there exists δ >  such that ∀t ∈ (, δ), we have



f(¯x) – ¯x, J(xn–¯x)≤f (x t ) – x t , J(x n – x t)

+

This implies that lim supn→∞f (¯x) – ¯x, J(x n–¯x) ≤ .

Finally, we show that x n → ¯x as n → ∞ Since  · is convex, we see that

y n–¯x =

N

i=

δ n ,iJ ri (x n + e n ,i) –¯x



≤

N

i=

δ n ,i J ri (x n + e n ,i) –¯x 

≤ x n–¯x+

N

i=

e n ,i e n ,i  + x n–¯x

It follows that

x n+–¯x = α n



f (x n) –¯x, J(x n+–¯x)+ β n



x n–¯x, J(x n+–¯x) + γ n



y n–¯x, J(x n+–¯x)

≤ α n α x n–¯xx n+–¯x + α n



f(¯x) – ¯x, J(xn+–¯x) + β n x n–¯xx n+–¯x + γ n y n–¯xx n+–¯x

≤α n α



x n–¯x+x n+–¯x + α n

f(¯x) – ¯x, J(x n+–¯x)

+β n



x n–¯x+x n+–¯x +γ n

 x n–¯x

+

N

i=

e n ,i e n ,i  + x n–¯x +γ n

x n+–¯x Hence, we have

x n+–¯x≤  – α n ( – α) x n–¯x+ α n



f(¯x) – ¯x, J(xn+–¯x)

+

N

i=

e n ,i e n ,i  + x n–¯x

Using Lemma ., we find x → ¯x as n → ∞ This completes the proof. 

Remark . There are many spaces satisfying the restriction in Theorem ., for example

L p , where p > .

Corollary . Let E be a Hilbert space and let N ≥  be some positive integer Let A m be a

maximal monotone operator in E for each m ∈ {, , , N} Assume that C :=N

m=D (A m)

is convex and has the normal structure Let f : C → C be an α-contractive mapping Let

{α n }, {β n }, and {γ n } be real number sequences in (, ) with the restriction α n + β n + γ n= 

Let {δ n ,i } be a real number sequence in (, ) with the restriction δ n,+ δ n,+· · · +δ n ,N =  Let

{r m } be a positive real numbers sequence and {e n ,i } a sequence in E for each i ∈ {, , , N}.

Assume thatN

i=A–

i () is not empty Let {x n } be a sequence generated in the following

manner:

x ∈ C, x n+= α n f (x n ) + β n x n + γ n

N

i=

δ n ,iJ ri (x n + e n ,i), ∀n ≥ ,

where J ri = (I + r i A i)– Assume that the control sequences {α n }, {β n }, {γ n }, and {δ n ,i } satisfy

the following restrictions:

(a) limn→∞α n= ,∞

n=α n=∞;

(b)  < lim infn→∞β n≤ lim supn→∞β n< ;

(c) ∞

n=e n ,m < ∞;

(d) limn→∞δ n ,i = δ i∈ (, )

Then the sequence {x n } converges strongly to ¯x, which is the unique solution to the following

variational inequality:f (¯x) – ¯x, p – ¯x ≤ , ∀p ∈N

i=A–i ()

3 Applications

In this section, we consider a variational inequality problem Let A : C → E∗be a single

valued monotone operator which is hemicontinuous; that is, continuous along each line

segment in C with respect to the weak∗topology of E∗ Consider the following variational

inequality:

find x ∈ C such that y – x, Ax ≥ , ∀y ∈ C.

The solution set of the variational inequality is denoted by VI(C, A) Recall that the normal

cone N C (x) for C at a point x ∈ C is defined by

N C (x) =

x∗∈ E∗:

y – x, x∗

≤ , ∀y ∈ C Now, we are in a position to give the convergence theorem

Theorem . Let E be a real reflexive , strictly convex Banach space with the uniformly

Gâteaux differentiable norm Let N ≥  be some positive integer and let C be nonempty

closed and convex subset of E Let A i : C → E∗a single valued , monotone and

hemicontinu-ous operator Assume thatN

i=VI(C, Ai ) is not empty and C has the normal structure Let

f : C → C be an α-contractive mapping Let {α n }, {β n }, and {γ n } be real number sequences

in (, ) with the restriction α n + β n + γ n =  Let {δ n ,i } be a real number sequence in (, )

with the restriction δ + δ +· · · + δ =  Let {r } be a positive real numbers sequence

and {e n ,i } a sequence in E for each i ∈ {, , , N} Let {x n } be a sequence generated in the

following manner:

x ∈ C, x n+= α n f (x n ) + β n x n + γ n

N

i=

δ n ,iVI



C , A i+ 

r i

(I – x n)

 , ∀n ≥ .

Assume that the control sequences {α n }, {β n }, {γ n }, and {δ n ,i } satisfy the following

restric-tions:

(a) limn→∞α n= ,∞

n=α n=∞;

(b)  < lim infn→∞β n≤ lim supn→∞β n< ;

(c) ∞

n=e n ,m < ∞;

(d) limn→∞δ n ,i = δ i∈ (, )

Then the sequence {x n } converges strongly to ¯x, which is the unique solution to the following

variational inequality:f (¯x) – ¯x, J(p – ¯x) ≤ , ∀p ∈N

i=VI(C, Ai)

Proof Define a mapping T i ⊂ E × E∗by

T i x:=

⎧

⎨

⎩

A i x + N C x, x ∈ C,

∅, x ∈ C./

From Rockafellar [], we find that T i is maximal monotone with T–

i () = VI(C, A i) For

each r i > , and x n ∈ E, we see that there exists a unique x ri ∈ D(T i ) such that x n ∈ x ri+

r i T i (x r i ), where x r i = (I + r i T i)–x n Notice that

y n ,i= VI



C , A i+ 

r i (I – x n)

 ,

which is equivalent to



y – y n ,i , A i y n ,i+ 

r i

(y n ,i – x n)



≥ , ∀y ∈ C,

that is, –A i y n ,i+ri(x n – y n ,i)∈ N C (y n ,i ) This implies that y n ,i = (I + r i T i)–x n Using

Theo-rem ., we find the desired conclusion immediately 

From Theorem ., the following result is not hard to derive

Corollary . Let E be a real reflexive , strictly convex Banach space with the uniformly

Gâteaux differentiable norm Let C be nonempty closed and convex subset of E Let A : C→

E∗a single valued , monotone and hemicontinuous operator with VI(C, A) Assume that C

has the normal structure Let f : C → C be an α-contractive mapping Let {α n }, {β n }, and

{γ n } be real number sequences in (, ) with the restriction α n + β n + γ n =  Let {x n } be a

sequence generated in the following manner:

x∈ C, x n+= α n f (x n ) + β n x n + γ nVI



C , A +

r (I – x n)

 , ∀n ≥ .

Assume that the control sequences {α }, {β }, and {γ } satisfy the following restrictions:

(a) limn→∞α n= ,∞

n=α n=∞;

(b)  < lim infn→∞β n≤ lim supn→∞β n< 

Then the sequence {x n } converges strongly to ¯x, which is the unique solution to the following

variational inequality:f (¯x) – ¯x, J(p – ¯x) ≤ , ∀p ∈ VI(C, A i)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1 School of Business and Administration, Henan University, Kaifeng, Henan, China 2 School of Mathematics and

Information Science, Shangqiu Normal University, Shangqiu, Henan, China 3 Vietnam National University, Hanoi, Vietnam.

Acknowledgements

The authors are grateful to the editor and the reviewers for useful suggestions which improved the contents of the article.

Received: 16 January 2014 Accepted: 30 April 2014 Published: 14 May 2014

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we have

y n – y n–... ,i – δ n –,i|,

where Mis an appropriate constant such that

M= max sup

n≥ J...

e j ,i

This proves that the sequence{x n } is bounded, and so is {y n} Since

y n – y n–=