A new explicit iteration method for variational inequalities on the set of common fixed points for a finite family of nonexpansive mappings

R E S E A R C H

Open Access

A new explicit iteration method for

variational inequalities on the set of common

fixed points for a finite family of

nonexpansive mappings

Jong Kyu Kim

_{and Nguyen Buong}

*_{Correspondence:} [email protected] 1_{Department of Mathematics} Education, Kyungnam University, Changwon, 631-701, Korea Full list of author information is available at the end of the article

Abstract

In this paper, we introduce a new explicit iteration method based on the steepest descent method and Krasnoselskii-Mann type method for finding a solution of a variational inequality involving a Lipschitz continuous and strongly monotone mapping on the set of common fixed points for a finite family of nonexpansive mappings in a real Hilbert space.

MSC: 41A65; 47H17; 47H20

Keywords: contraction; common fixed points; hybrid steepest descent method; nonexpansive mappings; monotone mappings

1 Introduction and preliminaries

LetCbe a nonempty closed and convex subset of a real Hilbert spaceHwith the inner product·,·and the norm·, and letF:H→Hbe a nonlinear mapping. The variational inequality problem is to find a pointp∗∈Csuch that

Fp∗,p–p∗≥, ∀p∈C. (.)

Variational inequalities were initially studied by Kinderlehrer and Stampacchia in [], and since then have been widely investigated. They cover partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance (see [–]).

It is well known that ifFis anL-Lipschitz continuous andη-strongly monotone,i.e.,F

satisfies the following conditions:

F(x) –F(y)≤Lx–y,

F(x) –F(y),x–y≥ηx–y,

whereLandηare fixed positive numbers, then (.) has a unique solution. It is also known that (.) is equivalent to the fixed point equation

p=PC

p–μF(p), (.)

wherePCdenotes the metric projection fromx∈HontoCandμis an arbitrarily positive

constant.

The fixed point formulation (.) involves the metric projectionPC. To overcome the

complexity caused byPC, Yamada [] introduced a hybrid steepest descent method for

solving (.). His idea is stated as follows. Assume thatC=N_i=Fix(Ti), the set of common

fixed points of a finite family of nonexpansive mappingsTionHwith an integerN≥.

Recall thatT:H→His nonexpansive if

Tx–Ty ≤ x–y, ∀x,y∈H

and

Fix(T) ={x∈H:x=Tx}

denotes the fixed point set ofT. Yamada proposed the following algorithm in []

uk+=T[k+]uk–λk+μF(T[k+]uk), (.)

where T[n]=TnmodN, for integern≥, with the mod-function taking values in the set

{, , . . . ,N}, μ∈(, η/L) and{λk} ⊂(, ), and proved that the sequence{uk}in (.)

converges strongly top∗under the following conditions:

(L) limλk= ;

(L) λk=∞;

(L) |λk–λk+N|<∞.

Further, Zeng and Yao [] proved the same result with (L) replaced by

(L) lim(λk–λk+N)/λk+N = .

Theorem .[] Let H be a real Hilbert space,and let F:H→H be an L-Lipschitz con-tinuous and η-strongly monotone mapping for some constants L,η> .Let{Ti}Ni= be N

nonexpansive self-maps of H such that C=N_i=Fix(Ti)=∅,μ∈(, η/L),and let

condi-tions(L), (L), (L)be satisfied.Assume in addition that

C=

N

i=

Fix(Ti) =Fix(TT· · ·TN)

=Fix(TNTT· · ·TN–)

=· · ·

=Fix(TT· · ·TNT). (.)

Then the sequence{uk}defined by(.)converges strongly to the unique element p∗in(.).

It is not difficult to show that (L) implies (L) iflimλk/λk+Nexists. However, in general,

conditions (L) and (L) are not comparable,i.e., neither one of them implies the other (see [] for details).

Recently, Zenget al.[] proposed the following iterative scheme:

uk+=T[k+]uk–λk+μk+F(T[k+]uk), (.)

Theorem .[] Let H be a real Hilbert space,and let F:H→H be an L-Lipschitz con-tinuous and η-strongly monotone mapping for some constants L,η> .Let{Ti}Ni= be N

nonexpansive self-maps of H such that

C=

N

i=

Fix(Ti)=∅,

and letμk∈(, η/L).Assume that the following conditions hold:

(i) λk=∞,where{λk} ⊂(, );

(ii) |μk–η/L| ≤

η_–cL_/L_{for somec∈(,η}_/L_); (iii) lim(μk+N– (λk/λk+N)μk) = .

Assume in addition that(.)holds.If

lim sup

k→∞

T[k+N]· · ·T[k+]uk–uk+N,T[k+N]· · ·T[k+]uk–uk ≤, (.)

then the sequence{uk}defined by(.)converges strongly to the unique element p∗in(.).

They also showed that conditions (L), (L) and (L) are sufficient for{uk}to be bounded

and

lim

k→∞uk–T[k+N]· · ·T[k+]uk= .

So, (.) is satisfied. They did not give another sufficient condition different from (L), (L) and (L).

LetFx=Ax–u, whereAis a self-adjoint bounded linear mapping such thatAis strongly positive,i.e.,

Ax,x ≥ηx, ∀x∈H

anduis some fixed element inH. Xu [] introduced the following iteration process:

uk+= (I–λk+A)Tk+uk+λk+u, (.)

whereIis the identity mapping ofH, and proved the following result.

Theorem .[] Let conditions(L), (L)and(L)or(L)be satisfied.Assume in addition that(.)holds.Then the sequence{uk}generated by algorithm(.)converges strongly to

the unique solution of(.)with Fx=Ax–u.

Very recently, Liu and Cui [] showed that the condition

C=

N

i=

Fix(Ti) =Fix(TT· · ·TN) (.)

is sufficient for (.) ifC=∅.

point problems to solve (.) withC=N_i=Fix(Ti), whereTiis a nonexpansive mapping

onHfor eachi.

Given a starting pointx∈H, the iteration is defined by ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈H,

y

k= (I–λkμF)xk,

y_ki = ( –β_ki)xk+βkiTiyik–, i= , . . . ,N,

xk+=yN_k,

(.)

and the sequences of parameters{λk}and{βki} ⊂(, ) satisfy the following conditions:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

λk→ ask→ ∞ and

λk=∞;

β_ki→βi,  <βi< ,i= , . . . ,N– ;  <lim infk→∞βkN≤lim supk→∞βkN< .

(.)

In Section , we prove the strong convergence theorem for (.)-(.) without condi-tions (L), (L) and (.). An application to the case thatTiis aγi-strictly

pseudocontrac-tive mapping is given in Section .

2 Main results

We need the following lemmas for the proof of our main result.

Lemma .[]

(i) x+y_{≤ x}_{+ y,x+y,∀x,y∈H.}

(ii) ( –t)x+ty_{= ( –t)x}_+ty_{– ( –t)tx–y}_{,∀x,y∈H,and for any fixed}

t∈[, ].

From [], we have the following lemma.

Lemma . [] Tλ_x–Tλ_{y ≤( –λτ)x–y,∀x,y∈H and for a fixed numberμ∈} (, η/L),λ∈(, ),whereτ= – –μ(η–μL_{)∈(, )and T}λ_{x= (I–λμF)x for x∈H.}

Lemma .[, ] Assume that T is a nonexpansive self-map of a closed convex subset K of a real Hilbert space H.If T has a fixed point,then I–T is demiclosed;that is,whenever

{xk}is a sequence in K weakly converging to some x∈K and the sequence{(I–T)xk}strongly

converges to some y,it follows that(I–T)x=y.

Lemma .[] Let{xk}and{zk}be bounded sequences in a Banach space E such that

xk+= ( –βk)xk+βkzk

for k≥,where{βk}is in[, ]such that

 <lim inf

Assume that

lim sup

k→∞ zk+–zk–xk+–xk ≤.

Thenlimk→∞xk–zk= .

Lemma .[] Let{ak}be a sequence of nonnegative real numbers satisfying the condition

ak+≤( –bk)ak+bkck,

where{bk}and{ck}are sequences of real numbers such that

(i) bk∈[, ]and ∞

k=bk=∞;

(ii) lim sup_k→∞ck≤.

Thenlimk→∞ak= .

Now, we are in a position to prove the following main result.

Theorem . Let H be a real Hilbert space,and let F:H→H be an L-Lipschitz con-tinuous and η-strongly monotone mapping for some constants L,η> .Let{Ti}Ni= be N

nonexpansive self-maps of H such that

C=

N

i=

Fix(Ti)=∅.

Then the sequence{xk}defined by(.)-(.)converges strongly to the unique element p∗

in(.).

Proof First, we prove that{xk}is bounded. By Lemma ., we have, for anyp∈C, from

(.) that

y_k–p=(I–λkμF)xk–p=(I–λkμF)xk– (I–λkμF)p–λkμF(p)

≤( –λkτ)xk–p+λkμF(p).

PutMp=max{x–p,μτF(p)}. Thenx–p ≤Mp. So, ifxk–p ≤Mp, theny

 k–p ≤

Mp. This conclusion has a place for{yik}withi= , . . . ,N– . Indeed, yi_k–p= –β_ki(xk–p) +βki

Tiyik––Tip

≤ –β_kixk–p+βkiyik––p

≤ –β_kiMp+βkiMp

=Mp.

Then

xk+–p= –βkN

(xk–p) +βkN

Tiyik––Tip

≤ –β_kiMp+βkiMp

=Mp.

Therefore, the sequence{xk}is bounded. So, the sequences{F(xk)},{yik}, and{Tiyik–}(i=

, , . . . ,N) are also bounded. Without loss of generality, we assume that they are bounded by a positive constantM.

Letzk=TNyNk–. Then we have from (.) that

xk+=

 –β_kNxk+βkNzk

and

zk+–zk=TNyNk+––TNykN–≤yNk+––yNk–

= –β_kN_+–xk++βkN+–TN–yNk+–

– –β_kN–xk+βkN–TN–yNk–

≤ –β_kN_+–xk+–xk+ MβkN+––βkN–

+β_kN_+–TN–yNk+––TN–yNk–

≤ –β_kN_+–xk+–xk+ MβkN+––βkN–

+β_kN_+–yN_k+––yN_k–

≤ –β_kN_+–xk+–xk+ MβkN+––βkN–

+β_kN_+– –β_kN_+–xk+–xk+ MβkN+––βkN–

+β_kN_+–yN_k+––yN_k–

≤ –β_kN_+–β_kN_+–xk+–xk+ MβkN+––βkN–

+β_kN_+––β_kN–+β_kN_+–β_kN_+–yN_k+––yN_k–

≤ · · ·

≤

 –

N–

i=

β_ki_+

xk+–xk+

N–

i=

β_ki_+y_k+–y_k

+ M N–

i=

β_ki_+–β_ki.

On the other hand,

y_k+–y_k=(I–λk+μF)xk+– (I–λkμF)xk

≤ xk+–xk+M(λk++λk).

So, we obtain that

zk+–zk–xk+–xk ≤M(λk++λk)

N–

i=

β_ki_++ M N–

i=

Sinceλk→ andβki→βi,i= , . . . ,N– , we have

lim sup

k→∞

zk+–zk–xk+–xk ≤.

By Lemma .,xk–zk → ask→ ∞,i.e.,xk–TNyNk– →.

Now, we prove that xk –Tiy_ki– → for i = , . . . ,N – . First, we prove xk –

TN–yNk– →. Let{xkn}be a subsequence of{xk}such that

lim sup

k→∞

xk–TN–yNk–=_nlim_→∞xkn–TN–yNkn–

and let{xkj}be a subsequence of{xkn}such that

lim sup

n→∞ xkn–p=jlim→∞xkj–p.

Further,

xkj–p ≤xkj–TNy

N–

kj +TNy N– kj –TNp

≤xkj–TNy

N– kj +y

N– kj –p

≤xkj–TNy

N– kj +

 –β_kN–xkj–p

+β_kN–

j TN–y N–

kj –TN–p

≤xkj–TNy

N– kj +

 –β_kN–xkj–p+β N– kj y

N– kj –p

≤xkj–TNy

N– kj +

 –β_kN–xkj–p

+β_kN_j– –β_kN–xkj–p+β N– kj TN–y

N–

kj –TN–p

≤xkj–TNy

N– kj +

 –β_kN–β_kN–

j

xkj–p

+β_kN_j–β_kN_j–yN_kj––p

≤ · · ·

≤xkj–TNy

N– kj +

 –

N–

i=

β_ki_j

xkj–p+

N–

i=

β_ki_jy_kj–p.

Since

y_kj–p≤( –λkjτ)xkj–p+λkjμF(p),

we have

xkj–p ≤xkj–TNy

N– kj +y

N– kj –p

≤xkj–TNy

N–

kj +xkj–p+ N–

i=

β_ki

Therefore,

lim

j→∞xkj–p=jlim→∞ _yN–

kj –p. (.)

Next, by Lemma . we have

yN_kj––p= –β_kN–

j

xkj–p

_+βN– kj TN–y

N– kj –p



– –β_kN–

j

β_kN–

j xkj–TN–y N– kj



≤ –β_kN_j–xkj–p

_+βN– kj y

N– kj –p



– –β_kN–

j

β_kN–

j xkj–TN–y N– kj



≤ –β_kN_j–xkj–p



+β_kN–

j  –β N– kj

(xkj–p) +β N– kj

TN–yNkj––p 

– –β_kN–

j

β_kN–

j xkj–TN–y N– kj



≤ –β_kN–

j β N– kj

xkj–p

_+βN– kj β

N– kj y

N– kj –p



– –β_kN–

j

β_kN–

j xkj–TN–y N– kj  ≤ · · · ≤  – N– i=

β_ki

j

xkj–p

₊ N–

i=

β_ki

jy  kj–p



– –β_kN–

j

β_kN–

j xkj–TN–y N– kj



.

On the other hand, by Lemma . we get

y_k–p=(I–λkμF)xk–p 

=xk–p– (I–λkμF)xk

≤ xk–p– λkμ

F(xk),xk–p

≤ xk–p+ λkμMMp.

Without loss of generality, assume that α≤β_ki ≤β,k≥ fori= , . . . ,N–  and some α,β∈(, ). Then we obtain that

α( –β)xkj–TN–y N– kj



≤ xkj–p

_–yN– kj –p



+ λkjμMMp N–

i=

β_ki_j,

which with λk→ and (.) implies thatxkj –TN–y N–

kj → as j→ ∞. So,xk–

TN–yN_k– → ask→ ∞.

Similarly, we obtain thatxk–TN–yNk– →, . . . ,xk–Tyk → ask→ ∞.

Further, we prove thatxk–Tixk → ask→ ∞fori= , . . . ,N. First, note thatyk–

Tiyik– → becausexk–Tiyik– →, fori= , . . . ,N. Now, from

xk–Tixk ≤xk–Tiyki–+Tiyik––Tixk

≤xk–Tiyik–+yik––xk

andxk–Tiyki–,yik––xk →, it follows thatxk–Tixk → fori= , . . . ,N.

Further, we have

lim sup

k→∞

Fp∗,p∗–xk

≤. (.)

Indeed, let{xkj}be a subsequence of{xk}that converges weakly top˜such that

lim sup

k→∞

Fp∗,p∗–xk

=lim

j→∞

Fp∗,p∗–xkj

.

Thenxkj–Tixkj →. So, by Lemma .,p˜∈C. Therefore, from (.) it implies (.).

Finally, we estimate the value

xk+–p∗ 

≤ –β_kNxk–p∗



+β_kNTNykN––TNp∗ 

≤ –β_kNxk–p∗



+β_kNyN_k––TNp∗ 

≤ –β_kNxk–p∗

+β_kN –β_kN–xk–p∗+βkN–TN–ykN––TN–p∗

≤ –β_kNβ_kN–xk–p∗+βkNβkN–ykN––TNp∗

≤ –β_kNβ_kN–xk–p∗



+β_kNβ_kN– –β_kN–xk–p∗ 

+β_kN–TN–ykN––TN–p∗  ≤ · · · ≤  – N i=

β_ki

xk–p∗ 

+

N

i=

β_kiy_k–p∗.

On the other hand, since

y_k–p∗=(I–λkμF)xk–p∗ 

=(I–λkμF)

xk–p∗

–λkμF

p∗

≤( –λkτ)xk–p∗– λkμ

F(xk),yk–p∗

,

we have

xk+–p∗  ≤  – N i=

β_ki

xk–p∗ 

+

N

i=

β_ki( –λkτ)xk–p∗ 

+λkμ

Fp∗,p∗–xk

+ λkμ

Fp∗,xk–yk N

i=

≤

 –λkτ N

i=

β_ki

xk–p∗ 

+λkτ N

i=

β_ki

μ τ

Fp∗,p∗–xk

+μ τ

Fp∗,xk–yk

.

Using Lemma . with

ak=xk–p∗,

bk=λkτ N

i=

β_ki,

ck=

μ τ

Fp∗,p∗–xk

+μ τ

Fp∗,xk–yk

,

xk–y_k → and (.), we have thatxk–p∗ →. This completes the proof.

3 Application

Recall that a mappingS:H→His calledγ-strictly pseudocontractive if there exists a constantγ ∈[, ) such that

Sx–Sy≤ x–y+γ(I–S)x– (I–S)y, ∀x,y∈H.

It is well known [] that a mappingT:H→HbyTx=αx+ ( –α)Sxwith a fixedα∈[γ, ) for allx∈His a nonexpansive mapping andFix(T) =Fix(S).

Using this fact, we can extend our result to the case C=N_i=Fix(Si), where Si is

γi-strictly pseudocontractive as follows.

Letαi∈[γi, ) be fixed numbers. ThenC= N

i=Fix(T˜i) with

˜

Tiy=αiy+ ( –αi)Siy, (.)

a nonexpansive mapping, for eachi= , . . . ,N. So, we have the following result.

Theorem . Let H be a real Hilbert space,and let F:H→H be an L-Lipschitzian and

η-strongly monotone mapping for some constants L,η> .Let{Si}Ni=be Nγi-strictly

pseu-docontractive self-maps of H such that

C=

N

i=

Fix(Si)=∅.

Letαi∈[γi, ),μ∈(, η/L).Assume that{λk},{β_ki} ⊂(, )satisfy(.).Then the

se-quence{xk}defined by(.)with Tireplaced byT˜iof(.)converges strongly to the unique

element p∗of(.).

4 Numerical example

Consider the following optimization problem: find an element

p∗∈C:ϕp∗=min

Table 1 Iterations of scheme (1.9), where starting pointx0= (4; 15)

kth xkby algorithm (1.9)

0 (4.00000;15.00000) 10 (4.10755;13.434117) 20 (3.89572; 12.58288) 1,000 (1.52757; 3.110295) 1,500 (1.28179; 2.12718) 2,000 (1.13269; 1.53076) 2,500 (1.03367; 1.13468) 3,000 (0.99933; 0.99738) 3,500 (0.99938; 0.99517) 4,000 (0.99942; 0.99768) 4,500 (0.99945; 0.99781) 5,000 (0.99948; 0.99792)

whereϕ(x) =x/,x= (x,x)∈E, Euclid space, andC=C∩C, defined by

C=

(x,x)∈E:x– x+ ≤

,

C=

(x,x)∈E: x–x– ≥

.

Clearly, the above problem possesses a unique solution p∗ = (; ) and F, the Fréchet derivative ofϕ, is -Lipschitz continuous and (/)-strongly monotone. Starting with the pointx_{= (x}

;x) = (; ),μ= /∈(; η/L) andλk= (k+ )/, we obtained the result

in Table .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by JKK. JKK and NB prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics Education, Kyungnam University, Changwon, 631-701, Korea.}2_{Vietnamese Academy of} Science and Technology, Institute of Information Technology, 18, Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam.

Acknowledgements

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138).

Received: 8 April 2013 Accepted: 18 August 2013 Published: 3 September 2013 References

1. Kinderlehrer, D, Stampacchia, G: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

2. Glowinski, R: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984) 3. Zeidler, E: Nonlinear Functional Analysis and Its Applications. Springer, New York (1985)

4. Yamada, Y: The hybrid steepest-descent method for variational inequalities problems over the intersection of the fixed point sets of nonexpansive mappings. In: Butnariu, D, Censor, Y, Reich, S (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473-504. North-Holland, Amsterdam (2001)

5. Zeng, LC, Yao, JC: Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings. Nonlinear Anal. TMA64, 2507-2515 (2006)

6. Xu, HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl.116, 659-678 (2003)

7. Zeng, LC, Wong, NC, Yao, JC: Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. J. Optim. Theory Appl.132, 51-69 (2007)

8. Liu, X, Cui, Y: The common minimal-norm fixed point of a finite family of nonexpansive mappings. Nonlinear Anal. TMA73, 76-83 (2010)

9. Marino, G, Xu, HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl.329, 336-346 (2007)

11. Li, G, Kim, JK: Demiclosedness principle and asymptotic behavior for nonexpansive mappings in metric spaces. Appl. Math. Lett.14(5), 645-649 (2001)

12. Suzuki, T: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc.135, 99-106 (2007)

doi:10.1186/1029-242X-2013-419