Strong convergence theorems for solving a general system of finite variational inequalities for finite accretive operators and fixed points of nonexpansive semigroups with weak contraction mappings

Abstract

In this paper, we prove a strong convergence theorem for finding a common solution of a general system of finite variational inequalities for finite different inverse-strongly accretive operators and solutions of fixed point problems for a nonexpansive semigroup in a Banach space based on a viscosity approximation method by using weak contraction mappings. Moreover, we can apply the above results to find the solutions of the class of k-strictly pseudocontractive mappings and apply a general system of finite variational inequalities into a Hilbert space. The results presented in this paper extend and improve the corresponding results of Ceng et al. (2008), Katchang and Kumam (2011), Wangkeeree and Preechasilp (2012), Yao et al. (2010) and many other authors.

MSC: 47H05, 47H10, 47J25.[PUBLICATION ABSTRACT]

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R E S E A R C H Open Access

Strong convergence theorems for solving a general system of nite variational inequalities for nite accretive operators and xed points of nonexpansive semigroups with weak contraction mappings

Nawitcha Onjai-uea1, Phayap Katchang2 and Poom Kumam1*

*Correspondence: mailto:[email protected]

Web End [email protected]

1Department of Mathematics, Faculty of Science, King Mongkuts University of Technology Thonburi (KMUTT), Bangmod, Bangkok, 10140, ThailandFull list of author information is available at the end of the article

Abstract

In this paper, we prove a strong convergence theorem for nding a common solution of a general system of nite variational inequalities for nite dierent inverse-strongly accretive operators and solutions of xed point problems for a nonexpansive semigroup in a Banach space based on a viscosity approximation method by using weak contraction mappings. Moreover, we can apply the above results to nd the solutions of the class of k-strictly pseudocontractive mappings and apply a general system of nite variational inequalities into a Hilbert space. The results presented in this paper extend and improve the corresponding results of Ceng et al. (2008), Katchang and Kumam (2011), Wangkeeree and Preechasilp (2012), Yao et al. (2010) and many other authors.

MSC: Primary 47H05; 47H10; 47J25

Keywords: inverse-strongly accretive operator; xed point; general system of nite variational inequalities; sunny nonexpansive retraction; weak contraction; nonexpansive semigroups

1 Introduction

Let E be a real Banach space with norm and C be a nonempty closed convex subset of E. Let E* be the dual space of E and , denote the pairing between E and E*. For q > , the generalized duality mapping Jq : E E* is dened by Jq(x) = {f E* : x, f = x q, f = x q} for all x E. In particular, if q = , the mapping J is called the normalized duality mapping and, usually, write J = J. Further, we have the following properties of the generalized duality mapping Jq: (i) Jq(x) = x qJ(x) for all x E with x = ;

(ii) Jq(tx) = tqJq(x) for all x E and t [, ); and (iii) Jq(x) = Jq(x) for all x E. It is known that if E is smooth, then J is single-valued, which is denoted by j. Recall that the duality mapping j is said to be weakly sequentially continuous if for each xn x weakly, we have j(xn) j(x) weakly-*. We know that if E admits a weakly sequentially continuous duality mapping, then E is smooth (for the details, see [, , ]).

Let f : C C be a k-contraction mapping if there exists k [, ) such that f (x) f (y) k x y , x, y C. Let S : C C a nonlinear mapping. We use F(S) to denote the

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set of xed points of S, that is, F(S) = {x C : Sx = x}. A mapping S is called nonexpansive if Sx Sy x y , x, y C. A mapping f is called weakly contractive on a closed convex set C in the Banach space E if there exists : [, ) [, ) is a continuous and strictly increasing function such that is positive on (, ), () = , limt (t) =

and x, y C

f

(.)

If (t) = ( k)t, then f is called to be contractive with the contractive coecient k. If (t) , then f is said to be nonexpansive.

A family S = {T(t) : t } of mappings of C into itself is called a nonexpansive semigroup (see also []) on C if it satises the following conditions:(i) T()x = x for all x C;(ii) T(s + t) = T(s)T(t) for all s, t ;(iii) T(s)x T(s)y x y for all x, y C and s ;(iv) for all x C, s T(s)x is continuous.

We denote by F(S) the set of all common xed points of S, that is,

F(S) =

N, (.)

where f : C C is a xed contractive mapping. Recall that an operator A : C E is said to be accretive if there exists j(x y) J(x y) such that

Ax Ay, j(x y)

for all x, y C. A mapping A : C E is said to be -strongly accretive if there exists a constant > such that

Ax Ay, j(x y) x y , x, y C.

An operator A : C E is said to be -inverse strongly accretive if, for any > ,

Ax Ay, j(x y) Ax Ay

(x) f (y)

x y x

y .

x C : T(t)x = x, t < .

It is known that F(S) is closed and convex. Moreover, for the study of nonexpansive semi-group mapping, see [, , ] for more details.

In , Suzuki [] was the rst one to introduce the following implicit iteration process in Hilbert spaces:

xn = nu + ( n)T(tn)(xn), n (.)

for the nonexpansive semigroup. In , Xu [] established a Banach space version of the sequence (.) of Suzuki []. In [], Chen and He considered the viscosity approximation process for a nonexpansive semigroup and proved another strong convergence theorems for a nonexpansive semigroup in Banach spaces, which is dened by

xn+ = nf (xn) + ( n)T(tn)xn, n

F

T(t) =

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for all x, y C. Evidently, the denition of the inverse strongly accretive operator is based on that of the inverse strongly monotone operator. To convey an idea of the variational inequality, let C be a closed and convex set in a real Hilbert space H. For a given operator A, we consider the problem of nding x* C such that

Ax*, x x*

for all x C, which is known as the variational inequality, introduced and studied by Stampacchia [] in in the eld of potential theory. In , Aoyama et al. [] rst considered the following generalized variational inequality problem in a smooth Banach space. Let A be an accretive operator of C into E. Find a point x C such that

Ax, j(y x) (.)

for all y C. This problem is connected with the xed point problem for nonlinear mappings, the problem of nding a zero point of an accretive operator and so on. For the problem of nding a zero point of an accretive operator by the proximal point algorithm, see Kamimura and Takahashi [, ]. In order to nd a solution of the variational inequality (.), Aoyama et al. [] proved the strong convergence theorem in the framework of Banach spaces which is generalized by Iiduka et al. [] from Hilbert spaces.

Motivated by Aoyama et al. [] and also Ceng et al. [], Qin et al. [] and Yao et al. [] rst considered the following new general system of variational inequalities in Banach spaces:

Let A : C E be a -inverse strongly accretive mapping. Find (x*, y*) C C such that

(.)

Let C be nonempty closed convex subset of a real Banach space E. For two given operators A, B : C E, consider the problem of nding (x*, y*) C C such that

(.)

where and are two positive real numbers. This system is called the general system of variational inequalities in a real Banach spaces. If we add up the requirement that A = B, then the problem (.) is reduced to the system (.).

By the following general system of variational inequalities, we extend into the general system of nite variational inequalities which is to nd (x*, x*, . . . , x*M) C C C and is dened by

Ay* + x* y*, j(x x*) , x C,

Ax* + y* x*, j(x y*) , x C.

Ay* + x* y*, j(x x*) , x C,

Bx* + y* x*, j(x y*) , x C,

MAMx*M + x* x*M, j(x x*) , x C,

MAMx*M + x*M x*M, j(x x*M) , x C, ...

Ax* + x* x*, j(x x*) , x C,

Ax* + x* x*, j(x x*) , x C,

(.)

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where {Al}Ml= : C E is a family of mappings, l , l {, , . . . , M}. The set of solutions of (.) is denoted by GSVI(C, Al). In particular, if M = , A = B, A = A, = , = , x* = x* and x* = y*, then the problem (.) is reduced to the problem (.).

In this paper, motivated and inspired by the idea of Ceng et al. [], Katchang and Kumam [] and Yao et al. [], we introduce a new iterative scheme with weak contraction for nding solutions of a new general system of nite variational inequalities (.) for nite dierent inverse-strongly accretive operators and solutions of xed point problems for nonexpansive semigroups in a Banach space. Consequently, we obtain new strong convergence theorems for xed point problems which solve the general system of variational inequalities (.). Moreover, we can apply the above theorem to nding solutions of zeros of accretive operators and the class of k-strictly pseudocontractive mappings. The results presented in this paper extend and improve the corresponding results of Ceng et al. [], Katchang and Kumam [], Wangkeeree and Preechasilp [], Yao et al. [] and many other authors.

2 Preliminaries

We always assume that E is a real Banach space and C is a nonempty closed convex subset of E.

Let U = {x E : x = }. A Banach space E is said to be uniformly convex if, for any

(, ], there exists > such that, for any x, y U, x y implies x+y . It is known that a uniformly convex Banach space is reexive and strictly convex. A Banach space E is said to be smooth if the limit limt x+ty x t exists for all x, y U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y U. The modulus of smoothness of E is dened by

() = sup

x + y + x y : x, y E, x = , y = ,

where : [, ) [, ) is a function. It is known that E is uniformly smooth if and only if lim () = . Let q be a xed real number with < q . A Banach space E is said to be q-uniformly smooth if there exists a constant c > such that () cq for all > : see, for instance, [, ].

We note that E is a uniformly smooth Banach space if and only if Jq is single-valued and uniformly continuous on any bounded subset of E. Typical examples of both uniformly convex and uniformly smooth Banach spaces are Lp, where p > . More precisely, Lp is min{p, }-uniformly smooth for every p > . Note also that no Banach space is q-uniformly smooth for q > ; see [, ] for more details.

Let D be a subset of C and Q : C D. Then Q is said to be sunny if

Q

Qx + t(x Qx) = Qx,

whenever Qx + t(x Qx) C for x C and t . A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction Q of C onto D. A mapping Q : C C is called a retraction if Q = Q. If a mapping Q : C C is a retraction, then Qz = z for all z in the range of Q. For example, see [, ] for more details. The following result describes a characterization of sunny nonexpansive retractions on a smooth Banach space.

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Proposition . ([]) Let E be a smooth Banach space and let C be a nonempty subset of E. Let Q : E C be a retraction and let J be the normalized duality mapping on E. Then the following are equivalent:(i) Q is sunny and nonexpansive;(ii) Qx Qy x y, J(Qx Qy) , x, y E;(iii) x Qx, J(y Qx) , x E, y C.

Proposition . ([]) Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let T be a nonexpansive mapping of C into itself with F(T) = . Then the set F(T) is a sunny nonexpansive retract of C.

A Banach space E is said to satisfy Opials condition if for any sequence {xn} in E, xn x (n ) implies

lim sup

n

xn x < lim sup

n

xn y , y E with x = y.

By [, Theorem ], it is well known that, if E admits a weakly sequentially continuous duality mapping, then E satises Opials condition and E is smooth.

We need the following lemmas for proving our main results.

Lemma . ([]) Let E be a real -uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:

x + y x + y, Jx + Ky , x, y E.

Lemma . ([]) Let {xn} and {yn} be bounded sequences in a Banach space X and let {n} be a sequence in [, ] with < lim infn n lim supn n < . Suppose xn+ = ( n)yn + nxn for all integers n and lim supn( yn+ yn xn+ xn ) . Then,

limn yn xn = .

Lemma . (Lemma . in []) Let {an} and {bn} be two nonnegative real number sequences and {n} a positive real number sequence satisfying the conditions:

n= n =

and limn bnn = . Let the recursive inequality

an+ an n(an) + bn, n ,

where (a) is a continuous and strict increasing function for all a with () = . Then

limn an = .

Lemma . ([]) Let E be a uniformly convex Banach space and Br() := {x E : x r} be a closed ball of E. Then there exists a continuous strictly increasing convex function g : [, ) [, ) with g() = such that

x + y + z x + y + z g x y

for all x, y, z Br() and , , [, ] with + + = .

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Lemma . ([]) Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E and let T be nonexpansive mapping of C into itself. If {xn} is a sequence of

C such that xn x weakly and xn Txn strongly, then x is a xed point of T.

Lemma . (Yao et al. [, Lemma .]; see also [, Lemma .]) Let C be a nonempty closed convex subset of a real -uniformly smooth Banach space E. Let the mapping A : C E be -inverse-strongly accretive. Then, we have

(I

A)x (I A)y

x y + K

Ax Ay .

If K, then I A is nonexpansive.

3 Main results

In this section, we prove a strong convergence theorem. In order to prove our main results, we need the following two lemmas.

Lemma . Let C be a nonempty closed convex subset of a real -uniformly smooth Banach space E. Let QC be the sunny nonexpansive retraction from E onto C. Let the mapping Al : C E be a l-inverse-strongly accretive such that l lK where l {, , . . . , M}. If

Q : C C is a mapping dened by

Q(x) = QC(I MAM)QC(I MAM) QC(I A)QC(I A)x, x C,

then Q is nonexpansive.

Proof Taking QlC = QC(I lAl)QC(I lAl) QC(I A)QC(I A), l {, , , . . . , M} and QC = I, where I is the identity mapping on E, we have Q = QMC. For any x, y C, we have

Q(x) Q(y)

= QMCx QMCy

= QC(I MAM)QMCx QC(I MAM)QMCy (I

MAM)QMCx (I MAM)QMCy

QMCx QMCy

...

QCx QCy

= x y .

Therefore, Q is nonexpansive.

Lemma . Let C be a nonempty closed convex subset of a real smooth Banach space E. Let QC be the sunny nonexpansive retraction from E onto C. Let Al : C E be nonlinear mapping, where l {, , . . . , M}. For x*l C, l {, , . . . , M}, (x*, x*, . . . , x*M) is a solution of

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problem (.) if and only if

x* = QC(I MAM)x*M,

x* = QC(I A)x*,

x* = QC(I A)x*,

...

x*M = QC(I MAM)x*M,

(.)

that is

x* = QC(I MAM)QC(I MAM) QC(I A)QC(I A)x*.

Proof From (.), we rewrite as

x* (x*M MAMx*M), j(x x*) , x C,

x*M (x*M MAMx*M), j(x x*M) , x C, ...

x* (x* Ax*), j(x x*) , x C,

x* (x* Ax*), j(x x*) , x C.

(.)

Using Proposition .(iii), the system (.) is equivalent to (.).

Throughout this paper, the set of xed points of the mapping Q is denoted by F(Q). The next result states the main result of this work.

Theorem . Let E be a uniformly convex and -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. Let S = {T(t) : t } be a nonexpansive semigroup on C and QC be a sunny nonexpansive retraction from E onto C. Let Al : C E be a l-inverse-strongly accretive such that l lK, where l {, , . . . , M}, and K be the best smooth constant.

Let f be a weakly contractive mapping on C into itself with function . Suppose F := F(Q) F(S) = , where Q is dened by Lemma .. For arbitrary given x = x C, the sequence {xn} is generated by

yn = QC(I MAM)QC(I MAM) QC(I A)QC(I A)xn, xn+ = nf (xn) + nxn + nT(n)yn,

(.)

where the sequences {n}, {n} and {n} are in (, ) and satisfy {n} + {n} + {n} = , n , {n} (, ), and l, l = , , . . . , M are positive real numbers. The following conditions are satised:

(C) limn n = and

n= n = ; (C) < lim infn n lim supn n < ;

(C) limn n = ;

(C) limn supyC T(n+)y T(n)y = , C bounded subset of C.

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Then {xn} converges strongly to x = QFf (x) and (x, x, . . . , xM) is a solution of the problem (.) where QF is the sunny nonexpansive retraction of C onto F.

Proof First, we prove that {xn} is bounded. Let p F, taking

QlC = QC(I lAl)QC(I lAl) QC(I A)QC(I A), l {, , , . . . , M},

QC = I, where I is the identity mapping on E. From the denition of QC is nonexpansive then QlC, l {, , , . . . , M} also. We note that

yn p =

QlCxn QlCp

xn p . (.)

From (.) and (.), we also have

xn+ p =

nf (xn) + nxn + nT(n)yn p

n f

(xn) p

+ n xn p + n T(

n)yn T(n)p

n x

n p x

n p

+ n

f

(p) p

+ n xn p + n yn p

xn p n x

n p

+ n

f

(p) p

max x

p , x

p , f

(p) p

.

(.)

This implies that {xn} is bounded, so are {f (xn)}, {yn}, and {T(n)yn}.Next, we show that limn xn+ xn = . Notice that

yn+ yn =

QMCxn+ QMCxn

= C(I MAM)QMCxn+ QC(I MAM)QMCxn (I

Q

MAM)QMCxn+ (I MAM)QMCxn

QMCxn+ QMCxn

...

QCxn+ QCxn

= xn+ xn .

Setting xn+ = ( n)zn + nxn for all n , we see that zn = xn+nxnn . Then we have

zn+ zn =

xn+ n+xn+

n+

xn+ nxn

n

=

n+f (xn+) + n+T(n+)yn+ n+

nf (xn) + nT(n)yn n

=

n+f (xn+) + n+T(n+)yn+

n+

n+f (xn)

n+ +

n+f (xn)

n+

n+T(n)yn

n+ +

n+T(n)yn

n+

nf (xn) + nT(n)yn n

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n+

= n+

f (xn+) f (xn)

n+

+ n+

T(n+)yn+ T(n)yn

n+

+ n+

n n

f (xn) +

n+

n+

n n

T(n)yn

n+

n+

f

(xn+) f (xn)

n+

+ n+

T(

n+)yn+ T(n)yn

n+

+ n+

n n

f

(xn)

n+

+ n+

n n

T(

n)yn

n+

n+

f

(xn+) f (xn)

+

n+ n+

n+

yn+ yn + T(n+)yn T(n)yn

n+

+ n+

n n

f

(xn)

+

n+ n+

n+

n n

n

T(

n)yn

n+

n+

xn+ xn xn+ xn

+

n+

n+

yn+ yn + T(

n+)yn T(n)yn

n+

+ n+

n n

f

(xn)

n+

+ n+

n n

T(

n)yn

n+ xn+ xn + yn+ yn + T(

n+

n+)yn T(n)yn

n+

+ n+

n n

f

(xn)

+

T(

n)yn

n+

n+ xn+ xn + xn+ xn +

sup

y{yn}

T(

n+)y T(n)y

n+

+ n+

n n

f

(xn)

+

T(

n)yn

.

Therefore,

zn+ zn xn+ xn

n+

n+ xn+ xn +

sup

y{yn}

T(

n+)y T(n)y

n+

+ n+

n n

f

(xn)

+

T(

n)yn

.

It follows from the conditions (C), (C) and (C), which implies that

lim sup

n

zn+ zn xn+ xn .

Applying Lemma ., we obtain limn zn xn = and also

xn+ xn = ( n) zn xn

as n . Therefore, we have

lim

n xn+ xn = . (.)

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Next, we show that limn T(n)yn yn = . Since p F, from Lemma ., we obtain

xn+ p =

nf (xn) + nxn + nT(n)yn p

n f

(xn) p

+ ( n n) xn p + n yn p

= n

f

(xn) p

+ ( n) xn p n x

n p yn p

= n

f

(xn) p

+ ( n) xn p

n

xn p yn p xn p + yn p

n f

(xn) p

+ xn p n xn yn .

Therefore, we have

n xn yn n f

(xn) p

+ xn p xn+ p

+ xn p + xn+ p xn xn+ .

From the condition (C) and (.), this implies that xn yn as n . Now, we note that

x

n T(n)yn

n f

(xn) p

xn xn+ + x

n+ T(n)yn

= xn xn+ +

nf (xn) + nxn + nT(n)yn T(n)yn

= xn xn+ +

n

f (xn) T(n)yn + n

xn T(n)yn

xn xn+ + n f

(xn) T(n)yn

+ n

x

n T(n)yn

.

Therefore, we get

x

n T(n)yn

n xn xn+ +

n n

f

(xn) T(n)yn

.

From the conditions (C), (C) and (.), this implies that xn T(n)yn as n . Since

x

n T(n)xn

x

n T(n)yn

+

T(

n)yn T(n)xn

x

n T(n)yn

+ yn xn ,

and hence it follows that limn T(n)xn xn = .

Next, we prove that z F := F(Q) F(S). By the reexivity of E and boundedness of the sequence {xn}, we may assume that xni z for some z C.

(a) First, we show that z F(S). Put xi = xni, i = ni, i = ni, i = ni and i = ni for i

N, let ti be such that

i and

T(i)xi xi

i , i .

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Fix t > . Notice that

x

i T(t)p

[t/i]

k=

T (k

+ )i

xi T(ki)xi

+

T [t/

i]i

xi T

[t/i]i

z

+

T [t/

i]i

z T(t)z

[t/i] T(

i)xi xi

+ xi p + T t

[t/i]i

z z

t

T(i)xi xi i + xi p + T t

[t/i]i

z z

t

T(i)xi xi i + xi p +

max

T(s)z

z

: s i .

For all i

N, we have

lim sup

i

x

i T(t)z

lim sup

i

xi z .

Since the Banach space E with a weakly sequentially continuous duality mapping satises Opials condition, this implies T(t)z = z. Therefore, z F(S).

(b) Next, we show that z F(Q). From Lemma ., we know that Q = QMC is nonexpansive; it follows that

yn Qyn =

QMCxn QMCyn

xn yn .

Thus limn yn Qyn = . Since Q is nonexpansive, we get

xn Qxn xn yn + yn Qyn + Qyn Qxn

xn yn + yn Qyn ,

and so

lim

n xn Qxn = . (.)

By Lemma . and (.), we have z F(Q). Therefore, z F.

Next, we show that lim supn (f I)x, J(xn x) , where x = QFf (x). Since {xn} is bounded, we can choose a sequence {xni} of {xn} where xni z such that

lim sup

n

(f I)x, J(xn x) = lim i (f I)x, J(xni x) .

(.)

Now, from (.), Proposition .(iii) and the weakly sequential continuity of the duality mapping J, we have

lim sup

n

(f I)x, J(xn x) = lim i (f I)x, J(xni x)

= (f I)x, J(z x) . (.)

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From (.), it follows that

lim sup

n

(f I)x, J(xn+ x) . (.)

Finally, we show that {xn} converges strongly to x = QFf (x). We compute that

xn+ x = x

n+ x, J(xn+ x) =

nf (xn) + nxn + nT(n)yn x, J(xn+ x) =

n

f (xn) x + n(xn x) + n T(n)yn x ,J(xn+ x)

= n

f (xn) f (x), J(xn+ x) + n

f (x) x, J(xn+ x)

+ n

xn x, J(xn+ x) + n

T(n)yn x, J(xn+ x)

n+ x + n

f (x) x, J(xn+ x) + n xn x xn+ x + n yn x xn+ x

n xn x xn+ x n x

n x x

n+ x

n

xn x x

n x x

f (x) x, J(xn+ x) + n xn x xn+ x + n xn x xn+ x

= xn x xn+ x n x

n x x

n+ x

+ n

f (x) x, J(xn+ x) =

xn x + xn+ x n

+ n

xn x xn+ x

f (x) x, J(xn+ x) .

By (.) and since {xn+ x} is bounded, i.e., there exists M > such that xn+ x M, which implies that

xn+ x xn x nM x

n x

+ n

f (x) x, J(xn+ x) .(.)

Now, from (C) and applying Lemma . to (.), we get xn x as n . This completes the proof.

Corollary . Let E be a uniformly convex and -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. Let S = {T(t) : t } be a nonexpansive semigroup on C and QC be a sunny nonexpansive retraction from E onto C. Let A : C E be a -inverse-strongly accretive such that K where K is the best smooth constant. Let f be a weakly contractive mapping of C into itself with function . Let the sequences {n}, {n} and {n} be in (, )

with {n} + {n} + {n} = , n , {n} (, ) and satisfy the conditions (C)-(C) in Theorem .. Suppose F := F(Q) F(S) = , where Q is dened by

Q(x) = QC(I A)QC(I A) QC(I A)x, x C,

+ n

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and be a positive real number. For arbitrary given x = x C, the sequences {xn} are generated by

Then {xn} converges strongly to x = QFf (x), where QF is the sunny nonexpansive retraction of C onto F.

Proof Putting A = AM = AM = = A = A, = M = M = = = and = M =

M = = = in Theorem ., we can conclude the desired conclusion easily. This completes the proof.

Corollary . Let E be a uniformly convex and -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. Let S = {T(t) : t } be a nonexpansive semigroup on C and QC be a sunny nonexpansive retraction from E onto C. Let Al : C E be a l-inverse-strongly accretive such that l lK, where l {, } and K be the best smooth constant. Let f be a weakly contractive mapping of C into itself with function . Let the sequences {n}, {n} and

{n} be in (, ) with {n} + {n} + {n} = , n , {n} (, ) and satisfy the conditions (C)-(C) in Theorem .. Suppose F := F(Q) F(S) = , where Q is dened by

Q(x) = QC(I A)QC(I A)x, x C,

and , are positive real numbers. For arbitrary given x = x C, the sequences {xn} are generated by

Then {xn} converges strongly to x = QFf (x) and (x, x) is a solution of the problem (.), where QF is the sunny nonexpansive retraction of C onto F.

Proof Taking M = in Theorem ., we can conclude the desired conclusion easily. This completes the proof.

4 Some applications4.1 (I) Application to strictly pseudocontractive mappings

Let E be a Banach space and let C be a subset of E. Recall that a mapping T : C C is said to be k-strictly pseudocontractive if there exist k [, ) and j(x y) J(x y) such that

Tx Ty, j(x y) x y k

for all x, y C. Then (.) can be written in the following form:

(I T)x (I T)y, j(x y) k

yn = QC(I A)QC(I A) QC(I A)xn, xn+ = nf (xn) + nxn + nT(n)yn.

(.)

yn = QC(I A)QC(I A)xn,

xn+ = nf (xn) + nxn + nT(n)yn.

(.)

(I

T)x (I T)y

(.)

(I T)x (I T)y

.

(.)

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Moreover, we know that A is k -inverse strongly monotone and A = F(T) (see also []).

Theorem . Let E be a uniformly convex and -uniformly smooth Banach space and C be a nonempty closed convex subset of E. Let S = {T(t) : t } be a nonexpansive semi-group on C and Tl : C C be a kl-strictly pseudocontractive mapping with l (kl)K, l {, , . . . , M}. Let f be a weakly contractive mapping of C into itself with function and suppose the sequences {n}, {n} and {n} in (, ) satisfy {n} + {n} + {n} = , n and {n} (, ). Suppose F := F(S) ( Ml= F(Tl)) = and let l, l = , , . . . , M be positive

real numbers. If the following conditions are satised:(i) limn n = and

n= n = ;(ii) < lim infn n lim supn n < ;(iii) limn n = ;

(iv) limn supy C T(n+)y T(n)y = , C bounded subset of C. Then the sequence {xn} is generated by x = x C and

converges strongly to QF, where QF is the sunny nonexpansive retraction of E onto F.

Proof Putting Al = I Tl, l {, , . . . , M}. From (.), we get Al is kl -inverse strongly accretive operator. It follows that GSVI(C, Al) = GSVI(C, I Tl) = F(Tl) = and (

Ml= GSVI(C, I Tl)) = F(Q) is the solution of the problem (.) (see also Ceng et al.

[, Theorem ., pp.-] and Aoyama et al. [, Theorem ., p.]).

( ) + T xn = QC ( ) + T

...

( M) + MTM

= QC

Therefore, by Theorem ., {xn} converges strongly to some element x of F.

4.2 (II) Application to Hilbert spaces

In real Hilbert spaces H, by Lemma ., we obtain the following lemma:

Lemma . For given (x*, x*, . . . , x*M), a solution of the problem is as follows:

yn = (( M) + MTM)(( M) + MTM) (( ) + T)

(( ) + T)xn,xn+ = nf (xn) + nxn + nT(n)yn

(.)

xn

(

) + T

xn

( M) + MTM QC

( ) + T

xn.

MAMx*M + x* x*M, x x* , x C,

MAMx*M + x*M x*M, x x*M , x C, ...

Ax* + x* x*, x x* , x C,

Ax* + x* x*, x x* , x C,

(.)

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if and only if

x* = PC(I MAM)PC(I MAM) PC(I A)PC(I A)x*

is a xed point of the mapping P : C C dened by

P(x) = PC(I MAM)PC(I MAM) PC(I A)PC(I A)x, x C,

where PC is a metric projection H onto C.

It is well known that the smooth constant K =

in Hilbert spaces. From Theorem ., we can obtain the following result immediately.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H. Let Al : C H be a l-inverse-strongly monotone mapping with l (, l), l {, , . . . , M}.

Let S = {T(t) : t } be a nonexpansive semigroup on C and f be a weakly contractive mapping of C into itself with function . Assume that F := F(P) F(S) = , where P is dened by Lemma . and let l, l = , , . . . , M be positive real numbers. Let the sequences

{n}, {n} and {n} in (, ) with {n} + {n} + {n} = , n and the following conditions be satised:(i) limn n = and

n= n = ;(ii) < lim infn n lim supn n < ;(iii) limn n = ;

(iv) limn supy C T(n+)y T(n)y = , C bounded subset of C. For arbitrary given x = x C, the sequences {xn} are generated by

Then {xn} converges strongly to x = PFf (x) and (x, x, . . . , xM) is a solution of the problem (.).

Remark . We can replace a contraction mapping f to a weak contractive mapping by setting (t) = ( k)t. Hence, our results can be obtained immediately.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors read and approved the nal manuscript.

Author details

1Department of Mathematics, Faculty of Science, King Mongkuts University of Technology Thonburi (KMUTT), Bangmod, Bangkok, 10140, Thailand. 2Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak, 63000, Thailand.

Acknowledgements

The authors thank the Hands-on Research and Development Project, Rajamangala University of Technology Lanna (RMUTL), Tak, Thailand (under grant No. UR1-005). Moreover, the authors would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Oce of the Higher Education Commission

yn = PC(I MAM)PC(I MAM) PC(I A)PC(I A)xn, xn+ = nf (xn) + nxn + nT(n)yn.

(.)

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(NRU-CSEC No. 55000613) for nancial support. Finally, the authors are grateful to the reviewers for careful reading of the paper and for the suggestions which improved the quality of this work.

Received: 26 April 2012 Accepted: 27 June 2012 Published: 20 July 2012

References

1. Aoyama, K, Iiduka, H, Takahashi, W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006, Article ID 35390 (2006)

2. Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 18, 78-81 (1976)

3. Ceng, L-C, Wang, C-Y, Yao, J-C: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 67, 375-390 (2008)

4. Chen, R, He, H: Viscosity approximation of common xed points of nonexpansive semigroups in Banach spaces. Appl. Math. Lett. 20, 751-757 (2007)

5. Chen, RD, He, HM, Noor, MA: Modied iterations for nonexpansive semigroups in Banach space. Acta Math. Sin. Engl. Ser. 26(1), 193-202 (2010)

6. Cho, YJ, Zhou, HY, Guo, G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 47, 707-717 (2004)

7. Gossez, JP, Dozo, EL: Some geometric properties related to the xed point theory for nonexpansive mappings. Pac. J. Math. 40, 565-573 (1972)

8. Iiduka, H, Takahashi, W, Toyoda, M: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J. 14(2), 49-61 (2004)

9. Iiduka, H, Takahashi, W: Strong convergence theorems for nonexpansive mapping and inverse-strong monotone mappings. Nonlinear Anal. 61, 341-350 (2005)

10. Kamimura, S, Takahashi, W: Approximating solutions of maximal monotone operators in Hilbert space. J. Approx. Theory 106, 226-240 (2000)

11. Kamimura, S, Takahashi, W: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set-Valued Anal. 8(4), 361-374 (2000)

12. Katchang, P, Kumam, P: An iterative algorithm for nding a common solution of xed points and a general system of variational inequalities for two inverse strongly accretive operators. Positivity 15, 281-295 (2011)

13. Kitahara, S, Takahashi, W: Image recovery by convex combinations of sunny nonexpansive retractions. Topol. Methods Nonlinear Anal. 2, 333-342 (1993)

14. Lau, AT-M: Amenability and xed point property for semigroup of nonexpansive mapping. In: Thera, MA, Baillon, JB (eds.) Fixed Point Theory and Applications. Pitman Res. Notes Math. Ser., vol. 252, pp. 303-313. Longman, Harlow (1991)

15. Lau, AT-M: Invariant means and xed point properties of semigroup of nonexpansive mappings. Taiwan. J. Math. 12, 1525-1542 (2008)

16. Lau, AT-M, Miyake, H, Takahashi, W: Approximation of xed points for amenable semigroups of nonexpansive mappings in Banach spaces. Nonlinear Anal. 67, 1211-1225 (2007)

17. Li, S, Su, Y, Zhang, L, Zhao, H, Li, L: Viscosity approximation methods with weak contraction for L-Lipschitzian pseudocontractive self-mapping. Nonlinear Anal. 74, 1031-1039 (2011)

18. Qin, X, Cho, SY, Kang, SM: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math. 233, 231-240 (2009)

19. Reich, S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44(1), 57-70 (1973)20. Suzuki, T: Strong convergence of Krasnoselskii and Manns type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305(1), 227-239 (2005)

21. Suzuki, T: On strong convergence to common xed points of nonexpansive semigroups in Hilbert spaces. Proc. Am. Math. Soc. 131, 2133-2136 (2002)

22. Stampacchi, G: Formes bilineaires coercivites sur les ensembles convexes. C. R. Math. Acad. Sci. Paris 258, 4413-4416 (1964)

23. Takahashi, W: Convex Analysis and Approximation Fixed Points. Yokohama Publishers, Yokohama (2000) (Japanese)24. Takahashi, W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000)

25. Takahashi, W: Viscosity approximation methods for resolvents of accretive operators in Banach spaces. J. Fixed Point Theory Appl. 1, 135-147 (2007)

26. Wangkeeree, R, Preechasilp, P: Modied Noor iterations for nonexpansive semigroups with generalized contraction in Banach spaces. J. Inequal. Appl. 2012, 6 (2012). doi:10.1186/1029-242X-2012-6

27. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127-1138 (1991)28. Xu, HK: A strong convergence theorem for contraction semigroups in Banach spaces. Bull. Aust. Math. Soc. 72,

371-379 (2005)

29. Yao, Y, Noor, MA, Noor, KI, Liou, Y-C, Yaqoob, H: Modied extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl. Math. 110(3), 1211-1224 (2010)

doi:10.1186/1687-1812-2012-114Cite this article as: Onjai-uea et al.: Strong convergence theorems for solving a general system of nite variational inequalities for nite accretive operators and xed points of nonexpansive semigroups with weak contraction mappings. Fixed Point Theory and Applications 2012 2012:114.