Approximating fixed points of amenable semigroup and infinite family of nonexpansive mappings and solving systems of variational inequalities and systems of equilibrium problems

Abstract

We introduce an iterative scheme for finding a common element of the set of solutions for systems of equilibrium problems and systems of variational inequalities and the set of common fixed points for an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. The results presented in this paper mainly extend and improved some well-known results in the literature.

Mathematics Subject Classification (2000): 47H09; 47H10; 47H20; 43A07; 47J25.[PUBLICATION ABSTRACT]

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RESEARCH Open Access

Approximating fixed points of amenable semigroup and infinite family of nonexpansive mappings and solving systems of variational inequalities and systems of equilibrium problems

Hossein Piri

Correspondence: mailto:[email protected]

Web End [email protected] Department of Mathematics, University of Bonab 55517-61167 Bonab, Iran

Abstract

We introduce an iterative scheme for finding a common element of the set of solutions for systems of equilibrium problems and systems of variational inequalities and the set of common fixed points for an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. The results presented in this paper mainly extend and improved some well-known results in the literature. Mathematics Subject Classification (2000): 47H09; 47H10; 47H20; 43A07; 47J25.

Keywords: common fixed point, strong convergence, amenable semigroup, explicit iterative, system of equilibrium problem.

1. Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.

Let A: C H be a nonlinear mapping. The classical variational inequality problem is to fined x C such that

Ax, y x 0, y C. (1)

The set of solution of (1) is denoted by VI(C, A), i.e.,

VI(C, A) = {x C : Ax, y x 0, y C}. (2)

Recall that the following definitions:

(1) A is called monotone if

Ax Ay, x y 0, x, y C.

(2) A is called a-strongly monotone if there exists a positive constant a such that

Ax Ay, x y x

y

2,

x, y C.

2012 Piri; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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(3) A is called -Lipschitzian if there exist a positive constant such that

Ax

, x, y C.

(4) A is called a-inverse strongly monotone, if there exists a positive real number a >0

such that

Ax Ay, x y Ax

x, y C.

It is obvious that any a-inverse strongly monotone mapping B is 1-Lipschitzian.

(5) A mapping T : C C is called nonexpansive if Tx - Ty x - y for all x, y C. Next, we denote by Fix(T) the set of fixed point of T.(6) A mapping f : C C is said to be contraction if there exists a coefficient a (0, 1) such that

f

, x, y C.

(7) A set-valued mapping U : H 2H is called monotone if for all x, y H, f Ux and g Uy imply x - y, f - g 0.(8) A monotone mapping U : H 2H is maximal if the graph G(U) of U is not properly contained in the graph of any other monotone mapping.

It is known that a monotone mapping U is maximal if and only if for (x, f) H H, x - y, f - g 0 for every (y, g) G(U) implies that f Ux. Let B be a monotone mapping of C into H and let NCx be the normal cone to C at x C, that is, NCx = {y H : x - z, y 0, z C} and define

Ux =

Bx + NCx, x C,

x C.

Then U is the maximal monotone and 0 Ux if and only if x VI(C, B); see [1]. Let F be a bi-function of CC into , where is the set of real numbers. The equilibrium problem for F : C C is to determine its equilibrium points, i.e the set

EP(F) = {x C : F(x, y) 0, y C}.

Let J = {Fi}iI be a family of bi-functions from C C into . The system of equilibrium problems for J = {Fi}iI is to determine common equilibrium points for

J = {Fi}iI , i.e the set

EP(J ) = {x C : Fi(x, y) 0, y C, i I}. (3)

Numerous problems in physics, optimization, and economics reduce into finding some element of EP(F). Some method have been proposed to solve the equilibrium problem; see, for instance [2-5]. The formulation (3), extend this formalism to systems of such problems, covering in particular various forms of feasibility problems [6,7].

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Ay

x

y

Ay

2,

(x) f (y)

x

y

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Page 3 of 22

Given any r >0 the operator JFr : H C defined by

JFr(x) = {z C : F(z, y) +

1r y z, z x 0, y C},

is called the resolvent of F, see [3]. It is shown [3] that under suitable hypotheses on

F (to be stated precisely in Sect. 2), JFr : H C is single- valued and firmly nonexpansive andsatisfies

Fix(JFr) = EP(F), r > 0.

Using this result, in 2007, Yao et al. [8], proposed the following explicit scheme with respect to W-mappings for an infinite family of nonexpansive mappings:

xn+1 = nf (xn) + nxn + nWnJFrnxn (4)

They proved that if the sequences {an}, {bn}, {gn} and {rn} of parameters satisfy appro

priate conditions, then, the sequences {xn} and {JFrnxn} both converge strongly to the unique x i=1Fix(Ti) EP(F), where x P

i=1Fix(Ti)EP(F) f (x). Their results extend

and improve the corresponding results announced by Combettes and Hirstoaga [3] and Takahashi and Takahashi [5].

Very recently, Jitpeera et al. [9], introduced the iterative scheme based on viscosity and Cesro mean

(un, y) + (y) (un) + 1rn y un, un xn 0, y C, yn = nun + (1 n)PC(un nBun),xn+1 = n f (xn) + nxn + ((1 n)I nA)

1 n+1

ni=0 Tiyn, n 0,

where B : C H is b-inverse strongly monotone, : C {} is a proper lower semi-continuous and convex function, Ti : C C is a nonexpansive mapping for all i = 1, 2, ..., n, {an}, {bn}, {n} (0, 1), {ln} (0, 2b) and {rn} (0, ) satisfy the following conditions

(i) limn an = 0,

n=1 n = ,(ii) limn n = 0(iii) 0 <lim infn bn lim supn bn <1.(iv) {ln} [a, b] (0, 2b) and lim infn | ln+1 - ln |= 0,(v) lim infn rn >0 and lim infn | rn+1 - rn |= 0.

They show that if = ni=1Fix(Ti) VI(C, B) MEP(, ) is nonempty, then the sequence {xn} converges strongly to the z = P(I - A + gf )z which is the unique solution of the variational inequality

( f A)z, x z 0 y .

In this paper, motivated and inspired by Yao et al. [8,10-15], Lau et al. [16], Jitpeera et al. [9], Kangtunyakarn [17] and Kim [18], Atsushiba and Takahashi [19], Saeidi [20], Piri [21-23] and Piri and Badali [24], we introduce the following iterative scheme for finding a common element of the set of solutions for a system of equilibrium problems

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J = {Fk : k = 1, 2, 3, . . . , M} for a family J = {Fk : k = 1, 2, 3, . . . , M} of equilibrium bi-

functions, systems of variational inequalities, the set of common fixed points for an infinite family = {Ti, i = 1, 2, ...} of nonexpansive mappings and a left amenable semigroup = {Tt : t S} of nonexpansive mappings, with respect to W-mappings and a left regular sequence {n} of means defined on an appropriate space of bounded real-valued functions of the semigroup

zn = JFMrM,n . . . JF2r2,nJF1r1,nxn,yn = nPC(zn nAzn) + (1 n)PC(zn nBzn),xn+1 = nf (TnWnyn) + nxn + nTnWnyn, n 1,

(5)

where A: C H be b-inverse monotone map and B : C H be -inverse monotone map. We prove that under mild assumptions on parameters like that in Yao et al. [8],

the sequences {xn} and {JFkrk,nxn}Mk=1 converge strongly to x F = i=1Fix(Ti) Fix() EP(J ) VI(C, A) VI(C, B), where x = PFf (x).

Compared to the similar works, our results have the merit of studying the solutions of systems of equilibrium problems, systems of variational inequalities and fixed point problems of amenable semigroup of nonexpansive mappings. Consequence for nonnegative integer numbers is also presented.

2. Preliminaries

Let S be a semigroup and let B(S) be the space of all bounded real valued functions defined on S with supremum norm. For s S and f B(S), we define elements lsf and rsf in B(S) by

(lsf )(t) = f (st), (rsf )(t) = f (ts), t S.

Let X be a subspace of B(S) containing 1 and let X* be its topological dual. An element of X* is said to be a mean on X if = (1) = 1. We often write t(f(t))

instead of (f) for X* and f X. Let X be left invariant (respectively right invariant), i.e., ls(X) X (respectively rs(X) X) for each s S. A mean on X is said to be left invariant (respectively right invariant) if (lsf) = (f) (respectively (rsf) = (f)) for each s S and f X. X is said to be left (respectively right) amenable if X has a left (respectively right) invariant mean. X is amenable if X is both left and right amenable. As is well known, B(S) is amenable when S is a commutative semigroup, see [25]. A net {a} of means on X is said to be strongly left regular if

lim

l

s

= 0,

for each s S, where ls is the adjoint operator of ls.

Let S be a semigroup and let C be a nonempty closed and convex subset of a reflexive Banach space E. A family = {Tt : t S} of mapping from C into itself is said to be a nonexpansive semigroup on C if Tt is nonexpansive and Tts = TtTs for each t, s S. By Fix() we denote the set of common fixed points of , i.e.

Fix() =

tS

{x C : Ttx = x}.

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Lemma 2.1. [25]Let S be a semigroup and C be a nonempty closed convex subset of a reflexive Banach space E. Let = {Tt : t S} be a nonexpansive semigroup on H such that {Ttx : t S} is bounded for some x C, let X be a subspace of B(S) such that 1 X and the mapping t Ttx, y* is an element of X for each x C and y* E*, and is a mean on X. If we write Tx instead of Ttxd(t), then the followings hold.(i) T is nonexpansive mapping from C into C.(ii) Tx = x for each x Fix().(iii) Tx co{Ttx : t S}for each x C.

Let C be a nonempty subset of a Hilbert space H and T : C H a mapping. Then T is said to be demiclosed at v H if, for any sequence {xn} in C, the following implication holds:

xn u C, Txn v imply Tu = v,

where (respectively ) denotes strong (respectively weak) convergence.

Lemma 2.2. [26]Let C be a nonempty closed convex subset of a Hilbert space H and suppose that T : C H is nonexpansive. then, the mapping I - T is demiclosed at zero.

Lemma 2.3. [27]For a given x H, y C,

y = PCx y x, z y 0, z C.

It is well known that PC is a firmly nonexpansive mapping of H onto C and satisfies

P

Cx PCy

PCx PCy, x y , x, y H. (6)

Moreover, PC is characterized by the following properties: PCx C and for all x H, y C,

x PCx, y PCx 0. (7)

It is easy to see that (7) is equivalent to the following inequality

x

y

2

2

x PCx 2 +

y

PCx

2.

(8)

Using Lemma 2.3, one can see that the variational inequality (1) is equivalent to a fixed point problem. It is easy to see that the following is true:

u VI(C, A) u = PC(u Au), > 0. (9)

Lemma 2.4. [28]Let {xn} and {yn} be bounded sequences in a Banach space E and let

{an} be a sequence in [0, 1] with 0 < lim inf

n

n lim sup

n

n < 1. Suppose xn+1 = anxn

+(1-an)yn for all integers n 0 and

lim sup

n

(

y

n+1

yn

xn+1 xn ) 1.

Then, lim

n

y

n

xn

= 0.

Let F : C C be a bi-function. Given any r >0, the operator JFr : H C defined by

JFrx =

z C : F(z, y) +1r y z, z x 0, y C

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is called the resolvent of F, see [3]. The equilibrium problem for F is to determine its equilibrium points, i.e., the set

EP(F) = {x C : F(x, y) 0, y C}.

Let J = {Fi}iI be a family of bi-functions from C C into . The system of equilibrium problems for J is to determine common equilibrium points for J = {Fi}iI . i.e,

the set

EP(J ) = {x C : Fi(x, y) 0, y C, i I}.

Lemma 2.5. [3]Let C be a nonempty closed convex subset of H and F : C C satisfy

(A1) F (x, x) = 0 for all x C,(A2) F is monotone, i.e, F(x, y) + F(y, x) 0 for all x, y C,(A3) for all x, y, z C, limt0 F(tz + (1 - t)x, y) F (x, y),(A4) for all x C, y F(x, y) is convex and lower semi-continuous.

Given r >0, define the operator JFr : H C , the resolvent of F, by

JFr(x) = {z C : F(z, y) +

1r y z, z x 0, y C}.

Then,

(1) JFr is single valued,

(2) JFr is firmly nonexpansive, i.e,

JF

r x JFry

2

JFrx JFry, x y for all x, y H,

(3) Fix(JFr) = EP(F) ,(4) EP(F) is closed and convex.

Let T1, T2, ... be an infinite family of mappings of C into itself and let l1, l2, ... be a real numbers such that 0 li <1 for every i N. For any n N, define a mapping Wn of C into C as follows:

Un,n+1 = I,

Un,n = nTnUn,n+1 + (1 n)I, Un,n1 = n1Tn1Un,n + (1 n1)I,

...

Un,k = kTkUn,k+1 + (1 k)I, Un,k1 = k1Tk1Un,k + (1 k1)I,

...

Un,2 = 2T2Un,3 + (1 2)I,Wn = Un,1 = 1T1Un,2 + (1 1)I.

(10)

Such a mapping Wn is called the W-mapping generated by T1, T2, ..., Tn and l1, l2, ..., ln.

Lemma 2.6. [29]Let C be a nonempty closed convex subset of a Hilbert space H, {Ti : C C} be an infinite family of nonexpansive mappings with i=1Fix(Ti) = , {li} be a real sequence such that 0 < li b <1, i 1. Then(1) Wn is nonexpansive and Fix(Wn) = ni=1Fix(Ti)for each n 1,

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(2) for each x C and for each positive integer j, the limit limn Un,j exists.(3) The mapping W : C C defined by

Wx := lim

n

Wnx = lim

n

Un,1x, x C,

is a nonexpansive mapping satisfying Fix(W) = i=1Fix(Ti)and it is called the W-mapping generated by T1, T2, ... and l1, l2, ....

Lemma 2.7. [30]Let C be a nonempty closed convex subset of a Hilbert space H, {Ti : C C} be a countable family of nonexpansive mappings with i=1Fix(Ti) = ,{li} be a real sequence such that 0 < li b <1, i 1. If D is any bounded subset of C, then

lim

n

sup

xD

Wx Wnx = 0.

Lemma 2.8. [31]Let {an} be a sequence of nonnegative real numbers such that

an+1 (1 bn)an + bncn, n 0,

where {bn} and {cn} are sequences of real numbers satisfying the following conditions:

(i) {bn} [0, 1],

n=0 bn = ,

(ii) either lim sup

n

cn 0 or

|bncn| < .

Then, lim

n

an = 0.

Lemma 2.9. [32]Let (E, ., .) be an inner product space. Then for all x, y, z E and a, b, g, [0, 1] such that a + b + g = 1, we have

x

+ y + z

2

= x 2 +

x

y

y 2

+ z 2

2

x z 2

y

z

2.

Notation Throughout the rest of this paper the open ball of radius r centered at 0 is denoted by Br. For a subset A of H we denote by coA the closed convex hull of A. For >0 and a mapping T : D H, we let F[notdef](T; D) be the set of [notdef]-approximate fixed points of T, i.e., F[notdef](T ; D) = {x D : x - Tx [notdef]}. Weak convergence is denoted by and strong convergence is denoted by .

3. Strong convergence

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H, A: C H a b-inverse strongly monotone, B : C H a g-inverse strongly monotone, S a semigroup and = {Tt : t S} be a nonexpansive semigroup from C into C such that Fix() = tSFix(Tt) = . Let X be a left invariant subspace of B(S) such that 1 X, and the function t Ttx, y is an element of X for each x C and y H, {n} a left regular sequence of means on X such that limn n+1 - n = 0. Let J = {Fk : k = 1, 2, . . . , M}be a finite family of bi-functions from C C into which satisfy (A1)-(A4) and {Ti}i=1 an infinite family of nonexpansive mappings of C into C such that Ti(Fix() EP(J )) Fix()

for each i N and F = i=1Fix(Ti) Fix() EP(J ) VI(C, A) VI(C, B) = . Let {an}, {bn}, {gn} and {hn} be a sequences in (0, 1). Let {n} a sequence in (0, 2b), {n} a

sequence in (0, 2g), {rk,n}Mk=1 be sequences in (0, ) and {ln} a sequence of real numbers such

that 0 < ln b <1. Assume that,

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n=1 n = ,(B2) 0 <lim infn bn lim supn bn <1,(B3) an + bn + gn = 1,(B4) limn | n+1 - n |= limn | n+1 - n |= 0,(B5) lim infn rk,n >0 and limn (rk,n+1 - rk,n) = 0 for k {1, 2, , M}.

Let f be a contraction of C into itself with coefficient a (0, 1) and given x1 C arbitrarily. If the sequences {xn}, {yn} and {zn} are generated iteratively by x1 C and

zn = JFMrM,n . . . JF2r2,nJF1r1,nxn,yn = nPC(zn nAzn) + (1 n)PC(zn nBzn),xn+1 = nf (TnWnyn) + nxn + nTnWnyn, n 1,

(B1) limn hn = h (0, 1), limn an = 0 and

(11)

then, the sequences {xn}, {yn} and {JFkrk,nxn}Mk=1converge strongly to x F , which is the

unique solution of the system of variational inequalities:

f (x) x, x y 0, y F,

Bx, y x 0 y C,

Ax, y x 0 y C.

Proof. Since A is a b-inverse strongly monotone map, for any x, y C, we have

(I

nA)x (I nA)y

(x

y) n(Ax Ay)

2

2

=

|x

y

2

2n x y, Ax Ay + 2n

Ax

Ay

2

=

x

y

2

2n

Ax

Ay

2

+ 2n

Ax

Ay

2

x

y

2

+ n(n 2)

Ax

Ay

2

=

x

y

2

It follows that

(I

nA)x (I nA)y

. (12)

Since B is a b-inverse strongly monotone map, repeating the same argument as above, we can deduce that

(I

nB)x (I nB)y

x

y

. (13)

Let p F , in the context of the variational inequality problem the characterization

of projection (9) implies that p = PC(p - nAp) and p = PC(p - nBp). Using (12) and(13), we get

y

n

x

y

p

= n[PC(zn nAzn) PC(p nAp)]+(1 n)[PC(zn nBzn) PC(p nBp)]

= nPC(zn nAzn) + (1 n)PC(zn nBzn) p

(14)

n

P

C(zn nAzn) PC(p nAp)

+ (1 n)

P

C(zn nBzn) PC(p nBp)

n

z

+ (1 n)

z

n

n

p

p

z

n

=

p

.

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By taking vn = PC(zn - nAzn), wn = PC(zn - nBzn) and J kn = JFkrk,n JF2r2,nJF1r1,n for k {1, 2, ..., M} and J 0n = I for all n N, we shall equivalently write scheme (11) as follows:

zn = J Mnxn,yn = nvn + (1 n)wn,xn+1 = nf (TnWnyn) + nxn + nTnWnyn, n 1.

We shall divide the proof into several steps.

Step 1. The sequence {xn} is bounded.

Proof of Step 1. Let p F . Since for each k {1, 2, ..., M}, JFkrk,n is nonexpansive we have

J knxn p

= J knxn J knp

x

, k {1, 2, . . . , M}. (15)

Thus, by Lemmas 2.1, 2.5 and (14), we have

x

n+1

n

p

p

n

f

(TnWnyn) p

+ n

x

n

p

f

(TnWnyn) f (p)

+ n

T

n WnJ Mnyn p

+ n

f

(p) p

n[

f

+ (p) p

x

n

y

n

+ (n + n)

] +

n

p

+ n

p

n

x

x

n

n

p

p

= [1 n(1 )]

x

+ n

f

(p) p

n

p

max

xn

p

, 1

1

f

(p) p

.

By induction,

x

n

p

max

x1

p

, 1

1

f

(p) p

, n 1.

Step 2. Let {un} be a bounded sequence in H. Then

lim

n

J kn+1un J knun

= 0, (16)

for every k {1, 2, ..., M}.

Proof of Step 2. This assertion is proved in [27, Step 2]. Step 3. Let {un} be a bounded sequence in H. Then

lim

n

Wn+1un Wnun = 0 and limn

T

n+1 un Tnun

= 0.

This assertion is proved in [21, Step 3].

Step 4. limn xn+1 - xn = 0.

Proof of Step 4. Setting xn+1 = bnxn + (1 - bn)tn for all n 1, we have

tn+1 tn

= 1

1 n+1

[xn+2 n+1xn+1]

11 n

[xn+1 nxn]

= n+1

1 n+1

[f (Tn+1Wn+1yn+1) f (TnWnyn)] +

n+11 n+1

n1 n

f (TnWnyn)

+ n+1

1 n+1

[Tn+1Wn+1yn+1 TnWnyn] +

n+11 n+1

n1 n

TnWnyn.

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Page 10 of 22

Therefore, we have

tn+1 tn

n+1

1 n+1

[

T

n+1 Wn+1yn+1 TnWnyn

T

+ n+1 Wn+1yn+1 TnWnyn

]

n+1

+ 1 n+1

n1 n

T

+ n+1 Wn+1yn+1 TnWnyn

[

f

(TnWnyn)

.

T

+ nWnyn

]

On the other hand

T

n+1 Wn+1yn+1 TnWnyn

T

n+1 Wn+1yn+1 Tn+1Wn+1yn

T

+ n+1 Wnyn TnWnyn

T

+ n+1 Wn+1yn Tn+1Wnyn

W

+ n+1yn Wnyn

y

T

+ n+1 Wnyn TnWnyn

n+1

yn

Observing that zn = J Mnxn , zn+1 = J Mn+1xn+1 and J Mnxn = JFMrM,nJ M1nxn we get

1rM,n y zn, zn J M1nxn + FM(zn, y) 0, y C, (17)

and

1rM,n+1 y zn+1, zn+1 J M1n+1xn+1 + FM(zn+1, y) 0, y C, (18)

Take y = zn+1 in (17) and y = zn in (18), by using (A2), it follows that

zn+1 zn, zn J M1nxn rM,nrM,n+1 (zn+1 J M1n+1xn+1) 0,

and hence

zn+1 zn, zn J M1nxn zn+1 + J M1n+1xn+1

+ 1 rM,n rM,n+1

(zn+1 J M1n+1xn+1) 0,

Thus, we have

zn+1 zn

J M1n+1xn+1 J M1nxn

+

1

rM,n

rM,n+1

+ J M1n+1xn J M1nxn

z

n+1

J M1n+1xn+1

J M1n+1xn+1 J M1n+1xn

rM,n

rM,n+1

z

+

1

n+1

J M1n+1xn+1

xn+1 xn +

J M1n+1xn J M1nxn

.

Since vn = PC(zn - nAzn) and wn = PC(zn - nBzn), it follows from the definition of {yn} that

z

+

1

rM,n

rM,n+1

n+1

J M1n+1xn+1

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y

n+1

= n+1vn+1 + (1 n+1)wn+1 nvn (1 n)wn

yn

= n+1(vn+1 vn) + (n+1 n)vn + (1 n+1)wn+1

(1 n+1)wn + (n n+1)wn

n+1 vn+1 vn + |n+1 n| ( vn + wn )

+(1 n+1) wn+1 wn = n+1

P

C(zn+1 n+1Azn+1) PC(zn nAzn)

+ |n+1 n| ( vn + wn ) +(1 n+1)

P

C(zn+1 n+1Bzn+1) PC(zn nBzn)

= n+1

P

C(zn+1 n+1Azn+1) PC(zn n+1Azn) +PC(zn n+1Azn) PC(zn nAzn)

+ |n+1 n| ( vn + wn ) +(1 n+1)

P

C(zn+1 n+1Bzn+1) PC(zn n+1Bzn) +PC(zn n+1Bzn) PC(zn nBzn)

n+1 zn+1 zn + n+1 |n+1 n| Azn + |n+1 n| ( vn + wn ) + (1 n+1) zn+1 zn +(1 n+1) |n+1 n| Bzn

zn+1 zn + n+1 |n+1 n| Azn + |n+1 n| ( vn + wn ) + |n+1 n| Bzn .

Therefore,

tn+1 tn xn+1 xn

n+1

1 n+1

[

y

n+1

yn

T

+ n+1 Wn+1yn+1 TnWnyn

]

n+1

+ 1 n+1

n1 n

[

f

(TnWnyn)

T

+ nWnyn

]

+ J M1n+1xn J M1nxn

+

1

rM,n

rM,n+1

z

n+1

J M1n+1xn+1

+n+1 |n+1 n| Azn + |n+1 n| ( vn + wn ) + |n+1 n| Bzn +

W

n+1yn Wnyn

T

+ n+1 Wnyn TnWnyn

.

This together with conditions (B1), (B4), Steps 2 and 3 imply that

lim sup

n

( tn+1 tn xn+1 xn ) 0.

Hence by Lemma 2.4, we obtain limn tn - xn = 0. Consequently,

lim

n

xn+1 xn = (1 n) tn xn = 0.

Step 5. limn

J k+1nxn J knxn

= 0, k {0, 1, 2, ..., M - 1}.

Proof of Step 5. Let p F and k {1, 2, ..., M - 1}. Since JFk+1rk+1,n is firmly nonexpansive, we obtain

J k+1nxn p

2

= JFk+1rk+1,nJ knxn JFk+1rk+1,np

2 = JFk+1rk+1,nJ knxn p, J knxn p

= 1

2 JFk+1rk+1,nJ knxn p 2

JFk+1rk+1,nJ knxn J knxn

2

+

J knxn p 2

.

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Page 12 of 22

It follows that

J k+1nxn p

2

x

2

J k+1nxn J knxn

2.

n

p

(19)

Using Lemma 2.9, (14) and (19), we obtain

x

n+1

p

2

n

f

(TnWnyn) p

2

+ n

p

2

+ n

T

nWnyn p

x

n

x

n

n

f

(TnWnyn) p

2

+ n

2

2

p

2

+ n

y

n

p

n

f

(TnWnyn) p

2

+ n

p

2

+ n

z

n

p

x

n

= n

f

(TnWnyn) p

2

+ n

x

n

2

p

2

+ n

J Mnxn p

2

= n

f

(TnWnyn) p

p

2

JFMrM,n . . . JFk+2rk+2,nJ k+1nyn JFMrM,n . . . JFk+2rk+2,np

2

2

+ n

x

n

+n

n

f (TnWnyn) p

2

+ n

xn p

2

+ n

J k+1nyn p

2

n

f

(TnWnyn) p

2

+ n

x

n

p

2

+ n

J k+1nxn J knxn

].

n

f

(TnWnyn) p

x

n

2

+ n

x

n

J k+1nxn p

2

p

2

p

2

+n[

Then, we have

n

J k+1nxn J knxn

2

n

f

(TnWnyn) p

2

2

+ n

x

n

p

+(1 n n)

x

n

p

n+1

p

f

(TnWnyn) p

2

x

2

x

2] + x

n

2

2

= n[

n

p

p

2

x

n+1

p

n[

f

(TnWnyn) p

n

p

+ xn xn+1 [

x

n

2

x

p

+

x

n+1

2]

p

].

It is easily seen that lim infn gn >0. So we have

lim

n

J k+1nxn J knxn

= 0.

Step 6. limn

x

= 0.

n

TnWnJ Mnyn

Proof of Step 6. Observe that

x

n

TnWnJ Mnyn

xn xn+1 +

x

n+1

TnWnJ Mnyn

n[f (TnWnyn) TnWnJ Mnyn] +n[xn TnWnJ Mnyn]

xn xn+1 + n[

= xn xn+1 +

f

(TnWnyn)

,

T

+ nWnJ Mnyn

]

+n

xn TnWnJ Mnyn

hence

x

n

TnWnJ Mnyn

11 n

xn+1 xn +

n1 n

[

f

T

+ nWnJ Mnyn

].

(TnWnyn)

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Page 13 of 22

It follows from conditions (B1), (B2) and Step 4, that

lim

n x

n

TnWnJ Mnyn

= 0.

Step 7. limn xn - Ttxn = 0, for all t S. Proof of Step 7. Let p F and set M0 = max{

x

and D = {y

H : y - p M0}, we remark that D is bounded closed convex set, {yn} D and it

is invariant under {JFkrk,n : k = 1, 2, ..., M, n

N

, 1

1

f

(p) p

}

1

p

} , and Wn for all n N. We will show

that

lim sup

n

sup

yD

T

n y TtTny

= 0, t S (20)

Let [notdef] >0. By [33, Theorem 1.2], there exists >0 such that

coF(Tt; D) + B F(Tt; D), t S. (21)

Also by [33, Corollary 1.1], there exists a natural number N such that

1 N + 1

1 N + 1

N

i=0Ttisy Tt

N

i=0Ttisy

, (22)

for all t, s S and y D. Let t S. Since {n} is strongly left regular, there exists N0

N such that

n

ltin

(M0+||p||) for n N0 and i = 1, 2, ..., N. Then, we have

sup

yD

Tny

1 N + 1

N

i=0 Ttisydn(s)

= sup

yD

sup

z =1

Tny, z

1N + 1N

i=0Ttisydn(s), z

= sup

yD

sup

z =1

1 N + 1

i=0(n)s Tsy, z 1 N + 1

N

i=0(n)s Ttisy, z

N

(23)

N

i=0supyDsup

z =1 |

1 N + 1

(n)s Tsy, z (ltin)s Tsy, z |

max

i=1,2,...,N

n

ltin

(M0 +

p )

, n N0.

By Lemma 2.1 we have

1 N + 1

1 N + 1

N

i=0TtiS ydn(s) co

N

i=0Tti (Tsy) : s S

. (24)

It follows from (21), (22), (23) and (24) that

Tny co

1 N + 1

+ B

N

i=0TtiS y : s S

coF(Tt; D) + B F(Tt; D),

Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99

Page 14 of 22

for all y D and n N0. Therefore,

lim sup

n

sup

yD

T

t(Tny) Tny

.

Since [notdef] >0 is arbitrary, we get (20).

Let t S and [notdef] >0. Then, there exists >0, which satisfies (21). From condition (B1),(20) and Step 6, there exists N1 N such that n <

4M0 , Tny F(Tt, D) for all y

D and

x

n

TnWnyn

< 2 for all n N1. We note that

n

f

(TnWnyn) TnWnyn

n[

f

(TnWnyn) f (p)

f

+ (p) p

f

+ (p) p

n[

y

+

p

yn

]

2M0n

p

+ TnWnyn

]

n

p

2,

for all n N1. Therefore, we have

xn+1 = TnWnyn + n(f (TnWnyn) TnWnyn)

+ n(xn TnWnyn)

F(Tt; D) + B

2 + B

2

F(Tt; D) + B F(Tt; D),

for all n N1. This shows that

xn Ttxn , n N1.

Since [notdef] >0 is arbitrary, we get limn xn - Tt(xn) = 0.

Step 8. The weak -limit set of {xn}, {xn}, is a subset of F .

Proof of Step 8. Let z {xn} and let {xnm} be a subsequence of {xn} weakly converging to z, we need to show that z F . Noting Step 5, with no loss of generality, we may assume that J knmxnm z, k {1, 2, . . . , M}. At first, note that by (A2) and given y C and k {1, 2, ..., M}, we have

y J k+1nmxnm,1rk+1,nm (J k+1nmxnm J knmxnm)

Fk+1(y, J k+1nmxnm).

Step 5 and condition(B5) imply that

J k+1nmxnm J knmxnm rk+1,nm 0.

Since J knmxnm z , from the lower semi-continuity of Fk+1 on the second variable, we have Fk+1(y, z) 0 for all y C and for all k {0, 1, 2, ..., M - 1}. For t with 0 < t 1 and y C, let yt = ty + (1 - t)z. Since y C and z C, we have yt C and hence Fk

+1(yt, z) 0. So from the convexity of Fk+1 on second variable, we have

0 = Fk+1(yt, yt) tFk+1(yt, y) + (1 t)Fk+1(yt, z) tFk+1(yt, y) Fk+1(yt, y).

hence Fk+1(yt, y) 0. therefore, we have Fk+1(z, y) 0 for all y C and k {0, 1, 2,

..., M-1}. Therefore z Mk=1EP(Fk) = EP(J ).

Since xnm z , it follows by Step 7 and Lemma 2.2 that z Fix(Tt) for all t S.

Therefore, z Fix(). We will show z Fix(W). Assume z Fix(W) Since

Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99

Page 15 of 22

z Fix() EP(J ) , by our assumption, we have Tiz Fix(),i N and then Wnz Fix(). Hence by Lemma 2.1, TnWnz = Wnz , therefore by Lemma 2.5, we get

TnWnJ Mnz = Wnz, n

N. (25)

Now, by (25), Step 6, Lemma 2.6 and Opials condition, we have

lim inf

n x

nm

z

< lim inf

n

x

nm

Wz

lim inf

n

x

nm

Tnm WnmJ Mnmxnm

T

+ nm WnmJ Mnmxnm Tnm WnmJ Mnmz

+

x

nm

T

+ nm WnmJ Mnmz Wz

W

+ nm z Wz

lim inf

n

x

nm

Tnm WnmJ Mnmxnm

z

lim inf

n

x

.

nm

z

This is a contradiction. So we get z Fix(W) =

i=1 Fix(Ti).

Now, let us show that z VI(C, A) VI(C, B). Observe that,

x

n+1

p

2

n

f

(TnWnyn) p

2

+ n

x

n

p

2

n WnJ Mnyn p

2

+ n

T

n

f

(TnWnyn) p

2

+ n

x

2

+ n

2

+ n

y

n

p

2

n

p

= n

f

(TnWnyn) p

2

nPC(zn nAzn)

+ n

x

n

p

+(1 n)PC(zn nBzn) p

2

n[PC(zn nAzn)

PC(p nAp)] + (1 n)[PC(zn nBzn) PC(p nBp)] 2.

(26)

2

+ n

= n

f

(TnWnyn) p

2

+ n

x

n

p

From (26), we have

x

n+1

p

2

n

f

(TnWnyn) p

2

2

+ n

x

n

p

+n[n

(z

n

p) n(Azn Ap)

p

2]

2

+ n

xn p

2

+ (1 n)

z

n

= n

f (TnWnyn) p

+n(1 n)

z

2

+ nn[

z

n

2

n

p

p

2

+ 2n

Az

2

n

Ap

2n Azn Ap, zn p ]

n

f

(TnWnyn) p

2

+ nn[

2

+ n

x

n

p

2

+n(1 n)

z

z

n

p

2

+ 2n

Az

2

n

p

n

Ap

2]

2n

Az

n

Ap, zn p

= n

f

(TnWnyn) p

2

+ n

x

n

p

2

+ n(n 2)

Az

n

2

+n

z

n

p

Ap

2

x

n

p

2

f

(TnWnyn) p

2

x

2]

+ n[

n

p

+n(n 2)

Az

2,

n

Ap

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Page 16 of 22

which implies that

n(n 2)

Az

2

n

Ap

[

x

n

p

]

xn xn+1

+

x

n+1

p

f

(TnWnyn) p

2

x

2].

+n[

n

p

Therefore, from step 4 and condition B1, we obtain

lim

n Az

n

Ap

= 0. (27)

On the other hand from (26), we have

x

n+1

p

2

n

f

(TnWnyn) p

2

+ n

x

n

p

2

+ n

2

n

z

n

p

+(1 n)

(z

n

p) n(Bzn Bp)

2

2

+ n

= n

f (TnWnyn) p

xn p

2

+ n

n

zn p

+(1 n)(

z

2

2n Bzn Bp, zn p + 2n

Bz

n

2

n

p

Bp

2)

n

f

(TnWnyn) p

2

2

+ n

2

+ n

x

n

p

n

z

n

p

+(1 n)(

z

2

2n

Bz

n

Bp

2

+ 2n

Bz

n

n

p

Bp

(TnWnyn) p

2

+ n

x

n

2

+ n

2)

= n

f

p

z

n

p

2

+n(n 2 )n(1 n)

Bz

n

Bp

x

n

2

+ n[

f

(TnWnyn) p

2

p

2

x

2]

n

p

+n(n 2 )

Bz

2

n

Bp

which implies that

n(n 2 )

Bz

x

n

2

n

Bp

[

p

]

xn xn+1

+

x

n+1

p

f

(TnWnyn) p

2

x

2].

+n[

n

p

Therefore, from step 4 and condition B1, we obtain

lim

n

Bz

n

Bp

= 0. (28)

From (6) and (12), we have

v

n

p

P

= C(zn nAzn) PC(p nAp)

2

2

(zn nAzn) (p nAp), vn p = 1

2

(zn nAzn) (p nAp)

(z

n

2

+

v

n

p

2

nAzn) (p nAp) (vn p)

2

= 1

2

zn p

2

+

v

n

p

2

zn vn 2

+2n zn vn, Azn Ap 2n

Az

n

Ap

2

.

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Page 17 of 22

So we obtain

v

n

p

2

zn vn 2

+ 2n zn vn, Azn Ap 2n

Az

z

n

p

2

(29)

2.

n

Ap

By using the same method as (29), we have

w

n

p

2

zn wn 2

+ 2n zn wn, Bzn Bp 2n

Bz

z

n

p

2

(30)

2.

n

Bp

From (29), (30) and definition of yn, we have,

y

n

= n[PC(zn nAzn) p]+(1 n)[PC(zn nBzn) p]

p

2

= n(vn p) + (1 n)(wn p)

2

2

n

v

n

p

2

+ (1 n)

p

n[

z

n

2

zn vn 2 + 2n zn vn, Azn Ap

2n

w

n

2

p

(31)

Az

n

Ap

2

] + (1 n)[

z

n

p

2

zn wn 2

+2n zn wn, Bzn Bp 2n

Bz

2]

n

Bp

z

2

+ n[ zn vn 2 + 2n zn vn

Ap

n

p

] + (1 n)[ zn wn 2 +2n zn wn

Bz

n

2n

Az

2

Az

n

n

Ap

Bp

2n

Bz

n

Bp

2]

By (31), we have

||xn+1 p||2

n||f (TnWnyn) p||2 + n||xn p||2 + n||TnWnJ Mnyn p||2

n||f (TnWnyn) p||2 + n||xn p||2 + n||zn p||2

+nn[||zn vn||2 + 2n||zn vn|| ||Azn Ap||

2n||Azn Ap||2] + n(1 n)[||zn wn||2 +2n|| ||zn wn|| ||Bzn Bp|| 2n||Bzn Bp||2]

n||f (TnWnyn) p||2 + n||xn p||2 + n||xn p||2

nn||zn vn||2 + nn[2n||zn vn|| ||Azn Ap||

2n||Azn Ap||2] n(1 n)||zn wn||2+n(1 n)[2n|| ||zn wn|| ||Bzn Bp|| 2n||Bzn Bp||2]

= ||xn p||2 + n[||f (TnWnyn) p||2 ||xn p||2]

nn||zn vn||2 + nn[2n||zn vn|| ||Azn Ap||

2n||Azn Ap||2] n(1 n)||zn wn||2+n(1 n)[2n|| ||zn wn|| ||Bzn Bp|| 2n||Bzn Bp||2],

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Page 18 of 22

which implies that

nn||zn vn||2

[||xn p|| + ||xn+1 p||]||xn+1 xn||+n[||f (TnWnyn) p||2 ||xn p||2] +nn[2n||zn vn|| ||Azn Ap|| 2n||Azn Ap||2]

+n(1 n)[2n|| ||zn wn|| ||Bzn Bp|| 2n||Bzn Bp||2],

and

n(1 n)||zn wn||2

[||xn p|| + ||xn+1 p||]||xn+1 xn||+nn[2n||zn vn|| ||Azn Ap|| 2n||Azn Ap||2]+n(1 n)[2n|| ||zn wn|| ||Bzn Bp|| 2n||Bzn Bp||2].

Therefore, from 0 <lim infn gn lim supn gn <1, condition B1, step 4, (27) and(28) we get

lim

n ||

zn vn|| = 0 and lim

n ||

zn wn|| = 0. (32)

Let U : H 2H be a set-valued mapping is defined by

Ux = Ax + NCx, x C,

, x C,

where NCx is the normal cone to C at x C. Since A is monotone. Thus U is maximal monotone see [1]. Let (x, y) G(U), hence y - Ax NCx and since vn = PC(zn -nAzn) therefore, x - vn, y - Ax 0. On the other hand from (7), we have

x vn, vn (zn nAzn) 0,

i.e.,

x vn,vn znn + Azn 0

Therefore, we have

x vni, y

x vni, Ax

x vni, Ax

x vni,vni znini + Azni

= x vni, Ax vni znini Azni

= x vni, Ax Avni + x vni, Avni Azni

x vni,vni zni ni

x vni, Avni Azni

x vni,vni zni ni

x vni, Avni Azni ||x vni||

vni zni ni

.

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Page 19 of 22

From (32), we get limi||vni zni|| = 0 . Noting that xni z and A is 1 -lipschitzian, we obtain

x z, y 0. (33) Since U is maximal monotone, we have z U-10, and hence z VI(C, A). Let V : H 2H be a set-valued mapping is defined by

Vx = Bx + NCx, x C,

, x C,where NCx is the normal cone to C at x C. Since B is monotone. Thus U is maximal monotone see [1]. Repeating the same argument as above, we can derive z VI (C, B). Therefore, z F .

Step 9. There exists a unique x* C such that

lim sup

n

f (x) x, xn x 0.

Proof of Step 9. Note that f is a contraction mapping with coefficient a (0, 1). Then

||PFf (x) PFf (y)|| ||f (x) f (y)|| ||x y|| for all x, y H. Therefore PF is a contraction of H into itself, which implies that there exists a unique element x* H

such that x = PFf (x). at the same time, we note that x* C. Using Lemma 2.3, we have

f (x) x, x z 0, z F. (34)

We can choose a subsequence {xnk} of {xn} such that lim sup

n

f (x) x, xn x = lim

k

f (x) x, xnk x .

Since {xnk} is bounded, therefore, {xnk} has subsequence {xnkj } such that xnkj z. With no loss of generality, we may assume that xnk z . Applying Step 8 and (34), we have

lim sup

n

f (x) x, xn x = f (x) x, z x 0.

Step 10, The sequences {xn} converges strongly to x*, which is obtained in Steep 9. Proof of Step 10. We have

||xn+1 x||2

= ||n(f (TnWnyn) x) + n(xn x) + n(TnWnyn x)||2

||n(xn x) + n(TnWnyn x)||2 + 2n f (TnWnyn) x, xn+1 x

[n||xn x|| + n||TnWnyn x||]2

+2n f (TnWnyn) f (x), xn+1 x + 2n f (x) x, xn+1 x

[n||xn x|| + n||yn x||]2

+2n||yn x|| ||xn+1 x|| + 2n f (x) x, xn+1 x

[n||xn x|| + n||xn x||]2

+2n||xn x|| ||xn+1 x|| + 2n f (x) x, xn+1 x = (1 n)2||xn x||2 + n[||xn x||2 + ||xn+1 x||2]

+2n f (x) x, xn+1 x

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Page 20 of 22

Which implies that

||xn+1 x||2

(1 n)2 + n 1 n ||

xn x||2 +

2n1 n

f (x) x, xn+1 x

(35)

= 1 n2 2 1 n

||xn x||2 + nn,

where

n = n

1 n ||

xn x||2 +

21 n

f (x) x, xn+1 x ,

By Step 9, and condition (B1), we get lim supn n 0. Now applying Lemma 2.8 to

(35), we conclude that xn x*. Consequently, from ||JFkrk,nxn x|| ||xn x||, we have JFkrk,nxn x, for all k {1, 2, ..., M}.

Corollary 3.2. (see Yao et al. [8]) Let C be a nonempty closed convex subset of a real Hilbert space H, F a bi-functions from CC into which satisfy (A1) - (A4) and

{Ti}i=1 an infinite family of nonexpansive mapping of C into C such that

i=1Fix(Ti) EP(F) = . Let {an}, {bn} and {gn} are three sequences in (0, 1) such that an + bn + gn = 1 and {rn} (0, ). Suppose the following conditions are satisfied:

(B1) limn an = 0 and

n=1 n = ,(B2) 0 <lim infn bn lim supn bn <1,(B3) lim infn rn >0 and limn (rn+1 - rn) = 0.

Let f be a contraction of C into itself with coefficient a (0, 1) and given x1 C arbitrarily. Then the sequence {xn} generated by

xn+1 = nf (xn) + nxn + nWnJFrnxn, n 1.

converge strongly to x i=1Fix(Ti) EP(F), where x P

i=1Fix(Ti)EP(F)f (x) .

Proof. Take A = B = 0, = {I}, F1 = F and Fk = 0 for k {2, ..., M} in Theorem 3.1,

then we have Tn = I and yn = zn = Jkrmxn . So from Theorem 3.1 the sequence {xn} con

verges strongly to x i=1Fix(Ti) EP(F), where x P

i=1Fix(Ti)EP(F)f (x) .

Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H,

J = {Fk : k = 1, 2, . . . , M}be a finite family of bi-functions from C C into which satisfy (A1)-(A4), T a nonexpansive mappings on C such that Fix(T) EP(J ) = . Let

{an}, {bn} and {gn} are three sequences in (0, 1) such that an + bn + gn = 1 and

{rk,n}Mk=1 be sequences in (0, ). Suppose the following conditions are satisfied:

(B1) limn an = 0 and

n=1 n = ,(B2) 0 <lim infn bn lim supn bn <1,(B3) lim infn rk,n >0 and limn (rk,n+1 - rk,n) = 0 for k {1, 2, ..., M}.

Let f be a contraction of H into itself and given x1 H arbitrarily. If the sequences {xn} generated iteratively by

xn+1 = nf (xn) + nxn + n 1

n

n 1 n

kTkJFMrM,n . . . JF2r2n, JF1r1n, xn, n 1.

Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99

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Then, sequences {xn} and {JFkrk,nxn}Mk=1converge strongly to x Fix(T) EP(J ), where x = PFix(T)EP(J)f (x).

Proof. Let S = {0, 1, ...}, = {Ti : i S} and T0 = I. For f = (x0, x1, ...) B(S), define

n(f ) = 1

n

kxk, n

N.

Then {n} is a regular sequence of means on B(S) such that limn n+ - n = 0; for more details, see [34]. Next for each x H and n N, we have

Tnx = 1 n

n 1 n

n 1 n

k Tkx.

Take A = B = 0, Ti = I for all i N in Theorem 3.1 then we have yn = zn and Wn = I for all n N. Therefore, it follows from Theorem 3.1 that the sequences {xn} and

{JFkrk,nxn}Mk=1 converge strongly, as n to a point x Fix(T) EP(J ), where x = PFix(T)EP(J)f (x).

Remark 3.4. Theorem 3.1 improve [8, Theorem 1.2] in the following aspects.(a) Our iterative process (11) is more general than Yao et al. process (14) because it can be applied to solving the problem of finding a common element of the set of solutions of systems of equilibrium problems and systems of variational inequalities.(b) Our iterative process (11) is very diffident from Yao et al. process (14) because there are left amenable semigroup of nonexpansive mappings.(c) Our method of proof is very different from the on in Yao et al. [8] for example we use Corollary 1.1 and Theorem 1.2 of Bruck [33] fore the proof of Theorem 3.1.

AcknowledgementsThe authors are extremely grateful to the referees for useful suggestions that improved the contents of the paper.

Competing interestsThe authors declare that they have no competing interests.

Received: 31 December 2011 Accepted: 16 June 2012 Published: 16 June 2012

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doi:10.1186/1687-1812-2012-99Cite this article as: Piri: Approximating fixed points of amenable semigroup and infinite family of nonexpansive mappings and solving systems of variational inequalities and systems of equilibrium problems. Fixed Point Theory and Applications 2012 2012:99.

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