Iterative approximation of common element of solution sets of various nonlinear operator problems

R E S E A R C H

Open Access

Iterative approximation of common element

of solution sets of various nonlinear operator

problems

Atid Kangtunyakarn

*

*_{Correspondence:} [email protected] Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Abstract

In this paper, we prove strong convergence theorem for finding a common element of the set of fixed point of a finite family of nonexpansive mappings and a finite family of

κ

i-strictly pseudocontractive mappings and the set of a finite family of the set of

solution of equilibrium problems by using the new mapping generated by a finite family of nonexpansive mappings and a finite family of

κ

i-strictly pseudocontractive

mappings and a sequences of positive real numbers. Furthermore, by using our main result, we obtain two interesting theorems involving variational inequality problems and variational inclusion problems. In the last section, we give numerical examples to support our main results.

Keywords: nonexpansive mapping; strictly pseudocontractive mapping; equilibrium problem; variational inequality problem; variational inclusion problem

1 Introduction

LetH be a real Hilbert space andC be a nonempty closed convex subset ofH. A self mappingf :C→C is acontractiononC if there exists a constantk∈[, ) such that

f(x) –f(y) ≤kx–y,∀x,y∈C. LetT:C→Cbe a mapping, a pointx∈Cis called a fixed point ofT if and only ifTx=x. In this paper, we useF(T) to denote the set of fixed point ofT. Recall the following definitions.

Definition . A mappingT:C→Cis called nonexpansive if and only if for allx,y∈C,

Tx–Ty ≤ x–y.

Definition . A mappingT:C→Cis calledκ-strictly pseudocontractive [] if and only if there exists a constantκ∈[, ) such that for allx,y∈C,

Tx–Ty≤ x–y+κ(I–T)x– (I–T)y. (.)

For such case,Tis also said to be aκ-strictly pseudo contraction.

Note that the class of κ-strict pseudo-contractions strictly includes the class of non-expansive mappings, that is T is nonexpansive if and only if T is -strict pseudo-contractive.

LetF:C×C→Rbe a bifunction. The equilibrium problem forFis to determine its equilibrium points,i.e., the set

EP(F) =x∈C:F(x,y)≥,∀y∈C. (.)

Given T :C→H, letF(x,y) =Tx,y–x for allx,y∈C. Thenz∈EP(F) if and only if

Tz,y–z ≥ for ally∈C, that is,zis a solution of the variational inequality.

Equilibrium problems, which were introduced in [] in , have had a great impact and influence in the development of several branches of pure and applied sciences. Numerous problems in physics, minimization problems, Nash equilibria in noncooperative games, optimization and economics reduce to find a solution ofEP(F) (see, for example, [–]). Some methods have been proposed to solve the equilibrium problem (see, for example, [–]).

In , Takahashi and Takahashi [] proved the following theorem.

Theorem . Let C be a nonempty closed convex subset of H.Let F be a bifunction from C×C toRsatisfying

(A) F(x,x) = ,∀x∈C;

(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤,∀x,y∈C; (A) ∀x,y,z∈C,

lim

t→+F

tz+ ( –t)x,y≤F(x,y);

(A) ∀x∈C,y→F(x,y)is convex and lower semicontinuous;

and let S be a nonexpansive mapping of C into H such that F(S)∩EP(G)=∅.Let f be a contraction of H into itself,and let{xn}and{un}be sequences generated by x∈H and

F(un,y) +

 rn

y–un,un–xn ≥, ∀y∈C,

xn+=αnf(xn) + ( –αn)Sun

for all n∈N,where{αn} ⊂[, ]and{rn} ⊂(, )satisfy(C)-(C)as follows:

(C) αn→; (C) ∞_n=αn=∞;

(C) either∞_n=|αn+–αn|<∞orlimn→∞ααn+n = , andlim inf_n→∞rn> and

∞

n=|rn+–rn|<∞.

Then{xn}and{un}converge strongly to z∈F(S)∩EP(F),where z=PF(S)∩EP(F)f(z).

In , Kangtunyakarn and Suantai [] proved the strong convergence theorem by us-ing theS-mapping generated by a finite family of strictly pseudocontractive mappings and a finite family of real number as follows.

Theorem . Let H be a Hilbert space,let f be anα-contraction on H,and let A be a strongly positive linear bounded self-adjoint operator with coefficientγ > .Assume that  <γ < γ_α.Let{Ti}N_i= be a finite family ofκi-strictly pseudo contraction of H into itself

for someκi∈[, )andκ =max{κi:i= , , . . . ,N} with

N

i=F(Ti)=∅.Let Sn be the

S-mappings generated by T,T, . . . ,TN andα(n),α (n)  , . . . ,α

(n)

N ,whereα

(n)

j = (α n,j

 ,α

n,j

 ,α

n,j

I×I×I,I= [, ],α_n,j+αn_,j+αn_,j= andκ<a≤α_n,j,α_n,j≤b< for all j= , , . . . ,N– ,

κ<c≤αn_,N ≤,κ≤α_n,N ≤d< ,κ≤α_n,j≤e< for all j= , , . . . ,N.For a point u∈H and x∈H,let{xn}and{yn}be the sequences defined iteratively by

⎧ ⎨ ⎩

yn=βnxn+ ( –βn)Snxn,

xn+=αnγ(anu+ ( –an)f(xn)) + (I–αnA)yn, n≥,

(.)

where{βn},{αn}and{an}are sequences in[, ].Assume that the following conditions hold:

(i) limn→∞αn= ,∞n=αn=∞andlimn→∞an= ;

(ii) ∞_n=|α_n+,j–αn_,j|<∞,_n∞_=|α_n+,j–α_n,j|<∞for allj∈ {, , , . . . ,N}and

∞

n=|αn+–αn|<∞,

∞

n=|βn+–βn|<∞and

∞

n=|an+–an|<∞;

(iii) ≤κ≤βn<θ< for alln≥for someθ∈(, ).

Then both{xn}and{yn}strongly converge to q∈N_i=F(Ti),which solves the following

vari-ational inequality

γf(q) –Aq,p–q≤, ∀p∈

N

i= F(Ti).

Question Can we prove a strong convergence theorem for finding a common solution of the set of fixed point of a finite family of nonexpansive mappings and a finite family of strictly pseudocontractive mappings and a finite family of the set of solution of equilibrium problems?

LetC be a nonempty closed convex subset of Hilbert spaceH. Let{Ti}N_i= be a finite family of κi-strict pseudo-contractions ofC into itself, and let{Si}Ni= be a finite family of nonexpansive mappings ofC into itself. For eachn∈Nandj= , , . . . ,N, letα_j(n)= (αn_,j,α_n,j,α_n,j)∈I×I×I, whereI= [, ],αn_,j+α_n,j+αn_,j= . We define the mapping SA_n:C→Cas follows:

Un,=I,

Un,=S

α_n,TUn,+αn,Un,+αn,I

,

Un,=S

α_n,TUn,+αn,Un,+αn,I

,

Un,=S

α_n,TUn,+αn,Un,+αn,I

,

.. .

Un,N–=SN–

α_n,N–TN–Un,N–+αn,N–Un,N–+αn,N–I

,

SA_n=Un,N=SN

α_n,NTNUn,N–+αn,NUn,N–+αn,NI

. (.)

In Lemma ., under suitable conditions of the real sequences{α_n,j},{α_n,j}and{αn_,j}for everyj= , , . . . ,N, we show thatF(SA

n) =

N

i=F(Si)∩

N

i=F(Ti) andSAnis a nonexpansive

mapping.

mappings and a finite family of the set of solution of equilibrium problems by using the mapping defined by (.). Furthermore, in the last section, we prove two interesting theo-rems involving a finite family of the set of solutions of variational inequality problem and variational inclusion problem. In the last section, we give numerical examples to support our main results.

2 Preliminaries

In this section, we need the following lemmas to prove our main result. LetCbe a closed convex subset of a real Hilbert spaceH, letPCbe the metric projection ofHontoC,i.e.,

forx∈H,PCxsatisfies the property

x–PCx=min y∈Cx–y.

The following characterizes the projectionPC.

Lemma .(See []) Given x∈H and y∈C.Then PCx=y if and only if the following

inequality holds

x–y,y–z ≥, ∀z∈C.

Lemma .(See []) Let{sn}be a sequence of nonnegative real numbers satisfying

sn+= ( –αn)sn+αnβn, ∀n≥,

where{αn},{βn}satisfy the conditions

() {αn} ⊂[, ],

∞

n=

αn=∞;

() lim sup

n→∞

βn≤ or

∞

n=

|αnβn|<∞.

Thenlimn→∞sn= .

Lemma .(See []) Let{sn}be a sequence of nonnegative real numbers satisfying

sn+= ( –αn)sn+δn, ∀n≥,

where{αn}is a sequence in(, )and{δn}is a sequence such that

()

∞

n=

αn=∞;

() lim sup

n→∞

δn αn≤

 or

∞

n=

|δn|<∞.

Lemma .(See []) Let C be a nonempty closed convex subset of a real Hilbert space H, and let S:C→C be a self-mapping of C.If S is aκ-strict pseudo-contraction mapping, then S satisfies the Lipschitz condition

Sx–Sy ≤  +κ

 –κx–y, ∀x,y∈C.

For solving the equilibrium problem for a bifunctionF:C×C→R, let us assume that Fsatisfies the following conditions:

(A) F(x,x) = ,∀x∈C;

(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤,∀x,y∈C; (A) ∀x,y,z∈C,

lim

t→+F

tz+ ( –t)x,y≤F(x,y);

(A) ∀x∈C,y→F(x,y)is convex and lower semicontinuous.

The following lemma appears implicitly in [].

Lemma .(See []) Let C be a nonempty closed convex subset of H,and let F be a bi-function of C×C intoRsatisfying(A)-(A).Let r> and x∈H.Then there exists z∈C such that

F(z,y) +

ry–z,z–x (.)

for all y∈C.

Lemma .(See []) Assume that F:C×C→Rsatisfies(A)-(A).For r> and x∈H, define a mapping Tr:H→C as follows:

Tr(x) =

z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

for all z∈H.Then the following hold:

() Tris single-valued;

() Tris firmly nonexpansive,i.e., Tr(x) –Tr(y)



≤Tr(x) –Tr(y),x–y

, ∀x,y∈H;

() F(Tr) =EP(F);

() EP(F)is closed and convex.

Lemma .(See []) Let E be a uniformly convex Banach space,C be a nonempty closed convex subset of E,and S:C→C be a nonexpansive mapping.Then I–S is demi-closed at zero.

Definition . LetCbe a nonempty convex subset of real Hilbert space. Let{Ti}N i=be a finite family ofκi-strict pseudo-contractions ofCinto itself, and let{Si}Ni=be a finite family of nonexpansive mappings ofC into itself. For eachj= , , . . . ,N, letαj= (αj,α

j

,α

j

I×I×I, whereI∈[, ] andαj_+αj_+αj_= . We define the mappingSA_:C→Cas

follows:

U=I,

U=S

α_TU+αU+αI

,

U=S

α_TU+αU+αI

,

U=S

α_TU+αU+αI

,

.. .

UN–=SN–

α_N–TN–UN–+αN–UN–+αN–I

,

SA=UN=SN

α_NTNUN–+αNUN–+αNI

.

This mapping is called the SA_{-mapping generated byS}

,S, . . . ,SN,T,T, . . . ,TN and α,α, . . . ,αN.

Lemma . Let C be a nonempty closed convex subset of a real Hilbert space.Let{Ti}N i= be a finite family ofκi-strict pseudo-contractions of C into itself,and let{Si}Ni=be a finite family of nonexpansive mappings of C into itself withN_i=F(Si)∩

N

i=F(Ti)=∅andκ=

max{κi:i= , , . . . ,N},and letαj= (αj,α

j

,α

j

)∈I×I×I,j= , , , . . . ,N,where I= [, ], αj_+α_j+αj_= ,α_j,α_j∈(κ, )for all j= , , . . . ,N– andα_N∈(κ, ],α_N∈[κ, ),α_j∈(κ, ) for all j= , , . . . ,N.Let SA_{be the S}A_{-mapping generated by S}

,S, . . . ,SN,T,T, . . . ,TNand α,α, . . . ,αN.Then F(SA) =

N

i=F(Si)∩ N

i=F(Ti),and SAis a nonexpansive mapping.

Proof It is easy to see that N_i=F(Si)∩Ni=F(Ti)⊆ F(SA). Let x∈F(SA) and x∗ ∈

N

i=F(Si)∩

N

i=F(Ti). Then we have _SA_x

–x∗ 

=SN

α_NTNUN–x+αNUN–x+αNx

–x∗

≤αN_TNUN–x+αNUN–x+αNx–x∗ 

=αN_TNUN–x–x∗

+α_NUN–x–x∗

+α_Nx–x∗

=α_NTNUN–x–x∗ 

+α_NUN–x–x∗ 

+α_Nx–x∗ 

–α_NαN_TNUN–x–UN–x–αNαNTNUN–x–x

–α_NαN_UN–x–x

≤α_NUN–x–x∗+κ(I–TN)UN–x– (I–TN)x∗

+α_NUN–x–x∗ 

+α_Nx–x∗ 

–αN_αN_TNUN–x–UN–x

–α_NαN_TNUN–x–x–αNαNUN–x–x

= –αN_UN–x–x∗+αN

κ–α_N(I–TN)UN–x

+ – –αN_x–x∗–αN αNTNUN–x–x

–α_NαN_UN–x–x

≤ –αN_UN–x–x∗ 

+ – –αN_x–x∗ 

–αN_α_NUN–x–x

≤ N

j=N–

 –α_jUN–x–x∗ 

+ –α_Nα_N–κ–α_N–(I–TN–)UN–x 

– –α_Nα_N–α_N–UN–x–x

+

 –

N

j=N–

 –αj_

x–x∗ 

≤ N

j=N–

 –α_jUN–x–x∗  +  – N

j=N–

 –αj_

x–x∗ 

≤ N

j=N–

 –α_jUN–x–x∗

+

N

j=N–

 –αj_α_N–κ–α_N–(I–TN–)UN–x 

–

N

j=N–

 –α_jα_N–α_N–UN–x–x

+

 –

N

j=N–

 –αj_

x–x∗

≤ N

j=N–

 –α_jUN–x–x∗  +  – N

j=N–

 –α_j

x–x∗  .. . ≤ N j=

 –α_jUx–x∗

+

N

j=

 –αj_α_κ–α_(I–T)Ux  – N j=

 –αj_α_α_Ux–x

+  – N j=

 –α_j

x–x∗ (.)

≤ N

j=

 –α_jUx–x∗+

 – N j=

 –αj_

x–x∗

≤ N

j=

 –α_jUx–x∗  + N j=

 –αj_α_κ–α_(I–T)Ux  – N j=

+  – N j=

 –α_j

x–x∗  (.) ≤ N j=

 –α_jUx–x∗  +  – N j=

 –αj_

x–x∗  ≤ N j=

 –α_jUx–x∗  + N j=

 –αj_α_κ–α_(I–T)Ux  – N j=

 –αj_α_α_Ux–x+

 – N j=

 –αj_

x–x∗ 

≤x–x∗+

N

j=

 –α_jα_κ–α_(I–T)x. (.)

By (.), we have

N

j=

 –αj_α_α_–κ(I–T)x 

≤x–x∗ 

–x–x∗ 

= ,

which implies thatTx=x, that is,x∈F(T). It implies that

Ux=S

α_TUx+αUx+αx

=Sx. (.)

By (.) and (.), we have

_SA_x

–x∗  ≤ N j=

 –α_jSx–x∗  + N j=

 –αj_α_κ–α_(I–T)Ux  – N j=

 –αj_α_α_Ux–x

+  – N j=

 –α_j

x–x∗

≤x–x∗–

N

j=

 –αj_α_α_Ux–x. (.)

By (.), we have

N

j=

It implies that

x=Ux. (.)

By (.) and (.), we havex∈F(S). Hence, we have

x∈F(S)∩F(T). (.)

Sincex=Uxand (.), we have

SAx–x∗ 

≤ N

j=

 –α_jUx–x∗ 

+

N

j=

 –αj_α_κ–α_(I–T)Ux 

–

N

j=

 –αj_α_α_Ux–x

+

 –

N

j=

 –α_j

x–x∗ 

=

N

j=

 –α_jx–x∗ 

+

N

j=

 –αj_α_κ–α_(I–T)x

+

 –

N

j=

 –α_j

x–x∗.

It follows that

N

j=

 –αj_α_α_–κ(I–T)x 

≤,

which implies thatx=Tx, that is,x∈F(T). Sincex=Ux=Tx, we have

Ux=S

α_TUx+αUx+αx

=Sx. (.)

By (.), we have

SAx–x∗ 

≤ N

j=

 –α_jUx–x∗ 

+

N

j=

–

N

j=

 –αj_α_α_Ux–x

+

 –

N

j=

 –α_j

x–x∗ 

≤ N

j=

 –α_jSx–x∗ 

–

N

j=

 –αj_α_α_Ux–x

+

 –

N

j=

 –α_j

x–x∗ 

≤x–x∗–

N

j=

 –αj_α_α_Ux–x.

It follows that

N

j=

 –αj_α_α_Ux–x≤.

It implies that

x=Ux. (.)

By (.) and (.), we havex∈F(S). Hence, we have

x∈F(S)∩F(T). (.)

By continuing in this way, we can show thatx∈F(Si)∩F(Ti) andx=Uixfor alli= , , . . . ,N– . Finally, we shall show thatx∈F(SN)∩F(TN). Since

SAx–x∗≤

 –αN_UN–x–x∗

+α_Nκ–α_N(I–TN)UN–x 

+ – –αN_x–x∗ 

= –αN_SN–x–x∗ 

+αN_ κ–α_N(I–TN)x 

+ – –αN_x–x∗ 

≤x–x∗ 

+α_Nκ–α_N(I–TN)x 

.

It implies that

αN_αN_ –κ(I–TN)x 

which implies thatx=TNx, that is,x∈F(TN). It implies that

x=SAx=SN

αN_TNUN–x+αNUN–x+αNx

=SNx. (.)

Then we havex∈F(SN)∩F(TN). HenceF(SA)⊆ N

i=F(Si)∩ N

i=F(Ti).

Applying (.), we have that the mappingSAis a nonexpansive.

Lemma . Let C be a nonempty closed convex subset of a real Hilbert space.Let{Ti}N i= be a finite family ofκi-strict pseudo-contractions of C into itself,and let{Si}Ni=be a finite family of nonexpansive mappings of C into itself withκ=max{κi:i= , , . . . ,N},and let α(_jn)= (α_n,j,α_n,j,αn_,j),αj= (αj,α

j

,α

j

)∈I×I×I,where I= [, ],α

n,j

 +α

n,j

 +α

n,j

 = andα

j

+ αj_+α_j= such thatα_in,j→αj_i∈[, ]as n→ ∞for i= , and j= , , , . . . ,N.Moreover, for every n∈N,let SA_{and S}A

n be the SA-mapping generated by S,S, . . . ,SN,T,T, . . . ,TN

andα,α, . . . ,αNand S,S, . . . ,SN,T,T, . . . ,TN andα(n),α (n)  , . . . ,α

(n)

N ,respectively.Then

lim_n→∞SA

nxn–SAxn= for every bounded sequence{xn}in C.

Proof Let{xn} be a bounded sequence inC,Uk andUn,k be generated byS,S, . . . ,SN,

T,T, . . . ,TN andα,α, . . . ,αN andS,S, . . . ,SN,T,T, . . . ,TN andα(n),α (n)  , . . . ,α

(n)

N ,

re-spectively. For eachn∈N, we have

Un,xn–Uxn=S

αn_,Txn+

 –αn_,xn

–S

α_Txn+

 –α_xn

≤αn_,Txn+

 –α_n,xn–αTxn–

 –α_xn

=α_n,–α_Txn–xn, (.)

and fork∈ {, , . . . ,N}, by using Lemma ., we obtain

Un,kxn–Ukxn=Sk

α_n,kTkUn,k–xn+αn,kUn,k–xn+αn,kxn

–Sk

α_kTkUk–xn+αkUk–xn+αkxn

≤α_n,kTkUn,k–xn+αn,kUn,k–xn+αn,kxn

–α_kTkUk–xn–αkUk–xn–αkxn

=α_n,k(TkUn,k–xn–TkUk–xn) +

α_n,k–α_kTkUk–xn

+αn_,k–α_kxn+αn,k(Un,k–xn–Uk–xn)

+αn_,k–α_kUk–xn

≤αn_,kTkUn,k–xn–TkUk–xn+αn,k–αkTkUk–xn

+αn_,k–α_kxn+α_n,kUn,k–xn–Uk–xn

+αn_,k–α_kUk–xn

=αn_,kTkUn,k–xn–TkUk–xn+αn,k–αkTkUk–xn

+αn_,kUn,k–xn–Uk–xn+ –αn,k–αn,k– 

+αk_+α_kUk–xn+αn,k–αkxn

≤αn_,k +κ

+αn_,k–α_kTkUk–xn+αn,kUn,k–xn–Uk–xn

+αk_–α_n,k+αn_,k–α_kUk–xn+αn,k–αkxn

≤ +κ

 –κUn,k–xn–Uk–xn+α n,k

 –αkTkUk–xn

+ –κ

 –κUn,k–xn–Uk–xn+α k

 –α

n,k



+αn_,k–α_kUk–xn+αn,k–αkxn

≤ 

 –κUn,k–xn–Uk–xn+α n,k

 –αkTkUk–xn+Uk–xn

+αn_,k–α_kUk–xn+xn

. (.)

By (.) and (.), we have

S_nAxn–SAxn=Un,Nxn–UNxn

≤ 

 –κUn,N–xn–UN–xn+α n,N

 –αNTNUN–xn

+UN–xn

+α_n,N–α_NUN–xn+xn

≤ 

 –κ



 –κUn,N–xn–UN–xn

+α_n,N––αN_–TN–UN–xn+UN–xn

+α_n,N––α_N–UN–xn+xn

+α_n,N–α_NTNUN–xn+UN–xn

+α_n,N–α_NUN–xn+xn

=

  –κ



Un,N–xn–UN–xn

+

N

j=N–

  –κ

N–j

αn_,j–αj_TjUj–xn+Uj–xn

+

N

j=N–

  –κ

N–j

αn_,j–αj_Uj–xn+xn

≤ · · ·

≤

  –κ

N–

Un,xn–Uxn

+

N

j=

  –κ

N–j

αn_,j–αj_TjUj–xn+Uj–xn

+

N

j=

  –κ

N–j

αn_,j–αj_Uj–xn+xn

=

  –κ

N–

+

N

j=

  –κ

N–j

αn_,j–αj_TjUj–xn+Uj–xn

+

N

j=

  –κ

N–j

αn_,j–αj_Uj–xn+xn

. (.)

This together with the assumption α_in,j→α_ijasn→ ∞(i= , ,j= , , . . . ,N), we can conclude that

lim

n→∞S A

nxn–SAxn= .

Lemma . Let C be a nonempty closed convex subset of a real Hilbert space.Let{Ti}N_i= be a finite family ofκi-strict pseudo-contractions of C into itself,and let{Si}Ni=be a finite family of nonexpansive mappings of C into itself withκ=max{κi:i= , , . . . ,N},and let α(_jn)= (α_n,j,α_n,j,αn_,j),αj= (αj,α

j

,α

j

)∈I×I×I,where I= [, ],α

n,j

 +α

n,j

 +α

n,j

 = andα

j

+ αj_+α_j= such that∞_n=|αn_+,j–α_n,j|<∞,∞_n=|α_n+,j–α_n,j|<∞for all j∈ {, , , . . . ,N}. For every n∈N,let SA_n be the SA-mapping generated by S,S, . . . ,SN,T,T, . . . ,TN and α_(n),α_(n), . . . ,α_N(n).Then∞_n=SA

n+zn–SnAzn<∞for every bounded sequence{zn}in C.

Proof Let{zn}be a bounded sequence inC. For eachn∈Nand the definition ofSA_{, we}

have

Un+,zn–Un,zn =S

α_n+,Tzn+

 –αn_+,zn

–S

αn_,Tzn+

 –αn_,zn

≤α_n+,Tzn+

 –αn_+,zn–αn,Tzn–

 –α_n,zn

=α_n+,–α_n,Tzn–zn. (.)

Fork∈ {, , . . . ,N}, and using the same method as (.) in Lemma ., we have

Un+,kzn–Un,kzn ≤



 –κUn+,k–zn–Un,k–zn+α n+,k

 –αn,kTkUn,k–zn

+Un,k–zn

+α_n+,k–αn_,kUn,k–zn+zn

. (.)

From (.), (.), and using the same method as (.) in Lemma ., we have

S_nA_+zn–SAnzn≤

  –κ

N–

α_n+,–α_n,Tzn–zn

+

N

j=

  –κ

N–j

α_n+,j–α_n,jTjUn,j–zn+Un,j–zn

+

N

j=

  –κ

N–j

α_n+,j–α_n,jUn,j–zn+zn

.

It implies that

∞

n=

3 Main result

Theorem . Let C be a nonempty closed convex subset of Hilbert spaces H,and let f be anα-contraction on H.Let Fibe a bifunction from C×C intoR,for every i= , , . . . ,N

satisfying(A)-(A).Let{Ti}N_i=be a finite family ofκi-strict pseudo-contractions of C into

itself,and let{Si}N

i=be a finite family of nonexpansive mappings of C into itself withF≡

N

i=F(Si)∩ N

i=F(Ti)∩ N

i=EP(Fi)=∅andκ =max{κi:i= , , . . . ,N},and letαj(n)=

(αn_,j,α_n,j,α_n,j)∈I×I×I,j= , , , . . . ,N,where I= [, ],α_n,j+α_n,j+α_n,j= ,α_n,j,αn_,j,αn_,j∈

[a,b]⊂(κ, ) for all j= , , . . . ,N.Let SA

n be the SA-mapping generated by S,S, . . . ,SN,

T,T, . . . ,TNandα(n),α (n)  , . . . ,α

(n)

N .Let{xn}and{zn}be the sequences generated by x∈C and

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Fi(uin,y) +_ri ny–u

i

n,uin–xn ≥, ∀y∈C and i= , , . . . ,N,

zn= N

i=δniuin,

xn+=αnf(zn) + ( –αn)SAnzn, ∀n≥,

(.)

where{αn}is a sequence in[, ].Assume that the following conditions hold:

(i) limn→∞αn= ,

∞

n=αn=∞;

(ii) ∞_n=|α_n+,j–αn_,j|<∞,∞_n=|α_n+,j–α_n,j|<∞,for allj∈ {, , , . . . ,N}and

∞

n=|αn+–αn|<∞;

(iii) N_i=δi n= ,

∞

n=|δni+–δin|<∞andlimn→∞δin=δi∈(κ, ),for everyi= , , . . . ,N; (iv) κ<θ≤ri

n≤η,for everyi= , , . . . ,Nand

_∞

n=|rin+–rni|<∞.

Then the sequence{xn}converges strongly to x∗=PFf(x∗).

Proof Letp∈F, we havep∈N_i=EP(Fi) from Lemma ., we obtainp∈

N

i=F(Trin). Since

Fi

ui_n,y+  ri n

y–ui_n,ui_n–xn

≥, ∀y∈Candi= , , . . . ,N. (.)

Again from Lemma ., we haveui

n=Trinxnfor everyi= , , . . . ,N. By definition ofxn, we have

xn+–p ≤αnf(zn) –p+ ( –αn)SAnzn–p

≤αnf(zn) –f(p)+αnf(p) –p+ ( –αn)SAnzn–p ≤αnαzn–p+αnf(p) –p+ ( –αn)zn–p

=αnf(p) –p+

 –αn( –α)

zn–p

=αnf(p) –p+

 –αn( –α)

N

i=

δ_niui_n–p

≤αnf(p) –p+

 –αn( –α)

N

i=

δi_nui_n–p

≤αnf(p) –p+

 –αn( –α)

xn–p. (.)

Put K=max{x–p,f(_–p)–αp}. By (.), we can show by induction thatxn–p ≤K,

∀n∈N. This implies that{xn}is bounded, and so are{ui

Next, we will show that

lim

n→∞xn+–xn= . (.)

By nonexpansiveness ofxn, we have

xn+–xn=αnf(zn) + ( –αn)SAnzn–αn–f(zn–) – ( –αn–)SAn–zn–

=αn

f(zn) –f(zn–)

+ (αn–αn–)f(zn–) + ( –αn)

SA_nzn–SAn–zn–

+ (αn––αn)SAn–zn–

≤αnf(zn) –f(zn–)+|αn–αn–|f(zn–)+ ( –αn)SnAzn–SAn–zn–

+|αn––αn|SnA–zn–

≤αnαzn–zn–+|αn–αn–|f(zn–)

+ ( –αn)SnAzn–SAnzn–+SAnzn––SAn–zn–

+|αn––αn|SnA–zn–

≤ –αn( –α)

zn–zn–+|αn–αn–|f(zn–)

+ ( –αn)SnAzn––SnA–zn–+|αn––αn|SnA–zn–

= –αn( –α)

N

i= δ_niui_n–

N

i=

δ_ni_–ui_n–

+|αn–αn–|f(zn–)

+ ( –αn)SnAzn––SnA–zn–+|αn––αn|SnA–zn–

= –αn( –α)

N

i=

δ_niui_n–ui_n–+

N

i=

δ_ni–δ_ni_–ui_n–

+|αn–αn–|f(zn–)+ ( –αn)SAnzn––SnA–zn–

+|αn––αn|SnA–zn–

≤ –αn( –α)

N

i=

δi_nui_n–ui_n–+

N

i=

δi_n–δ_ni_–ui_n–

+|αn–αn–|f(zn–)+ ( –αn)SAnzn––SnA–zn–

+|αn––αn|SnA–zn–. (.)

From Lemma ., we have

∞

n=

SA_n+zn–SnAzn<∞. (.)

Sinceui_n=T_ri

nxnfor everyi= , , . . . ,N. By definition ofTrin, we have

F(T_ri nxn,y) +

 ri n

y–T_ri

nxn,Trinxn–xn ≥, ∀y∈C, (.)

similarly,

F(T_ri

n+xn+,y) +

 ri

n+

y–T_ri

From (.) and (.), we obtain

F(T_ri

nxn,Trin+xn+) +

 ri n

T_ri

n+xn+–Trinxn,Trinxn–xn ≥ (.)

and

F(T_ri

n+xn+,Trinxn) +  ri

n+

T_ri

nxn–Trin+xn+,Trin+xn+–xn+ ≥. (.)

By (.) and (.), we have

 ri n

T_ri

n+xn+–Trnixn,Trinxn–xn + 

r_ni_+Trnixn–Trni+xn+,Trin+xn+–xn+ ≥.

It follows that

T_ri

nxn–Trni+xn+,

T_ri

n+xn+–xn+

rni+

–Trinxn–xn ri

n

≥.

This implies that

≤

T_ri

n+xn+–Trinxn,Trnixn–Trin+xn++Trin+xn+–xn–

ri n

ri_n+(Trin+xn+–xn+)

.

It follows that

T_ri

n+xn+–Trinxn 

≤

T_ri

n+xn+–Trnixn,Trin+xn+–xn–

ri n

ri n+

(T_ri

n+xn+–xn+)

=

T_ri

n+xn+–Trinxn,xn+–xn+

 – r

i n

ri n+

(T_ri

n+xn+–xn+)

≤ T_ri

n+xn+–Trnixn

xn+–xn+

 – r

i n

ri n+

(T_ri

n+xn+–xn+)

≤ T_ri

n+xn+–Trnixn

xn+–xn+

 – r

i n

ri_n+

T_ri

n+xn+–xn+

=T_ri

n+xn+–Trinxn

xn+–xn+  ri_n+r

i

n+–rinTri_n+xn+–xn+

≤ T_ri

n+xn+–Trnixn

xn+–xn+  ar

i

n+–rinTrni+xn+–xn+

.

It follows that

ui_n+–ui_n≤ xn+–xn+  ar

i

n+–rniuin+–xn+ (.)

Substitute (.) into (.), we have

xn+–xn ≤

 –αn( –α)

N

i=

δi_nui_n–ui_n–+

N

i=

δi_n–δ_ni_–ui_n–

+|αn–αn–|f(zn–)+ ( –αn)SAnzn––SnA–zn–

+|αn––αn|SnA–zn–

≤ –αn( –α)

N

i= δi_n

xn+–xn+  ar

i

n+–rniuin+–xn+

+

N

i=

δ_ni–δ_ni_–ui_n–

+|αn–αn–|f(zn–)+ ( –αn)SAnzn––SnA–zn–

+|αn––αn|SnA–zn–

= –αn( –α)

xn+–xn+

N

i= δ_ni

ar

i

n+–rniuin+–xn+

+

N

i=

δ_ni–δ_ni_–ui_n–

+|αn–αn–|f(zn–)+ ( –αn)SAnzn––SnA–zn–

+|αn––αn|SnA–zn–

≤ –αn( –α)

xn+–xn+

N

i= δ_ni

ar

i

n+–rniuin+–xn+

+

N

i=

δ_ni–δ_ni_–ui_n–+|αn–αn–|f(zn–)

+S_nAzn––SnA–zn–+|αn––αn|SnA–zn–. (.)

By (.), (.), conditions (iii), (iv) and Lemma ., we have

lim

n→∞xn+–xn= . (.)

From (.), (.) and condition (iv), we have

lim

n→∞u

i

n+–uin= , ∀i= , , . . . ,N. (.)

Letp∈F. Fromui n=Tri

nxnfor everyi= , , . . . ,N, we have

_ui n–p

 =T_ri

nxn–Trinp 

≤ T_ri

nxn–Trinp,xn–p

=  u

i n–p



+xn–p–uin–xn

It implies that

ui n–p



≤ xn–p–uin–xn



. (.)

By definition of{xn}and (.), we have

xn+–p≤αnf(zn) –p



+ ( –αn)SAnzn–p



≤αnf(zn) –p



+ ( –αn)zn–p

=αnf(zn) –p



+ ( –αn)

N

i=

δi_nui_n–p



≤αnf(zn) –p+ ( –αn) N

i=

δ_niui_n–p

≤αnf(zn) –p



+ ( –αn) N

i=

δ_nixn–p–uin–xn



≤αnf(zn) –p



+xn–p– ( –αn) N

i=

δi_nui_n–xn

 .

It implies that

( –αn) N

i=

δ_niui_n–xn≤αnf(zn) –p+xn–p–xn+–p

≤αnf(zn) –p



+xn–p

+xn+–p

xn+–xn. (.)

From conditions (i), (iii) and (.), we have

lim

n→∞u i

n–xn= , ∀i= , , . . . ,N. (.)

Since

xn+–SAnzn=αn

f(zn) –SAnzn

,

from condition (i), we have

lim

n→∞xn+–S A

nzn= . (.)

From the definition ofzn, we have

zn–xn=

N

i= δ_niui

n–xn

≤ N

i=

From condition (iii) and (.), we have

lim

n→∞zn–xn= . (.)

Since

zn–SAnzn≤ zn–xn+xn–xn++xn+–SAnzn,

by (.), (.) and (.), we have

lim

n→∞zn–S A

nzn= . (.)

Next, we show that

lim sup

n→∞

f(z) –z,xn–z

≤, (.)

wherez=PFf(z). To show this inequality, take a subsequence{xnk}of{xn}such that

lim sup

n→∞

f(z) –z,xn–z

= lim

k→∞

f(z) –z,xnk–z

. (.)

Without loss of generality, we may assume that a subsequence{xnk}of {xn}converges weakly to someq∈H. From (.), we have that{znk}converges weakly toq.

Sinceκ<a≤αn_,j,α_n,j,α_n,j≤b<  for allj= , , . . . ,N. Without loss of generality, we may assume that

αnk,j

 →α

j

∈(κ, ), α

nk,j

 →α

j

∈(κ, ) and α

nk,j

 →α

j

∈(κ, ) ask→ ∞,

∀j= , , . . . ,N.

Let SA _{be the S}A_{-mapping generated by S}

,S, . . . ,SN, T,T, . . . ,TN and β,β, . . . ,βN,

whereβj= (αj,α

j

,α

j

),∀j= , , . . . ,N. By Lemma .,SAis a nonexpansive mapping, and F(SA_{) =}N

i=F(Si)∩

N

i=F(Ti).

By Lemma ., we have

lim

k→∞

S_nA_kznk–S

A_z

nk= . (.)

Since

znk–S

A_z

nk≤znk–S

A

nkznk+S

A nkznk–S

A_z nk,

by (.), (.), we have

lim

k→∞ znk–S

A_z

nk= . (.)

Since{znk}converges weakly toqask→ ∞(.) and Lemma ., we have

q∈FSA=

N

i= F(Si)∩

N

i=

Next, we show thatq∈N_i=EP(Fi). To show this, we may assume that

lim

k→∞r i nk=r

i_{∈[θ,η], ∀i= , , . . . ,N.}

By Lemmas . and ., for everyi= , , . . . ,N, we defineTri:H→Cby

Tri(x) =

z∈C:Fi(z,y) +



riy–z,z–x ≥,∀y∈C

, ∀x∈Handi= , , . . . ,N.

Then we have

Fi(Trix_n,y) + 

riy–Trixn,Trixn–xn ≥, ∀y∈Candi= , , . . . ,N.

From (.) andui

n=Trinxn, we have

Fi(Trinxn,y) +  ri n

y–T_ri

nxn,Trinxn–xn ≥, ∀y∈Candi= , , . . . ,N.

It implies that

Fi(Trixnk,Tr_nki xnk) + 

riTri_nkxnk–Trixnk,Trixnk–xnk ≥, ∀i= , , . . . ,N

and

Fi(T_ri

nkxnk,Trixnk) +  ri nk

T_rixnk–Tri_nkxnk,Tri_nkxnk–xnk ≥, ∀i= , , . . . ,N.

By (A), we have



riTr_nki xnk–Trixnk,Trixnk–xnk +  ri nk

T_rixnk–Tr_nki xnk,Tri_nkxnk–xnk ≥.

It implies that

T_ri

nkxnk–Trixnk,

T_rixnk–xnk

ri –

T_ri

nkxnk–xnk ri

nk

≥.

It follows that

T_ri

nkxnk–Trixnk,Trixnk–xnk– ri

ri nk

(T_ri

nkxnk–xnk)

≥.

Then

≤

T_ri

nkxnk–Trix_nk,T_rix_nk–T_ri

nkxnk+Tr_nki xnk–xnk– ri

ri nk

(T_ri

nkxnk–xnk)

=

T_ri

nkxnk–Trix_nk,T_rix_nk–T_ri nkxnk+

 – r

i

ri nk

(T_ri

nkxnk–xnk)

It follows that

T_ri

nkxnk–Trixnk _≤

T_ri

nkxnk–Trixnk,

 – r

i

ri nk

(T_ri

nkxnk–xnk)

≤ T_ri

nkxnk–Trixnk

 – r

i

ri nk

T_ri

nkxnk–xnk.

It implies that

T_ri

nkxnk–Trixnk ≤  ar

i nk–r

i_T ri

nkxnk–xnk.

Fromlimk→∞rink=r

i_{and (.), we have}

lim

k→∞Trinkxnk–Trixnk= , ∀i= , , . . . ,N. (.)

For everyi= , , . . . ,N, we have

xnk–Trixnk ≤ xnk–Tr_nki xnk+Tri_nkxnk–Trixnk

=xnk–u

i

nk+Tri_nkxnk–Trixnk,

by (.) and (.), we have

lim

k→∞xnk–Trixnk= , ∀i= , , . . . ,N. (.)

Since a subsequence{xnk}of{xn}converges weakly toqask→ ∞, from (.) and Lem-ma ., we have

q∈F(Tri), ∀i= , , . . . ,N.

Then

q∈

N

i=

F(Tri). (.)

From Lemma ., we haveEP(Fi) =F(Tri),∀i= , , . . . ,N. From (.), we have

q∈

N

i=

F(Tri) =

N

i=

EP(Fi). (.)

By (.) and (.), we have

q∈F. (.)

Sincexnk qask→ ∞andq∈Fand (.), we have

lim sup

n→∞

f(z) –z,xn–z

= lim

k→∞

f(z) –z,xnk–z