Weak convergence theorems for an infinite family of nonexpansive mappings and equilibrium problems (tt)

‘Weak Convergence Theorems for an Infinite Family .... ‘Weak Convergence Theorems for an Infinite Family ..... -‘Weak Convergence Theorems for an Infinite Family .... ‘Weak Convergence T

14 P.N Anh, L B Long, N V Quy and L Q Thuy

1 Introduction

Let # be areal [lilbert space with inner product (., -) and norm ||-|| Let

C be a closed convex subset of a real Hilbert space Ho and Pr be the

Frojection of 7 onto C When {x} is a sequence in H, then x” > = (resp x° — ¥} will denole strong (esp weak) convergence of the sequence

{&"} to ¥ asa > Amapping $:C > C is said to be nonexpansive if

||S(x)- #(y x~y|, À.yĂeC

Fad) is denoted by the set of fixed points of 8 Let f:CxC +R bea

‘bifunction such that f(x, x)= 0 forall xc C, We consider the equilibrium problems in the sense of Blum and Oettli [8| which are presented as follows:

Find x" c C suchthat f(x", y} 2 0 forall yc C EP(f,C)

The set of solutions of ZP(/, C) is denoted by Sof, C) The bifunction f

is called strongly monotone on C with B > 0 if

Se v)+ ZŒ, z) < =B|x= ylỄ Vx, ye monotone on Cif

pseuslomanotone om Cif

Lipschite-type continuous on C with constants cy > 0 and cy > 0 if

Fle, y+ fly 2) 2 fle zì—a[|x—+ ?2— |» V8 zCC,

In this paper, we are interested in the problem of Ñnding ä conunon element of the solntion set of the equilibrium problems ZP(f, () and the set

of Bxcd points ne Fe(Sy ) of an infinite fanily of uoncxpansive mappings

‘Weak Convergence Theorems for an Infinite Family 115

x), where 2°; C —> ?⁄ and this problem is reduced (o finding a

Motivated by the viscosity method in [16] and the approximation methad

in [7] via an duuprovernent sel of extragradient methods in [2-4], we introduce anew ileration algorilinu for finding, a common element of problem (1.1) Al cach main iteration m, we only solve two strongly convex prograns with ạ pseudomonotone ‘aud Lipschit:-iype continuous bifunclion, We show thai all

of Uie iterative scquences generated by this algonillun converge weakly to the conuuon clement in areal Hilbert space

This paper is organized us fullows: Scetion 2 recalls some concepts in equilibritan problems and fixed point probleras thai will be used in the sequel

116 Ð.N Anh, L B Long, N V Quy and L Q Thuy

and an iterative algorithm for solving problem (1.1) Section 3 investigates the convergence of the algorithms presented in Section 2 as the main results

of our paper

2 Preliminaries

In 1953, Mann [13] introduced a well-known classical iteration method

lu approximate a ñxed poinl of a nonexpansive mappmg SCC ina teal Hilbert space 7¢ This iteration is defined as

xf# = axŸ —(1—d„)8@Ÿ), VÉ >0,

where C 1s nonemply closed convex subsel of 7 and {q;}© [0, 1] Then

4x") converges weakly to x" c Fax(S)

For finding a conunon fixed point of an infinite family of noncxpansive

mappings {Š„}, Aoyama el a [7] introduced an iterative sequence {x*} of

C defined by +? = C and

xl x 4 day SO"), vez,

where C is anonemply closed convex subsel of a real Lilbert space, {u,.}

Jo, 1] and (2, Fads, }# 2 The anthors proved that the sequence {x*}

et opagtet ys Best 1 yy),

where g:C > ís contractive and Wy, is W-mapping of {8,} Under mild

120 P.N Anh, L B Long, N V Quy and L Q Thuy

Firm || u(x — x7 )4 (1 — ay Spf) — 2") | — Tim | xT

abit g(x — 2°) + (1 ony Meee") — 2°) | — J |

118 P.N, Anh, L.B Long, N V Quy and L Q Thuy

tx}, fy") of 1 such that

im sup] x* |e,

ae Tim sup|| yŸ

kao

Than

Lemma 2.3 [7] Jet C he a nonempty clased convex subset of a Banach

apace Let {S} be a sequence of nonexpansive mappings of C into itself and

Sis @ mapping of C into itself such that

[Shem seule) 8469 [sxe Che,

@ nonempiy closed comes subset C of a real Iiflbert space H if Fie(S}

Assume that 8 ís œ nonexpansive selfmapping of

#, then I S is demiclosed: that is, whenever {x*} is a sequence in C

weakly converging to some ¥ © C and the sequence {(1 - S)(x")} strongly

converging fo some y, il follows that UZ — 8)( 3 Here J is the identity

operator of TÍ

Lemma 2.5 (See [17]) Zet C be a nonempty closed comes subsel of a

real Hilbert space H Suppose thal, for all u € C, the sequence {x"} satisfies

Then the sequence (Pra (x* )} converges strongly to some x © C.

118 P.N, Anh, L.B Long, N V Quy and L Q Thuy

tx}, fy") of 1 such that

im sup] x* |e,

ae Tim sup|| yŸ

kao

Than

Lemma 2.3 [7] Jet C he a nonempty clased convex subset of a Banach

apace Let {S} be a sequence of nonexpansive mappings of C into itself and

Sis @ mapping of C into itself such that

[Shem seule) 8469 [sxe Che,

Assume that 8 ís œ nonexpansive selfmapping of

#, then I S is demiclosed: that is, whenever {x*} is a sequence in C

weakly converging to some ¥ © C and the sequence {(1 - S)(x")} strongly

converging fo some y, il follows that UZ — 8)( 3 Here J is the identity

operator of TÍ

Lemma 2.5 (See [17]) Zet C be a nonempty closed comes subsel of a

real Hilbert space H Suppose thal, for all u € C, the sequence {x"} satisfies

Then the sequence (Pra (x* )} converges strongly to some x © C.

‘Weak Convergence Theorems for an Infinite Family 123

another subsequence of {x”} such that x" — & as & —x co Then

= liming] x"* £|

ben

< liminf| x" - © koe

)-‘Weak Convergence Theorems for an Infinite Family uy

3 Convergence Results Now, we prove (he main convergence theorem,

Theorem 3.1 Suppose that Assumptions Al-A5 are satisfied, x° = C

and bo positive sequences {h,}, {ay} satisfy the folowiny restrictions:

fox} [ed] 0.05

;

fig) <a, Bl for some œ 6 ca +} where E = max{2ey,

Then the sequences {x*}, {v} and {PP} generated by (2.1) and (2.2)

converge weakly to the same patat x” © (2, Fil) SoK S.C), where

HE, Pop meegmnenty e9 }

The proof of this theorem is divided into several steps

In Steps | and 2, we will consider weak clusters of {x*} It follows Lom

Step 1 Claim that x € 1 Fats, }

Proof of Step 1 By Iemma 2.1 and (3.1), we have

(=a) tt - 26g) * - »* Ễ s 4— đj@Œ Daye] xf - x P

effoe Pope oe B

Oa kom,

122 P.N Anh, L B Long, N V Quy and L Q Thuy

where w & 62,f(x", »”) má 7 © Ne(y"), By the definition of the norrual

cone ÄV 2, we imply that

On the other tund, since f(x®, -) is subdifferentiable on C, by the well

known Moreau-Rockafellar theorem, there exists wc Ga f(x" y”) such thar

This means that x £ #øi(7, Œ)

Step 3 Claim that the sequences {x}, {yŸ} and {1°} converge weakly

to Lhe same point x", where

fim Pree mg naili.c *)

Proof of Step 3 It follows from Steps 2 and 3 that for every weak cluster

point x of the sequence {x"} swisfies x € (Y", Fils; )A Sof C) We

show that {x"} converges weakly to x Now, we aseume that {x”*} is an

‘Weak Convergence Theorems for an Infinite Family 121 Let S be a mapping of C into itself defined by

This follows thal

120 P.N Anh, L B Long, N V Quy and L Q Thuy

Firm || u(x — x7 )4 (1 — ay Spf) — 2") | — Tim | xT

abit g(x — 2°) + (1 ony Meee") — 2°) | — J |

‘Weak Convergence Theorems for an Infinite Family 7

aesumplions on parameters, the authors proved thal the sequences {x"} and

{)"} converge strongly to x”, where

In our scheme, the main steps are to solve two strongly convex problems

1

l — mg in| 4z 7P, +)+t>|y-## Poy ec}

|e = axg:min{ AS" vt |e fips cÌ

and compute the next iteration point by Mann-type fixed points,

To investigale Lhe convergence of this scheme, we 1

(echnical Jerumas which val be used in the sequel

call the Dollowing,

Lemma 2.1 (See [I]} Let C be a nonempty closed convex subset of a real Hilbert space H Let [Cx CR be a pseudomonvtone, Lipschitz-

type continuous Vifunction For each xc, let fix.) be convex and

subdifferentiable on C Suppose that the sequences {x}, {y*}, {(È} am

generated by scheme (2.1) and x" = Sol f, C) Then

FP <i ah a" PG ange) xy P

(L 2yedpeY AP whee

Lemma 2.2 [17] Let 1 be a real Hilbert space, {3,} be a sequence of real numbers such that {8,}c |o, B|< (0,1), © > 0 and two sequences

122 P.N Anh, L B Long, N V Quy and L Q Thuy

where w & 62,f(x", »”) má 7 © Ne(y"), By the definition of the norrual

cone ÄV 2, we imply that

On the other tund, since f(x®, -) is subdifferentiable on C, by the well

known Moreau-Rockafellar theorem, there exists wc Ga f(x" y”) such thar

This means that x £ #øi(7, Œ)

Step 3 Claim that the sequences {x}, {yŸ} and {1°} converge weakly

to Lhe same point x", where

fim Pree mg naili.c *)

Proof of Step 3 It follows from Steps 2 and 3 that for every weak cluster

point x of the sequence {x"} swisfies x € (Y", Fils; )A Sof C) We

show that {x"} converges weakly to x Now, we aseume that {x”*} is an

‘Weak Convergence Theorems for an Infinite Family uy

3 Convergence Results Now, we prove (he main convergence theorem,

Theorem 3.1 Suppose that Assumptions Al-A5 are satisfied, x° = C

and bo positive sequences {h,}, {ay} satisfy the folowiny restrictions:

fox} [ed] 0.05

;

fig) <a, Bl for some œ 6 ca +} where E = max{2ey,

Then the sequences {x*}, {v} and {PP} generated by (2.1) and (2.2)

converge weakly to the same patat x” © (2, Fil) SoK S.C), where

HE, Pop meegmnenty e9 }

The proof of this theorem is divided into several steps

In Steps | and 2, we will consider weak clusters of {x*} It follows Lom

Step 1 Claim that x € 1 Fats, }

Proof of Step 1 By Iemma 2.1 and (3.1), we have

(=a) tt - 26g) * - »* Ễ s 4— đj@Œ Daye] xf - x P

effoe Pope oe B

Oa kom,

‘Weak Convergence Theorems for an Infinite Family 123

another subsequence of {x”} such that x" — & as & —x co Then

= liming] x"* £|

ben

< liminf| x" - © koe

)-‘Weak Convergence Theorems for an Infinite Family 123

another subsequence of {x”} such that x" — & as & —x co Then

= liming] x"* £|

ben

< liminf| x" - © koe

)-122 P.N Anh, L B Long, N V Quy and L Q Thuy

where w & 62,f(x", »”) má 7 © Ne(y"), By the definition of the norrual

cone ÄV 2, we imply that

On the other tund, since f(x®, -) is subdifferentiable on C, by the well

known Moreau-Rockafellar theorem, there exists wc Ga f(x" y”) such thar

This means that x £ #øi(7, Œ)

Step 3 Claim that the sequences {x}, {yŸ} and {1°} converge weakly

to Lhe same point x", where

fim Pree mg naili.c *)

Proof of Step 3 It follows from Steps 2 and 3 that for every weak cluster

point x of the sequence {x"} swisfies x € (Y", Fils; )A Sof C) We

show that {x"} converges weakly to x Now, we aseume that {x”*} is an

‘Weak Convergence Theorems for an Infinite Family 7

aesumplions on parameters, the authors proved thal the sequences {x"} and

{)"} converge strongly to x”, where

In our scheme, the main steps are to solve two strongly convex problems

1

l — mg in| 4z 7P, +)+t>|y-## Poy ec}

|e = axg:min{ AS" vt |e fips cÌ

and compute the next iteration point by Mann-type fixed points,

To investigale Lhe convergence of this scheme, we 1

(echnical Jerumas which val be used in the sequel

call the Dollowing,

Lemma 2.1 (See [I]} Let C be a nonempty closed convex subset of a real Hilbert space H Let [Cx CR be a pseudomonvtone, Lipschitz-

type continuous Vifunction For each xc, let fix.) be convex and

subdifferentiable on C Suppose that the sequences {x}, {y*}, {(È} am

generated by scheme (2.1) and x" = Sol f, C) Then

FP <i ah a" PG ange) xy P

(L 2yedpeY AP whee

Lemma 2.2 [17] Let 1 be a real Hilbert space, {3,} be a sequence of real numbers such that {8,}c |o, B|< (0,1), © > 0 and two sequences

‘Weak Convergence Theorems for an Infinite Family 7

aesumplions on parameters, the authors proved thal the sequences {x"} and

{)"} converge strongly to x”, where

at — Pr Py Fly nancy, fe)

Methods for volving problem (1.1) have been well developed by many roscarchers (sce [5, 9-11, 18-21]) ‘These methods require solving approximation cqutlibrium problems with strongly monotone or monotone and Lipschits-type continuous bifuniclions

In our scheme, the main steps are to solve two strongly convex problems

1

l — mg in| 4z 7P, +)+t>|y-## Poy ec}

|e = axg:min{ AS" vt |e fips cÌ

and compute the next iteration point by Mann-type fixed points,

To investigale Lhe convergence of this scheme, we 1

(echnical Jerumas which val be used in the sequel call the Dollowing, Lemma 2.1 (See [I]} Let C be a nonempty closed convex subset of a real Hilbert space H Let [Cx CR be a pseudomonvtone, Lipschitz-

type continuous Vifunction For each xc, let fix.) be convex and

subdifferentiable on C Suppose that the sequences {x}, {y*}, {(È} am

generated by scheme (2.1) and x" = Sol f, C) Then

FP <i ah a" PG ange) xy P

(L 2yedpeY AP whee

Lemma 2.2 [17] Let 1 be a real Hilbert space, {3,} be a sequence of real numbers such that {8,}c |o, B|< (0,1), © > 0 and two sequences

‘Weak Convergence Theorems for an Infinite Family 123

another subsequence of {x”} such that x" — & as & —x co Then

= liming] x"* £|

ben

< liminf| x" - © koe

)-‘Weak Convergence Theorems for an Infinite Family 7

aesumplions on parameters, the authors proved thal the sequences {x"} and

{)"} converge strongly to x”, where

at — Pr Py Fly nancy, fe)

Methods for volving problem (1.1) have been well developed by many roscarchers (sce [5, 9-11, 18-21]) ‘These methods require solving approximation cqutlibrium problems with strongly monotone or monotone and Lipschits-type continuous bifuniclions

In our scheme, the main steps are to solve two strongly convex problems

1

l — mg in| 4z 7P, +)+t>|y-## Poy ec}

|e = axg:min{ AS" vt |e fips cÌ

and compute the next iteration point by Mann-type fixed points,

To investigale Lhe convergence of this scheme, we 1

(echnical Jerumas which val be used in the sequel call the Dollowing, Lemma 2.1 (See [I]} Let C be a nonempty closed convex subset of a real Hilbert space H Let [Cx CR be a pseudomonvtone, Lipschitz-

type continuous Vifunction For each xc, let fix.) be convex and

subdifferentiable on C Suppose that the sequences {x}, {y*}, {(È} am

generated by scheme (2.1) and x" = Sol f, C) Then

FP <i ah a" PG ange) xy P

(L 2yedpeY AP whee

Lemma 2.2 [17] Let 1 be a real Hilbert space, {3,} be a sequence of real numbers such that {8,}c |o, B|< (0,1), © > 0 and two sequences

120 P.N Anh, L B Long, N V Quy and L Q Thuy

Firm || u(x — x7 )4 (1 — ay Spf) — 2") | — Tim | xT

abit g(x — 2°) + (1 ony Meee") — 2°) | — J |

122 P.N Anh, L B Long, N V Quy and L Q Thuy

where w & 62,f(x", »”) má 7 © Ne(y"), By the definition of the norrual

cone ÄV 2, we imply that

On the other tund, since f(x®, -) is subdifferentiable on C, by the well

known Moreau-Rockafellar theorem, there exists wc Ga f(x" y”) such thar

This means that x £ #øi(7, Œ)

Step 3 Claim that the sequences {x}, {yŸ} and {1°} converge weakly

to Lhe same point x", where

fim Pree mg naili.c *)

Proof of Step 3 It follows from Steps 2 and 3 that for every weak cluster

point x of the sequence {x"} swisfies x € (Y", Fils; )A Sof C) We

show that {x"} converges weakly to x Now, we aseume that {x”*} is an

120 P.N Anh, L B Long, N V Quy and L Q Thuy

Firm || u(x — x7 )4 (1 — ay Spf) — 2") | — Tim | xT

abit g(x — 2°) + (1 ony Meee") — 2°) | — J |

‘Weak Convergence Theorems for an Infinite Family 123

another subsequence of {x”} such that x" — & as & —x co Then

= liming] x"* £|

ben

< liminf| x" - © koe

)-122 P.N Anh, L B Long, N V Quy and L Q Thuy

where w & 62,f(x", »”) má 7 © Ne(y"), By the definition of the norrual

cone ÄV 2, we imply that

On the other tund, since f(x®, -) is subdifferentiable on C, by the well

known Moreau-Rockafellar theorem, there exists wc Ga f(x" y”) such thar

This means that x £ #øi(7, Œ)

Step 3 Claim that the sequences {x}, {yŸ} and {1°} converge weakly

to Lhe same point x", where

fim Pree mg naili.c *)

Proof of Step 3 It follows from Steps 2 and 3 that for every weak cluster

point x of the sequence {x"} swisfies x € (Y", Fils; )A Sof C) We

show that {x"} converges weakly to x Now, we aseume that {x”*} is an

122 P.N Anh, L B Long, N V Quy and L Q Thuy

where w & 62,f(x", »”) má 7 © Ne(y"), By the definition of the norrual

cone ÄV 2, we imply that

On the other tund, since f(x®, -) is subdifferentiable on C, by the well

known Moreau-Rockafellar theorem, there exists wc Ga f(x" y”) such thar

This means that x £ #øi(7, Œ)

Step 3 Claim that the sequences {x}, {yŸ} and {1°} converge weakly

to Lhe same point x", where

fim Pree mg naili.c *)

Proof of Step 3 It follows from Steps 2 and 3 that for every weak cluster

point x of the sequence {x"} swisfies x € (Y", Fils; )A Sof C) We

show that {x"} converges weakly to x Now, we aseume that {x”*} is an

120 P.N Anh, L B Long, N V Quy and L Q Thuy

Firm || u(x — x7 )4 (1 — ay Spf) — 2") | — Tim | xT

abit g(x — 2°) + (1 ony Meee") — 2°) | — J |

‘Weak Convergence Theorems for an Infinite Family uy

3 Convergence Results Now, we prove (he main convergence theorem,

Theorem 3.1 Suppose that Assumptions Al-A5 are satisfied, x° = C

and bo positive sequences {h,}, {ay} satisfy the folowiny restrictions:

fig) <a, Bl for some œ 6 ca +} where E = max{2ey,

Then the sequences {x*}, {v} and {PP} generated by (2.1) and (2.2)

converge weakly to the same patat x” © (2, Fil) SoK S.C), where

HE, Pop meegmnenty e9 }

The proof of this theorem is divided into several steps

In Steps | and 2, we will consider weak clusters of {x*} It follows Lom

Step 1 Claim that x € 1 Fats, }

Proof of Step 1 By Iemma 2.1 and (3.1), we have

(=a) tt - 26g) * - »* Ễ s 4— đj@Œ Daye] xf - x P

effoe Pope oe B

Oa kom,

118 P.N, Anh, L.B Long, N V Quy and L Q Thuy

tx}, fy") of 1 such that

im sup] x* |e,

ae Tim sup|| yŸ

kao

Than

Lemma 2.3 [7] Jet C he a nonempty clased convex subset of a Banach

apace Let {S} be a sequence of nonexpansive mappings of C into itself and

Sis @ mapping of C into itself such that

[Shem seule) 8469 [sxe Che,

Assume that 8 ís œ nonexpansive selfmapping of

#, then I S is demiclosed: that is, whenever {x*} is a sequence in C

weakly converging to some ¥ © C and the sequence {(1 - S)(x")} strongly

converging fo some y, il follows that UZ — 8)( 3 Here J is the identity

operator of TÍ

Lemma 2.5 (See [17]) Zet C be a nonempty closed comes subsel of a

real Hilbert space H Suppose thal, for all u € C, the sequence {x"} satisfies

Then the sequence (Pra (x* )} converges strongly to some x © C.

‘Weak Convergence Theorems for an Infinite Family 121 Let S be a mapping of C into itself defined by

Step2 When x <x as j › s, we show thất # Sel(ƒ, C)

Pruol of Step 2 Sinci

This follows thal

118 P.N, Anh, L.B Long, N V Quy and L Q Thuy

tx}, fy") of 1 such that

im sup] x* |e,

ae Tim sup|| yŸ

kao

Than

Lemma 2.3 [7] Jet C he a nonempty clased convex subset of a Banach

apace Let {S} be a sequence of nonexpansive mappings of C into itself and

Sis @ mapping of C into itself such that

[Shem seule) 8469 [sxe Che,

Assume that 8 ís œ nonexpansive selfmapping of

#, then I S is demiclosed: that is, whenever {x*} is a sequence in C

weakly converging to some ¥ © C and the sequence {(1 - S)(x")} strongly

converging fo some y, il follows that UZ — 8)( 3 Here J is the identity

operator of TÍ

Lemma 2.5 (See [17]) Zet C be a nonempty closed comes subsel of a

real Hilbert space H Suppose thal, for all u € C, the sequence {x"} satisfies

Then the sequence (Pra (x* )} converges strongly to some x © C.

120 P.N Anh, L B Long, N V Quy and L Q Thuy

Firm || u(x — x7 )4 (1 — ay Spf) — 2") | — Tim | xT

abit g(x — 2°) + (1 ony Meee") — 2°) | — J |

‘Weak Convergence Theorems for an Infinite Family 7

aesumplions on parameters, the authors proved thal the sequences {x"} and

{)"} converge strongly to x”, where

In our scheme, the main steps are to solve two strongly convex problems

1

l — mg in| 4z 7P, +)+t>|y-## Poy ec}

|e = axg:min{ AS" vt |e fips cÌ

and compute the next iteration point by Mann-type fixed points,

To investigale Lhe convergence of this scheme, we 1

(echnical Jerumas which val be used in the sequel

call the Dollowing,

Lemma 2.1 (See [I]} Let C be a nonempty closed convex subset of a real Hilbert space H Let [Cx CR be a pseudomonvtone, Lipschitz-

type continuous Vifunction For each xc, let fix.) be convex and

subdifferentiable on C Suppose that the sequences {x}, {y*}, {(È} am

generated by scheme (2.1) and x" = Sol f, C) Then

FP <i ah a" PG ange) xy P

(L 2yedpeY AP whee

Lemma 2.2 [17] Let 1 be a real Hilbert space, {3,} be a sequence of real numbers such that {8,}c |o, B|< (0,1), © > 0 and two sequences

‘Weak Convergence Theorems for an Infinite Family 121 Let S be a mapping of C into itself defined by

Step2 When x <x as j › s, we show thất # Sel(ƒ, C)

Pruol of Step 2 Sinci

This follows thal

‘Weak Convergence Theorems for an Infinite Family uy

3 Convergence Results Now, we prove (he main convergence theorem,

Theorem 3.1 Suppose that Assumptions Al-A5 are satisfied, x° = C

and bo positive sequences {h,}, {ay} satisfy the folowiny restrictions:

fox} [ed] 0.05

;

fig) <a, Bl for some œ 6 ca +} where E = max{2ey,

Then the sequences {x*}, {v} and {PP} generated by (2.1) and (2.2)

converge weakly to the same patat x” © (2, Fil) SoK S.C), where

HE, Pop meegmnenty e9 }

The proof of this theorem is divided into several steps

In Steps | and 2, we will consider weak clusters of {x*} It follows Lom

Step 1 Claim that x € 1 Fats, }

Proof of Step 1 By Iemma 2.1 and (3.1), we have

(=a) tt - 26g) * - »* Ễ s 4— đj@Œ Daye] xf - x P

effoe Pope oe B

Oa kom,

118 P.N, Anh, L.B Long, N V Quy and L Q Thuy

tx}, fy") of 1 such that

im sup] x* |e,

ae Tim sup|| yŸ

kao

Than

Lemma 2.3 [7] Jet C he a nonempty clased convex subset of a Banach

apace Let {S} be a sequence of nonexpansive mappings of C into itself and

Sis @ mapping of C into itself such that

[Shem seule) 8469 [sxe Che,

@ nonempiy closed comes subset C of a real Iiflbert space H if Fie(S}

Assume that 8 ís œ nonexpansive selfmapping of

#, then I S is demiclosed: that is, whenever {x*} is a sequence in C

weakly converging to some ¥ © C and the sequence {(1 - S)(x")} strongly

converging fo some y, il follows that UZ — 8)( 3 Here J is the identity

operator of TÍ

Lemma 2.5 (See [17]) Zet C be a nonempty closed comes subsel of a

real Hilbert space H Suppose thal, for all u € C, the sequence {x"} satisfies

Then the sequence (Pra (x* )} converges strongly to some x © C.