Approximating fixed points for nonexpansive mappings and generalized mixed equilibrium problems in Banach spaces

ISSN: 2008-6822 (electronic)

http://www.ijnaa.semnan.ac.ir

Approximating Fixed Points for Nonexpansive

Mappings and Generalized Mixed Equilibrium

Problems in Banach Spaces

L. Cholamjiak, S. Suantai∗

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand.

(Communicated by M. Eshaghi Gordji)

Abstract

We introduce a new iterative scheme for finding a common element of the solutions set of a generalized mixed equilibrium problem and the fixed points set of an infinitely countable family of nonexpansive mappings in a Banach space setting. Strong convergence theorems of the proposed iterative scheme are also established by the generalized projection method. Our results generalize the corresponding results in the literature.

Keywords: Generalized Mixed Equilibrium Problem, Nonexpansive Mappings, Common Fixed Point, Strong Convergence, Generalized Projection.

2010 MSC: 47H09, 47H10.

1. Introduction

Let C be a closed convex subset of a Banach space E. A mapping T : C → C is said to be

nonexpansive if kT x−T yk ≤ kx−yk for all x, y ∈C. We denote by F(T) the set of a fixed point of T.

Let f : C ×C → _R be a bifunction, A : C → E∗ a mapping, and ϕ : C → _R a real-valued function. The generalized mixed equilibrium problem is to find x∈C such that

f(x, y) +hAx, y−xi+ϕ(y)≥ϕ(x), ∀y ∈C. (1.1)

The solutions set of (1.1) is denoted by GM EP(f, A, ϕ).

∗_{Corresponding author}

Email addresses: [email protected](L. Cholamjiak),[email protected](S. Suantai )

If A ≡ 0, then the generalized mixed equilibrium problem (1.1) reduces to the following mixed equilibrium problem: finding x∈C such that

f(x, y) +ϕ(y)≥ϕ(x), ∀y∈C. (1.2)

Problem (1.2) was introduced by Ceng and Yao [7]. The solutions set of (1.2) is denoted by

M EP(f, ϕ).

If f ≡ 0, then the generalized mixed equilibrium problem (1.1) reduces to the following mixed variational inequality problem: findingx∈C such that

hAx, y−xi+ϕ(y)≥ϕ(x), ∀y ∈C. (1.3)

The solutions set of (1.3) is denoted by V I(C, A, ϕ).

Ifϕ≡0, then the mixed equilibrium problem (1.2) reduces to the following equilibrium problem: findingx∈C such that

f(x, y)≥0, ∀y ∈C. (1.4)

The solutions set of (1.4) is denoted by EP(f).

Iff ≡0, then the mixed equilibrium problem (1.2) reduces to the following convex minimization problem: findingx∈C such that

ϕ(y)≥ϕ(x), ∀y∈C. (1.5)

The solutions set of (1.5) is denoted by CM P(ϕ).

The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others; see for instance, [5, 11, 13, 20].

For solving the equilibrium problem, let us assume that: (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 allx, y, z ∈C, lim sup_t↓0f(tz+ (1−t)x, y)≤f(x, y); (A4) for allx∈C, f(x, .) is convex and lower semicontinuous.

In 1953, Mann [19] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mappingT in a Hilbert space H:

xn+1 =αnxn+ (1−αn)T xn, n≥0, (1.6) where the initial pointx0 is taken in C arbitrarily and{αn} is a sequence in (0,1).

However, we note that Mann’s iteration process (1.6) has only weak convergence, in general; for instance, see [4, 14, 27].

LetC be a nonempty, closed and convex subset of a Banach space E and let{Tn}be sequence of mappings ofC into itself such thatT∞_n=1F(Tn)6=∅. Then {Tn} is said to satisfy the NST-condition if for each bounded sequence {zn} ⊂C,

lim

n→∞kzn−Tnznk= 0 impliesωw(zn)⊂

T∞

In 2008, Takahashi et al. [33] has adapted Nakajo and Takahashi [23]’s idea to modify the process (1.6) so that strong convergence is guaranteed. They proposed the following modification for nonexpansive mappings in a Hilbert space: x0 ∈H, C1 =C, u1 =PC1x0 and

 



yn=αnun+ (1−αn)Tnun,

Cn+1 ={z ∈Cn :kyn−zk ≤ kun−zk}, un+1 =PCn+1x0, n∈N,

(1.7)

where 0 ≤ αn ≤ a < 1 for all n ∈ N and PK is a metric projection from a Hilbert space H onto a nonempty, closed and convex subset K of H. They proved that if{Tn} satisfies the NST-condition, then {un} generated by (1.7) converges strongly to a common fixed point of{Tn}∞n=1.

Xu [36] introduced the following iterative scheme for finding a fixed point of a nonexpansive mapping in a Banach space: x0 =x∈C and

 



Cn=co{z ∈C:kz−T zk ≤tnkxn−T xnk}, Dn={z ∈C:hxn−z, J x−J xni ≥0}, xn+1 = ΠCn∩Dnx, n≥0,

(1.8)

where coD denotes the convex closure of the set D, {tn} is a sequence in (0,1) with tn → 0, and ΠCn∩Dn is a generalized projection from a Banach space E onto Cn∩Dn. Then, he proved that the

sequence {xn} generated by (1.8) converges strongly to a fixed point of T.

Very recently, Kimura and Nakajo [16], by using the Mosco convergence technique, obtained strong convergence theorems in a Banach space. They also proposed the following algorithm: x1 =x ∈C and

 



Cn=co{z ∈C:kz−Tnzk ≤tnkxn−Tnxnk}, Dn={z ∈C:hxn−z, J x−J xni ≥0}, xn+1 = ΠCn∩Dnx, n≥1,

(1.9)

where {tn} is a sequence in (0,1) with tn → 0 as n → ∞. They proved that if {Tn} satisfies the NST-condition, then the sequence {xn} generated by (1.9) converges strongly to a common fixed point of {Tn}∞n=1.

The problem of finding a common element of the fixed points set and the solutions set of an equilibrium problem in the framework of Hilbert spaces and Banach spaces has been studied by many authors; for instance, see [8, 9, 24, 25, 26, 29, 30, 32, 35, 37] and the references therein.

Motivated and inspired by Xu [36], Kimura and Nakajo [16], we introduce a new hybrid projection algorithm for finding a common element of the solutions set of a generalized mixed equilibrium problem and the fixed points set of an infinitely countable family of nonexpansive mappings in the framework of Banach spaces.

2. Preliminaries and lemmas

LetE be a real Banach space and let U ={x∈E :kxk= 1} be the unit sphere ofE. A Banach space E is said to be strictly convex if for any x, y ∈U,

x6=y implies

x+y

2 <1.

It is also said to be uniformly convex if for each ε ∈ (0,2], there exists δ > 0 such that for any

x, y ∈U,

kx−yk ≥ε implies

x+y

2

It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a function

δ: [0,2]→[0,1] called themodulus of convexity of E as follows:

δ(ε) = infn1−

x+y

2

: x, y ∈E, kxk=kyk= 1, kx−yk ≥ε o

.

Then E is uniformly convex if and only if δ(ε)>0 for allε∈(0,2]. A Banach spaceE is said to be

smooth if the limit

lim t→0

kx+tyk − kxk

t (2.1)

exists for all x, y ∈U. It is also said to be uniformly smooth if the limit (2.1) is attained uniformly for x, y ∈U. The normalized duality mapping J :E →2E∗ _{is defined by}

J(x) ={ x∗ ∈E∗ :hx, x∗i=kxk2 =kx∗k2}

for all x ∈ E. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E; see [31] for more details.

Let E be a smooth Banach space. The function φ :E×E →_R is defined by

φ(x, y) =kxk2_{−2hx, J yi+kyk}2

for all x, y ∈E. In a Hilbert space H, we have φ(x, y) =kx−yk2 _{for all x, y ∈H.}

Lemma 2.1 (Kamimura and Takahashi [15]). Let E be a uniformly convex and smooth Banach space and let {xn},{yn}be two sequences of E. If φ(xn, yn)→0and either {xn}or {yn} is bounded,

then kxn−ynk →0 as n→ ∞.

Let E be a reflexive, strictly convex and smooth Banach space and let C be a nonempty, closed and convex subset of E. The generalized projection mapping, introduced by Alber [1], is a mapping

ΠC :E →C, that assigns to an arbitrary point x∈E the minimum point of the functionalφ(y, x),

that is,ΠCx= ¯x, where ¯x is the solution to the minimization problem

φ(¯x, x) = min{φ(y, x) :y ∈C}.

In fact, we have the following result.

Lemma 2.2 (Alber [1]). Let C be a nonempty, closed and convex subset of a real reflexive, strictly convex, and smooth Banach spaceE and letx∈E. Then, there exists a unique element x0 ∈C such

that φ(x0, x) = min{φ(z, x) :z ∈C}.

The existence and uniqueness of the operatorΠC follows from the properties of the functional φ and

strict monotonicity of the duality mapping J; for instance, see [1, 2, 10, 15, 31]. In a Hilbert space,

ΠC is coincident with the metric projection.

Lemma 2.3 (Alber [1] and Kamimura and Takahashi [15]). Let C be a nonempty, closed and con-vex subset of a smooth Banach space E and x∈E. Then x0 =ΠCx if and only if

hx0−y, J x−J x0i ≥0, ∀y∈C.

Lemma 2.4 (Alber [1] and Kamimura and Takahashi [15]). Let C be a nonempty, closed and con-vex subset of a reflexive, strictly concon-vex and smooth Banach space E and let x∈E. Then

Lemma 2.5 (Bruck [6]). Let C be a bounded, closed and convex subset of a uniformly convex Ba-nach spaceE. Then, there exists a strictly increasing convex continuous function γ : [0,∞)→[0,∞)

such that γ(0) = 0 and

γ T

_Xn

i=1

λixi

−

n X

i=1

λiT xi

≤ max

1≤j≤k≤n(kxj−xkk − kT xj −T xkk)

for all n ∈ _N, {x1, x2, ..., xn} ⊂ C, {λ1, λ2, ..., λn} ⊂ [0,1] with Pn

i=1λi = 1 and nonexpansive

mapping T of C into E.

Lemma 2.6 (Blum and Oettli [5]). LetCbe a closed and convex subset of a smooth, strictly convex, and reflexive Banach space E, letf be a bifunction from C×C to _Rwhich satisfies conditions (A1)-(A4), and let r >0 and x∈E. Then there exists z ∈C such that

f(z, y) + 1

rhy−z, J z−J xi ≥0, ∀y∈C.

The following result can be found in [38].

Lemma 2.7 (Zhang [38]). Let C be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space E. Let A: C → E∗ be a continuous and monotone mapping, let

f be a bifunction from C×C to _R satisfying (A1)-(A4) and let ϕ be a lower semicontinuous and convex function from C to _R. For all r >0 and x∈E, there exists z ∈C such that

f(z, y) +hAz, y−zi+ϕ(y) + 1

rhy−z, J z−J xi ≥ϕ(z), ∀y ∈C.

Define a mapping Sr :E →2C as follows:

Sr(x) = {z∈C :f(z, y) +hAz, y−zi+ϕ(y) + 1

rhy−z, J z−J xi ≥ϕ(z), ∀y∈C}.

Then, the followings hold:

(1) Sr is single-valued;

(2) Sr is firmly nonexpansive-type mapping; [18], i.e., for allx, y ∈E,

hSrx−Sry, J Srx−J Sryi ≤ hSrx−Sry, J x−J yi; (3) F(Sr) = GM EP(f, A, ϕ);

(4) GM EP(f, A, ϕ) is closed and convex.

3. Main Results

In this section, we prove the strong convergence theorem for finding a common element of the fixed points set for nonexpansive mappings and the solutions set of a generalized mixed equilibrium problem in Banach spaces.

Theorem 3.1. Let E be a uniformly convex and uniformly smooth Banach space andC a nonempty, closed and convex subset of E. Let f be a bifunction from C ×C to _R satisfying (A1)-(A4), A :

from C to _R. Let {Ti}∞i=1 be a sequence of nonexpansive mappings of C into itself such that F := T∞

i=1F(Ti)∩GM EP(f, A, ϕ)6=∅. Let {xn} be a sequence generated by 

  

  

x0 ∈C, D0 =C,

Cn= T∞

i=1co{z ∈C :kz−Tizk ≤tnkxn−Tixnk}, n≥0,

Dn={z ∈Dn−1 :hSrnxn−z, J xn−J Srnxni ≥0}, n≥1,

xn+1 = ΠCn∩Dnx0, n≥0,

where {tn} and {rn} are sequences satisfying the conditions: (C1) {tn} ⊂(0,1) and limn→∞tn= 0;

(C2) {rn} ⊂(0,∞) and lim infn→∞rn >0.

Then, the sequence {xn} converges strongly to ΠFx0.

Proof . First, we show that the sequence {xn} is well-defined. It is easy to verify that Cn∩Dn is closed and convex and F ⊂ Cn for all n ≥ 0. Since D0 = C, we also have F ⊂ C0∩D0. Suppose that F ⊂Ck−1∩Dk−1 for k ≥2. It follows from Lemma 2.7 (2) that

hSrkxk−Srku, J xk−J Srkxk−(J u−J Srku)i ≥0,

for all u∈F. This implies that

hSrkxk−u, J xk−J Srkxki ≥0,

for all u ∈ F. Hence F ⊂ Dk. By the mathematical induction, we get that F ⊂ Cn∩Dn for each n≥0. By Lemma 2.7 (4), we know that F :=T∞_i=1F(Ti)∩GM EP(f, A, ϕ) is nonempty, closed and convex. Then there exists a unique element w ∈ F such that w = ΠFx0. Since F ⊂ Cn−1∩Dn−1 and xn= ΠCn−1∩Dn−1x0, we have

φ(xn, x0)≤φ(w, x0), n≥1. (3.1)

Since xn= ΠCn−1∩Dn−1x0 and xn+1 ∈Dn ⊂Dn−1, we have

φ(xn, x0)≤φ(xn+1, x0), n≥1. (3.2) From (3.1) and (3.2) we can conclude that limn→∞φ(xn, x0) exists.

Next, we show that limm,n→∞φ(xm, xn) = 0. From xn = ΠCn−1∩Dn−1x0 and xm ∈ Dm−1 ⊂ Dn−1

for m > n≥1, we have by Lemma 2.4

φ(xm, xn) +φ(xn, x0)≤φ(xm, x0). This implies that

φ(xm, xn)≤φ(xm, x0)−φ(xn, x0).

Hence limm,n→∞φ(xm, xn) = 0. By Lemma 2.1, we obtain lim

m,n→∞kxm−xnk= 0. In particular, we also have

lim

n→∞kxn+1−xnk= 0. (3.3)

Next, we show that v ∈ T∞_i=1F(Ti). Since xn+1 ∈ Cn and tn > 0, there exists m ∈ N,

{λ0, λ1, ..., λm} ⊂[0,1] and {y0, y1, ..., ym} ⊂C such that m

X

j=0

λj = 1,

xn+1− m X

j=0

λjyj

< tn, and kyj−Tiyjk ≤tnkxn−Tixnk

for each j = 0,1, ..., m and i ∈ _N. Put M = sup_n≥0kxn − wk. We note that kyj −Tiyjk ≤ tnkxn−Tixnk ≤2tnkxn−wk ≤ 2tnM for each j = 0,1, ..., m and i ∈N. Since {xn} is bounded, by Lemma 2.5, we have

kxn−Tixnk ≤ kxn−xn+1k+

xn+1− m X

j=0

λjyj + m X j=0

λjyj − m X

j=0

λjTiyj + m X j=0

λjTiyj−Ti( m X

j=0

λjyj) +

Ti(

m X

j=0

λjyj)−Tixn

≤ kxn−xn+1k+tn+ m X

j=0

λjkyj−Tiyjk

+γ−1 max

0≤j≤k≤m(kyj −ykk − kTiyj−Tiykk) + m X j=0

λjyj −xn

≤ kxn−xn+1k+tn+ (2tnM) m X

j=0

λj

+γ−1 max

0≤j≤k≤m(kyj −Tiyjk+kyk−Tiykk) + k m X j=0

λjyj −xn+1k+kxn−xn+1k

≤ 2kxn−xn+1k+tn+ 2tnM +γ−1(4M tn) +tn

= 2kxn−xn+1k+ (2 + 2M)tn+γ−1(4M tn). It follows from (3.3) and (C1) that

lim

n→∞kxn−Tixnk= 0,

for all i∈_N. Thus v ∈T∞_i=1F(Ti).

Next, we show that v ∈ GM EP(f, A, ϕ). By the construction of Dn, we see from Lemma 2.3 that Srnxn= ΠDn−1xn. Since xn+1 ∈Dn⊂Dn−1, we obtain

φ(Srnxn, xn)≤φ(xn+1, xn)→0,

asn → ∞. From Lemma 2.1, we have lim

n→∞kSrnxn−xnk= 0.

Since xn→v, we have Srnxn→v as n→ ∞. Since J is uniformly norm-to-norm continuous on the

bounded set, we have

lim

By (C2) we also have

lim n→∞

kJ Srnxn−J xnk

rn

= 0. (3.4)

For each y∈C, we see that

f(Srnxn, y) +hASrnxn, y −Srnxni+ϕ(y) +

1

rn

hy−Srnxn, J Srnxn−J xni ≥ϕ(Srnxn).

By using the same argument as in the proof of [28], we can verify that

f(v, y) +hAv, y−vi+ϕ(y)≥ϕ(v), ∀y∈C.

This shows thatv ∈GM EP(f, A, ϕ) and hencev ∈F :=T∞_i=1F(Ti)∩GM EP(f, A, ϕ). Finally, we show that v =w= ΠFx0. Sincexn+1 = ΠCn∩Dnx0, we have

hJ x0−J xn+1, xn+1−zi ≥0 ∀z ∈Cn∩Dn. Since F ⊂Cn∩Dn, we also have

hJ x0−J xn+1, xn+1−zi ≥0 ∀z ∈F. (3.5) By taking limit in (3.5), we obtain that

hJ x0−J v, v−zi ≥0 ∀z ∈F.

By Lemma 2.3, we can conclude thatv = ΠFx0 =w. This completes the proof. If we take Ti =I for all i∈Nin Theorem 3.1, then we obtain the following result.

Theorem 3.2. LetEbe a uniformly convex and uniformly smooth Banach space andCbe a nonempty, closed and convex subset of E. Let f be a bifunction from C × C to _R satisfying (A1)-(A4),

A : C → E∗ a continuous and monotone mapping, and ϕ a lower semicontinuous and convex function from C to _R such that GM EP(f, A, ϕ)6=∅. Let {xn} be a sequence generated by

 



x0 ∈C, D0 =C,

Dn={z ∈Dn−1 :hSrnxn−z, J xn−J Srnxni ≥0}, n≥1,

xn+1 = ΠDnx0, n≥0.

If{rn} ⊂(0,∞)andlim infn→∞rn>0, then the sequence {xn}converges strongly toΠGM EP(f,A,ϕ)x0. If we takef ≡0, A≡0 and ϕ≡0 in Theorem 3.1, we obtain the following result.

Theorem 3.3. LetEbe a uniformly convex and uniformly smooth Banach space andCbe a nonempty, closed and convex subset of E. Let {Ti}∞i=1 be a sequence of nonexpansive mappings of C into itself

such that F :=T∞_i=1F(Ti)6=∅. Let {xn} be a sequence generated by 





x0 ∈C,

Cn= T∞

i=1co{z ∈C :kz−Tizk ≤tnkxn−Tixnk},

xn+1 = ΠCnx0, n≥0.

If {tn} ⊂(0,1) and limn→∞tn= 0, then the sequence {xn} converges strongly to ΠFx0.

4. Acknowledgement

The authors would like to thank the referees for the valuable suggestions on the manuscript. The first author is supported by the Royal Golden Jubilee Grant PHD/0261/2551 and the Graduate School of Chiang Mai University, Thailand.

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