Afr. Mat.DOI 10.1007/s13370-013-0210-2

Strong convergence theorems of a finite familyof quasi-nonexpansive and Lipschitzmulti-valued mappings

Suthep Suantai · Watcharaporn Cholamjiak ·Prasit Cholamjiak

Received: 16 February 2013 / Accepted: 18 October 2013© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2013

Abstract In this paper, we study a mapping generated by a finite family of quasi-nonexpansive and Lipschitz multi-valued mappings. We also introduce a hybrid algorithmfor finding common fixed points of such mappings and prove a strong convergence theorem.

Keywords Quasi-nonexpansive multi-valued mapping · Projection method · Commonfixed point · Strong convergence

Mathematical Subject Classification: 47H10 · 54H25

1 Introduction

Let D be a nonempty convex subset of a Banach space E . The set D is called proximinalif for each x ∈ E , there exists an element y ∈ D such that ‖x − y‖ = d(x, D), whered(x, D) = inf{‖x − z‖ : z ∈ D}. Let C B(D), K (D) and P(D) be the family of nonemptyclosed bounded subsets, nonempty compact subsets, and nonempty proximinal boundedsubsets of D respectively. The Hausdorff metric on C B(D) is defined by

H(A, B) = max

{supx∈A

d(x, B), supy∈B

d(y, A)

}

for A, B ∈ C B(D). A single-valued mapping T : D → D is called nonexpansive if‖T x − T y‖ ≤ ‖x − y‖ for all x, y ∈ D. A multi-valued mapping T : D → C B(D) is said

S. SuantaiDepartment of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailande-mail: scmti005@chiangmai.ac.th

W. Cholamjiak · P. Cholamjiak (B)School of Science, University of Phayao, Phayao 56000, Thailande-mail: prasitch2008@yahoo.com

W. Cholamjiake-mail: c-wchp007@hotmail.com

123

S. Suantai et al.

to be nonexpansive if H(T x, T y) ≤ ‖x − y‖ for all x, y ∈ D. An element p ∈ D is calleda fixed point of T : D → D (respectively, T : D → C B(D)) if p = T p (respectively,p ∈ T p). The set of fixed points of T is denoted by F(T ).

The mapping T : D → C B(D) is called

(i) quasi-nonexpansive [19] if F(T ) �= ∅ and H(T x, T p) ≤ ‖x − p‖ for all x ∈ D and allp ∈ F(T );

(ii) L-Lipschitz if there exists a constant L > 0 such that H(T x, T y) ≤ L‖x − y‖ for allx, y ∈ D.

It is clear that every nonexpansive multi-valued mapping T with F(T ) �= ∅ is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are not nonexpansive (see[17]).

Mann [11] introduced the following iterative procedure to approximate a fixed point of anonexpansive mapping T in a Hilbert space H :

xn+1 = αn xn + (1 − αn)T xn, ∀n ≥ 0, (1.1)

where the initial point x0 is taken in C arbitrarily and {αn} is a sequence in (0,1).However, we note that Mann’s iteration process (1.1) has only weak convergence, in

general; for instance, see [4,7,15].Since 1974, the iterative schemes for a multi-valued mapping in a Banach space or a

Hilbert space have been studied by many authors (see [1,5,6,8,10,14,16,17,19–21]).Shahzad and Zegeye [17] extended and improved the results of Panyanak [14], Sastry and

Babu [16], and Song and Wang [20] to a quasi-nonexpansive multi-valued mapping. Theyalso relaxed compactness of the domain of T and constructed an iteration scheme whichremoves the restriction of T namely T p = {p} for any p ∈ F(T ). The results provided anaffirmative answer to Panyanak’s question [14] in a more general setting. They introducednew iterations as follows:Let D be a nonempty convex subset of a Banach space E and T : D → C B(D) and letαn, α′

n ∈ [0, 1],(A) The sequence of Ishikawa iterates is defined by x0 ∈ D,

yn = α′nz′

n + (1 − α′n)xn, n ≥ 0,

xn+1 = αnzn + (1 − αn)xn, n ≥ 0,

where z′n ∈ T xn and zn ∈ T yn .

(B) Let T : D → P(D) and PT x = {y ∈ T x : ‖x − y‖ = d(x, T x)}. The sequence ofIshikawa iterates is defined by x0 ∈ D,

yn = α′nz′

n + (1 − α′n)xn, n ≥ 0,

xn+1 = αnzn + (1 − αn)xn, n ≥ 0,

where z′n ∈ PT xn and zn ∈ PT yn .

Atsushiba and Takahashi [3] introduced the concept of the W -mapping as follows:

U1 = β1T1 + (1 − β1)I,

U2 = β2T2U1 + (1 − β2)I,...

UN−1 = βN−1TN−1UN−2 + (1 − βN−1)I,

W = UN = βN TN UN−1 + (1 − βN )I,

123

Strong convergence theorems of a finite family of quasi-nonexpansive...

where {Ti }Ni=1 is a finite mapping of C into itself and βi ∈ [0, 1] for all i = 1, 2, ..., N .

Such a mapping W is called the W-mapping generated by T1, T2, ..., TN and β1, β2, ..., βN ;see also [18,22,23].

Takahashi et al. [25] introduced a new method called the shrinking projection method fora nonexpansive mapping T in a Hilbert space:

Theorem TTK [25] Let H be a Hilbert space and let C be a nonempty closed convex subsetof H. Let T be a nonexpansive mapping of C into H such that F(T ) �= ∅ and let x0 ∈ H.For C1 = C and u1 = PC1 x0, we define a sequence {un} of C as follows:⎧⎨

⎩yn = αnun + (1 − αn)T un,

Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖un − z‖},un+1 = PCn+1 x0, n ∈ N,

(1.2)

where 0 ≤ αn ≤ a < 1 for all n ∈ N. Then {un} converges strongly to z0 = PF(T )x0.

By using this method, they can show that the sequence {xn} generated by (1.2) is a Cauchysequence, without the use of demiclosedness principle, Opial’s condition and the Kadec-Kleeproperty.

In this paper, motivated by Atsushiba and Takahashi [3] and Takahashi et al. [25], weintroduce a new approximation scheme for finding common fixed points of a finite familyof quasi-nonexpansive and Lipschitz multi-valued mappings in the framework of Hilbertspaces.

Let X be a real Banach space and D be a nonempty closed convex subset of X . LetT : D → C B(D) and S : D → C B(D) be multi-valued mappings and A ⊂ D. We defineT (A) = ∪x∈AT x and (ST )x = S(T x) for all x ∈ D. For a family of quasi-nonexpansiveand Lipschitz multi-valued mappings T1, T2, ..., TN and a sequence {βi,n}N

i=1 ⊂ [0, 1], weconsider the mapping Wn defined as follows:

U0,n = I,

U1,n = β1,nT1U0,n + (1 − β1,n)I,

U2,n = β2,nT2U1,n + (1 − β2,n)I,

...

UN−1,n = βN−1,nTN−1UN−2,n + (1 − βN−1,n)I,

Wn = UN ,n = βN ,nTN UN−1,n + (1 − βN ,n)I. (1.3)

This mapping is called the W -mapping generated by T1, T2, ..., TN and β1,n, β2,n, ..., βN ,n .

2 Preliminaries and lemmas

The following lemmas give some characterizations and a useful property of the metric pro-jection PD in a Hilbert space.

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖. Let D be a closedconvex subset of H . For every point x ∈ H , there exists a unique nearest point in D, denotedby PD x , such that

‖x − PD x‖ ≤ ‖x − y‖, ∀y ∈ D.

PD is called the metric projection of H onto D. We know that PD is a nonexpansive mappingof H onto D.

123

S. Suantai et al.

Lemma 2.1 [12] Let D be a closed convex subset of a real Hilbert space H and let PD bethe metric projection from H onto D. Given x ∈ H and z ∈ D. Then z = PD x if and only ifthere holds the relation:

〈x − z, y − z〉 ≤ 0, ∀y ∈ D.

Lemma 2.2 [13] Let D be a nonempty closed convex subset of a real Hilbert space H andPD : H → D be the matric projection from H onto D. Then the following inequality holds:

‖y − PD x‖2 + ‖x − PD x‖2 ≤ ‖x − y‖2, ∀x ∈ H, ∀y ∈ D.

Lemma 2.3 [12] Let H be a real Hilbert space. Then the following equations hold:

(i) ‖x − y‖2 = ‖x‖2 − ‖y‖2 − 2〈x − y, y〉, ∀x, y ∈ H;(ii) ‖t x + (1 − t)y‖2 = t‖x‖2 + (1 − t)‖y‖2 − t (1 − t)‖x − y‖2, ∀t ∈ [0, 1]

and x, y ∈ H.

Lemma 2.4 [9] Let D be a nonempty closed convex subset of a real Hilbert space H. Givenx, y, z ∈ H and also given a ∈ R, the set{

v ∈ D : ‖y − v‖2 ≤ ‖x − v‖2 + 〈z, v〉 + a}

is convex and closed.

Lemma 2.5 [26] Let X be a uniformly convex Banach space and Br (0) be a closed ball ofX. Then there exists a continuous strictly increasing convex function g : [0,∞) → [0,∞)

with g(0) = 0 such that

‖λx + (1 − λ)y‖2 ≤ λ‖x‖2 + (1 − λ)‖y‖2 − λ(1 − λ)g(‖x − y‖)for all x, y ∈ Br (0) and λ ∈ [0, 1].

In order to deal with a family of mappings, the following conditions are introduced. LetX be a Banach space and let D be a subset of X :

(i) Let {Tn} and τ be two families of mappings of D into itself with ∅ �= F(τ ) ⊂⋂∞n=1 F(Tn), where F(Tn) is the set of all fixed points of Tn , F(τ ) is the set of all

common fixed points of τ . {Tn} is said to satisfy the NST-condition [24] with respect toτ if for each bounded sequence {zn} in C ,

limn→∞ ‖zn − Tnzn‖ = 0 ⇒ lim

n→∞ ‖zn − T zn‖ = 0, ∀T ∈ τ.

(ii) Let {Tn} and τ be two families of multi-valued mappings of D into 2D . A family ofmulti-valued mappings {Tn} is said to satisfy the SC-condition with respect to τ if∅ �= F(τ ) ⊂ ⋂∞

n=1 F(Tn), where F(Tn) is the set of all fixed points of Tn , F(τ ) isthe set of all common fixed points of τ if for each bounded sequence {zn} in D andsn ∈ Tnzn ,

limn→∞ ‖sn − zn‖ = 0 ⇒ lim

n→∞ ‖cn − zn‖ = 0, ∃cn,T = cn ∈ T zn, ∀T ∈ τ.

It is easy to see that if the family of nonexpansive mappings {Tn} satisfies the N ST −condition, then {Tn} will satisfy SC−condition for single-valued mappings.

Lemma 2.6 [2] If A, B ∈ C B(D) and x ∈ A, then for each positive number α there existsy ∈ B such that

‖x − y‖ ≤ H(A, B) + α.

123

Strong convergence theorems of a finite family of quasi-nonexpansive...

Lemma 2.7 Let D be a closed and convex subset of a real Hilbert space H. Let T : D →C B(D) be a quasi-nonexpansive multi-valued mapping such that T p = {p} for each p ∈F(T ). Then F(T ) is a closed and convex subset of D.

Proof First, we will show that F(T ) is closed. Let {xn} be a sequence in F(T ) such thatxn → x as n → ∞. We have

d(x, T x) ≤ d(x, xn) + d(xn, T x)

≤ d(x, xn) + H(T xn, T x)

≤ 2d(x, xn).

It follows that d(x, T x) = 0, so x ∈ F(T ). Next, we show that F(T ) is convex. Letp = tp1 + (1 − t)p2 where p1, p2 ∈ F(T ) and t ∈ (0, 1). Let z ∈ T p, by Lemma 2.3, wehave

‖p − z‖2 = ‖t (z − p1) + (1 − t)(z − p2)‖2

= t‖z − p1‖2 + (1 − t)‖z − p2‖2 − t (1 − t)‖p1 − p2‖2

= td(z, T p1)2 + (1 − t)d(z, T p2)

2 − t (1 − t)‖p1 − p2‖2

≤ t H(T p, T p1)2 + (1 − t)H(T p, T p2)

2 − t (1 − t)‖p1 − p2‖2

≤ t‖p − p1‖2 + (1 − t)‖p − p2‖2 − t (1 − t)‖p1 − p2‖2

= t (1 − t)2‖p1 − p2‖2 + (1 − t)t2‖p1 − p2‖2 − t (1 − t)‖p1 − p2‖2

= 0

hence p = z. Therefore p ∈ F(T ). ��Lemma 2.8 Let D be a nonempty closed convex subset of a uniformly convex Banach spaceX. Let {Ti }N

i=1 be a finite family of quasi-nonexpansive multi-valued mappings of D into

C B(D) such that F := ⋂Ni=1 F(Ti ) �= ∅ and {βi }N

i=1 ⊂ [0, 1]. Let W be the W -mappinggenerated by T1, T2, ..., TN and β1, β2, ..., βN . For each i = 1, 2, ..., N, we assume thatTi p = {p}, ∀p ∈ F. Then, for each x ∈ D

H(W x, W p) ≤ ‖x − p‖, ∀p ∈ F.

Proof Let x ∈ D. For each u1 ∈ U1x and p ∈ F , we have

‖u1 − p‖ = ‖β1a1 + (1 − β1)x − p‖ for some a1 ∈ T1x

≤ β1‖a1 − p‖ + (1 − β1)‖x − p‖= β1d(a1, T1 p) + (1 − β1)‖x − p‖≤ β1 H(T1x, T1 p) + (1 − β1)‖x − p‖≤ ‖x − p‖. (2.1)

For each u2 ∈ U2x and p ∈ F , we have

‖u2 − p‖ = ‖β2a2 + (1 − β2)x − p‖ for some a2 ∈ T2U1x

≤ β2‖a2 − p‖ + (1 − β2)‖x − p‖= β2d(a2, T2 p) + (1 − β2)‖x − p‖≤ β2 H(T2u1, T2 p) + (1 − β2)‖x − p‖ for some u1 ∈ U1x

≤ β2‖u1 − p‖ + (1 − β2)‖x − p‖. (2.2)

123

S. Suantai et al.

It follows from (2.1) and (2.2) that

‖u2 − p‖ ≤ ‖x − p‖. (2.3)

Similarly, for each u3 ∈ U3x and p ∈ F , we have

‖u3 − p‖ = ‖β3a3 + (1 − β3)x − p‖ for some a3 ∈ T3U2x

≤ β3‖a3 − p‖ + (1 − β3)‖x − p‖= β3d(a3, T3 p) + (1 − β3)‖x − p‖≤ β3 H(T3u2, T3 p) + (1 − β3)‖x − p‖ for some u2 ∈ U2x

≤ β3‖u2 − p‖ + (1 − β3)‖x − p‖. (2.4)

From (2.3) and (2.4), we obtain

‖u3 − p‖ ≤ ‖x − p‖. (2.5)

By continuing in the way, for each uN ∈ UN x = W x and p ∈ F , we obtain

d(uN , W p) ≤ ‖uN − p‖ ≤ ‖x − p‖.This implies, for each p ∈ F , that

supuN ∈W x

d(uN , W p) ≤ ‖x − p‖. (2.6)

Since W p = {p}, it follows from (2.6) that

H(W x, W p) ≤ ‖x − p‖,for each p ∈ F . Since x is arbitrary, we complete the proof. ��

The assumption that T p = {p} for all p ∈ F(T ) was introduced by Shahzad and Zegeye[17]. We next give examples of quasi-nonexpansive multi-valued mapping T which satisfiesthat condition.

Example 2.9 Consider D = [0, 1] × [0, 1] with the usual norm. Define T : D → C B(D)

by

T (x, y) =

⎧⎪⎪⎨⎪⎪⎩

{(x, 0)}, x �= 0, y = 0{(0, y)}, x = 0, y �= 0{(x, 0), (0, y)}, x, y �= 0{(0, 0)}, x, y = 0.

Clearly, F(T ) = {(x, 0), (0, y) : x, y ∈ [0, 1]}. Since, for x = (x1, y1) ∈ D, p =(x2, y2) ∈ F(T ), we have

H(T x, T p) = max{|x1 − x2|, |y1 − y2|} ≤√

(x1 − x2)2 + (y1 − y2)2 = ‖x − p‖.Hence T is quasi-nonexpansive.

Example 2.10 Consider D = [0, 1] with the usual norm. Define T : D → C B(D) by

T x =[

x + 1

2, 1

].

123

Strong convergence theorems of a finite family of quasi-nonexpansive...

Clearly, F(T ) = {1}. Since, for x ∈ [0, 1], we have

H(T x, T 1) = |x − 1|2

≤ |x − 1|.Hence T is quasi-nonexpansive.

Example 2.11 Consider D = [0, 1] × [0, 1] with the usual norm. Define T : D → C B(D)

by

T (x, y) = {x} ×[

y + 1

2, 1

].

Clearly, F(T ) = {(x, 1) : x ∈ [0, 1]}. Since, for x = (x1, y1) ∈ D, p = (x2, 1) ∈ F(T ),we have

H(T x, T p) =√

(x1 − x2)2 + (y1 − 1)

2

2

≤√

(x1 − x2)2 + (y1 − 1)2 = ‖x − p‖.

This implies that T is quasi-nonexpansive.

Lemma 2.12 Let D be a nonempty closed convex subset of a uniformly convex Banachspace X. Let {Ti }N

i=1 be a finite family of quasi-nonexpansive and Li -Lipschitz multi-valued

mappings of D into C B(D) with F := ⋂Ni=1 F(Ti ) �= ∅ and let {βi,n}N

i=1 be sequencesin (0, 1) such that 0 < lim infn→∞ βi,n ≤ lim supn→∞ βi,n < 1, ∀i = 1, 2, ..., N. Forevery n ∈ N, let Wn be the W -mappings generated by T1, T2, ..., TN and β1,n, β2,n, ..., βN ,n.For each i = 1, 2, ..., N, we assume that Ti p = {p}, ∀p ∈ F. Then {Wn} satisfies theSC-condition.

Proof It is clear that F ⊂ F(Wn). Let {vn} ⊂ D be a bounded sequence such thatlimn→∞ ‖an − vn‖ = 0 where an ∈ Wnvn . Then, there exists aN ,n ∈ TN UN−1,nvn such thatan = βN ,naN ,n + (1 − βN ,n)vn . We observe that

‖vn − aN ,n‖ = 1

βN ,n‖an − vn‖ → 0, as n → ∞.

Let p ∈ F and M = supn∈N{‖vn −aN ,n‖+2‖aN ,n − p‖}. From (1.3), there exists aN−1,n ∈TN−1UN−2,nvn such that uN−1,n = βN−1,naN−1,n + (1 − βN−1,n)vn . It follows by Lemma2.5 and Lemma 2.8 that

‖vn − p‖2 ≤ (‖vn − aN ,n‖ + ‖aN ,n − p‖)2

≤ M‖vn − aN ,n‖ + ‖aN ,n − p‖2

= M‖vn − aN ,n‖ + d(aN ,n, TN p)2

≤ M‖vn − aN ,n‖ + H(TN uN−1,n, TN p)2 for some uN−1,n ∈ UN−1,nvn

≤ M‖vn − aN ,n‖ + ‖uN−1,n − p‖2

= M‖vn − aN ,n‖ + ‖βN−1,naN−1,n + (1 − βN−1,n)vn − p‖2

≤ M‖vn − aN ,n‖ + βN−1,n‖aN−1,n − p‖2 + (1 − βN−1,n)‖vn − p‖2

−βN−1,n(1 − βN−1,n)g(‖aN−1,n − vn‖)= M‖vn − aN ,n‖ + βN−1,nd(aN−1,n, TN−1 p)2 + (1 − βN−1,n)‖vn − p‖2

− βN−1,n(1 − βN−1,n)g(‖aN−1,n − vn‖)≤ M‖vn − aN ,n‖ + βN−1,n H(TN−1uN−2,n, TN−1 p)2

123

S. Suantai et al.

+ (1 − βN−1,n)‖vn − p‖2 − βN−1,n(1 − βN−1,n)g(‖aN−1,n − vn‖)for some uN−2,n ∈ UN−2,nvn

≤ M‖vn − aN ,n‖ + βN−1,n‖uN−2,n − p‖2

+ (1 − βN−1,n)‖vn − p‖2 − βN−1,n(1 − βN−1,n)g(‖aN−1,n − vn‖)≤ M‖vn − aN ,n‖ + ‖vn − p‖2 − βN−1,n(1 − βN−1,n)g(‖aN−1,n − vn‖).

It follows that

βN−1,n(1 − βN−1,n)g(‖aN−1,n − vn‖) ≤ M‖vn − aN ,n‖.This implies by the property of the function g that

limn→∞ ‖aN−1,n − vn‖ = 0.

By continuing in this way, we have

limn→∞ ‖aN ,n − vn‖ = lim

n→∞ ‖aN−1,n − vn‖ = ... = limn→∞ ‖a1,n − vn‖ = 0. (2.7)

We next show that there exists ci,n ∈ Tivn such that limn→∞ ‖ci,n − vn‖ = 0 for eachi = 1, 2, ..., N . If i = 1, then we choose c1,n = a1,n , ∀n ∈ N. If i = 2, by Lemma 2.6 and(2.7), there exists c2,n ∈ T2vn and L2 > 0 such that

‖a2,n − c2,n‖ ≤ H(T2u1,n, T2vn) + 1

n

≤ L2‖u1,n − vn‖ + 1

n

≤ L2‖a1,n − vn‖ + 1

n→ 0, as n → ∞. (2.8)

This implies that

‖vn − c2,n‖ ≤ ‖vn − a2,n‖ + ‖a2,n − c2,n‖ → 0, as n → ∞. (2.9)

It follows from (2.7), (2.8) and (2.9) that

‖vn − c2,n‖ → 0, as n → ∞.

By continuing in the way, there exists ci,n ∈ Tivn such that

‖vn − ci,n‖ → 0, as n → ∞,

for each i = 1, 2, ..., N . ��

3 Main results

In this section, we prove a strong convergence theorem of the iteration (3.1) to find commonfixed points of a family of quasi-nonexpansive and Lipschitz multi-valued mappings.

Theorem 3.1 Let D be a nonempty closed convex subset of a real Hilbert space H. Let{Ti }N

i=1 be a finite family of quasi-nonexpansive and Li -Lipschitz multi-valued mappings of

D into C B(D) with F := ⋂Ni=1 F(Ti ) �= ∅. Let {βi,n}N

i=1 be sequences in (0, 1) such that0 < lim infn→∞ βi,n ≤ lim supn→∞ βi,n < 1 for all i = 1, 2, ..., N and let {αn} ⊂ (0, 1)

123

Strong convergence theorems of a finite family of quasi-nonexpansive...

such that lim supn→∞ αn < 1. For each n ≥ 1, let Wn be the W -mapping generated byT1, T2, ..., TN and β1,n, β2,n, ..., βN ,n. For each i = 1, 2, ..., N, we assume that Ti p = {p},∀p ∈ F. For an initial point x0 ∈ D with C1 = D and x1 = PC1 x0, let {xn} be a sequencegenerated by ⎧⎨

⎩yn ∈ αn xn + (1 − αn)Wn xn,

Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖xn − z‖},xn+1 = PCn+1 x0, ∀n ≥ 1.

(3.1)

Then {xn} converges strongly to zo = PF x0.

Proof We divide the proof into five steps.Step 1. Show that PCn+1 x0 is well-defined.By Lemma 2.7, we know that F is closed and convex. By Lemma 2.4, we see that Cn is

closed and convex for all n ≥ 1. Let p ∈ F , by Lemma 2.8, we have

‖yn − p‖ = ‖αn xn + (1 − αn)an − p‖ for some an ∈ Wn xn

≤ αn‖xn − p‖ + (1 − αn)‖an − p‖= αn‖xn − p‖ + (1 − αn)d(an, Wn p)

≤ αn‖xn − p‖ + (1 − αn)H(Wn xn, Wn p)

≤ ‖xn − p‖.It follows that F ⊂ Cn+1. This shows that PCn+1 x0 is well-defined.

Step 2. Show that limn→∞ ‖xn − x0‖ exists.Since F is a nonempty closed convex subset of D, there exists a unique element z0 =

PF x0 ∈ F ⊂ Cn . From xn = PCn x0, we obtain

‖xn − x0‖ ≤ ‖z0 − x0‖. (3.2)

Hence {‖xn − x0‖} is bounded.Since xn+1 = PCn+1 x0 ∈ Cn+1 ⊂ Cn , we also have

‖xn − x0‖ ≤ ‖xn+1 − x0‖. (3.3)

From (3.2) and (3.3), we get that limn→∞ ‖xn − x0‖ exists.Step 3. Show that {xn} is a Cauchy sequence.By the construction of the set Cn , we know that xm = PCm x0 ∈ Cm ⊂ Cn for m > n.

From Step 2, it follows that

‖xm − xn‖2 ≤ ‖xm − x0‖2 − ‖xn − x0‖2 → 0 (3.4)

as m, n → ∞. From Step 2, we obtain {xn} is a Cauchy sequence. Hence, there exists q ∈ Csuch that xn → q as n → ∞.

Step 4. Show that q ∈ F .From (3.4), we get

‖xn+1 − xn‖ → 0

as n → ∞. Since xn+1 ∈ Cn+1, we have

‖yn − xn‖ ≤ ‖yn − xn+1‖ + ‖xn+1 − xn‖ ≤ 2‖xn+1 − xn‖ → 0, (3.5)

123

S. Suantai et al.

as n → ∞. Since ‖an − xn‖ = 11−αn

‖yn − xn‖ → 0 as n → ∞ for some an ∈ Wn xn .

From Lemma 2.12, {Wn} satisfies the SC-condition with respect to {Ti }Ni=1. Then there exists

ci,n ∈ Ti xn such that

‖ci,n − xn‖ → 0 (3.6)

as n → ∞ for each i = 1, 2, ..., N . So we have

d(q, Ti q) ≤ ‖q − xn‖ + ‖xn − ci,n‖ + d(ci,n, Ti q)

≤ ‖q − xn‖ + ‖xn − ci,n‖ + H(Ti xn, Ti q)

≤ ‖q − xn‖ + ‖xn − ci,n‖ + Li‖q − xn‖ → 0, (3.7)

as n → ∞. So we obtain d(q, Ti q) = 0 for each i = 1, 2, ..., N . Hence q ∈ F .Step 5. Show that q = PF x0.Since xn = PCn x0 and F ⊂ Cn , we obtain⟨

x0 − xn, xn − p⟩ ≥ 0 ∀p ∈ F. (3.8)

By taking n → ∞ in (3.8), we obtain⟨x0 − q, q − p

⟩ ≥ 0 ∀p ∈ F.

This shows that q = PF x0 by Lemma 2.1. ��Corollary 3.2 Let D be a nonempty closed convex subset of a real Hilbert space H. LetT : D → C B(D) be a quasi-nonexpansive and L-Lipschitz multi-valued mapping withF(T ) �= ∅. Let {αn} ⊂ (0, 1) such that lim supn→∞ αn < 1. Assume that T p = {p},∀p ∈ F(T ). For an initial point x0 ∈ D with C1 = D and x1 = PC1 x0, let {xn} be asequence generated by⎧⎨

⎩yn ∈ αn xn + (1 − αn)T xn,

Cn+1 = {z ∈ Cn : ‖yn − z‖ ≤ ‖xn − z‖},xn+1 = PCn+1 x0, ∀n ≥ 1.

(3.9)

Then {xn} converges strongly to zo = PF(T )x0.

Remark 3.3 Corollary 3.2 generalizes Theorem TTK to the case of quasi-nonexpansive andLipschitz multi-valued mappings.

Acknowledgments The authors wish to thank the referees and the editor for valuable comments. PrasitCholamjiak was supported by the Thailand Research Fund, the Commission on Higher Education, and Uni-versity of Phayao under Grant No. MRG5580016.

References

1. Abbas, M., Khan, S.H., Khan, A.R., Agarwal, R.P.: Common fixed points of two multivalued nonexpansivemappings by one-step iterative scheme. Appl. Math. Lett. 24, 97–102 (2011)

2. Assad, N.A., Kirk, W.A.: Fixed point theorems for set-valued mappings of contractive type. Pacific J.Math. 43, 553–562 (1972)

3. Atsushiba, S., Takahashi, W.: Strong convergence theorems for a finite family of nonexpansive mappingsand applications, in: B.N. Prasad Birth Centenary Commemoration Volume. Indian J. Math. 41, 435–453(1999)

4. Bauschke, H.H., Matouskova, E., Reich, S.: Projection and proximal point methods: convergence resultsand counterexamples. Nonlinear Anal. 56, 715–738 (2004)

123

Strong convergence theorems of a finite family of quasi-nonexpansive...

5. Cholamjiak, W., Suantai, S.: A hybrid method for a countable family of multivalued maps, equilibriumproblems, and variational inequality problems. Discrete Dyn. Nat. Soc. 2010 (2010, Art. ID 349158)

6. Cholamjiak, P., Cholamjiak, W., Cho, Y.J., Suantai, S.: Weak and strong convergence to common fixedpoints of a countable family of multi-valued mappings in Banach spaces. Thai J. Math. 9, 505–520 (2011)

7. Genal, A., Lindenstrass, J.: An example concerning fixed points. Israel J. Math. 22, 81–86 (1975)8. Hu, T., Huang, J.C., Rhoades, B.E.: A general principle for Ishikawa iterations for multivalued mappings.

Indian J. Pure Appl. Math. 28, 1091–1098 (1997)9. Kim, T.H., Xu, H.K.: Strongly convergence of modified Mann iterations for with asymptotically nonex-

pansive mappings and semigroups. Nonlinear Anal. 64, 1140–1152 (2006)10. Lim, T.C.: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach

space. Bull. Am. Math. Soc. 80, 1123–1126 (1974)11. Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)12. Marino, G., Xu, H.K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert

spaces. J. Math. Anal. Appl. 329, 336–346 (2007)13. Nakajo, K., Takahashi, W.: Strongly convergence theorems for nonexpansive mappings and nonexpansive

semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)14. Panyanak, B.: Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput.

Math. Appl. 54, 872–877 (2007)15. Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal.

Appl. 67, 274–276 (1979)16. Sastry, K.P.R., Babu, G.V.R.: Convergence of Ishikawa iterates for a multivalued mappings with a fixed

point. Czechoslovak Math. J. 55, 817–826 (2005)17. Shahzad, N., Zegeye, H.: On Mann and Ishikawa iteration schemes for multi-valued maps in Banach

spaces. Nonlinear Anal. 71, 838–844 (2009)18. Shimoji, K., Takahashi, W.: Strong convergence to common fixed points of infinite nonexpansive mappings

and applications. Taiwanese J. Math. 5, 387–404 (2001)19. Shiau, C., Tan, K.K., Wong, C.S.: Quasi-nonexpansive multi-valued maps and selection. Fund. Math. 87,

109–119 (1975)20. Song, Y., Wang, H.: Erratum to “Mann and Ishikawa iterative processes for multivalued mappings in

Banach spaces”[Comput. Math. Appl. 54, : 872–877]. Comput. Math. Appl. 55(2008), 2999–3002 (2007)21. Suantai, S., Cholamjiak, W., Cholamjiak, P.: An implicit iteration process for a finite family of multi-

valued mappings in Banach spaces. Appl. Math. Lett. 25, 1656–1660 (2012)22. Takahashi, W.: Weak and strong convergence theorems for families of nonexpansive mappings and their

applications. Ann. Univ. Mariae Curie-Sklodowska Sect. A 51, 277–292 (1997)23. Takahashi, W.: Convex Analysis and Approximation of Fixed Points. Yokohama Publishers, Yokohama

(2000)24. Nakajo, K., Shimoji, K., Takahashi, W.: Strong convergence theorems by the hybrid method for families

of mappings in Hilbert spaces. Taiwanese J. Math. 10, 339–360 (2006)25. Takahashi, W., Takeuchi, Y., Kubota, R.: Strong convergence theorems by hybrid methods for families of

nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)26. Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)

123