Weak and strong convergence of an implicit iterative process for a countable family of nonexpansive mappings in Banach spaces

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LVIII, 2004 SECTIO A 69–78

MISAKO KIKKAWA and WATARU TAKAHASHI

Weak and strong convergence of an implicit iterative process for a countable family of nonexpansive mappings in Banach spaces

Dedicated to W.A. Kirk on the occasion of His Honorary Doctorate of

Maria Curie-Skłodowska University

Abstract. In this paper, we introduce an implicit sequence for an infinite family of nonexpansive mappings in a uniformly convex Banach space and prove weak and strong convergence theorems for finding a common fixed point of the mappings.

1. Introduction. Let H be a Hilbert space and let C be a closed convex subset of H. Let {T₁, T2, . . . , TN} be nonexpansive mappings of C into itself such that TN

i=1F (T_i) is nonempty. In 2001, Xu and Ori [15] introduced an implicit iteration process {x_n} for a finite family of nonexpansive mappings as follows: x₀∈ C and

x₁ = t₁x₀+ (1 − t₁)T₁x₁, x₂ = t₂x₁+ (1 − t₂)T₂x₂,

...

x_N = t_Nx_{N −1}+ (1 − t_N)T_Nx_N,

xN +1= tN +1xN + (1 − tN +1)T1xN +1, xN +2= tN +2xN +1+ (1 − tN +2)T2xN +2,

...

where {t_n} is a real sequence in (0, 1) and they proved that this process converges weakly to a common fixed point of {T₁, T₂, . . . , T_N} in a Hilbert space setting. Further, Xu and Ori [15] pointed out that it is yet unclear what assumptions on the mappings {T₁, T₂, . . . , T_N} and/or the parameters {t_n} are sufficient to guarantee the strong convergence of {x_n}. In 2002, Liu [5] gave an affirmative answer to that question as follows (see also [10]): Let E be a uniformly convex Banach space and let C be a nonempty bounded closed convex subset of E. Let {T_i : i = 1, 2, . . . , N } be a finite family of nonexpansive mappings of C into itself such that TN

i=1F (T_i) is nonempty.

Let {x_n} be a sequence generated by implicit iteration process. If {t_n} and d satisfy 0 < d < 1 and 0 < t_n ≤ d < 1 and there exists some T ∈ {T_i : i = 1, 2, . . . , N } which is semi-compact, then, {x_n} converges strongly to z ∈ TN

i=1F (Ti). Further, in 2003, Sun [9] proved that the modified implicit iteration process for a finite family of asymptotically quasi- nonexpansive mappings converges strongly to a common fixed point of the mappings in a uniformly convex Banach space, requiring one member T in the family to be semi-compact.

In this paper, we introduce an implicit sequence for an infinite family of nonexpansive mappings in a uniformly convex Banach space and prove weak and strong convergence theorems for finding a common fixed point of the mappings.

2. Preliminaries and lemmas. Let E be a real Banach space. Let C be a nonempty closed convex subset of E. Then a mapping T of C into itself is called nonexpansive if kT x − T yk ≤ kx − yk for any x, y ∈ C. For a mapping T of C into itself, we denote by F (T ) the set of fixed points of T , i.e., F (T ) = {x ∈ C : T x = x}. We also denote by N the set of all natural numbers and by R and R⁺ the sets of all real numbers and all nonnegative real numbers, respectively. A Banach space E is called uniformly convex if for any two sequences {x_n}, {y_n} in E such that kx_nk = ky_nk = 1 and limn→∞kx_n+ ynk = 2, lim_n→∞kx_n− y_nk = 0 holds. E is said to satisfy Opial’s condition [6] if for any sequence {x_n} in E such that {x_n} converges weakly to z ∈ E, lim inf_n→∞kx_n− zk < lim inf_n→∞kx_n− yk holds for all y ∈ E with y 6= z. All Hilbert spaces and l^p (1 < p < ∞) satisfy Opial’s condition, while L^p (1 < p < ∞, p 6= 2) do not.

Let T₁, T2, . . . be an infinite sequence of mappings of C into itself and let λ₁, λ₂, . . . be real numbers such that 0 ≤ λ_i ≤ 1 for every i ∈ N. Then, for any n ∈ N, Takahashi [11] (see also [8], [13]) defined a mapping Wn of C

into itself as follows:

Un,n+1= I,

Un,n= λnTnUn,n+1+ (1 − λn)I, Un,n−1= λn−1Tn−1Un,n+ (1 − λn−1)I,

...

Un,k= λkTkUn,k+1+ (1 − λk)I, U_n,k−1= λ_k−1T_k−1U_n,k+ (1 − λ_k−1)I,

...

Un,2= λ2T2Un,3+ (1 − λ2)I, Wn= Un,1= λ1T1Un,2+ (1 − λ1)I.

Such a mapping W_n is called the W -mapping generated by T_n, Tn−1, . . . , T1

and λ_n, λ_n−1, . . . , λ₁.

Using [8] and [1], we obtain the following two lemmas.

Lemma 1. Let C be a nonempty closed convex subset of a Banach space E.

Let T₁, T2, . . . be nonexpansive mappings of C into itself such thatT∞ i=1F (Ti) is nonempty and let λ₁, λ₂, . . . be real numbers such that 0 < λ₁ ≤ 1 and 0 < λi ≤ b < 1 for any i = 2, 3, . . .. Then for every x ∈ C and k ∈ N, the lim_n→∞U_n,kx exists.

Using Lemma 1, for k ∈ N, we define mappings U∞,k and U of C into itself as follows:

U∞,kx = lim

n→∞U_n,kx and

U x = lim

n→∞W_nx = lim

n→∞U_n,1x

for every x ∈ C. Such a U is called the W -mapping generated by T₁, T2, . . ., and λ₁, λ₂, . . ..

Lemma 2. Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let T₁, T2, . . . be nonexpansive mappings of C into itself such that T∞

i=1F (T_i) is nonempty and let λ₁, λ₂, . . . be real numbers such that 0 < λ₁ ≤ 1 and 0 < λ_i ≤ b < 1 for any i = 2, 3, . . .. Let W_n (n = 1, 2, . . .) be the W -mappings of C into itself generated by Tn, Tn−1, . . . , T1

and λ_n, λ_n−1, . . . , λ₁ and let U be the W -mapping generated by T₁, T₂, . . . and λ₁, λ2, . . .. Then F (Wn) =Tn

i=1F (Ti) and F (U ) =T∞

i=1F (Ti).

The following lemma was proved by Xu [14].

Lemma 3. Let E be a uniformly convex Banach space and let r > 0. Then, there exists a continuous, strictly increasing and convex function g : R⁺→ R⁺ with g(0) = 0 such that

kλx + (1 − λ)yk² ≤ λkxk²+ (1 − λ)kyk²− λ(1 − λ)g(kx − yk) for all x, y ∈ B_r and 0 ≤ λ ≤ 1, where B_r= {x ∈ E : kxk ≤ r}.

We also know the following lemma proved by Schu [7].

Lemma 4. Let E be a uniformly convex Banach space, let {t_n} be a real sequence such that 0 < b ≤ t_n ≤ c < 1 for n ≥ 1 and let a ≥ 0. Sup- pose that {x_n} and {y_n} are sequences of E such that lim sup_n→∞kx_nk ≤ a, lim sup_n→∞ky_nk ≤ a and lim_n→∞kt_nx_n + (1 − t_n)y_nk = a. Then limn→∞kx_n− y_nk = 0.

The following lemma was proved by Browder [2].

Lemma 5. Let C be a nonempty bounded closed convex subset of a uni- formly convex Banach space E and let T be a nonexpansive mapping of C into itself. If {x_n} converges weakly to z ∈ C and {x_n− T x_n} converges strongly to 0, then T z = z.

3. Weak convergence theorem. In this section, we prove a weak conver- gence theorem of the implicit iteration process for finding a common fixed point of a countable family of nonexpansive mappings in a Banach space.

Theorem 6. Let E be a uniformly convex Banach space which satisfies Opial’s condition. Let C be a nonempty closed convex subset of E. Let {T_n} be a countable family of nonexpansive mappings of C into itself with a nonempty common fixed point set T∞

i=1F (T_i). Let b be a real number with 0 < b < 1 and let λ1, λ2, . . . be real numbers such that 0 < λ1 ≤ 1 and 0 < λ_i≤ b < 1 for every i = 2, 3, . . .. Let W_n (n = 1, 2, . . .) be W -mappings of C into itself generated by T_n, Tn−1, . . . , T1 and λ_n, λn−1, . . . , λ1. Let U be the W -mapping generated by T₁, T₂, . . . and λ₁, λ₂, . . ., i.e.,

U x = lim

n→∞Wnx = lim

n→∞Un,1x for every x ∈ C. Let {x_n} be a sequence generated by

x0= x ∈ C,

xn= αnxn−1+ (1 − αn)Wnxn, n = 1, 2, . . . ,

where {α_n} and d satisfy 0 < d < 1 and 0 < α_n ≤ d < 1. Then, {x_n} converges weakly to z ∈T∞

n=1F (Tn).

Proof. From Lemma 2, we obtain T∞

n=1F (Tn) = T∞

n=1F (Wn) = F (U ).

Let x ∈ C. Then for u ∈ T∞

n=1F (T_n), we obtain that D = {y ∈ C : ky − uk ≤ kx − uk} is a bounded closed convex subset of C and x ∈ D.

Further, for any y ∈ D, we have T_ny ∈ C and kT_ny − uk ≤ ky − uk

≤ kx − uk.

Then D is invariant under T_n for all n ∈ N. So, without loss of generality, we may assume that C is bounded.

Let x₀ ∈ C and define S₁ by S₁x = α₁x₀ + (1 − α₁)W₁x for all x ∈ C.

Then, we have, for all x, y ∈ C,

kS₁x − S₁yk ≤ (1 − α₁)kW₁x − W₁yk

≤ (1 − α₁)kx − yk.

So, we obtain that S₁ is a contraction mapping C into itself. By the Banach contraction principle, there exists a unique point x₁ such that x₁ = S₁x₁. Similarly, for n ∈ N, we define Snby S_nx = αnx0+(1−αn)Wnx for all x ∈ C and obtain a unique point x_n∈ C such that x_n= S_nx_n. Let u ∈ F (U ). By the definition of {x_n} and Lemma 3, we have

kx_n− uk²= kαn(xn−1− u) + (1 − α_n)(Wnxn− u)k²

≤ α_nkx_n−1− uk²+ (1 − α_n)kW_nx_n− uk²

− α_n(1 − α_n)g(kW_nx_n− x_n−1k)

≤ α_nkx_n−1− uk²+ (1 − αn)kxn− uk²

− α_n(1 − α_n)g(kW_nx_n− x_n−1k)

≤ α_nkx_n−1− uk²+ (1 − α_n)kx_n− uk²

for some g : R⁺→ R⁺, which is continuous, strictly increasing, convex and g(0) = 0. Therefore, we obtain kx_n− uk ≤ kx_n−1− uk, and hence the limit of {kx_n− uk} exists for u ∈ F (U ). Since

αn(1 − αn)g(kWnxn− x_n−1k) ≤ α_n(kxn−1− uk²− kx_n− uk²) for all n ∈ N and from 0 < αn≤ d < 1, we have

(1 − d)g(kWnxn− x_n−1k) ≤ kx_n−1− uk²− kx_n− uk² and hence lim_n→∞g(kWnxn− x_n−1k) = 0. This implies

(1) lim

n→∞kW_nx_n− x_n−1k = 0.

Therefore, from kx_n− x_n−1k ≤ (1 − α_n)kWnxn− x_n−1k, we have

(2) lim

n→∞kx_n− x_n−1k = 0.

Further, from kW_nxn− x_nk ≤ kW_nxn− x_n−1k + kx_n−1− x_nk, (1) and (2), we obtain

(3) lim

n→∞kW_nx_n− x_nk = 0.

Since {x_n} is bounded, we assume that there exists a subsequence {x_nj} ⊂ {x_n} such that {x_nj} converges weakly to w. Suppose that w 6= U w. From Opial’s condition, the definition of U and (3), we have

lim inf

j→∞ kx_nj− wk < lim inf

j→∞ kx_nj− U wk

≤ lim inf

j→∞ (kx_nj− W_njx_njk

+ kW_njx_nj− W_njwk + kW_njw − U wk)

≤ lim inf

j→∞ (kx_nj− W_njx_njk

+ kxnj− wk + kW_njw − U wk)

= lim inf

j→∞ kx_nj− wk.

This is a contradiction. Hence, we obtain w ∈ F (U ). To complete the proof, we prove that {x_n} has at most one weak subsequential limit. We assume that z₁and z₂are two distinct weak subsequential limits of the subsequences {x_ni} and {x_nj} of {x_n}, respectively. From Opial’s condition, we obtain

n→∞limkx_n− z₁k = lim

i→∞kx_ni− z₁k < lim

i→∞kx_ni− z₂k = lim

n→∞kx_n− z₂k

= lim

j→∞kx_nj− z₂k < lim

j→∞kx_nj− z₁k = lim

n→∞kx_n− z₁k.

This is a contradiction. So, {x_n} converges weakly to z ∈T∞

n=1F (T_n). This

completes the proof. 

As a direct consequence of Theorem 6, we obtain the following result.

Corollary 7. Let X be a Hilbert space. Let C be a nonempty closed convex subset of X. Let {T_n} be a countable family of nonexpansive mappings of C into itself such thatT∞

i=1F (T_i) is nonempty. Let b be a real number with 0 < b < 1 and let λ1, λ2, . . . be real numbers such that 0 < λ1 ≤ 1 and 0 < λ_i≤ b < 1 for every i = 2, 3, . . .. Let W_n (n = 1, 2, . . .) be W -mappings of C into itself generated by T_n, Tn−1, . . . , T1 and λ_n, λn−1, . . . , λ1. Let U be the W -mapping generated by T₁, T₂, . . . and λ₁, λ₂, . . ., i.e.,

U x = lim

n→∞W_nx = lim

n→∞U_n,1x

for every x ∈ C. Let {x_n} be a sequence generated by

x0= x ∈ C,

xn= αnxn−1+ (1 − αn)Wnxn, n = 1, 2, . . . ,

where {α_n} and d satisfy 0 < d < 1 and 0 < α_n ≤ d < 1. Then, {x_n} converges weakly to z ∈T∞

n=1F (Tn).

4. Strong convergence theorem. In this section, we consider the strong convergence of the implicit iterative process generated by a countable fam- ily of nonexpansive mappings in a Banach space. We need the following definition [3].

Definition 1. Let C be a closed subset of a Banach space E. A mapping T from C into itself is said to be semi-compact, if for any sequence {x_n} in C such that lim_n→∞kx_n− T x_nk = 0, then there exists a subsequence {x_ni} ⊂ {x_n} such that x_ni → x^∗ ∈ C, where → denotes the strong convergence.

Theorem 8. Let E be a uniformly convex Banach space. Let C be a nonempty closed convex subset of E. Let {T_n} be a countable family of non- expansive mappings of C into itself with a nonempty common fixed point set T∞

i=1F (Ti). Let a and b be real numbers with 0 < a ≤ b < 1 and let λ1, λ2, . . . be real numbers such that 0 < a ≤ λ_i≤ b < 1 for every i = 1, 2, . . .. Let W_n (n = 1, 2, . . .) be W -mappings of C into itself generated by Tn, Tn−1, . . . , T1

and λ_n, λn−1, . . . , λ1. Let U be the W -mapping generated by T₁, T2, . . . and λ₁, λ₂, . . ., i.e.,

U x = lim

n→∞W_nx = lim

n→∞U_n,1x for every x ∈ C. Let {x_n} be a sequence generated by

x₀= x ∈ C,

x_n= α_nx_n−1+ (1 − α_n)W_nx_n, n = 1, 2, . . . ,

where {α_n} and d satisfy 0 < d < 1 and 0 < α_n ≤ d < 1. If there exists some T ∈ {T_i : i ∈ N} which is semi-compact, then {xn} converges strongly to z ∈T∞

n=1F (T_n),

Proof. Since a uniformly convex Banach space is strictly convex, from Lemma 2, we have T∞

n=1F (Tn) =T∞

n=1F (Wn) = F (U ). As in the proof of Theorem 5, we may assume that C is bounded and obtain that the limit of {kx_n− uk} exists for any u ∈ F (U ). Let c = lim_n→∞kx_n− uk. Fix k ∈ N.

For all n ∈ N with n ≥ k, we have

kU_n,kxn− uk ≤ kx_n− uk.

So, we obtain lim sup_n→∞kU_n,kxn− uk ≤ c. By the definition of {x_n}, we have

kx_n− uk = kα_n(x_n−1− u) + (1 − α_n)(W_nx_n− u)k

≤ α_nkx_n−1− uk + (1 − α_n)kWnxn− uk

≤ α_nkx_n−1− uk

+ (1 − α_n){λ₁kT₁U_n,2x_n− uk + (1 − λ₁)kx_n− uk}

≤ α_nkx_n−1− uk

+ (1 − α_n){λ₁kU_n,2x_n− uk + (1 − λ₁)kx_n− uk}

≤ α_nkx_n−1− uk

+ (1 − αn){λ1λ2kU_n,3xn− uk + (1 − λ₁λ2)kxn− uk}

...

≤ α_nkx_n−1− uk + (1 − α_n)

k−1

Q

i=1

λikU_n,kxn− uk + (1 − α_n)(1 −

k−1Q

i=1

λ_i)kx_n− uk.

Therefore, we obtain kx_n− uk ≤ αn

(1 − αn)Qk−1 i=1 λi

(kxn−1− uk − kx_n− uk) + kU_n,kxn− uk

≤ d

(1 − d)Qk−1 i=1λi

(kx_n−1− uk − kx_n− uk) + kU_n,kx_n− uk.

Consequently, we have c ≤ lim inf_n→∞kU_n,kx_n− uk and hence

n→∞limkU_n,kx_n− uk = c

for all k ∈ N. Moreover, since c = lim

n→∞kU_n,kx_n− uk

= lim

n→∞kλ_k(T_kU_n,k+1x_n− u) + (1 − λ_k)(x_n− u)k and

lim sup

n→∞

kT_kU_n,k+1xn− uk ≤ lim sup

n→∞

kU_n,k+1xn− uk ≤ c,

we obtain lim_n→∞kT_kUn,k+1xn− x_nk = 0 by Lemma 4. For any k ∈ N, we have

kT_kx_n− x_nk ≤ kT_kx_n− T_kU_n,k+1x_nk + kT_kU_n,k+1x_n− x_nk

≤ kx_n− U_n,k+1x_nk + kT_kU_n,k+1x_n− x_nk

≤ λ_k+1kT_k+1Un,k+2xn− x_nk + kT_kUn,k+1xn− x_nk.

Hence we have lim sup_n→∞kT_kxn− x_nk ≤ 0. This implies

(4) lim

n→∞kT_kxn− x_nk = 0.

for all k ∈ N. By the assumption, there exists a subsequence {xni} of {x_n} such that x_ni → p ∈ C as i → ∞. From (4), we have

kp − T_kpk = lim

i→∞kx_ni− T_kxnik = 0

for all k ∈ N. This implies p ∈ F (Tk) for all k ∈ N. Therefore we have lim inf_n→∞d(x_n, F (U )) = 0. For any u ∈ F (U ), we have

kx_n− uk ≤ kx_n−1− uk and hence

d(xn, F (U )) ≤ d(xn−1, F (U )).

So, we obtain lim_n→∞d(x_n, F (U )) = 0. Let us prove that {x_n} is a Cauchy sequence. For any m, n ∈ N, we have

kx_n+m− uk ≤ kx_n− uk

for any u ∈ F (U ). Since lim_n→∞d(xn, F (U )) = 0, for any  > 0 there exists n₀∈ N such that d(xn, F (U )) < ₂^ for any n ≥ n₀. Hence there exists u1 ∈ F (U ) such that kx_n0 − u₁k < ₂^. So, for any n, m ≥ n₀, we have

kx_m− x_nk ≤ kx_m− u₁k + kx_n− u₁k

≤ kx_n0 − u₁k + kx_n0− u₁k

<  2 + 

2 = .

Then, {x_n} is a Cauchy sequence, and hence lim_n→∞xn exists in C. Let u = lim_n→∞x_n. From (4) and Lemma 5, we have u ∈ F (T_k) for all k ∈ N.

So, {x_n} converges strongly to u ∈ F (U ). 

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Misako Kikkawa Wataru Takahashi

Department of Mathematical Department of Mathematical and Computing Sciences and Computing Sciences Tokyo Institute of Technology Tokyo Institute of Technology Oh-okayama, Meguro-ku Oh-okayama, Meguro-ku

Tokyo, 152-8552 Tokyo, 152-8552

Japan Japan

e-mail: takahata@is.titech.ac.jp e-mail: wataru@is.titech.ac.jp Received October 4, 2004