Nonlinear Analysis 66 (2007) 591–603 www.elsevier.com/locate/na
Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces✩ Yisheng Song ∗ , Rudong Chen Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, PR China Received 17 November 2005; accepted 1 December 2005
Abstract Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E ∗ , and K be a nonempty closed convex subset of E. Suppose that {Tn } (n = 1, 2, . . .) is a uniformly asymptotically regular sequence of nonexpansive mappings from K into itself such that F(T F := ∞ n ) = ∅. For arbitrary initial value x0 ∈ K and fixed contractive mapping f : K → K , n=1 define iteratively a sequence {xn } as follows: xn+1 = λn+1 f (xn ) + (1 − λn+1 )Tn+1 xn ,
n ≥ 0, ∞ where {λn } ⊂ (0, 1) satisfies limn→∞ λn = 0 and n=1 λn = ∞. We prove that {xn } converges strongly to p ∈ F, as n → ∞, where p is the unique solution in F to the following variational inequality:
(I − f ) p, j ( p − u) ≤ 0 for all u ∈ F(T ). Our results extend and improve the corresponding ones given by O’Hara et al. [J.G. O’Hara, P. Pillay, H.-K. Xu, Iterative approaches to finding nearest common fixed point of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426], J.S. Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520], H.K. Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] and O’Hara et al. [J.G. O’Hara, P. Pillay, H.-K. Xu, Iterative approaches to convex feasibility problem in Banach space, Nonlinear Anal. Available online 20 October 2005. doi:10.1016/j.na.2005.07.36]. c 2005 Elsevier Ltd. All rights reserved.
✩ This work is supported by the National Natural Science Foundation of China, Grant (10471033) and Grant (10271011). ∗ Corresponding author. Tel.: +86 222 455 67 54. E-mail addresses:
[email protected] (Y. Song),
[email protected] (R. Chen).
c 2005 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2005.12.004
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MSC: 47H05; 47H10; 47H17 Keywords: Uniformly asymptotically regular mapping sequence; Viscosity approximation methods; A family of infinitely many nonexpansive maps; Weakly sequentially continuous duality mapping
1. Introduction Let K be a nonempty closed convex subset of a Banach space E and let T1 , T2 , . . . , TN be nonexpansive mappings from K into itself (recall that a mapping T : K → K is nonexpansive if T x − T y ≤ x − y ∀x, y ∈ K ). We consider an iteration scheme: for N nonexpansive mappings T1 , T2 , . . . , TN and y, x 0 ∈ K , x n+1 = λn+1 y + (1 − λn+1 )Tn+1 x n ,
n ≥ 0.
(1.1)
In 1967, Halpern [6] firstly introduced the iteration scheme (1.1) for y = 0, N = 1 (that is, he considered only one mapping T ). He pointed out that the control conditions (C1) limn→∞ λn = 0 and (C2) ∞ n=1 λn = ∞ are necessary for the convergence of the iteration scheme (1.1) to a fixed point of T . Subsequently, many mathematicians studied the iteration scheme (1.1) (see [1, 10,15,12,13,17]). Recently, O’Hara et al. [9] generalized the result of Shimizu and Takahashi [13] in the framework of a Hilbert space. Very recently, Jung [7] and O’Hara et al. [18] extended the main results of O’Hara et al. [9] to a uniformly smooth Banach space. Unfortunately, there are gaps in the proofs of [18, Theorem 3.1] and [7, Theorem 5] and [9, Theorem 3.3]. That is, in their proofs, they showed only that ∞
F(Tn ) = F ⊂ F(T ),
n=1
and ∞ n=1
F(Tn ) = F ⊂ F(V ) =
N
F(Vk ),
k=1
but it is not clear whether or not Q(y) (Q F (V ) a or Py) is a fixed point of Tn (n ≥ 0), where Q or P is the unique sunny nonexpansive retraction from C onto F(T ). Hence, the following inequality in step 7 in the proofs for [18, Theorem 3.1] is very questionable: Case 1: Φ(x n − Q(y)) = Φ((1 − λn )(Tn x n−1 − Q(y)) + λn (y − Q(y))) ≤ (1 − λn )Φ(Tn x n−1 − Q(y)) + λn y − Q(y), Jφ (x n − Q(y)) ≤ (1 − λn )Φ(x n−1 − Q(y)) + λn y − Q(y), Jφ (x n − Q(y)) . Case 2: x n − Q(y)2 ≤ (1 − λn )2 Tn x n−1 − Q(y)2 + 2λn y − Q(y), J (x n − Q(y)) ≤ (1 − λn )2 x n−1 − Q(y)2 + 2λn y − Q(y), J (x n − Q(y)) . Also, the proofs of [9, Theorem 3.3] and equation (8) of [7] is doubtful.
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For T : K → K a nonexpansive mapping with F(T ) = ∅, and f : K → K a fixed contractive mapping, Xu [8] propose the explicit iterative process {x n } given by x n+1 = αn f (x n ) + (1 − αn )T x n ,
(1.2)
and prove that the explicit iterative process {x n } converges to a fixed point p of T in a uniformly smooth Banach space. In this paper, we introduce the following iterative sequence {x n } defined by (1.3) for the nonexpansive mapping sequence {Tn } (if f (x n ) ≡ u ∈ K in (1.3), we can get (1.1)): x n+1 = λn+1 f (x n ) + (1 − λn+1 )Tn+1 x n ,
n ≥ 0.
(1.3)
We first introduce a new concept of a uniformly asymptotically regular sequence {Tn } of nonexpansive mappings, and then prove several strong convergence results by using this concept. The results presented in this paper not only overcome the gaps in [18] and [7] and [9], but also extend and improve the corresponding results of [7–9,18]. 2. Preliminaries Throughout this paper, it is assumed that E is a real Banach space with norm ·, and we let ∗ J denote the normalized duality mapping from E into 2 E given by J (x) = { f ∈ E ∗ , x, f = x f , x = f },
∀ x ∈ E,
where E ∗ denotes the dual space of E and ·, · denotes the generalized duality pairing. In the following, we shall denote the single-valued duality mapping by j , and define F(T ) = {x ∈ E; T x = x}. When {x n } is a sequence in E, then x n → x (respectively x n x, x n x) will denote strong (respectively weak, weak∗ ) convergence of the sequence {x n } to x. Let E be a real Banach space and T a mapping with domain D(T ) and range R(T ) in E. T is called nonexpansive (respectively, contractive) if for any x, y ∈ D(T ), such that T x − T y ≤ x − y , (respectively, T x − T y ≤ β x − y
for some 0 < β < 1).
Recall that a Banach space E is said to be smooth if the duality mapping J is single valued. In this case, the duality mapping J is strong–weak∗ continuous [14, Lemma 4.3.3]. A Banach space E is said to be strictly convex if x = y = 1,
x = y implies
x + y < 1. 2
A Banach space E is said to be uniformly convex if δ E (ε) > 0 for all ε > 0, where δ E (ε) is modulus of convexity of E defined by x + y δ E (ε) = inf 1 − : x ≤ 1, y ≤ 1, x − y ≥ ε , ε ∈ [0, 2]. 2 A uniformly convex Banach space E is reflexive and strictly convex [14, Theorem 4.1.6, Theorem 4.1.2]. The following lemma was proved by Bruck [2]:
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Lemma 2.1 ([2, Lemma 3]). Let K be a nonempty closed convex subset of a strictly convex Banach space E. For each n ≥ 0, Tn : K → E is a nonexpansive mapping. Then there exists a nonexpansive mapping T : K → E such that ∞
F(T ) =
F(Tn ).
n=0
∞ ∞ In particular, if βn Tn satisfies the above n=0 F(Tn ) = ∅, then the mapping T = n=0 requirement, where {βn } is a positive real number sequence such that ∞ n=0 βn = 1. If a Banach space E admits a sequentially continuous duality mapping J from weak topology to weak star topology, then by Lemma 1 of reference [5], we have that the duality mapping J is single-valued. In this case, the duality mapping J is also said to be weakly sequentially continuous, i.e., for each {x n } ⊂ E with x n x, we have J (x n ) J (x) [5,7,11]. A Banach space E is said to satisfy Opial’s condition if for any sequence {x n } in E, x n x (n → ∞) implies lim sup x n − x < lim sup x n − y , n→∞
n→∞
∀y ∈ E with x = y.
By Theorem 1 of [5], we know that if E admits a weakly sequentially continuous duality mapping, then E satisfies Opial’s condition, and E is smooth; for the details, see [5]. Now, we introduce the concepts of an asymptotically regular sequence of mappings and a uniformly asymptotically regular sequence of mappings. Let C be a nonempty closed convex subset of a Banach space E, and Tn : C → C, n ≥ 1; then the mapping sequence {Tn } is said to be asymptotically regular (for short, a.r.) if for all m ≥ 1 and any bounded subset K of C, lim Tm (Tn x) − Tn x = 0,
n→∞
∀x ∈ K .
The mapping sequence {Tn } is said to be uniformly asymptotically regular (for short, u.a.r.) on C if for all m ≥ 1 and any bounded subset K of C, lim sup Tm (Tn x) − Tn x = 0.
n→∞ x∈K
The following lemma was proved by Bruck in [3,4]. Lemma 2.2 (Bruck [3,4]). Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E and T : C → C be nonexpansive. For each x ∈ C, if we define j x, then T Tn x = n1 n−1 j =0 lim sup Tn x − T (Tn x) = 0.
n→∞ x∈C
Lemma 2.2 has been extended to a pair of mappings [13, Lemma 1]. Lemma 2.3 (Shimizu and Takahashi [13]). Let C be a nonempty bounded closed convex subset of a Hilbert space H and T, S : C → C be two nonexpansive mappings such that ST = T S. For each x ∈ C, we define Tn x =
n−1 2 S i T j x, n(n + 1) k=0 i+ j =k
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then lim sup Tn x − T (Tn x) = 0,
n→∞ x∈C
lim sup Tn x − S(Tn x) = 0.
n→∞ x∈C
We now give several examples of u.a.r. sequences of maps. Example 1. The mapping sequence {Tn } that appeared in Lemma 2.3 is u.a.r. Indeed, for each fixed j ≥ 1, we have T 2 (Tn x) − Tn x ≤ T 2 Tn x − T (Tn x) + T (Tn x) − Tn x ≤ 2T (Tn x) − Tn x T (Tn x) − Tn x ≤ T 3 (Tn x) − T 2 Tn x + T 2 (Tn x) − Tn x 3
≤ 3T (Tn x) − Tn x .. . T j (Tn x) − Tn x ≤ j T (Tn x) − Tn x. Thus, S i T j x(Tn x) − Tn x ≤ S i T j Tn x − S i (Tn x) + S i (Tn x) − Tn x ≤ T j (Tn x) − Tn x + S i (Tn x) − Tn x ≤ j T (Tn x) − Tn x + i S(Tn x) − Tn x. Therefore,
m−1 2 i j Tm (Tn x) − Tn x = S T (Tn x) − Tn x m(m + 1) k=0 i+ j =k ≤
m−1 2 S i T j (Tn x) − Tn x m(m + 1) k=0 i+ j =k
≤
m−1 2 ( j T (Tn x) − Tn x + i S(Tn x) − Tn x) m(m + 1) k=0 i+ j =k
=
m−1 k(k + 1) 2 (T (Tn x) − Tn x + S(Tn x) − Tn x) m(m + 1) k=0 2
m +2 (T (Tn x) − Tn x + S(Tn x) − Tn x). 3 Hence, using Lemma 2.3, we get m +2 lim sup Tm (Tn x) − Tn x ≤ lim sup T (Tn x) − Tn x n→∞ x∈C n→∞ x∈C 3
+ lim sup S(Tn x) − Tn x = 0. =
n→∞ x∈C
By using a reasoning similar to that of the Example 1, we can show that:
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Example 2. The mapping sequence {Tn } that appeared in Lemma 2.2 is u.a.r. Example 3. Let T be a contractive mapping on the nonempty bounded closed subset C of the Banach space E with Lipschitz constant β < 1. Put Sn = T n , n ≥ 1; then the sequence {Sn } is u.a.r. Indeed, ∀x ∈ C, we have Sm (Sn x) − Sn x = T n+m x − T n x ≤
m−1
T n+i+1 x − T n+i x
i=0
≤
m−1
β n+i T x − x ≤
i=0
Therefore, lim
n→∞
βn T x − x. 1−β
sup Sm (Sn x) − Sn x = 0. x∈C
In the following, we also need the following lemmas which can be found in [7] or [7–9,16] respectively. Lemma 2.4 (Demiclosedness Principle). Let C be a nonempty closed convex subset of a reflexive Banach space E which satisfies Opial’s condition, and suppose T : C → E is nonexpansive. Then the mapping I − T is demiclosed at zero, i.e., x n x, x n − T x n → 0
implies x = T x.
Lemma 2.5 (Xu [16]). Let {an } be a sequence of nonnegative real numbers satisfying the property an+1 ≤ (1 − γn )an + βn ,
n ≥ 0,
where {γn } ⊂ (0, 1) and {βn } is a real number sequence such that (i) ∞ n=0 γn = ∞; (ii) either lim supn→∞ βγnn ≤ 0 or ∞ n=0 |βn | < +∞. Then {an } converges to zero, as n → ∞. 3. Main results Let K be a nonempty closed convex subset of Banach space E, and T : K → K a nonexpansive mapping. For any fixed contractive mapping f : K → K with contractive coefficient β ∈ (0, 1) and each t ∈ (0, 1), it is obvious that the mapping t f + (1 − t)T is a contractive mapping from K to itself. By the Banach contractive principle, we can define the following iterative process for {Tm } (m = 1, 2, . . .) of a nonexpansive mapping sequence, from K into itself: z m = tm f (z m ) + (1 − tm )Tm z m .
(3.1)
First, we show that the sequence {z m } strongly converges to some common fixed point of {Tm } (m = 1, 2, . . .).
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Theorem 3.1. Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J from E to E ∗ . Suppose K is a nonempty closed convex subset of E, and {Tm } (m = 1, 2, . . .) is a u.a.r. sequence of nonexpansive mappings from K into itself such that F := ∞ n=1 F(Tn ) = ∅. Let {z m } be defined by (3.1) and tm ∈ (0, 1) such that limm→∞ tm = 0. Then as m → ∞, z m converges strongly to some fixed point p of T such that p is the unique solution in F to the following variational inequality:
( f − I ) p, j (y − p) ≤ 0
for all y ∈ F.
(3.2)
Proof. First, we show the uniqueness of the solution to the variational inequality (3.2) in F. In fact, suppose p, q ∈ F satisfy (3.2); we see that
( f − I ) p, j (q − p) ≤ 0
( f − I )q, j ( p − q) ≤ 0.
(3.3) (3.4)
Adding (3.3) and (3.4) up, we have that (1 − β) p − q2 ≤ (I − f ) p − (I − f )q, j ( p − q) ≤ 0. We must have p = q and the uniqueness is proved. Below we use p ∈ F to denote the unique solution of (3.2). Now we show that {z m } is bounded. In fact, for any fixed y ∈ F, it follow from the Eq. (3.1) that z m − y2 = tm ( f (z m ) − y) + (1 − tm )(Tm z m − y), j (z m − y) = tm f (z m ) − f (y) + f (y) − y, j (z m − y) + (1 − tm ) Tm z m − Tm y, j (z m − y) ≤ tm f (z m ) − f (y) j (z m − y) + tm f (y) − y, j (z m − y) + (1 − tm ) Tm z m − Tm y j (z m − y) ≤ (1 − (1 − β)tm ) z m − y2 + tm f (y) − y, j (z m − y) . Therefore, z m − y2 ≤
1 1 f (y) − y z m − y .
f (y) − y, j (z m − y) ≤ 1−β 1−β
(3.5)
Furthermore, z m − y ≤
1 f (y) − y . 1−β
Thus {z m } is bounded, and so are {Tm z m } and { f (z m )}. This implies that lim z m − Tm z m = lim tm Tm z m − f (z m ) = 0.
m→∞
m→∞
Since {Tm } (m = 1, 2, . . .) is a u.a.r. sequence of nonexpansive mappings, we have for all n≥1 lim Tn (Tm z m ) − Tm z m ≤ lim sup Tn (Tm x) − Tm x = 0,
m→∞
m→∞ x∈C
where C is any bounded subset of K containing {z m }. Hence, z m − Tn z m ≤ z m − Tm z m + Tm z m − Tn (Tm z m ) + Tn (Tm z m ) − Tn z m ≤ 2 z m − Tm z m + Tm z m − Tn (Tm z m ) → 0
(m → ∞).
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That is, for all n ≥ 1, lim z m − Tn z m = 0.
m→∞
We claim that the set {z m } is sequentially compact. Indeed, there exists a weakly convergence subsequence {z m k } ⊆ {z m } by reflexivity of E and boundedness of the sequence {z m }; now we suppose z m k q ∈ K (k → ∞). Since for all n ≥ 1, limk→∞ z m k − Tn z m k = 0, taken together with Banach space E with a weakly sequentially continuous duality mapping satisfying Opial’s condition [5, Theorem 1], it follows from Lemma 2.4 that q = Tn q. Since n is arbitrary, we have q ∈ F. In Eq. (3.5), interchange q and y to obtain z m − q 2 ≤ k
1
f (q) − q, j (z m k − q) . 1−β
Using that the duality map J is single-valued and weakly sequentially continuous from E to E ∗ , we get that 2 lim sup z m k − q ≤ lim k→∞
k→∞
1
f (q) − q, j (z m k − q) = 0, 1−β
namely, zmk → q
as k → ∞.
Hence, {z m } is sequentially compact, namely there is a subsequence {z m k } ⊆ {z m } that strongly converges to q ∈ F as k → ∞. Next we show that q is a solution in F to the variational inequality (3.2). In fact, for any y ∈ F, from the Eq. (3.1) and the inequality
Tm z m − y, j (z m − y) ≤ Tm z m − y z m − y ≤ z m − y2 , we obtain that z m − y2 = tm f (z m ) − z m , j (z m − y) + tm z m − y, j (z m − y) + (1 − tm ) Tm z m − Tm y, j (z m − y) ≤ tm f (z m ) − z m , j (z m − y) + z m − y2 . Therefore,
f (z m ) − z m , j (y − z m ) ≤ 0.
(3.6)
Since the set {z m − y} and {z m − f (z m )} is bounded and the duality map J is single-valued and weakly sequentially continuous from E to E ∗ , for any fixed y ∈ F, by z m k → q (k → ∞), we have (I − f )z m − (I − f )q → 0 (k → ∞) k | z m k − f (z m k ), j (z m k − y) − (I − f )q, j (q − y) | = | (I − f )z m k − (I − f )q, j (z m k − y) + (I − f )q, j (z m k − y) − j (q − y) | ≤ (I − f )z m k − (I − f )q z m k − y + | (I − f )q, j (z m k − y) − j (q − y) | → 0 as k → ∞.
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Therefore, noticing Eq. (3.6), for any y ∈ F, we get
f (q) − q, j (y − q) = lim f (z m k ) − z m k , j (y − z m k ) ≤ 0. k→∞
That is, q ∈ F is a solution of the variational inequality (3.2); hence q = p by uniqueness. In summary, we have proved that {z m } is sequentially compact and each cluster point of {z m } (as m → ∞) equals p. Therefore, z m → p as m → ∞. Remark 1. For a uniformly asymptotically regular sequence {Tn } (n = 1, 2, . . .), a u.a.r. sequence of nonexpansive self-mappings, it follows from Theorem 3.1 that there is a unique solution of the variational inequality (3.2) in F = ∞ n=1 F(Tn ) whenever F = ∅. Theorem 3.2. Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J from E to E ∗ . Suppose K is a nonempty closed convex subset of E, and 1, 2, . . .) is a u.a.r. sequence of nonexpansive mappings from K into itself such {Tn } (n = that F := ∞ ) = ∅. Let {λn } be a sequence of positive numbers in (0, 1) satisfying n=1 F(Tn limn→∞ λn = 0 and ∞ n=1 λn = ∞. Then, for an arbitrary x 0 ∈ K and a fixed contractive mapping f : K → K with contractive coefficient β ∈ (0, 1), the sequence {x n } is generated by the following iterative process: x n+1 = λn+1 f (x n ) + (1 − λn+1 )Tn+1 x n ,
n ≥ 0.
Then {x n } converges strongly to p such that p is the unique solution in F to the inequality (3.2). Proof. First we show that {x n } is bounded. Take u ∈ F. It follows that x n+1 − u ≤ (1 − λn+1 ) Tn+1 x n − u + λn+1 f (x n ) − f (u) + f (u) − u ≤ (1 − λn+1 ) x n − u + λn+1 (β x n − u + f (u) − u) = (1 − (1 − β)λn+1 ) x n − u + λn+1 f (u) − u 1 f (u) − u . ≤ max x n − u , 1−β By induction,
x n − u ≤ max x 0 − u ,
1 f (u) − u , 1−β
n ≥ 0,
and {x n } is bounded, which leads to the boundedness of { f (x n )} and {Tn+1 x n }. Using the assumption that limn→∞ λn = 0, we get that x n+1 − Tn+1 x n = λn+1 f (x n ) − Tn+1 x n → 0.
(3.7)
Since {Tn } (n = 1, 2, . . .) is a u.a.r. sequence of nonexpansive mappings, we get that for all m ≥ 1, lim Tm (Tn+1 x n ) − Tn+1 x n ≤ lim sup Tm (Tn+1 x) − Tn+1 x = 0,
n→∞
n→∞ x∈C
where C is any bounded subset of K containing {x n }. Hence, for all m ≥ 1, x n+1 − Tm x n+1 ≤ x n+1 − Tn+1 x n + Tn+1 x n − Tm (Tn+1 x n ) + Tm (Tn+1 x n ) − T x n+1 ≤ 2 x n+1 − Tn+1 x n + Tn+1 x n − Tm (Tn+1 x n ) .
(3.8)
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Combining (3.7) and (3.8), we get that for all m ≥ 1, lim x n+1 − Tm x n+1 = 0.
n→∞
It follows from Theorem 3.1 that there is a unique solution p ∈ F to the following variational inequality:
( f − I ) p, j (u − p) ≤ 0
for all u ∈ F.
(3.9)
We next show that lim sup ( f − I ) p, j (x n+1 − p) ≤ 0.
(3.10)
n→∞
Indeed, we can take a subsequence {x nk +1 } of {x n+1 } such that lim sup ( f − I ) p, j (x n+1 − p) = lim ( f − I ) p, j (x nk +1 − p) . k→∞
n→∞
x∗
We may assume that x nk +1 since E is reflexive and {x n+1 } bounded. Since for all m ≥ 1, limk→∞ x nk +1 − Tm x nk +1 = 0, taken together with Banach space E with a weakly sequentially continuous duality mapping satisfying Opial’s condition [5, Theorem 1], it follows from Lemma 2.4 that x ∗ ∈ F(Tm ). Since m is arbitrary, we have x ∗ ∈ F. In view of the variational inequality (3.9) and the assumption that the duality mapping J is weakly sequentially continuous, we conclude lim sup ( f − I ) p, j (x n+1 − p) = lim ( f − I ) p, j (x nk +1 − p) k→∞
n→∞
= ( f − I ) p, j (x ∗ − p) ≤ 0.
The Eq. (3.10) is proved. Finally, we show that x n → p. As a matter of fact, x n+1 − p2 = λn+1 f (x n ) − p, j (x n+1 − p) + (1 − λn+1 ) Tn+1 x n − p, j (x n+1 − p) ≤ λn+1 f ( p) − p, j (x n+1 − p) + λn+1 f (x n ) − f ( p), j (x n+1 − p) + (1 − λn+1 ) Tn+1 x n − p x n+1 − p ≤ λn+1 f ( p) − p, j (x n+1 − p) + λn+1 f (x n ) − f ( p) x n+1 − p + (1 − λn+1 ) x n − p x n+1 − p ≤ λn+1 f ( p) − p, j (x n+1 − p) + λn+1 + (1 − λn+1 )
β 2 x n − p2 + x n+1 − p2 2
x n − p2 + x n+1 − p2 . 2
And furthermore, x n+1 − p2 ≤ (1 − λn+1 (1 − β 2 )) x n − p2 + 2λn+1 f ( p) − p, j (x n+1 − p) , that is x n+1 − p2 = (1 − γn ) x n − p2 + γn αn , where γn = λn+1 (1 − β 2 ) and αn =
2
f ( p) − 1−β 2
(3.11) p, j (x n+1 − p) .
Y. Song, R. Chen / Nonlinear Analysis 66 (2007) 591–603
It is easily seen that γn → 0, lim sup αn ≤ lim sup n→∞
n→∞
∞
n=1 γn
601
= ∞, and (noting (3.10))
2
f ( p) − p, j (x n+1 − p) ≤ 0. 1 − β2
Using Lemma 2.5 on (3.11), we conclude that x n → p. This completes the proof of the theorem. Remark 2. In Theorem 3.2, Taking f (x) ≡ y ∈ K , ∀x ∈ K , this result not only overcomes the gaps in [7,9,18], but also extends and improves the corresponding results of [7,9,18]. Corollary 3.3. Let {λn } ⊂ (0, 1) satisfy that limn→∞ λn = 0 and ∞ n=1 λn = ∞. Suppose K is a nonempty closed convex subset of a Banach space E, and let S, T : K → K be nonexpansive mappings with fixed points and f : K → K be a fixed contractive mapping with contractive coefficient β ∈ (0, 1). j (a) Set Tn x = n1 n−1 j =0 T x and x ∈ K , for x 0 ∈ K ; define x n+1 = λn+1 f (x n ) + (1 − λn+1 )Tn+1 x n ,
n ≥ 0.
If E is a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping J from E to E ∗ and F(T ) = ∅, then {x n } converges strongly to p such that p is the unique solution in F(T ) to the following variational inequality:
( f − I ) p, j (u − p) ≤ 0 for all u ∈ F(T ). (3.12) n−1 2 i j (b) Set Tn x = n(n+1) i+ j =k S T x for n ≥ 1 and x ∈ K . For x 0 ∈ K and u ∈ K define k=0 x n+1 = λn+1 f (x n ) + (1 − λn+1 )Tn+1 x n , n ≥ 0. Suppose that ST = T S and F(T ) F(S) = ∅, and E is a Hilbert space. Then {x n } converges strongly to p such that p is the unique solution in F = F(T ) F(S) to the following variational inequality:
( f − I ) p, u − p ≤ 0 for all u ∈ F. Proof. In case (a), take w ∈ F(T ) and define a subset D of K by D = {x ∈ K : x − w ≤ r }, 1 where r = max{w − x 0 , 1−β w − f (w), w − v}, where v is the unique solution in F(T ) to the variational inequality (3.12). Then D is a nonempty closed bounded convex subset of K and T (D) ⊂ D and {x n } ⊂ D. Also Lemma 2.2 implies
lim sup Tn x − T (Tn x) = 0,
n→∞ x∈D
∞ j along with Tn = n1 n−1 n=0 FD (Tn ), where FD (T ) = {x ∈ j =0 T ; then clearly FD (T ) = D : T x = x}. And furthermore, {Tn } is a u.a.r. sequence of nonexpansive mappings on D (see Example 2 in the Preliminaries); also since every uniformly convex Banach space E is reflexive and strictly convex [14, Theorem 4.1.6, Theorem 4.1.2], it follows from Theorem 3.2 that x n converges strongly to p, where p is the unique solution in FD (T ) to the following variational inequality:
( f − I ) p, j (u − p) ≤ 0 for all u ∈ FD (T ).
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Since p ∈ FD (T ) ⊂ F(T ) and the variational inequality (3.12) have a unique solution in F(T ), we get v = p. As for case (b), taking A = γ T + (1 − γ )S, γ ∈ (0, 1), by Lemma 2.1 we obtain that F(A) = F(T ) F(S) and A is a nonexpansive self-mapping on K . Let w ∈ F(A) and v be the unique solution in F(A) to the variational following inequality:
( f − I ) p, u − p ≤ 0
for all u ∈ F.
Using an argument similar to that of case (a) we find a nonempty closed bounded convex subset D of K and A(D) ⊂ D. Also Lemma 2.3 implies lim sup Tn x − T (Tn x) = 0,
n→∞ x∈D
lim sup Tn x − S(Tn x) = 0,
n→∞ x∈D
together with Tn =
2 n(n+1)
FD (A) = FD (T )
n−1
k=0
i+ j =k
FD (S) =
∞
S i T j , so it is clear that
FD (Tn ).
n=0
And furthermore, {Tn } is a u.a.r. sequence of nonexpansive mappings on D (see Example 1 in the Preliminaries). The remainder of the proof is the same as in case (a), and the conclusion is proved. Acknowledgment The authors would like to thank Prof. W.A. Kirk for valuable suggestions which helped to improve this manuscript. References [1] H. Bauschke, The approximation of fixed points of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150–159. [2] R.E. Bruck Jr., Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc. 179 (1973) 251–262. [3] R.E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math. 32 (1979) 107–116. [4] R.E. Bruck, On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. Math. 38 (1981) 304–314. [5] J.P. Gossez, E. Lami Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40 (1972) 565–573. [6] B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961. [7] J.S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520. [8] H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291. [9] J.G. O’Hara, P. Pillay, H.-K. Xu, Iterative approaches to finding nearest common fixed point of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426. [10] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292. [11] J. Schu, Approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 112 (1991) 143–151. [12] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997) 3641–3645.
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