- Email: [email protected]
Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces
Nonlinear Analysis 73 (2010) 1562–1568
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces Yu Kurokawa a , Wataru Takahashi a,b,∗ a
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, Tokyo 152-8552, Japan
b
Applied Mathematics, National Sun Yay-sen University, Taiwan
article
info
Article history: Received 27 September 2009 Accepted 28 April 2010 MSC: primary 47A35 47J05 47J25
abstract In this paper, we first obtain a weak mean convergence theorem of Baillon’s type for nonspreading mappings in a Hilbert space. Further, using an idea of mean convergence, we prove a strong convergence theorem for nonspreading mappings in a Hilbert space. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Nonspreading mapping Fixed point Mean ergodic theorem Weak convergence Strong convergence
1. Introduction Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Then a mapping T : C → C is said to be nonexpansive if kTx − Tyk ≤ kx − yk for all x, y ∈ C . We denote by F (T ) the set of fixed points of T . Let T : C → C be a nonexpansive mapping with F (T ) 6= ∅. From [1] we know the first nonlinear ergodic theorem in a Hilbert space: For any x ∈ C, Sn x =
n−1 1X
n k=0
T kx
converges weakly to a fixed point of T . We also know the following weak convergence theorem of Mann’s type: For any x1 = x ∈ C , define a sequence {xn } in C by xn+1 = αn xn + (1 − αn )Txn for n = 1, 2, . . . , where {αn } ⊂ [0, 1] satisfies see, for instance, [2–4].
P∞
n=1
αn (1 − αn ) = ∞. Then {xn } converges weakly to a fixed point of T ;
∗ Corresponding author at: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, Tokyo 152-8552, Japan. Fax: +81 03 5734 3208. E-mail addresses: [email protected] (Y. Kurokawa), [email protected] (W. Takahashi). 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.04.060
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
1563
The following strong convergence theorem of Halpern’s type was proved by Wittmann: For any x1 = x ∈ C , define a sequence {xn } in C by xn+1 = αn x + (1 − αn )Txn for n = 1, 2, . . . , where {αn } ⊂ [0, 1] satisfies αn → 0, ∞ X
P∞
n =1
αn = ∞ and
|αn − αn+1 | < ∞.
n =1
Then {xn } converges strongly to a fixed point of T ; see [5–8,4]. Nakajo and Takahashi [9] proved the following strong convergence theorem of the hybrid type: Let {xn } be a sequence in C defined by x1 = x ∈ C and
y = αn xn + (1 − αn )Txn , n Cn = {z ∈ C : kyn − z k ≤ kxn − z k}, Qn = {z ∈ C : hxn − z , x − xn i ≥ 0}, xn+1 = PCn ∩Qn x for n = 1, 2, . . . , where {αn } ⊂ [0, 1] satisfies lim supn→∞ αn < 1. Then {xn } converges strongly to a fixed point of T ; see also [10]. An important example of nonexpansive mappings in a Hilbert space is a firmly nonexpansive mapping. A mapping F : C → C is said to be firmly nonexpansive if
kFx − Fyk2 ≤ hx − y, Fx − Fyi for all x, y ∈ C ; see, for instance, Browder [11] and Goebel and Kirk [12]. It is also known that a firmly nonexpansive mapping F is deduced from an equilibrium problem in a Hilbert space as follows: Let C be a nonempty closed convex subset of H and let f : C × C → R be a bifunction satisfying the following conditions: (A1) (A2) (A3) (A4)
f (x, x) = 0, ∀x ∈ C ; f is monotone, i.e., f (x, y) + f (y, x) ≤ 0, ∀x, y ∈ C ; limt ↓0 f (tz + (1 − t )x, y) ≤ f (x, y), ∀x, y, z ∈ C ; for each x ∈ C , y 7→ f (x, y) is convex and lower semicontinuous.
We know the following theorem; see, for instance, [13,14]. Theorem 1.1. Let C be a nonempty closed convex subset of H and let f be a bifunction from C × C into R satisfying (A1)–(A4). Then, for any r > 0 and x ∈ H, there exists a unique point z ∈ C such that f (z , y) +
1 r
hy − z , z − xi ≥ 0,
∀y ∈ C .
Further, for any r > 0 and x ∈ H, define Tr : H → C by z = Tr x. Then, Tr is firmly nonexpansive, i.e.,
kTr x − Tr yk2 ≤ hTr x − Tr y, x − yi,
∀ x, y ∈ H .
Recently, Kohsaka and Takahashi [15,16] introduced the following nonlinear mapping: Let E be a smooth, strictly convex and reflexive Banach space, let J be the duality mapping of E and let C be a nonempty closed convex subset of E. Then, a mapping S : C → C is said to be nonspreading if
φ(Sx, Sy) + φ(Sy, Sx) ≤ φ(Sx, y) + φ(Sy, x) for all x, y ∈ C , where φ(x, y) = kxk2 − 2hx, Jyi+kyk2 for all x, y ∈ E. They considered such a mapping to study the resolvents of a maximal monotone operator in the Banach space. In the case when E is a Hilbert space, we know that φ(x, y) = kx − yk2 for all x, y ∈ E. So, a nonspreading mapping S : C → C in a Hilbert space H is defined as follows: 2kSx − Syk2 ≤ kSx − yk2 + kx − Syk2 for all x, y ∈ C . We also know that a nonspreading mapping is deduced from a firmly nonexpansive mapping; see [15,17, 18]. A weak convergence theorem of Mann’s type and a strong convergence theorem of the hybrid type for nonspreading mappings have been proved by Matsushita and Takahashi [19,20]. In this paper, we first obtain a nonlinear mean ergodic theorem of Baillon’s type for nonspreading mappings in a Hilbert space. Further, using an idea of mean convergence, we prove a strong convergence theorem of Halpern’s type for nonspreading mappings in a Hilbert space.
1564
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
2. Preliminaries Throughout this paper, we denote by H a real Hilbert space with inner product h·, ·i and norm k · k. We also denote by N the set of natural numbers. In a Hilbert space, it is known that
kyk2 − kxk2 ≤ 2hy − x, yi
(2.1)
for all x, y ∈ H; see, for instance, [21]. Let {xn } be a sequence in H and x ∈ H. Weak convergence of {xn } to x is denoted by xn * x and strong convergence by xn → x. Let C be a nonempty closed convex subset of H. We can define the metric projection of H onto C : For each x ∈ H, there exists a unique point z ∈ C such that
kx − z k = min{kx − yk : y ∈ C }. For each x ∈ H, such a point z is denoted by Px and P is called the metric projection of H onto C . It is known that
hx − Px, Px − yi ≥ 0
(2.2)
for all x ∈ H and y ∈ C ; see [22] for more details. Let T be a mapping from C into itself. The set of fixed points of T is denoted by F (T ). A mapping T is said to be nonspreading [15] if 2kTx − Tyk2 ≤ kTx − yk2 + kx − Tyk2 for all x, y ∈ C . Iemoto and Takahashi [23] proved that T : C → C is nonspreading if and only if
kTx − Tyk2 ≤ kx − yk2 + 2hx − Tx, y − Tyi
(2.3)
for all x, y ∈ C . A mapping T : C → C is called quasi-nonexpansive if F (T ) 6= ∅ and kTx − uk ≤ kx − uk for all x ∈ C and u ∈ F (T ). If T is a nonspreading mapping from C into itself and F (T ) is nonempty, then T is quasi-nonexpansive. Further, we know that the set of fixed points of a quasi-nonexpansive mapping is closed and convex; see [24]. Then we can define the metric projection of H onto F (T ). To prove our main results, we need the following lemmas: Lemma 2.1 (Aoyama–Kimura–Takahashi–Toyoda [25]). Let {sn } be a sequence of nonnegative realP numbers, let {αn } be a P∞ ∞ sequence of [0, 1] with n=1 αn = ∞, let {βn } be a sequence of nonnegative real numbers with n=1 βn < ∞, and let {γn } be a sequence of real numbers with lim supn→∞ γn ≤ 0. Suppose that sn+1 ≤ (1 − αn )sn + αn γn + βn for all n = 1, 2, . . .. Then limn→∞ sn = 0. Lemma 2.2 (Takahashi–Toyoda [26]). Let D be a nonempty closed convex subset of a real Hilbert space H. Let P be the metric projection of H onto D and let {xn } be a sequence in H. If kxn+1 − uk ≤ kxn − uk for all u ∈ D and n ∈ N, then {Pxn } converges strongly. 3. Weak convergence theorem In this section, motivated by [27] and [28], we prove the following weak convergence theorem for nonspreading mappings in a Hilbert space. Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonspreading mapping from C into itself. Define two sequences {xn } and {zn } in C as follows: x1 = x ∈ C and
xn+1 = αnn xn + (1 − αn )Txn , 1X xk zn = n k=1
for all n ∈ N, where 0 ≤ αn < 1 and αn → 0. If F (T ) 6= ∅, then {zn } converges weakly to z ∈ F (T ), where z = limn→∞ Pxn and P is the metric projection of H onto F (T ). In particular, for any x ∈ C , define Sn x =
n−1 1X
n k=0
T k x.
Then {Sn x} converges weakly to z ∈ F (T ), where z = limn→∞ PT n x.
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
1565
Proof. Since F (T ) is nonempty and a nonspreading mapping T with a fixed point is quasi-nonexpansive, we have that for all u ∈ F (T ),
kxn+1 − uk = kαn xn + (1 − αn )Txn − uk ≤ αn kxn − uk + (1 − αn )kTxn − uk ≤ kxn − uk.
(3.1)
So, we have that limn→∞ kxn − uk exists and {xn } is bounded. We have from (2.3) that for all y ∈ C and k ∈ N,
kxk+1 − Tyk2 = kαk xk + (1 − αk )Txk − Tyk2 ≤ αk kxk − Tyk2 + (1 − αk )kTxk − Tyk2 ≤ αk kxk − Tyk2 + (1 − αk ) kxk − yk2 + 2hxk − Txk , y − Tyi . Since
kxk − yk2 = kxk − Tyk2 + kTy − yk2 + 2hxk − Ty, Ty − yi, we have
kxk+1 − Tyk2 ≤ αk kxk − Tyk2 + 2(1 − αk )hxk − Txk , y − Tyi + (1 − αk ) kxk − Tyk2 + kTy − yk2 + 2hxk − Ty, Ty − yi = kxk − Tyk2 + 2h(1 − αk )(xk − Txk ), y − Tyi + (1 − αk )kTy − yk2 + 2(1 − αk )hxk − Ty, Ty − yi. Since (1 − αk )(xk − Txk ) = xk − xk+1 and 0 ≤ αk < 1, we have
kxk+1 − Tyk2 ≤ kxk − Tyk2 + 2hxk − xk+1 , y − Tyi + kTy − yk2 + 2hxk − Ty, Ty − yi − 2αk hxk − Ty, Ty − yi. Summing these inequalities from k = 1 to n and dividing by n, we have 1 n
1
kxn+1 − Tyk2 ≤
n
kx1 − Tyk2 + 2
Dx
1
n
−
+ 2hzn − Ty, Ty − yi −
xn+1
E , y − Ty + kTy − yk2
n n 2X n k=1
αk hxk − Ty, Ty − yi.
Since {xn } is bounded, {zn } is also bounded. Then, there exists a subsequence {zni } of {zn } such that zni * w ∈ C . Further, replacing n by ni , we have 1 ni
2
kxni +1 − Tyk ≤
1 ni
2
kx1 − Tyk + 2
x1 ni
−
+ 2hzni − Ty, Ty − yi −
xni +1 ni
, y − Ty + kTy − yk2
ni 2 X
ni k=1
αk hxk − Ty, Ty − yi.
Since {xn } is bounded, we have that αk hxk − Ty, Ty − yi → 0 as k → ∞. Further, since zni * w and 1/ni Ty, Ty − yi → 0 as i → ∞, we have
Pni
k=1
αk hxk −
0 ≤ kTy − yk2 + 2hw − Ty, Ty − yi. Putting y = w , we have 0 ≤ kT w − wk2 + 2hw − T w, T w − wi
= −kT w − wk2 . Hence, w ∈ F (T ). To show that {zn } converges weakly to a fixed point of T , we first show that limn→∞ Pxn exists. Since F (T ) 6= ∅, from (3.1) we have that for all u ∈ F (T ),
kxn+1 − uk ≤ kxn − uk. On the other hand, since T is nonspreading, F (T ) is closed and convex. So, we can define the metric projection P of H onto F (T ). Putting D = F (T ) in Lemma 2.2, we have that limn→∞ Pxn converges strongly. Put z = limn→∞ Pxn . Then we can prove zn * z. In fact, let {zni } be a subsequence of {zn } such that zni * w . From the above argument, we have w ∈ F (T ). To complete the proof of the first part, it is sufficient to prove z = w . From w ∈ F (T ) and (2.2), we have
hw − z , xk − Pxk i = hw − Pxk , xk − Pxk i + hPxk − z , xk − Pxk i ≤ hPxk − z , xk − Pxk i
1566
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
≤ kPxk − z kkxk − Pxk k ≤ kPxk − z kM for all k ∈ N, where M = sup{kxk − Pxk k : k ∈ N}. Summing these inequalities from k = 1 to ni and dividing by ni , we have
* w − z , zni −
ni 1 X
ni k=1
+ Pxk
≤
ni 1 X
ni k=1
kPxk − z kM .
Since zni * w as i → ∞ and Pxn → z as n → ∞, we have hw − z , w − z i ≤ 0. This implies z = w . This completes Pn the proof of the first part. In particular, putting αn = 0 for all n ∈ N, we see that xn+1 = T n x and zn = 1/n k=1 T k−1 x for all n ∈ N. So, we obtain Sn x = zn . Therefore, {Sn x} converges weakly to z ∈ F (T ), where z = limn→∞ PT n x. So, we get the desired result. 4. Strong convergence theorem In this section, motivated by [29] and [30], we prove the following strong convergence theorem for nonspreading mappings in a Hilbert space. Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonspreading mapping of C into itself. Let u ∈ C and define two sequences {xn } and {zn } in C as follows: x1 = x ∈ C and
xn+1 = αn u + (1 − αn )zn , n −1 1X k z = T xn n n k=0
for all n = 1, 2, . . ., where 0 ≤ αn ≤ 1, αn → 0 and to Pu, where P is the metric projection of H onto F (T ).
P∞
n =1
αn = ∞. If F (T ) is nonempty, then {xn } and {zn } converge strongly
Proof. Since F (T ) 6= ∅, T is quasi-nonexpansive. So, we have that for all q ∈ F (T ) and n = 1, 2, 3, . . .,
n −1 n−1
1 X
1X
k kzn − qk = T xn − q ≤ kT k xn − qk
n k=0
n k=0 ≤
n−1 1X
n k=0
kxn − qk = kxn − qk.
(4.1)
Then we have
kxn+1 − qk = kαn u + (1 − αn )zn − qk ≤ αn ku − qk + (1 − αn )kzn − qk ≤ αn ku − qk + (1 − αn )kxn − qk. Hence, by induction, we obtain
kxn − qk ≤ max {ku − qk, kx − qk} for all n ∈ N. This implies that {xn } and {zn } are bounded. Since kT n xn − qk ≤ kxn − qk, we have also that {T n xn } is bounded. Let n ∈ N. Since T is nonspreading, we have from (2.3) that for all y ∈ C and k = 0, 1, 2, . . . , n − 1,
kT k+1 xn − Tyk2 ≤ kT k xn − yk2 + 2hT k xn − T k+1 xn , y − Tyi = kT k xn − Tyk2 + kTy − yk2 + 2hT k xn − Ty, Ty − yi + 2hT k xn − T k+1 xn , y − Tyi. Summing these inequalities from k = 0 to n − 1 and dividing by n, we have 1 n
kT n xn − Tyk2 ≤
1 n
2
kxn − Tyk2 + kTy − yk2 + 2hzn − Ty, Ty − yi + hxn − T n xn , y − Tyi. n
Since {zn } is bounded, there exists a subsequence {zni } of {zn } such that zni * w ∈ C . Replacing n by ni , we have 1 ni
kT ni xni − Tyk2 ≤
1 ni
kxni − Tyk2 + kTy − yk2 + 2hzni − Ty, Ty − yi +
Since {xn } and {T n xn } are bounded, we have that 0 ≤ kTy − yk2 + 2hw − Ty, Ty − yi
2 ni
hxni − T ni xni , y − Tyi.
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
1567
as i → ∞. Putting y = w , we have 0 ≤ kT w − wk2 + 2hw − T w, T w − wi = −kT w − wk2 . Hence, w ∈ F (T ). On the other hand, since xn+1 − zn = αn (u − zn ), {zn } is bounded and αn → 0, we have limn→∞ kxn+1 − zn k = 0. Let us show lim supn→∞ hu−Pu, xn+1 −Pui ≤ 0. We may assume without loss of generality that there exists a subsequence {xni +1 } of {xn+1 } such that lim suphu − Pu, xn+1 − Pui = lim hu − Pu, xni +1 − Pui n→∞
i→∞
and xni +1 * v . From kxn+1 − zn k → 0, we have zni * v . From the above argument, we have v ∈ F (T ). Since P is the metric projection of H onto F (T ), we have lim hu − Pu, xni +1 − Pui = hu − Pu, v − Pui ≤ 0.
i→∞
This implies lim suphu − Pu, xn+1 − Pui ≤ 0.
(4.2)
n→∞
Since xn+1 − Pu = (1 − αn )(zn − Pu) + αn (u − Pu), from (2.1) and (4.1) we have
kxn+1 − Puk2 = k(1 − αn )(zn − Pu) + αn (u − Pu)k2 ≤ (1 − αn )2 kzn − Puk2 + 2αn hu − Pu, xn+1 − Pui ≤ (1 − αn )kxn − Puk2 + 2αn hu − Pu, xn+1 − Pui. Putting sn = kxn − Puk2 , βn = 0 and γn = 2hu − Pu, xn+1 − Pui in Lemma 2.1, from
P∞
n =1
αn = ∞ and (4.2) we have
lim kxn − Puk = 0.
n→∞
By limn→∞ kxn − zn k = 0, we also obtain zn → Pu as n → ∞.
Remark. We do not know whether a strong convergence theorem of Halpern’s type for nonspreading mappings holds or not. References [1] J.-B. Baillon, Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér. A-B 280 (1975) Aii, A1511–A1514 (in French). [2] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–510. [3] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979) 274–276. [4] W. Takahashi, Convex Analysis and Approximation of Fixed Points, Yokohama Publishers, Yokohama, 2000 (in Japanese). [5] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 957–961. [6] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491. [7] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997) 3641–3645. [8] N. Shioji, W. Takahashi, A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Arch. Math. (Basel) 72 (1999) 354–359. [9] K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–378. [10] S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002) 938–945. [11] F.E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z. 100 (1967) 201–225. [12] K. Goebel, W.A. Kirk, Topics in metric fixed point theory, in: Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. [13] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994) 123–145. [14] P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 117–136. [15] F. Kohsaka, W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel) 91 (2008) 166–177. [16] F. Kohsaka, W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19 (2008) 824–835. [17] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11 (2010) 79–88. [18] T. Igarashi, W. Takahashi, K. Tanaka, Weak convergence theorems for nonspreading mappings and equilibrium problems, in: S. Akashi, W. Takahashi, T. Tanaka (Eds.), Nonlinear Analysis and Optimization, Yokohama Publishers, Yokohama, 2009, pp. 75–85. [19] S. Matsushita, W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. (2004) 37–47. [20] S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257–266. [21] W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, 2009. [22] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000. [23] S. Iemoto, W. Takahashi, Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear Anal. 71 (2009) 2082–2089. [24] S. Itoh, W. Takahashi, The common fixed point theory of singlevalued mappings and multivalued mappings, Pacific J. Math. 79 (1978) 493–508.
1568
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
[25] K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350–2360. [26] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003) 417–428. [27] M. Akatsuka, K. Aoyama, W. Takahashi, Mean ergodic theorems for a sequence of nonexpansive mappings in Hilbert spaces, Sci. Math. Jpn. 68 (2008) 233–239. [28] W. Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981) 253–256. [29] T. Shimizu, W. Takahashi, Strong convergence theorem for asymptotically nonexpansive mappings, Nonlinear Anal. 26 (1996) 265–272. [30] T. Shimizu, W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (1997) 71–83.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces Yu Kurokawa a , Wataru Takahashi a,b,∗ a
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, Tokyo 152-8552, Japan
b
Applied Mathematics, National Sun Yay-sen University, Taiwan
article
info
Article history: Received 27 September 2009 Accepted 28 April 2010 MSC: primary 47A35 47J05 47J25
abstract In this paper, we first obtain a weak mean convergence theorem of Baillon’s type for nonspreading mappings in a Hilbert space. Further, using an idea of mean convergence, we prove a strong convergence theorem for nonspreading mappings in a Hilbert space. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Nonspreading mapping Fixed point Mean ergodic theorem Weak convergence Strong convergence
1. Introduction Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Then a mapping T : C → C is said to be nonexpansive if kTx − Tyk ≤ kx − yk for all x, y ∈ C . We denote by F (T ) the set of fixed points of T . Let T : C → C be a nonexpansive mapping with F (T ) 6= ∅. From [1] we know the first nonlinear ergodic theorem in a Hilbert space: For any x ∈ C, Sn x =
n−1 1X
n k=0
T kx
converges weakly to a fixed point of T . We also know the following weak convergence theorem of Mann’s type: For any x1 = x ∈ C , define a sequence {xn } in C by xn+1 = αn xn + (1 − αn )Txn for n = 1, 2, . . . , where {αn } ⊂ [0, 1] satisfies see, for instance, [2–4].
P∞
n=1
αn (1 − αn ) = ∞. Then {xn } converges weakly to a fixed point of T ;
∗ Corresponding author at: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, Tokyo 152-8552, Japan. Fax: +81 03 5734 3208. E-mail addresses: [email protected] (Y. Kurokawa), [email protected] (W. Takahashi). 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.04.060
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
1563
The following strong convergence theorem of Halpern’s type was proved by Wittmann: For any x1 = x ∈ C , define a sequence {xn } in C by xn+1 = αn x + (1 − αn )Txn for n = 1, 2, . . . , where {αn } ⊂ [0, 1] satisfies αn → 0, ∞ X
P∞
n =1
αn = ∞ and
|αn − αn+1 | < ∞.
n =1
Then {xn } converges strongly to a fixed point of T ; see [5–8,4]. Nakajo and Takahashi [9] proved the following strong convergence theorem of the hybrid type: Let {xn } be a sequence in C defined by x1 = x ∈ C and
y = αn xn + (1 − αn )Txn , n Cn = {z ∈ C : kyn − z k ≤ kxn − z k}, Qn = {z ∈ C : hxn − z , x − xn i ≥ 0}, xn+1 = PCn ∩Qn x for n = 1, 2, . . . , where {αn } ⊂ [0, 1] satisfies lim supn→∞ αn < 1. Then {xn } converges strongly to a fixed point of T ; see also [10]. An important example of nonexpansive mappings in a Hilbert space is a firmly nonexpansive mapping. A mapping F : C → C is said to be firmly nonexpansive if
kFx − Fyk2 ≤ hx − y, Fx − Fyi for all x, y ∈ C ; see, for instance, Browder [11] and Goebel and Kirk [12]. It is also known that a firmly nonexpansive mapping F is deduced from an equilibrium problem in a Hilbert space as follows: Let C be a nonempty closed convex subset of H and let f : C × C → R be a bifunction satisfying the following conditions: (A1) (A2) (A3) (A4)
f (x, x) = 0, ∀x ∈ C ; f is monotone, i.e., f (x, y) + f (y, x) ≤ 0, ∀x, y ∈ C ; limt ↓0 f (tz + (1 − t )x, y) ≤ f (x, y), ∀x, y, z ∈ C ; for each x ∈ C , y 7→ f (x, y) is convex and lower semicontinuous.
We know the following theorem; see, for instance, [13,14]. Theorem 1.1. Let C be a nonempty closed convex subset of H and let f be a bifunction from C × C into R satisfying (A1)–(A4). Then, for any r > 0 and x ∈ H, there exists a unique point z ∈ C such that f (z , y) +
1 r
hy − z , z − xi ≥ 0,
∀y ∈ C .
Further, for any r > 0 and x ∈ H, define Tr : H → C by z = Tr x. Then, Tr is firmly nonexpansive, i.e.,
kTr x − Tr yk2 ≤ hTr x − Tr y, x − yi,
∀ x, y ∈ H .
Recently, Kohsaka and Takahashi [15,16] introduced the following nonlinear mapping: Let E be a smooth, strictly convex and reflexive Banach space, let J be the duality mapping of E and let C be a nonempty closed convex subset of E. Then, a mapping S : C → C is said to be nonspreading if
φ(Sx, Sy) + φ(Sy, Sx) ≤ φ(Sx, y) + φ(Sy, x) for all x, y ∈ C , where φ(x, y) = kxk2 − 2hx, Jyi+kyk2 for all x, y ∈ E. They considered such a mapping to study the resolvents of a maximal monotone operator in the Banach space. In the case when E is a Hilbert space, we know that φ(x, y) = kx − yk2 for all x, y ∈ E. So, a nonspreading mapping S : C → C in a Hilbert space H is defined as follows: 2kSx − Syk2 ≤ kSx − yk2 + kx − Syk2 for all x, y ∈ C . We also know that a nonspreading mapping is deduced from a firmly nonexpansive mapping; see [15,17, 18]. A weak convergence theorem of Mann’s type and a strong convergence theorem of the hybrid type for nonspreading mappings have been proved by Matsushita and Takahashi [19,20]. In this paper, we first obtain a nonlinear mean ergodic theorem of Baillon’s type for nonspreading mappings in a Hilbert space. Further, using an idea of mean convergence, we prove a strong convergence theorem of Halpern’s type for nonspreading mappings in a Hilbert space.
1564
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
2. Preliminaries Throughout this paper, we denote by H a real Hilbert space with inner product h·, ·i and norm k · k. We also denote by N the set of natural numbers. In a Hilbert space, it is known that
kyk2 − kxk2 ≤ 2hy − x, yi
(2.1)
for all x, y ∈ H; see, for instance, [21]. Let {xn } be a sequence in H and x ∈ H. Weak convergence of {xn } to x is denoted by xn * x and strong convergence by xn → x. Let C be a nonempty closed convex subset of H. We can define the metric projection of H onto C : For each x ∈ H, there exists a unique point z ∈ C such that
kx − z k = min{kx − yk : y ∈ C }. For each x ∈ H, such a point z is denoted by Px and P is called the metric projection of H onto C . It is known that
hx − Px, Px − yi ≥ 0
(2.2)
for all x ∈ H and y ∈ C ; see [22] for more details. Let T be a mapping from C into itself. The set of fixed points of T is denoted by F (T ). A mapping T is said to be nonspreading [15] if 2kTx − Tyk2 ≤ kTx − yk2 + kx − Tyk2 for all x, y ∈ C . Iemoto and Takahashi [23] proved that T : C → C is nonspreading if and only if
kTx − Tyk2 ≤ kx − yk2 + 2hx − Tx, y − Tyi
(2.3)
for all x, y ∈ C . A mapping T : C → C is called quasi-nonexpansive if F (T ) 6= ∅ and kTx − uk ≤ kx − uk for all x ∈ C and u ∈ F (T ). If T is a nonspreading mapping from C into itself and F (T ) is nonempty, then T is quasi-nonexpansive. Further, we know that the set of fixed points of a quasi-nonexpansive mapping is closed and convex; see [24]. Then we can define the metric projection of H onto F (T ). To prove our main results, we need the following lemmas: Lemma 2.1 (Aoyama–Kimura–Takahashi–Toyoda [25]). Let {sn } be a sequence of nonnegative realP numbers, let {αn } be a P∞ ∞ sequence of [0, 1] with n=1 αn = ∞, let {βn } be a sequence of nonnegative real numbers with n=1 βn < ∞, and let {γn } be a sequence of real numbers with lim supn→∞ γn ≤ 0. Suppose that sn+1 ≤ (1 − αn )sn + αn γn + βn for all n = 1, 2, . . .. Then limn→∞ sn = 0. Lemma 2.2 (Takahashi–Toyoda [26]). Let D be a nonempty closed convex subset of a real Hilbert space H. Let P be the metric projection of H onto D and let {xn } be a sequence in H. If kxn+1 − uk ≤ kxn − uk for all u ∈ D and n ∈ N, then {Pxn } converges strongly. 3. Weak convergence theorem In this section, motivated by [27] and [28], we prove the following weak convergence theorem for nonspreading mappings in a Hilbert space. Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonspreading mapping from C into itself. Define two sequences {xn } and {zn } in C as follows: x1 = x ∈ C and
xn+1 = αnn xn + (1 − αn )Txn , 1X xk zn = n k=1
for all n ∈ N, where 0 ≤ αn < 1 and αn → 0. If F (T ) 6= ∅, then {zn } converges weakly to z ∈ F (T ), where z = limn→∞ Pxn and P is the metric projection of H onto F (T ). In particular, for any x ∈ C , define Sn x =
n−1 1X
n k=0
T k x.
Then {Sn x} converges weakly to z ∈ F (T ), where z = limn→∞ PT n x.
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
1565
Proof. Since F (T ) is nonempty and a nonspreading mapping T with a fixed point is quasi-nonexpansive, we have that for all u ∈ F (T ),
kxn+1 − uk = kαn xn + (1 − αn )Txn − uk ≤ αn kxn − uk + (1 − αn )kTxn − uk ≤ kxn − uk.
(3.1)
So, we have that limn→∞ kxn − uk exists and {xn } is bounded. We have from (2.3) that for all y ∈ C and k ∈ N,
kxk+1 − Tyk2 = kαk xk + (1 − αk )Txk − Tyk2 ≤ αk kxk − Tyk2 + (1 − αk )kTxk − Tyk2 ≤ αk kxk − Tyk2 + (1 − αk ) kxk − yk2 + 2hxk − Txk , y − Tyi . Since
kxk − yk2 = kxk − Tyk2 + kTy − yk2 + 2hxk − Ty, Ty − yi, we have
kxk+1 − Tyk2 ≤ αk kxk − Tyk2 + 2(1 − αk )hxk − Txk , y − Tyi + (1 − αk ) kxk − Tyk2 + kTy − yk2 + 2hxk − Ty, Ty − yi = kxk − Tyk2 + 2h(1 − αk )(xk − Txk ), y − Tyi + (1 − αk )kTy − yk2 + 2(1 − αk )hxk − Ty, Ty − yi. Since (1 − αk )(xk − Txk ) = xk − xk+1 and 0 ≤ αk < 1, we have
kxk+1 − Tyk2 ≤ kxk − Tyk2 + 2hxk − xk+1 , y − Tyi + kTy − yk2 + 2hxk − Ty, Ty − yi − 2αk hxk − Ty, Ty − yi. Summing these inequalities from k = 1 to n and dividing by n, we have 1 n
1
kxn+1 − Tyk2 ≤
n
kx1 − Tyk2 + 2
Dx
1
n
−
+ 2hzn − Ty, Ty − yi −
xn+1
E , y − Ty + kTy − yk2
n n 2X n k=1
αk hxk − Ty, Ty − yi.
Since {xn } is bounded, {zn } is also bounded. Then, there exists a subsequence {zni } of {zn } such that zni * w ∈ C . Further, replacing n by ni , we have 1 ni
2
kxni +1 − Tyk ≤
1 ni
2
kx1 − Tyk + 2
x1 ni
−
+ 2hzni − Ty, Ty − yi −
xni +1 ni
, y − Ty + kTy − yk2
ni 2 X
ni k=1
αk hxk − Ty, Ty − yi.
Since {xn } is bounded, we have that αk hxk − Ty, Ty − yi → 0 as k → ∞. Further, since zni * w and 1/ni Ty, Ty − yi → 0 as i → ∞, we have
Pni
k=1
αk hxk −
0 ≤ kTy − yk2 + 2hw − Ty, Ty − yi. Putting y = w , we have 0 ≤ kT w − wk2 + 2hw − T w, T w − wi
= −kT w − wk2 . Hence, w ∈ F (T ). To show that {zn } converges weakly to a fixed point of T , we first show that limn→∞ Pxn exists. Since F (T ) 6= ∅, from (3.1) we have that for all u ∈ F (T ),
kxn+1 − uk ≤ kxn − uk. On the other hand, since T is nonspreading, F (T ) is closed and convex. So, we can define the metric projection P of H onto F (T ). Putting D = F (T ) in Lemma 2.2, we have that limn→∞ Pxn converges strongly. Put z = limn→∞ Pxn . Then we can prove zn * z. In fact, let {zni } be a subsequence of {zn } such that zni * w . From the above argument, we have w ∈ F (T ). To complete the proof of the first part, it is sufficient to prove z = w . From w ∈ F (T ) and (2.2), we have
hw − z , xk − Pxk i = hw − Pxk , xk − Pxk i + hPxk − z , xk − Pxk i ≤ hPxk − z , xk − Pxk i
1566
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
≤ kPxk − z kkxk − Pxk k ≤ kPxk − z kM for all k ∈ N, where M = sup{kxk − Pxk k : k ∈ N}. Summing these inequalities from k = 1 to ni and dividing by ni , we have
* w − z , zni −
ni 1 X
ni k=1
+ Pxk
≤
ni 1 X
ni k=1
kPxk − z kM .
Since zni * w as i → ∞ and Pxn → z as n → ∞, we have hw − z , w − z i ≤ 0. This implies z = w . This completes Pn the proof of the first part. In particular, putting αn = 0 for all n ∈ N, we see that xn+1 = T n x and zn = 1/n k=1 T k−1 x for all n ∈ N. So, we obtain Sn x = zn . Therefore, {Sn x} converges weakly to z ∈ F (T ), where z = limn→∞ PT n x. So, we get the desired result. 4. Strong convergence theorem In this section, motivated by [29] and [30], we prove the following strong convergence theorem for nonspreading mappings in a Hilbert space. Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonspreading mapping of C into itself. Let u ∈ C and define two sequences {xn } and {zn } in C as follows: x1 = x ∈ C and
xn+1 = αn u + (1 − αn )zn , n −1 1X k z = T xn n n k=0
for all n = 1, 2, . . ., where 0 ≤ αn ≤ 1, αn → 0 and to Pu, where P is the metric projection of H onto F (T ).
P∞
n =1
αn = ∞. If F (T ) is nonempty, then {xn } and {zn } converge strongly
Proof. Since F (T ) 6= ∅, T is quasi-nonexpansive. So, we have that for all q ∈ F (T ) and n = 1, 2, 3, . . .,
n −1 n−1
1 X
1X
k kzn − qk = T xn − q ≤ kT k xn − qk
n k=0
n k=0 ≤
n−1 1X
n k=0
kxn − qk = kxn − qk.
(4.1)
Then we have
kxn+1 − qk = kαn u + (1 − αn )zn − qk ≤ αn ku − qk + (1 − αn )kzn − qk ≤ αn ku − qk + (1 − αn )kxn − qk. Hence, by induction, we obtain
kxn − qk ≤ max {ku − qk, kx − qk} for all n ∈ N. This implies that {xn } and {zn } are bounded. Since kT n xn − qk ≤ kxn − qk, we have also that {T n xn } is bounded. Let n ∈ N. Since T is nonspreading, we have from (2.3) that for all y ∈ C and k = 0, 1, 2, . . . , n − 1,
kT k+1 xn − Tyk2 ≤ kT k xn − yk2 + 2hT k xn − T k+1 xn , y − Tyi = kT k xn − Tyk2 + kTy − yk2 + 2hT k xn − Ty, Ty − yi + 2hT k xn − T k+1 xn , y − Tyi. Summing these inequalities from k = 0 to n − 1 and dividing by n, we have 1 n
kT n xn − Tyk2 ≤
1 n
2
kxn − Tyk2 + kTy − yk2 + 2hzn − Ty, Ty − yi + hxn − T n xn , y − Tyi. n
Since {zn } is bounded, there exists a subsequence {zni } of {zn } such that zni * w ∈ C . Replacing n by ni , we have 1 ni
kT ni xni − Tyk2 ≤
1 ni
kxni − Tyk2 + kTy − yk2 + 2hzni − Ty, Ty − yi +
Since {xn } and {T n xn } are bounded, we have that 0 ≤ kTy − yk2 + 2hw − Ty, Ty − yi
2 ni
hxni − T ni xni , y − Tyi.
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
1567
as i → ∞. Putting y = w , we have 0 ≤ kT w − wk2 + 2hw − T w, T w − wi = −kT w − wk2 . Hence, w ∈ F (T ). On the other hand, since xn+1 − zn = αn (u − zn ), {zn } is bounded and αn → 0, we have limn→∞ kxn+1 − zn k = 0. Let us show lim supn→∞ hu−Pu, xn+1 −Pui ≤ 0. We may assume without loss of generality that there exists a subsequence {xni +1 } of {xn+1 } such that lim suphu − Pu, xn+1 − Pui = lim hu − Pu, xni +1 − Pui n→∞
i→∞
and xni +1 * v . From kxn+1 − zn k → 0, we have zni * v . From the above argument, we have v ∈ F (T ). Since P is the metric projection of H onto F (T ), we have lim hu − Pu, xni +1 − Pui = hu − Pu, v − Pui ≤ 0.
i→∞
This implies lim suphu − Pu, xn+1 − Pui ≤ 0.
(4.2)
n→∞
Since xn+1 − Pu = (1 − αn )(zn − Pu) + αn (u − Pu), from (2.1) and (4.1) we have
kxn+1 − Puk2 = k(1 − αn )(zn − Pu) + αn (u − Pu)k2 ≤ (1 − αn )2 kzn − Puk2 + 2αn hu − Pu, xn+1 − Pui ≤ (1 − αn )kxn − Puk2 + 2αn hu − Pu, xn+1 − Pui. Putting sn = kxn − Puk2 , βn = 0 and γn = 2hu − Pu, xn+1 − Pui in Lemma 2.1, from
P∞
n =1
αn = ∞ and (4.2) we have
lim kxn − Puk = 0.
n→∞
By limn→∞ kxn − zn k = 0, we also obtain zn → Pu as n → ∞.
Remark. We do not know whether a strong convergence theorem of Halpern’s type for nonspreading mappings holds or not. References [1] J.-B. Baillon, Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér. A-B 280 (1975) Aii, A1511–A1514 (in French). [2] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–510. [3] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979) 274–276. [4] W. Takahashi, Convex Analysis and Approximation of Fixed Points, Yokohama Publishers, Yokohama, 2000 (in Japanese). [5] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 957–961. [6] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491. [7] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997) 3641–3645. [8] N. Shioji, W. Takahashi, A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Arch. Math. (Basel) 72 (1999) 354–359. [9] K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–378. [10] S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002) 938–945. [11] F.E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z. 100 (1967) 201–225. [12] K. Goebel, W.A. Kirk, Topics in metric fixed point theory, in: Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. [13] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994) 123–145. [14] P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 117–136. [15] F. Kohsaka, W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel) 91 (2008) 166–177. [16] F. Kohsaka, W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19 (2008) 824–835. [17] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11 (2010) 79–88. [18] T. Igarashi, W. Takahashi, K. Tanaka, Weak convergence theorems for nonspreading mappings and equilibrium problems, in: S. Akashi, W. Takahashi, T. Tanaka (Eds.), Nonlinear Analysis and Optimization, Yokohama Publishers, Yokohama, 2009, pp. 75–85. [19] S. Matsushita, W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. (2004) 37–47. [20] S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257–266. [21] W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, 2009. [22] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000. [23] S. Iemoto, W. Takahashi, Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear Anal. 71 (2009) 2082–2089. [24] S. Itoh, W. Takahashi, The common fixed point theory of singlevalued mappings and multivalued mappings, Pacific J. Math. 79 (1978) 493–508.
1568
Y. Kurokawa, W. Takahashi / Nonlinear Analysis 73 (2010) 1562–1568
[25] K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350–2360. [26] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003) 417–428. [27] M. Akatsuka, K. Aoyama, W. Takahashi, Mean ergodic theorems for a sequence of nonexpansive mappings in Hilbert spaces, Sci. Math. Jpn. 68 (2008) 233–239. [28] W. Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981) 253–256. [29] T. Shimizu, W. Takahashi, Strong convergence theorem for asymptotically nonexpansive mappings, Nonlinear Anal. 26 (1996) 265–272. [30] T. Shimizu, W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (1997) 71–83.