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Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings
Nonlinear Analysis 61 (2005) 341 – 350 www.elsevier.com/locate/na
Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings Hideaki Iiduka∗ , Wataru Takahashi Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152-8522, Japan Received 4 August 2002; accepted 29 July 2003
Abstract In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inversestrongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: primary 47H05; 47J05; 47J25 Keywords: Metric projection; Inverse-strongly monotone mapping; Nonexpansive mapping; Variational inequality; Strong convergence
1. Introduction Let C be a closed convex subset of a real Hilbert space H and let PC be the metric projection of H onto C. A mapping A of C into H is called monotone if for all x, y ∈ C,
∗ Corresponding author.
E-mail addresses: [email protected], [email protected] (H. Iiduka), [email protected] (W. Takahashi). 0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2003.07.023
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x − y, Ax − Ay 0. The variational inequality problem is to find a u ∈ C such that v − u, Au 0 for all v ∈ C; see [1,2,4,7,14]. The set of solutions of the variational inequality is denoted by V I (C, A). A mapping A of C into H is called inverse-strongly monotone if there exists a positive real number such that x − y, Ax − Ay Ax − Ay2 for all x, y ∈ C; see [3,6,8,9]. For such a case, A is called -inverse-strongly monotone. A mapping S of C into itself is called nonexpansive if Sx − Sy x − y for all x, y ∈ C; see [13,15,16] for the results of nonexpansive mappings. We denote by F (S) the set of fixed points of S. In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a real Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we first obtain a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. 2. Preliminaries Let H be a real Hilbert space with inner product ·, · and norm · , and let C be a closed convex subset of H . We write xn x to indicate that the sequence {xn } converges weakly to x. xn → x implies that {xn } converges strongly to x. For every point x ∈ H , there exists a unique nearest point in C, denoted by PC x, such that x − PC x x − y for all y ∈ C. PC is called the metric projection of H onto C. We know that PC is a nonexpansive mapping of H onto C. It is also known that PC satisfies x − y, PC x − PC y PC x − PC y2
(2.1)
for every x, y ∈ H . Moreover, PC x is characterized by the properties: PC x ∈ C and x − PC x, PC x − y 0 for all y ∈ C. In the context of the variational inequality problem, this implies that u ∈ V I (C, A) ⇐⇒ u = PC (u − Au),
∀ > 0.
(2.2)
It is also known that H satisfies Opial’s condition [10], i.e., for any sequence {xn } with xn x, the inequality lim inf xn − x < lim inf xn − y n→∞
n→∞
holds for every y ∈ H with y = x.
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We state some examples for inverse-strongly monotone mappings. If A = I − T , where T is a nonexpansive mapping of C into itself and I is the identity mapping of H , then A is 1/2-inverse-strongly monotone and V I (C, A) = F (T ). A mapping A of C into H is called strongly monotone if there exists a positive real number such that x − y, Ax − Ay x − y2 for all x, y ∈ C. In such a case, we say that A is -strongly monotone. If A is -strongly monotone and -Lipschitz continuous, i.e., Ax − Ay x − y for all x, y ∈ C, then A is /2 -inverse-strongly monotone. If A is an -inverse-strongly monotone mapping of C into H , then it is obvious that A is 1/-Lipschitz continuous. We also have that for all x, y ∈ C and > 0, (I − A)x − (I − A)y2 = (x − y) − (Ax − Ay)2 = x − y2 − 2x − y, Ax − Ay + 2 Ax − Ay2 (2.3) x − y2 + ( − 2)Ax − Ay2 . So, if 2, then I − A is a nonexpansive mapping of C into H . A set-valued mapping T : H → 2H is called monotone if for all x, y ∈ H , f ∈ T x and g ∈ T y imply x − y, f − g 0. A monotone mapping T : H → 2H is maximal if the graph G(T ) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f ) ∈ H × H , x − y, f − g 0 for every (y, g) ∈ G(T ) implies f ∈ T x. Let A be an inverse-strongly monotone mapping of C into H and let NC v be the normal cone to C at v ∈ C, i.e., NC v = {w ∈ H : v − u, w 0, ∀u ∈ C}, and define Av + NC v, v ∈ C, Tv = ∅, v∈ / C. Then T is maximal monotone and 0 ∈ T v if and only if v ∈ V I (C, A); see [11,12]. 3. Strong convergence theorem In this section, we prove a strong convergence theorem for nonexpansive mappings and inverse-strongly monotone mappings. Theorem 3.1. Let C be a closed convex subset of a real Hilbert space H. Let A be an -inverse-strongly monotone mapping of C into H and let S be a nonexpansive mapping of C into itself such that F (S) ∩ V I (C, A) = ∅. Suppose x1 = x ∈ C and {xn } is given by xn+1 = n x + (1 − n )SP C (xn − n Ax n ) for every n = 1, 2, . . . , where {n } is a sequence in [0, 1) and {n } is a sequence in [0, 2]. If {n } and {n } are chosen so that n ∈ [a, b] for some a, b with 0 < a < b < 2, lim n =0,
n→∞
∞ n=1
n =∞,
∞
|n+1 −n | < ∞
n=1
then {xn } converges strongly to PF (S)∩V I (C,A) x.
and
∞ n=1
|n+1 − n | < ∞,
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Proof. Put yn = PC (xn − n Ax n ) for every n = 1, 2, . . . . Let u ∈ F (S) ∩ V I (C, A). Since I − n A is nonexpansive and u = PC (u − n Au) from (2.2), we have yn − u = PC (xn − n Ax n ) − PC (u − n Au) (xn − n Ax n ) − (u − n Au) xn − u for every n = 1, 2, . . . . Then we have x2 − u = 1 x + (1 − 1 )Sy 1 − u 1 x − u + (1 − 1 )Sy 1 − u 1 x − u + (1 − 1 )y1 − u 1 x − u + (1 − 1 )x − u = x − u. If xk − u x − u holds for some k ∈ N, we can similarly show xk+1 − u x − u. Therefore, {xn } is bounded. Hence {yn }, {Sy n } and {Ax n } are also bounded. Since I − n A is nonexpansive, we also have yn+1 − yn = PC (xn+1 − n+1 Ax n+1 ) − PC (xn − n Ax n ) (xn+1 − n+1 Ax n+1 ) − (xn − n Ax n ) = (xn+1 − n+1 Ax n+1 ) − (xn − n+1 Ax n ) + (n − n+1 )Ax n (xn+1 − n+1 Ax n+1 ) − (xn − n+1 Ax n ) + |n − n+1 |Ax n xn+1 − x = n + |n − n+1 |Ax n (3.1) for every n = 1, 2, . . . . So, we obtain xn+1 − xn = (n x + (1 − n )Sy n ) − (n−1 x + (1 − n−1 )Sy n−1 ) = (n − n−1 )(x − Sy n−1 ) + (1 − n )(Sy n − Sy n−1 ) |n − n−1 |x − Sy n−1 + (1 − n )Sy n − Sy n−1 |n − n−1 |x − Sy n−1 + (1 − n )yn − yn−1 |n − n−1 |x − Sy n−1 + (1 − n )(xn − xn−1 + |n − n−1 |Ax n−1 ) (1 − n )xn − xn−1 + M|n − n−1 | + L|n − n−1 | for every n = 1, 2, . . . , where L = sup{x − Sy n : n ∈ N} and M = sup{Ax n : n ∈ N}. By mathematical induction, we have xn+m+1 − xn+m
n+m−1
n+m−1
k=m
k=m
(1 − k+1 )xm+1 − xm + M
+L
n+m−1 k=m
|k+1 − k |
|k+1 − k |
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for every n, m = 1, 2, . . . . So, we obtain lim sup xn+1 − xn = lim sup xn+m+1 − xn+m n→∞
n→∞ ∞
M
|k+1 − k | + L
∞
|k+1 − k |
k=m
k=m
for every m = 1, 2, . . . . Since obtain
∞
n=1 |n+1
− n | < ∞ and
∞
n=1 |n+1
− n | < ∞, we
lim sup xn+1 − xn 0 n→∞
and hence lim xn+1 − xn = 0.
n→∞
From (3.1) and
∞
n=1 |n+1
− n | < ∞, we also obtain yn+1 − yn → 0. Since
xn − Sy n xn − Sy n−1 + Sy n−1 − Sy n n−1 x − Sy n−1 + yn−1 − yn , we have xn − Sy n → 0. For u ∈ F (S) ∩ V I (C, A), from (2.3), we obtain xn+1 − u2 = n x + (1 − n )Sy n − u2
n x − u2 + (1 − n )Sy n − u2 n x − u2 + (1 − n )yn − u2 n x − u2 + (1 − n ){xn − u2 + n (n − 2)Ax n − Au2 } n x − u2 + xn − u2 + (1 − n )a(b − 2)Ax n − Au2 . Therefore, we have −(1 − n )a(b − 2)Ax n − Au2 n x − u2 + xn − u2 − xn+1 − u2 = n x − u2 + (xn − u + xn+1 − u) × (xn − u − xn+1 − u)
n x − u2 + (xn − u + xn+1 − u) × xn − xn+1 . Since n → 0 and xn+1 − xn → 0, we obtain Ax n − Au → 0. From (2.1), we have yn − u2 = PC (xn − n Ax n ) − PC (u − n Au)2 (xn − n Ax n ) − (u − n Au), yn − u = 21 {(xn − n Ax n ) − (u − n Au)2 + yn − u2 − (xn − n Ax n ) − (u − n Au) − (yn − u)2 }
21 {xn − u2 + yn − u2 − (xn − yn ) − n (Ax n − Au)2 } = 21 {xn − u2 + yn − u2 − xn − yn 2 + 2n xn − yn , Ax n − Au − 2n Ax n − Au2 }.
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So, we obtain yn − u2 xn − u2 − xn − yn 2 + 2n xn − yn , Ax n − Au − 2n Ax n − Au2 and hence xn+1 − u2 = n x + (1 − n )Sy n − u2
n x − u2 + (1 − n )Sy n − u2 n x − u2 + (1 − n )yn − u2 n x − u2 + xn − u2 − (1 − n )xn − yn 2 + 2(1 − n )n xn − yn , Ax n − Au − (1 − n )2n Ax n − Au2 . Since n → 0, xn+1 − xn → 0 and Ax n − Au → 0, we obtain xn − yn → 0. Since Sy n − yn Sy n − xn + xn − yn , we obtain Sy n − yn → 0. Next we show that lim supx − z0 , Sy n − z0 0, n→∞
where z0 = PF (S)∩V I (C,A) x. To show it, choose a subsequence {yni } of {yn } such that lim supx − z0 , Sy n − z0 = lim x − z0 , Sy ni − z0 . i→∞
n→∞
As {yni } is bounded, we have that a subsequence {yni j } of {yni } converges weakly to z. We may assume without loss of generality that yni z. Since Sy n − yn → 0, we obtain Sy ni z. Then we can obtain z ∈ F (S) ∩ V I (C, A). In fact, let us first show that z ∈ V I (C, A). Let Tv =
Av + NC v, ∅,
v ∈ C, v∈ / C.
Then T is maximal monotone. Let (v, w) ∈ G(T ). Since w − Av ∈ NC v and yn ∈ C, we have v − yn , w − Av 0. On the other hand, from yn = PC (xn − n Ax n ), we have v − yn , yn − (xn − n Ax n ) 0 and hence yn − xn v − yn , + Ax n 0. n
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347
Therefore, we have v − yni , w v − yni , Av
yni − xni + Ax ni v − yni , Av − v − yni , ni yn − xni = v − yni , Av − Ax ni − i ni = v − yni , Av − Ay ni + v − yni , Ay ni − Ax ni yn − xni − v − yni , i ni yni − xni . v − yni , Ay ni − Ax ni − v − yni , ni
Hence we obtain v − z, w 0 as i → ∞. Since T is maximal monotone, we have z ∈ T −1 0 and hence z ∈ V I (C, A). Let us show that z ∈ F (S). Assume z ∈ / F (S). From Opial’s condition, we have lim inf yni − z < lim inf yni − Sz i→∞
i→∞
= lim inf yni − Sy ni + Sy ni − Sz i→∞
= lim inf Sy ni − Sz i→∞
lim inf yni − z. i→∞
This is a contradiction. Thus, we obtain z ∈ F (S). Then we have lim supx − z0 , Sy n − z0 = lim x − z0 , Sy ni − z0 i→∞
n→∞
= x − z0 , z − z0 0. Therefore, for any > 0, there exists m ∈ N such that x − z0 , Sy n − z0 ,
n x − z0 2
for all n m. For all n m, we have xn+1 − z0 2 = n x + (1 − n )Sy n − z0 2 = 2n x − z0 2 + 2n (1 − n )x − z0 , Sy n − z0 + (1 − n )2 Sy n − z0 2
n + 2n (1 − n ) + (1 − n )Sy n − z0 2 3n + (1 − n )Sy n − z0 2 3n + (1 − n )xn − z0 2 = 3(1 − (1 − n )) + (1 − n )xn − z0 2 . By mathematical induction, we obtain n n 2 xn+1 − z0 3 1 − (1 − k ) + (1 − k )xm − z0 2 . k=m
k=m
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H. Iiduka, W. Takahashi / Nonlinear Analysis 61 (2005) 341 – 350
Therefore, we have lim sup xn+1 − z0 2 3. n→∞
Since > 0 is arbitrary, we have lim supn→∞ xn+1 − z0 2 0 and hence xn → z0 .
Remark. We obtain Wittmann’s theorem [18] if A = 0 in Theorem 3.1; see also [5]. Takahashi and Toyoda [17] considered Mann’s type iteration: x1 = x ∈ C, xn+1 = n xn + (1 − n )SP C (xn − n Ax n ), and obtained that the sequence {xn } converges weakly to z ∈ F (S) ∩ V I (C, A). As a direct consequence of Theorem 3.1, we obtain the following: Corollary 3.2. Let C be a closed convex subset of a real Hilbert space H. Let A be an inverse-strongly monotone mapping of C into H such that V I (C, A) = ∅. Suppose x1 = x ∈ C and {xn } is given by xn+1 = n x + (1 − n )PC (xn − n Ax n ) for every n = 1, 2, . . . , where {n } is a sequence in [0, 1) and {n } is a sequence in [0, 2]. If {n } and {n } are chosen so that n ∈ [a, b] for some a, b with 0 < a < b < 2, lim n =0,
n→∞
∞ n=1
n =∞,
∞
|n+1 −n | < ∞
and
n=1
∞
|n+1 − n | < ∞,
n=1
then {xn } converges strongly to PV I (C,A) x. 4. Applications In this section, we prove two theorems in a real Hilbert space by using Theorem 3.1. A mapping T : C → C is called strictly pseudocontractive if there exists k with 0 k < 1 such that T x − T y2 x − y2 + k(I − T )x − (I − T )y2 for all x, y ∈ C. If k = 0, then T is nonexpansive. Put A = I − T , where T : C → C is a strictly pseudocontractive mapping with k. Then A is (1 − k)/2-inverse-strongly monotone; see [3]. Actually, we have, for all x, y ∈ C, (I − A)x − (I − A)y2 x − y2 + kAx − Ay2 . On the other hand, since H is a real Hilbert space, we have (I − A)x − (I − A)y2 = x − y2 + Ax − Ay2 − 2x − y, Ax − Ay.
H. Iiduka, W. Takahashi / Nonlinear Analysis 61 (2005) 341 – 350
349
Hence we have x − y, Ax − Ay
1−k Ax − Ay2 . 2
Using Theorem 3.1, we first prove a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Theorem 4.1. Let C be a closed convex subset of a real Hilbert space H. Let S be a nonexpansive mapping of C into itself and let T be a k-strictly pseudocontractive mapping of C into itself such that F (S) ∩ F (T ) = ∅. Suppose x1 = x ∈ C and {xn } is given by xn+1 =n x+(1−n )S((1−n )xn +n T x n ) for every n=1, 2, . . . , where {n } is a sequence in [0, 1) and {n } is a sequence in [0, 1−k]. If {n } and {n } are chosen so that n ∈ [a, b] for some a, b with 0 < a < b < 1 − k, lim n =0,
n→∞
∞
n =∞,
n=1
∞
|n+1 −n | < ∞
and
n=1
∞
|n+1 −n | < ∞,
n=1
then {xn } converges strongly to PF (S)∩F (T ) x. Proof. Put A = I − T . Then A is (1 − k)/2-inverse-strongly monotone. We have F (T ) = V I (C, A) and PC (xn − n Ax n ) = (1 − n )xn + n T x n . So, by Theorem 3.1, we obtain the desired result. Using Theorem 3.1, we also have the following: Theorem 4.2. Let H be a real Hilbert space. Let A be an -inverse-strongly monotone mapping of H into itself and let S be a nonexpansive mapping of H into itself such that F (S) ∩ A−1 0 = ∅. Suppose x1 = x ∈ H and {xn } is given by xn+1 = n x + (1 − n )S(xn − n Ax n ) for every n = 1, 2, . . . , where {n } is a sequence in [0, 1) and {n } is a sequence in [0, 2]. If {n } and {n } are chosen so that n ∈ [a, b] for some a, b with 0 < a < b < 2, lim n =0,
n→∞
∞ n=1
n =∞,
∞
|n+1 − n | < ∞
n=1
and
∞
|n+1 − n | < ∞,
n=1
then {xn } converges strongly to PF (S)∩A−1 0 x. Proof. We have A−1 0 = V I (H, A). So, putting PH = I , by Theorem 3.1, we obtain the desired result. Remark. See [19] for the case when S is a nonexpansive mapping of a real Hilbert space H into itself and A is a strongly monotone and Lipschitz continuous mapping of H into itself.
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References [1] F.E. Browder, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc. 71 (1965) 780–785. [2] F.E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968) 283–301. [3] F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967) 197–228. [4] R.E. Bruck, On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61 (1977) 159–164. [5] B.R. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 957–961. [6] H. Iiduka, W. Takahashi, M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, PanAmer. Math. J. 14 (2004) 49–61. [7] J.L. Lions, G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967) 493–517. [8] F. Liu, M.Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal. 6 (1998) 313–344. [9] K. Nakajo, W. Takahashi, Strong and weak convergence theorems by an improved splitting method, Comm. Appl. Nonlinear Anal. 9 (2002) 99–107. [10] Z. Opial, Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591–597. [11] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970) 75–88. [12] R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976) 877–898. [13] W. Takahashi, Nonlinear variational inequalities and fixed point theorems, J. Math. Soc. Japan 28 (1976) 168–181. [14] W. Takahashi, Nonlinear complementarity problem and systems of convex inequalities, J. Optim. Theory Appl. 24 (1978) 493–508. [15] W. Takahashi, Convex Analysis and Approximation of Fixed Points, Yokohama Publishers, Yokohama, 2000 (Japanese). [16] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000. [17] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003) 417–428. [18] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491. [19] I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, in: D. Butnariu, Y. Censor, S. Reich (Eds.), Inherently Parallel Algorithms for Feasibility and Optimization and Their Applications, Elsevier, Amsterdam, 2001, pp. 473–504.
Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings Hideaki Iiduka∗ , Wataru Takahashi Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152-8522, Japan Received 4 August 2002; accepted 29 July 2003
Abstract In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inversestrongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: primary 47H05; 47J05; 47J25 Keywords: Metric projection; Inverse-strongly monotone mapping; Nonexpansive mapping; Variational inequality; Strong convergence
1. Introduction Let C be a closed convex subset of a real Hilbert space H and let PC be the metric projection of H onto C. A mapping A of C into H is called monotone if for all x, y ∈ C,
∗ Corresponding author.
E-mail addresses: [email protected], [email protected] (H. Iiduka), [email protected] (W. Takahashi). 0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2003.07.023
342
H. Iiduka, W. Takahashi / Nonlinear Analysis 61 (2005) 341 – 350
x − y, Ax − Ay 0. The variational inequality problem is to find a u ∈ C such that v − u, Au 0 for all v ∈ C; see [1,2,4,7,14]. The set of solutions of the variational inequality is denoted by V I (C, A). A mapping A of C into H is called inverse-strongly monotone if there exists a positive real number such that x − y, Ax − Ay Ax − Ay2 for all x, y ∈ C; see [3,6,8,9]. For such a case, A is called -inverse-strongly monotone. A mapping S of C into itself is called nonexpansive if Sx − Sy x − y for all x, y ∈ C; see [13,15,16] for the results of nonexpansive mappings. We denote by F (S) the set of fixed points of S. In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a real Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we first obtain a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. 2. Preliminaries Let H be a real Hilbert space with inner product ·, · and norm · , and let C be a closed convex subset of H . We write xn x to indicate that the sequence {xn } converges weakly to x. xn → x implies that {xn } converges strongly to x. For every point x ∈ H , there exists a unique nearest point in C, denoted by PC x, such that x − PC x x − y for all y ∈ C. PC is called the metric projection of H onto C. We know that PC is a nonexpansive mapping of H onto C. It is also known that PC satisfies x − y, PC x − PC y PC x − PC y2
(2.1)
for every x, y ∈ H . Moreover, PC x is characterized by the properties: PC x ∈ C and x − PC x, PC x − y 0 for all y ∈ C. In the context of the variational inequality problem, this implies that u ∈ V I (C, A) ⇐⇒ u = PC (u − Au),
∀ > 0.
(2.2)
It is also known that H satisfies Opial’s condition [10], i.e., for any sequence {xn } with xn x, the inequality lim inf xn − x < lim inf xn − y n→∞
n→∞
holds for every y ∈ H with y = x.
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We state some examples for inverse-strongly monotone mappings. If A = I − T , where T is a nonexpansive mapping of C into itself and I is the identity mapping of H , then A is 1/2-inverse-strongly monotone and V I (C, A) = F (T ). A mapping A of C into H is called strongly monotone if there exists a positive real number such that x − y, Ax − Ay x − y2 for all x, y ∈ C. In such a case, we say that A is -strongly monotone. If A is -strongly monotone and -Lipschitz continuous, i.e., Ax − Ay x − y for all x, y ∈ C, then A is /2 -inverse-strongly monotone. If A is an -inverse-strongly monotone mapping of C into H , then it is obvious that A is 1/-Lipschitz continuous. We also have that for all x, y ∈ C and > 0, (I − A)x − (I − A)y2 = (x − y) − (Ax − Ay)2 = x − y2 − 2x − y, Ax − Ay + 2 Ax − Ay2 (2.3) x − y2 + ( − 2)Ax − Ay2 . So, if 2, then I − A is a nonexpansive mapping of C into H . A set-valued mapping T : H → 2H is called monotone if for all x, y ∈ H , f ∈ T x and g ∈ T y imply x − y, f − g 0. A monotone mapping T : H → 2H is maximal if the graph G(T ) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f ) ∈ H × H , x − y, f − g 0 for every (y, g) ∈ G(T ) implies f ∈ T x. Let A be an inverse-strongly monotone mapping of C into H and let NC v be the normal cone to C at v ∈ C, i.e., NC v = {w ∈ H : v − u, w 0, ∀u ∈ C}, and define Av + NC v, v ∈ C, Tv = ∅, v∈ / C. Then T is maximal monotone and 0 ∈ T v if and only if v ∈ V I (C, A); see [11,12]. 3. Strong convergence theorem In this section, we prove a strong convergence theorem for nonexpansive mappings and inverse-strongly monotone mappings. Theorem 3.1. Let C be a closed convex subset of a real Hilbert space H. Let A be an -inverse-strongly monotone mapping of C into H and let S be a nonexpansive mapping of C into itself such that F (S) ∩ V I (C, A) = ∅. Suppose x1 = x ∈ C and {xn } is given by xn+1 = n x + (1 − n )SP C (xn − n Ax n ) for every n = 1, 2, . . . , where {n } is a sequence in [0, 1) and {n } is a sequence in [0, 2]. If {n } and {n } are chosen so that n ∈ [a, b] for some a, b with 0 < a < b < 2, lim n =0,
n→∞
∞ n=1
n =∞,
∞
|n+1 −n | < ∞
n=1
then {xn } converges strongly to PF (S)∩V I (C,A) x.
and
∞ n=1
|n+1 − n | < ∞,
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Proof. Put yn = PC (xn − n Ax n ) for every n = 1, 2, . . . . Let u ∈ F (S) ∩ V I (C, A). Since I − n A is nonexpansive and u = PC (u − n Au) from (2.2), we have yn − u = PC (xn − n Ax n ) − PC (u − n Au) (xn − n Ax n ) − (u − n Au) xn − u for every n = 1, 2, . . . . Then we have x2 − u = 1 x + (1 − 1 )Sy 1 − u 1 x − u + (1 − 1 )Sy 1 − u 1 x − u + (1 − 1 )y1 − u 1 x − u + (1 − 1 )x − u = x − u. If xk − u x − u holds for some k ∈ N, we can similarly show xk+1 − u x − u. Therefore, {xn } is bounded. Hence {yn }, {Sy n } and {Ax n } are also bounded. Since I − n A is nonexpansive, we also have yn+1 − yn = PC (xn+1 − n+1 Ax n+1 ) − PC (xn − n Ax n ) (xn+1 − n+1 Ax n+1 ) − (xn − n Ax n ) = (xn+1 − n+1 Ax n+1 ) − (xn − n+1 Ax n ) + (n − n+1 )Ax n (xn+1 − n+1 Ax n+1 ) − (xn − n+1 Ax n ) + |n − n+1 |Ax n xn+1 − x = n + |n − n+1 |Ax n (3.1) for every n = 1, 2, . . . . So, we obtain xn+1 − xn = (n x + (1 − n )Sy n ) − (n−1 x + (1 − n−1 )Sy n−1 ) = (n − n−1 )(x − Sy n−1 ) + (1 − n )(Sy n − Sy n−1 ) |n − n−1 |x − Sy n−1 + (1 − n )Sy n − Sy n−1 |n − n−1 |x − Sy n−1 + (1 − n )yn − yn−1 |n − n−1 |x − Sy n−1 + (1 − n )(xn − xn−1 + |n − n−1 |Ax n−1 ) (1 − n )xn − xn−1 + M|n − n−1 | + L|n − n−1 | for every n = 1, 2, . . . , where L = sup{x − Sy n : n ∈ N} and M = sup{Ax n : n ∈ N}. By mathematical induction, we have xn+m+1 − xn+m
n+m−1
n+m−1
k=m
k=m
(1 − k+1 )xm+1 − xm + M
+L
n+m−1 k=m
|k+1 − k |
|k+1 − k |
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for every n, m = 1, 2, . . . . So, we obtain lim sup xn+1 − xn = lim sup xn+m+1 − xn+m n→∞
n→∞ ∞
M
|k+1 − k | + L
∞
|k+1 − k |
k=m
k=m
for every m = 1, 2, . . . . Since obtain
∞
n=1 |n+1
− n | < ∞ and
∞
n=1 |n+1
− n | < ∞, we
lim sup xn+1 − xn 0 n→∞
and hence lim xn+1 − xn = 0.
n→∞
From (3.1) and
∞
n=1 |n+1
− n | < ∞, we also obtain yn+1 − yn → 0. Since
xn − Sy n xn − Sy n−1 + Sy n−1 − Sy n n−1 x − Sy n−1 + yn−1 − yn , we have xn − Sy n → 0. For u ∈ F (S) ∩ V I (C, A), from (2.3), we obtain xn+1 − u2 = n x + (1 − n )Sy n − u2
n x − u2 + (1 − n )Sy n − u2 n x − u2 + (1 − n )yn − u2 n x − u2 + (1 − n ){xn − u2 + n (n − 2)Ax n − Au2 } n x − u2 + xn − u2 + (1 − n )a(b − 2)Ax n − Au2 . Therefore, we have −(1 − n )a(b − 2)Ax n − Au2 n x − u2 + xn − u2 − xn+1 − u2 = n x − u2 + (xn − u + xn+1 − u) × (xn − u − xn+1 − u)
n x − u2 + (xn − u + xn+1 − u) × xn − xn+1 . Since n → 0 and xn+1 − xn → 0, we obtain Ax n − Au → 0. From (2.1), we have yn − u2 = PC (xn − n Ax n ) − PC (u − n Au)2 (xn − n Ax n ) − (u − n Au), yn − u = 21 {(xn − n Ax n ) − (u − n Au)2 + yn − u2 − (xn − n Ax n ) − (u − n Au) − (yn − u)2 }
21 {xn − u2 + yn − u2 − (xn − yn ) − n (Ax n − Au)2 } = 21 {xn − u2 + yn − u2 − xn − yn 2 + 2n xn − yn , Ax n − Au − 2n Ax n − Au2 }.
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So, we obtain yn − u2 xn − u2 − xn − yn 2 + 2n xn − yn , Ax n − Au − 2n Ax n − Au2 and hence xn+1 − u2 = n x + (1 − n )Sy n − u2
n x − u2 + (1 − n )Sy n − u2 n x − u2 + (1 − n )yn − u2 n x − u2 + xn − u2 − (1 − n )xn − yn 2 + 2(1 − n )n xn − yn , Ax n − Au − (1 − n )2n Ax n − Au2 . Since n → 0, xn+1 − xn → 0 and Ax n − Au → 0, we obtain xn − yn → 0. Since Sy n − yn Sy n − xn + xn − yn , we obtain Sy n − yn → 0. Next we show that lim supx − z0 , Sy n − z0 0, n→∞
where z0 = PF (S)∩V I (C,A) x. To show it, choose a subsequence {yni } of {yn } such that lim supx − z0 , Sy n − z0 = lim x − z0 , Sy ni − z0 . i→∞
n→∞
As {yni } is bounded, we have that a subsequence {yni j } of {yni } converges weakly to z. We may assume without loss of generality that yni z. Since Sy n − yn → 0, we obtain Sy ni z. Then we can obtain z ∈ F (S) ∩ V I (C, A). In fact, let us first show that z ∈ V I (C, A). Let Tv =
Av + NC v, ∅,
v ∈ C, v∈ / C.
Then T is maximal monotone. Let (v, w) ∈ G(T ). Since w − Av ∈ NC v and yn ∈ C, we have v − yn , w − Av 0. On the other hand, from yn = PC (xn − n Ax n ), we have v − yn , yn − (xn − n Ax n ) 0 and hence yn − xn v − yn , + Ax n 0. n
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Therefore, we have v − yni , w v − yni , Av
yni − xni + Ax ni v − yni , Av − v − yni , ni yn − xni = v − yni , Av − Ax ni − i ni = v − yni , Av − Ay ni + v − yni , Ay ni − Ax ni yn − xni − v − yni , i ni yni − xni . v − yni , Ay ni − Ax ni − v − yni , ni
Hence we obtain v − z, w 0 as i → ∞. Since T is maximal monotone, we have z ∈ T −1 0 and hence z ∈ V I (C, A). Let us show that z ∈ F (S). Assume z ∈ / F (S). From Opial’s condition, we have lim inf yni − z < lim inf yni − Sz i→∞
i→∞
= lim inf yni − Sy ni + Sy ni − Sz i→∞
= lim inf Sy ni − Sz i→∞
lim inf yni − z. i→∞
This is a contradiction. Thus, we obtain z ∈ F (S). Then we have lim supx − z0 , Sy n − z0 = lim x − z0 , Sy ni − z0 i→∞
n→∞
= x − z0 , z − z0 0. Therefore, for any > 0, there exists m ∈ N such that x − z0 , Sy n − z0 ,
n x − z0 2
for all n m. For all n m, we have xn+1 − z0 2 = n x + (1 − n )Sy n − z0 2 = 2n x − z0 2 + 2n (1 − n )x − z0 , Sy n − z0 + (1 − n )2 Sy n − z0 2
n + 2n (1 − n ) + (1 − n )Sy n − z0 2 3n + (1 − n )Sy n − z0 2 3n + (1 − n )xn − z0 2 = 3(1 − (1 − n )) + (1 − n )xn − z0 2 . By mathematical induction, we obtain n n 2 xn+1 − z0 3 1 − (1 − k ) + (1 − k )xm − z0 2 . k=m
k=m
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H. Iiduka, W. Takahashi / Nonlinear Analysis 61 (2005) 341 – 350
Therefore, we have lim sup xn+1 − z0 2 3. n→∞
Since > 0 is arbitrary, we have lim supn→∞ xn+1 − z0 2 0 and hence xn → z0 .
Remark. We obtain Wittmann’s theorem [18] if A = 0 in Theorem 3.1; see also [5]. Takahashi and Toyoda [17] considered Mann’s type iteration: x1 = x ∈ C, xn+1 = n xn + (1 − n )SP C (xn − n Ax n ), and obtained that the sequence {xn } converges weakly to z ∈ F (S) ∩ V I (C, A). As a direct consequence of Theorem 3.1, we obtain the following: Corollary 3.2. Let C be a closed convex subset of a real Hilbert space H. Let A be an inverse-strongly monotone mapping of C into H such that V I (C, A) = ∅. Suppose x1 = x ∈ C and {xn } is given by xn+1 = n x + (1 − n )PC (xn − n Ax n ) for every n = 1, 2, . . . , where {n } is a sequence in [0, 1) and {n } is a sequence in [0, 2]. If {n } and {n } are chosen so that n ∈ [a, b] for some a, b with 0 < a < b < 2, lim n =0,
n→∞
∞ n=1
n =∞,
∞
|n+1 −n | < ∞
and
n=1
∞
|n+1 − n | < ∞,
n=1
then {xn } converges strongly to PV I (C,A) x. 4. Applications In this section, we prove two theorems in a real Hilbert space by using Theorem 3.1. A mapping T : C → C is called strictly pseudocontractive if there exists k with 0 k < 1 such that T x − T y2 x − y2 + k(I − T )x − (I − T )y2 for all x, y ∈ C. If k = 0, then T is nonexpansive. Put A = I − T , where T : C → C is a strictly pseudocontractive mapping with k. Then A is (1 − k)/2-inverse-strongly monotone; see [3]. Actually, we have, for all x, y ∈ C, (I − A)x − (I − A)y2 x − y2 + kAx − Ay2 . On the other hand, since H is a real Hilbert space, we have (I − A)x − (I − A)y2 = x − y2 + Ax − Ay2 − 2x − y, Ax − Ay.
H. Iiduka, W. Takahashi / Nonlinear Analysis 61 (2005) 341 – 350
349
Hence we have x − y, Ax − Ay
1−k Ax − Ay2 . 2
Using Theorem 3.1, we first prove a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Theorem 4.1. Let C be a closed convex subset of a real Hilbert space H. Let S be a nonexpansive mapping of C into itself and let T be a k-strictly pseudocontractive mapping of C into itself such that F (S) ∩ F (T ) = ∅. Suppose x1 = x ∈ C and {xn } is given by xn+1 =n x+(1−n )S((1−n )xn +n T x n ) for every n=1, 2, . . . , where {n } is a sequence in [0, 1) and {n } is a sequence in [0, 1−k]. If {n } and {n } are chosen so that n ∈ [a, b] for some a, b with 0 < a < b < 1 − k, lim n =0,
n→∞
∞
n =∞,
n=1
∞
|n+1 −n | < ∞
and
n=1
∞
|n+1 −n | < ∞,
n=1
then {xn } converges strongly to PF (S)∩F (T ) x. Proof. Put A = I − T . Then A is (1 − k)/2-inverse-strongly monotone. We have F (T ) = V I (C, A) and PC (xn − n Ax n ) = (1 − n )xn + n T x n . So, by Theorem 3.1, we obtain the desired result. Using Theorem 3.1, we also have the following: Theorem 4.2. Let H be a real Hilbert space. Let A be an -inverse-strongly monotone mapping of H into itself and let S be a nonexpansive mapping of H into itself such that F (S) ∩ A−1 0 = ∅. Suppose x1 = x ∈ H and {xn } is given by xn+1 = n x + (1 − n )S(xn − n Ax n ) for every n = 1, 2, . . . , where {n } is a sequence in [0, 1) and {n } is a sequence in [0, 2]. If {n } and {n } are chosen so that n ∈ [a, b] for some a, b with 0 < a < b < 2, lim n =0,
n→∞
∞ n=1
n =∞,
∞
|n+1 − n | < ∞
n=1
and
∞
|n+1 − n | < ∞,
n=1
then {xn } converges strongly to PF (S)∩A−1 0 x. Proof. We have A−1 0 = V I (H, A). So, putting PH = I , by Theorem 3.1, we obtain the desired result. Remark. See [19] for the case when S is a nonexpansive mapping of a real Hilbert space H into itself and A is a strongly monotone and Lipschitz continuous mapping of H into itself.
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