STRONG CONVERGENCE THEOREMS BY A HYBRID STEEPEST DESCENT METHOD FOR COUNTABLE NONEXPANSIVE MAPPINGS IN HILBERT SPACES

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1 Scientiae Mathematicae Japonicae Online, e-2008, STRONG CONVERGENCE THEOREMS BY A HYBRID STEEPEST DESCENT METHOD FOR COUNTABLE NONEXPANSIVE MAPPINGS IN HILBERT SPACES SHIGERU IEMOTO AND WATARU TAKAHASHI Received October 10, 2008; revised October 31, 2008 Abstract. In this paper, we introduce a new iterative procedure for finding a solution of the variational inequality problem over the intersection of fixed point sets of infinite nonexpansive mappings in a Hilbert space and then discuss the strong convergence of the iterative procedure. 1. Introduction Let C be a closed convex subset of a real Hilbert space H. A mapping A of C into H is called monotone if x y, Ax Ay 0 for all x, y C. The variational inequality problem for A is to find z C such that y z,az 0 for all y C. The set of solutions of the variational inequality is denoted by VIC, A). A mapping A of C into H is called strongly monotone if there exists a positive real number α such that x y, Ax Ay α x y 2 for all x, y C. Such A is called α-strongly monotone. A mapping T of C into itself is called nonexpansive if Tx Ty x y for all x, y C. We denote by F T ) the set of all fixed points of T. The well-known iterative procedure for finding a solution of the variational inequality problem may be the projected gradient method [3, 14]: x 1 C and 1) x n1 = P C I ρa)x n for n =1, 2,..., where P C is the metric projection of H onto C and ρ is a positive real number. Indeed, when A is strongly monotone and Lipschitzian, the sequence {x n } generated by 1) converges strongly to a unique solution of VIC, A). However, the projected gradient method requires the use of the metric projection P C of which the closed form expression is not known. In order to reduce the complexity which is caused by P C, Yamada [13] introduced the following iterative procedure called the hybrid steepest descent method: x 1 H and 2) x n1 =I λ n ρa)tx n 2000 Mathematics Subject Classification. Primary 49J40; Secondary 47J20. Key words and phrases. variational inequality, hybrid steepest descent method, Hilbert space, strongly monotone, Lipschitzian, nonexpansive mapping.

2 558 SHIGERU IEMOTO AND WATARU TAKAHASHI for all n =1, 2,..., where {λ n } is a sequence in 0, 1] and ρ is a positive real number. He showed that the sequence {x n } generated by 2) converges strongly to a unique solution of VIF T ),A). On the other hand, Kimura and Takahashi [4] established a weak convergence theorem for an infinite family of nonexpansive mappings which is connected with the feasibility problem and generalizes the result of Takahashi and Shimoji [11]. Shimoji and Takahashi [5] also proved a strong convergence theorem for an infinite family of nonexpansive mappings by using the methods of proofs of Shioji and Takahashi [6] and Atsushiba and Takahashi [2]. The purpose of the present paper is to prove a strong convergence theorem for finding a solution of the variational inequality problem over the intersection of fixed point sets of infinite nonexpansive mappings {T n } in a real Hilbert space. We deal with the following iterative scheme: { x 1 H, x n1 =I λ n ρa)w n x n for all n =1, 2,..., where {W n } is a sequence of W -mappings generated by nonexpansive mappings T n,t n 1,...,T 1 of H into itself, A is a strongly monotone and Lipschitzian mapping of H into itself, {λ n } 0, 1] and ρ>0. 2. Preliminaries Throughout this paper, we denote by N the set of positive integers and by R the set of real numbers. Let H be a real Hilbert space with inner product, and norm and let C be a closed convex subset of H. We denote the strong convergence and the weak convergence of x n to x H by x n x and x n x, respectively. A mapping T of C into itself is nonexpansive if Tx Ty x y for all x, y C. We denote by F T ) the set of all fixed points of T, that is, F T )={z H : Tz = z} and by RT ) the range of T. We know that if C is a bounded closed convex subset of H and T is a nonexpansive mapping of C into itself, F T ) is nonempty. It is also well-known that F T ) is a closed convex subset of H. For every point x H, there exists a unique nearest point in C, denoted by P C x, such that x P C x x y for all y C. P C is called the metric projection of H onto C. We know that P C is a nonexpansive mapping of H onto C and 3) x P C x, P C x y 0 for all y C. In the context of the variational inequality problem, this implies that u VIC, A) u = P C u ρau) for all ρ>0, where A is a monotone mapping of C into H. A mapping A of C into H is called strongly monotone if there exists a positive real number α such that x y, Ax Ay α x y 2 for all x, y C. Such A is called α-strongly monotone. If A : C H is α-strongly monotone and β-lipschitzian, then without loss of generality we can assume α<β. The following lemma is in [14].

3 HYBRID STEEPEST DESCENT METHOD 559 Lemma 2.1 [14]). Let H be a real Hilbert space, let C be a nonempty closed convex subset of H and let α, β, ρ > 0. Suppose that A is an α-strongly monotone and β-lipschitzian mapping of C into H and ρ 0, 2α/β 2 ). Then P C I ρa)x P C I ρa)y 1 ρ2α ρβ 2 ) x y for all x, y C. In particular, P C I ρa) is a contraction of C into itself. Remark 1. In this lemma, ρ2α ρβ 2 ) is actually in the interval 0, 1). In fact, it is easy that ρ2α ρβ 2 ) > 0. We have that α<β= α 2 β 2 < 0 = β 2 ρ 2 2αρ 1> 0 = ρ2α ρβ 2 ) < 1. The following theorem is due to [14]; see also [3]. Theorem 2.1 Projected gradient method). Let H be a real Hilbert space, let C be a nonempty closed convex subset of H and let α, β > 0. Suppose that A is an α-strongly monotone and β-lipschitzian mapping of C into H. Then the following hold: i) VIC, A) has its unique solution u C; ii) for any x 1 C and ρ 0, 2α/β 2 ), the sequence {x n } generated by x n1 = P C I ρa)x n for n N converges strongly to a unique solution u of VIC, A). Motivated by Theorem 2.1, Yamada [13] proved the following theorem. Theorem 2.2 Hybrid steepest descent method). Let H be a real Hilbert space, let T be a nonexpansive mapping on H such that F T ) is nonempty and let α, β > 0. Suppose that A is an α-strongly monotone and β-lipschitzian mapping of RT ) into H. Then, VIF T ),A) has its unique solution u C. Further, for any x 1 H and ρ 0, 2α/β 2 ), let {x n } be the sequence generated by x n1 =I λ n ρa)tx n for all n N. Then, the sequence {x n } converges strongly to a unique solution u of VIF T ),A), where {λ n } is a sequence of 0, 1] satisfying C1) lim λ n =0; C2) n=1 λ n = ; λ C3) lim n =0. λ 2 n1 The above condition C3) can be generalized to the following condition by Xu [12]: λ C4) lim n =1. Let T 1,T 2,... be mappings on H and let γ 1,γ 2,... be real numbers such that 0 γ i 1 for every i N. Then, for any n N, we define a mapping W n on H as follows: U n,n1 = I, U n,n = γ n T n U n,n1 1 γ n )I, U n,n 1 = γ n 1 T n 1 U n,n 1 γ n 1 )I,. U n,k = γ k T k U n,k1 1 γ k )I, U n,k 1 = γ k 1 T k 1 U n,k 1 γ k 1 )I,

4 560 SHIGERU IEMOTO AND WATARU TAKAHASHI. U n,2 = γ 2 T 2 U n,3 1 γ 2 )I, W n = U n,1 = γ 1 T 1 U n,2 1 γ 1 )I. Such a mapping W n is called the W -mapping generated by T n,t n 1,...,T 1 and γ n, γ n 1,..., γ 1 ; see [8], [10] and [11]. The following lemma was proved in [11]. Lemma 2.2 [11]). Let H be a real Hilbert space. Let T 1,T 2,...,T n be nonexpansive mappings on H such that n i=1 F T i) is nonempty and let γ 1,γ 2,...,γ n be real numbers such that 0 <γ i < 1 for i =1, 2,...,n. For any n N, let W n be the W -mapping generated by T n,t n 1,...,T 1 and γ n,γ n 1,...,γ 1. Then W n is nonexpansive and F W n )= n i=1 F T i). For k N, from Lemma 3.2 in [5], we define mappings U,k and W on H as follows: U,k x = lim U n,kx and Wx = lim W nx = lim U n,1x for all x H. Such a mapping W is called the W -mapping generated by T 1,T 2,... and γ 1,γ 2,... We know the following two lemmas: Lemma 2.3 [5]). Let H be a real Hilbert space. Let T 1,T 2,... be nonexpansive mappings on H such that n=1 F T n) is nonempty and let γ 1,γ 2,... be real numbers such that 0 < γ i < 1 for all i N. LetW be the W -mapping generated by T 1,T 2,... and γ 1,γ 2,... Then W is nonexpansive and F W )= n=1 F T n). Lemma 2.4 [7]). Let {x n } and {y n } be bounded sequences in H and let {β n } be a sequence in [0, 1] with 0 < lim inf β n lim sup β n < 1. Suppose that x n1 = β n y n 1 β n )x n for all n N and lim sup y n1 y n x n1 x n ) 0. Then lim y n x n =0. Let µ be a mean on N, i.e., a continuous linear functional on l satisfying µ =1=µ1). We know that µ is a mean on N if and only if inf a n µf) sup a n n N for each f =a 1,a 2,...) l. Occasionally, we use µ n a n ) instead of µf). So, a Banach limit µ is a mean on N satisfying µ n a n )=µ n a n1 ). Let f =a 1,a 2,...) l with a n a as n and let µ be a Banach limit on N. Then µf) =µ n a n )=a. We also know the following lemma [6]. Lemma 2.5 [6]). Let a be a real number and let a 1,a 2,...) l such that µ n a n ) a for all Banach limit µ and lim sup a n1 a n ) 0. Then, lim sup a n a. The following lemma is proved in [1]. Lemma 2.6 [1]). Let {s n } be a sequence of nonnegative real numbers, let {α n } be a sequence of [0, 1] with n=1 α n =, let {β n } be a sequence of real numbers with lim sup n β n 0, and let {γ n } be a sequence of real numbers with n=1 γ n <. Suppose that s n1 1 α n )s n α n β n γ n for all n N. Then lim s n =0. n N

5 HYBRID STEEPEST DESCENT METHOD 561 The following lemma is in [9, 10]. Lemma 2.7. In a real Hilbert space H, the following inequality holds: x y 2 x 2 2 x y, y for all x, y H. The following theorem plays a crucial role for our main theorem. Theorem 2.3 [13]). Let H be a real Hilbert space and α, β > 0. LetW be a nonexpansive mapping on H such that F W ) is nonempty and let A be an α-strongly monotone and β- Lipschitzian mapping of RW ) into H. For ρ 0, 2α/β 2 ), define S n : H H and C f by S n =I λ n ρa)w for all n N, { C f = x H : x f ρawf) } for all f F W ), r where {λ n } 0, 1] and r =1 1 ρ2α ρβ 2 ) 0, 1). Then the following holds: i) For each n N, S n is a contraction which has a unique fixed point u n f F W ) C f. ii) Suppose that the sequence {λ n } 0, 1] satisfies lim λ n =0.Letu n be a unique fixed point of S n, that is, u n = S n u n = Wu n λ n ρawu n ). Then the sequence {u n } converges strongly to a unique solution u of VIF W ),A). 3. Main theorems In this section, we show a strong convergence theorem for finding a solution of the variational inequality problem over the intersection of fixed point sets of infinite nonexpansive mappings. Before proving the theorem, we need the following lemma which is essentially used in the proof. Lemma 3.1. Let H be a real Hilbert space and let α, β > 0. LetT 1,T 2,... be nonexpansive mappings on H such that n=1 F T n) is nonempty and let γ 1,γ 2,... be real numbers such that 0 <a γ i b<1 for all i =1, 2,... and some a, b 0, 1) with a b. For any n N, let W n be the W-mapping generated by T n,t n 1,...,T 1 and γ n,γ n 1,...,γ 1 and let A be an α-inverse strongly monotone and β-lipschitzian mapping on H. Suppose that {x n } is a sequence generated by x 1 H and 4) x n1 =I λ n ρa)w n x n for all n N, where ρ 0, 2α/β 2 ), and {λ n } 0, 1] satisfies lim λ n =0. Then lim x n1 x n =0. Proof. Putting T n = I λ n ρa)w n, we can rewrite 4) to x n1 = T n x n. Let u n=1 F T n). It follows from Lemma 2.2 that x n1 u = T n x n u T n x n T n u T n u u 1 λ n r) x n u I λ n ρa)w n u W n u =1 λ n r) x n u λ n ρ Au =1 λ n r) x n u λ n r ρ r Au max { x n u, ρ } r Au,

6 562 SHIGERU IEMOTO AND WATARU TAKAHASHI where r =1 1 ρ2α ρβ 2 ) 0, 1). By induction, we get x n u max { x 1 u, ρ } r Au =: K and hence {x n } is bounded. So, {T n x n } is also bounded. From 4), we note that x n1 =I λ n ρa)w n x n = λ n I ρa)w n x n 1 λ n )W n x n = λ n I ρa)w n x n 1 λ n )γ 1 T 1 U n,2 x n 1 γ 1 )x n ). Put y n = λ ni ρa)w n x n 1 λ n )γ 1 T 1 U n,2 x n. λ n 1 λ n )γ 1 Then, we have y n u = λ n {I ρa)w n x n u} 1 λ n )γ 1 T 1 U n,2 x n u) λ n 1 λ n )γ 1 λ n I ρa)w n x n u 1 λ n )γ 1 T 1 U n,2 x n u λ n 1 λ n )γ 1 1 {λ n I ρa)w n x n I ρa)w n u λ n 1 λ n )γ 1 λ n I ρa)w n u u 1 λ n )γ 1 T 1 U n,2 x n u } 1 {λ n 1 r) x n u λ n ρ Au λ n 1 λ n )γ 1 1 λ n )γ 1 U n,2 x n u } λ n1 r)k λ n rk 1 λ n )γ 1 K λ n 1 λ n )γ 1 = K. So, the sequence {y n } is also bounded. Furthermore, we have that lim sup y n1 y n x n1 x n ) I ρa)w n1 x n1 1 )γ 1 T 1 U n1,2 x n1 = lim sup 1 )γ 1 λ ) ni ρa)w n x n 1 λ n )γ 1 T 1 U n,2 x n λ n 1 λ n )γ 1 x n1 x n I ρa)w n1 x n1 I ρa)w n1 x n lim sup 1 )γ 1 I ρa)w n1 x n I ρa)w n x n 1 )γ 1 λ n 1 )γ 1 λ n 1 λ n )γ 1 1 ρa)w nx n T 1 U n1,2 x n1 T 1 U n,2 x n1 1 )γ 1 T 1 U n,2 x n1 T 1 U n,2 x n 1 )γ 1 1 λ ) n)γ 1 1 )γ 1 λ n 1 λ n )γ 1 T 1U n,2 x n x n1 x n

7 HYBRID STEEPEST DESCENT METHOD 563 lim sup x n1 x n 1 )γ 1 W n1 x n W n x n 1 )γ 1 λ n 1 )γ 1 λ n 1 λ n )γ 1 1 ρa)w nx n T 1 U n1,2 x n1 T 1 U n,2 x n1 1 )γ 1 x n1 x n 1 )γ 1 1 λ ) n)γ 1 1 )γ 1 λ n 1 λ n )γ 1 T 1U n,2 x n x n1 x n = lim sup W n1 x n W n x n 1 )γ 1 λ n 1 )γ 1 λ n 1 λ n )γ 1 1 ρa)w nx n T 1 U n1,2 x n1 T 1 U n,2 x n1 1 )γ 1 1 λ ) n)γ 1 1 )γ 1 λ n 1 λ n )γ 1 T 1U n,2 x n and that W n1 x n W n x n = U n1,1 x n U n,1 x n = γ 1 T 1 U n1,2 x n 1 γ 1 )x n {γ 1 T 1 U n,2 x n 1 γ 1 )x n } = γ 1 T 1 u n1,2 x n T 1 U n,2 x n γ 1 U n1,2 x n U n,2 x n = γ 1 γ 2 T 2 U n1,3 x n 1 γ 2 )x n {γ 2 T 2 U n,3 x n 1 γ 2 )x n )} = γ 1 γ 2 T 2 U n1,3 x n T 2 U n,3 x n γ 1 γ 2 U n1,3 x n U n,3 x n.. n ) γ i U n1,n1 x n U n,n1 x n Similarly, we have that i=1 n ) = γ i γ n1 T n1 U n1,n2 x n 1 γ n1 )x n x n = i=1 n1 i=1 γ i ) T n1 x n x n b n1 T n1 x n x n. T 1 U n1,2 x n1 T 1 U n,2 x n1 b n T n1 x n1 x n1.

8 564 SHIGERU IEMOTO AND WATARU TAKAHASHI So, we have lim sup y n1 y n x n1 x n ) lim sup b n1 T n1 x n x n 1 )γ 1 λ n 1 )γ 1 λ n 1 λ n )γ 1 1 ρa)w nx n b n T n1 x n1 x n1 1 )γ 1 1 λ ) n)γ 1 1 )γ 1 λ n 1 λ n )γ 1 T 1U n,2 x n lim sup b n1 1 )γ 1 λ n 1 )γ 1 λ n 1 λ n )γ 1 b n 1 )γ 1 1 λ ) n)γ 1 1 )γ 1 λ n 1 λ n )γ 1 L, where L = max{2k, K I ρa)u } with u n=1 F T n). Since lim λ n = 0 and γ n b<1 for all n N, we obtain lim sup y n1 y n x n1 x n ) 0. Now, we note that x n1 =1 λ n )1 γ 1 )x n {1 1 λ n )1 γ 1 )}y n for all n N and that 0 < lim inf 1 λ n)1 γ 1 ) lim sup1 λ n )1 γ 1 ) < 1. From Lemma 2.4, we get lim y n x n = 0. Therefore we have lim x n1 x n = lim {1 1 λ n)1 γ 1 )} y n x n =0. This completes the proof. We are now in a position to prove our main theorem. Theorem 3.1. Let H be a real Hilbert space and let α, β > 0. LetT 1,T 2,... be nonexpansive mappings on H such that n=1 F T n) is nonempty and let γ 1,γ 2,... be real numbers such that 0 <a γ i b<1 for all i =1, 2,... and some a, b 0, 1) with a b. For any n N, let W n be the W-mapping generated by T n,t n 1,...,T 1 and γ n,γ n 1,...,γ 1 and let A be an α-inverse strongly monotone and β-lipschitzian mapping on H. Suppose that {x n } is a sequence generated by x 1 H and x n1 =I λ n ρa)w n x n for all n N, where ρ 0, 2α/β 2 ), and {λ n } 0, 1] satisfies C1) and C2), that is, lim λ n =0and λ n =. n=1

9 HYBRID STEEPEST DESCENT METHOD 565 Then the sequence {x n } converges strongly to a unique solution u of VI n=1 F T n),a). Proof. As in the proof of Lemma 3.1, {x n } and {AW n x n } are bounded. Let W be the W -mapping generated by T 1,T 2,... and γ 1,γ 2,... such that Wx = lim W n x for each x H. Using the mapping W and λ k =1/k in Theorem 2.3, we have from Lemma 2.3 that there exists the sequence {u k } such that {u k } converges strongly to a unique solution u of VI n=1 F T n),a). Then we have that for all n, k N, x n1 Wu k = W n x n λ n ρaw n x n Wu k W n x n W n u k W n u k Wu k λ n ρ AW n x n x n u k W n u k Wu k λ n ρ AW n x n. Since lim λ n = 0 and Wu k = lim W n u k for each k N, for any Banach limit µ, we obtain 5) µ n x n Wu k 2 = µ n x n1 Wu k 2 µ n x n u k 2. From the definition of {u k }, we have x n u k = x n Wu k 1 k ρaw u k) = 1 1 ) x n Wu k ) 1 k k x n Wu k ρaw u k ) and hence So we have 1 1 ) x n Wu k )=x n u k ) 1 k k x n Wu k ρaw u k ). 1 1 ) 2 x n Wu k 2 k x n u k 2 2 k x n u k,x n Wu k ρaw u k = x n u k 2 2 k x n u k,x n u k u k Wu k ρaw u k = 1 2 ) x n u k 2 2 k k x n u k, u k Wu k ρaw u k. From 5), we have 1 1 ) 2 µ n x n u k 2 k = 1 1 k 1 2 k ) 2 µ n x n Wu k 2 ) µ n x n u k 2 2 k µ n x n u k, u k Wu k ρaw u k and hence 1 2k µ n x n u k 2 µ n x n u k, u k Wu k ρaw u k for all k N. Letting k, from u VIF W ),A) we obtain 0 µ n x n u, u Wu ρaw u = µ n x n u, ρaw u

10 566 SHIGERU IEMOTO AND WATARU TAKAHASHI and hence 0 µ n x n u, AW u. In addition, from Lemma 3.1 we have lim x n1 u, AW u x n u, AW u = lim x n1 x n, AW u =0. Therefore, it follows from Lemma 2.5 that 6) 0 lim sup x n u, AW u. Finally, we show that lim x n u = 0. In fact, from Lemma 2.7 we have x n1 u 2 = T n x n u 2 = T n x n T n u )T n u u ) 2 T n x n T n u 2 2 x n1 u,t n u u = T n x n T n u 2 2 x n1 u,w n u λ n ρaw n u u 1 λ n r) x n u 2 2λ n ρ x n1 u, AW n u { =1 λ n r) x n u 2 2ρ λ n r r x n1 u, AW u } x n1 u,awu AW n u ) { 1 λ n r) x n u 2 2ρ λ n r r x n1 u, AW u } M Wu W n u ), where M = β sup n N x n u. From Lemma 2.6 and 6), we obtain that the sequence {x n } converges strongly to a unique solution u of VI n=1 F T n),a). Using Theorem 3.1, we obtain the following theorem for finding a solution of the variational inequality problem over the intersection of fixed point sets of finite nonexpansive mappings. Theorem 3.2. Let H be a real Hilbert space and let α, β > 0. Let T 1,T 2,...,T r be nonexpansive mappings on H such that r i=1 F T i) is nonempty and let γ 1,γ 2,...,γ r be real numbers such that 0 <a γ i b<1 for all i =1, 2,...,r and some a, b 0, 1) with a b. LetW be the W-mapping generated by T 1,T 2,...,T r and γ 1,γ 2,...,γ r and let A be an α-inverse strongly monotone and β-lipschitzian mapping on H. Suppose that {x n } is a sequence generated by x 1 H and x n1 =I λ n ρa)wx n for all n N, where ρ 0, 2α/β 2 ), and {λ n } 0, 1] satisfies C1) and C2), that is, lim λ n =0and λ n =. Then, {x n } converges strongly to a unique solution u of VI r i=1 F T i),a). The following theorem is connected with the projected gradient method; see Theorem 2.1. n=1

11 HYBRID STEEPEST DESCENT METHOD 567 Theorem 3.3. Let H be a real Hilbert space and let α, β > 0. LetC 1,C 2,... be nonempty closed convex subsets of H such that C := n=1 C n is nonempty, let P Cn be the metric projections of H onto C n for each n N and let γ 1,γ 2,... be real numbers such that 0 <a γ i b<1 for all i =1, 2,... and some a, b 0, 1) with a b. For any n N, let W n be the W-mapping generated by P C1,P C2,... and γ 1,γ 2,... and let A be an α-inverse strongly monotone and β-lipschitzian mapping on H. Suppose that {x n } is a sequence generated by x 1 H and x n1 =I λ n ρa)w n x n for all n N, where ρ 0, 2α/β 2 ), and {λ n } 0, 1] satisfies C1) and C2), that is, lim λ n =0and λ n =. Then, the sequence {x n } converges strongly to a unique solution u of VIC, A). Proof. Since F W )= n=1 F P C n )= n=1 C n = C, from Theorem 3.1 we obtain the conclusion. 4. Application Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C H be a bounded linear operator and let AC) be the range of A. Given an element b H, consider the following convexly constrained linear inverse problem: 7) Find z C such that z argmin Ax b 2. x C We denote the set of solutions 7) by S b. Then it is known that S b is nonempty if and only if P AC) b) AC), where P C is the metric projection of H onto C. Indeed, the necessary part is trivial. Suppose that there exists z C such that z argmin x C Ax b 2. Then we have 8) Az b 2 y b 2 for all y AC). Let y 0 be in AC). Then there exists a sequence {y n } AC) such that y n y 0 as n. So, from 8) we have Az b 2 y n b 2. Letting n, we get Az b 2 y 0 b 2. Since y 0 AC) is arbitrary, we obtain P AC) b) =Az AC). If S b is nonempty, then we know that S b is closed and convex because of the continuity of A. In this case, S b has a unique element z S b with minimum norm, that is, z S b satisfies 9) z 2 = min{ x 2 : x S b }. The C-constrained pseudoinverse of A denoted by A C ) is defined as DA C )={b H : P AC) b) AC)}, A C b) = z, b DA C ), where z S b is a unique solution to 9). n=1

12 568 SHIGERU IEMOTO AND WATARU TAKAHASHI We now introduce the C-constrained generalized pseudoinverse of A; see [13]. Let f be afréchet differentiable convex function from H to R such that f is a k-lipschitzian and α-strongly monotone operator for some k>0 and α>0. Under these assumptions, there exists a unique element z S b for b DA C ) such that 10) f z ) = min{fx) :x S b }. The C-constrained generalized pseudoinverse of A associated with f denoted by A C,f )is defined as DA C,f )=DA C ), A C,f b) = z, b DA C,f ), where z S b is a unique solution to 10). We now apply our main theorem to construct the C-constrained generalized pseudoinverse A C,f of A. We know from 3) that z S b Az = P AC) b) AC) b P AC) b),p AC) b) Ax 0 for all x C b Az, Az Ax 0 for all x C A Az b),x z 0 for all x C, where A is the adjoint of A. This means that for each r>0, {ra b I ra A)z z},x z 0 for all x C, and hence 11) P C ra b I ra A)z) =z. Now, assume that {Sb 1,S2 b...} is a family of the solution sets of 7) for {C1,C 2,...} such that S b := Sb i. For each i N, define a mapping T i : H H by 12) T i x = P C ira b I ra A)x) for all x H, i=1 where P C i is the metric projection of H onto C i. The following lemma was shown by Xu and Kim [12]. Lemma 4.1. Let H be a real Hilbert space and b H. Let Sb i be a family of the solution sets of 7) such that i=1 Si b and let T i be a mapping of H onto C i which is defined by 12) for each i N. If r 0, 2/ A 2 ) and b DA C ), then T i is nonexpansive and F T i )=Sb i for all i N. Using these settings, we obtain the following theorem. Theorem 4.1. Let H be a real Hilbert space. Let T 1,T 2,... be mappings on H defined by 12) such that S b := n=1 F T n) is nonempty and let γ 1,γ 2,... be real numbers such that 0 <a γ i b<1 for all i N and some a, b with a b. For any n N, let W n be the W-mapping generated by T n,t n 1,...,T 1 and γ n,γ n 1,...,γ 1 and let f be a Fréchet differentiable convex function from H to R such that f is a k-lipschitzian and α-strongly

13 HYBRID STEEPEST DESCENT METHOD 569 monotone mapping on H for some k, α > 0. Suppose that {x n } is a sequence generated by x 1 H and 13) x n1 =I λ n ρ f)w n x n for all n N, where ρ 0, 2α/k 2 ), and {λ n } 0, 1] satisfies C1) and C2), that is, lim λ n =0and λ n =. Then, the sequence {x n } converges strongly to A S b,f b) which is a unique solution of 10). n=1 Proof. Put A C b) = z for b DA C ). Then we have that z argmin fx) 0 f z )N Sb z ) x S b f z ),x z 0 for all x S b z VIS b, f), where N Sb is the normal cone to S b ; see [10] for more details. So, it follows from Theorem 3.1 and Lemma 4.1 that the sequence {x n } generated by 13) converges strongly to A S b,f b). References [1] K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal ), [2] S. Atsushiba and W. Takahashi, Strong convergence theorems for a finite family of nonexpansive mappings and applications, Indian J. Math ), [3] A. A. Goldstein, Convex programming in Hilbert space, Bull. Amer. Math. Soc ), [4] Y. Kimura and W. Takahashi, Weak convergence to common fixed points of countable nonexpansive mappings and its applications, J. Korean Math. Soc ), [5] K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math ), [6] N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc ), [7] T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl ), [8] W. Takahashi, Weak and strong convergence theorems for families of nonexpansive mappings and their applications, Ann. Univ. Mariae Curie-Sk lodowska Sect. A 51, 1997, pp [9], Nonlinear Functional Analysis. Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, [10], Convex Analysis and Approximation of Fixed Points, Yokohama Publishers, Yokohama, 2000 Japanese). [11] W. Takahashi and K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Modelling ), [12] H. K. Xu and T. H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theory Appl ), [13] I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications D. Butnariu, Y. Censor and S. Reich, Eds.), North-Holland, Amsterdam, 2001, pp [14] E. Zeidler, Nonlinear Functional Analysis and its Applications. III:Variational methods and optimization, Springer-Verlag, New York, 1985.

14 570 SHIGERU IEMOTO AND WATARU TAKAHASHI S. Iemoto) Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo , Japan address: W. Takahashi) Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo , Japan address: