International Mathematical Forum, 5, 2010, no. 44, 2165-2172 Strong Convergence of Two Iterative Algorithms for a Countable Family of Nonexpansive Mappings in Hilbert Spaces Jintana Joomwong Division of Mathematics, Faculty of Science Maejo University, Chiang Mai, Thailand, 50290 jintana@mju.ac.th Abstract In this paper, we study two iterative algorithms for a countable family of nonexpansive mappings in Hilbert Spaces.We prove that the proposed algorithms converge strongly to a fixed point of nonexpansive mappings {T n }.The results of this paper extend and improve the results of Yonghong Yoa, et al. [14]. Mathematics Subject Classification: 47H05, 47H10 Keywords: Nonexpansive mapping; Fixed Point; Two iterative algorithms; Hilbert space 1 Introduction Let H be a real Hilbert space with norm and inner product,. And let C be a nonempty closed convex subset of H. A mapping T of C into itself is said to be nonexpansive if Tx Ty x y for each x, y C. We denote by F (T ) the set of fixed points of T. In 2003, for finding an element of F (S) VI(A, C), Takahashi and Toyoda [10] introduced the following iterative scheme: x n+1 = α n x n +(1 α n )SPc(x n λ n Ax n ) for every n =0, 1, 2,..., where x 0 = x C,{α n }is a sequence in (0, 1), and {λ n } is a sequence in (0, 2α). Let A is a strongly positive bounded linear operator on H. That is, there is a constant γ >0 with property Ax, x γ x 2 for all x H.
2166 J. Joomwong A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbeert space H. min x C 1 Ax, x x, b 2 where C is the fixed point set of a nonexpansive mapping T on H and b is a given point in H. Recently, Xu [12] proved that the sequence {x n } defined by the iterative method below, with the initial guess x 0 H chosen arbitrarily: x n+1 =(I α n A)Tx n + α n u, n 0, (1.1) converges strongly to the unique solution of the minimization problem provided the sequence {α n } satisfies certain conditions. On the other hand, Aoyama, et al.,[1] introduce a Halpern type iterative sequence for finding a common fixed point of a countable family of nonexpansive mappings. Let x 1 = x Cand x n+1 = α n x +(1 α n )T n x n (1.2) for all n N, where C is a nonempty closed convex subset of a Banach space, {α n } is a sequence in [0, 1] and {T n } is a sequence of nonexpansive mappings with some condition. They proved that {x n } defined by (1.2) converges strongly to a common fixed point of {T n }. Very recently, Yonghong Yoa,et al.[14] introduced two iterative algorithms defined by, for given x 0 C arbitrarily and let the sequence {x n },n 0be generated by { y n = P C [(1 α n )x n ], (1.3) x n+1 =(1 β n )x n + β n Ty n, and they prove that the algorithms strongly converge to a fixed point of nonexpansive mappings T. The research in this field, iterative algorithms for finding fixed points of nonlinear mappings, is important and find applications in a variety of applied areas of inverse problem, partial differential equations,image recovery and signal processing, see [2] [3] [4] [6]. In this paper motivate by result of Yonghong Yoa,et al.,and the ongoing research in this field, we introduced the two iterative algorithms in Hilbert space defined by, for given x 0 C arbitrarily and let the sequence {x n },n 0 be generated by { y n = P C [(1 α n )x n ], (1.4) x n+1 =(1 β n )x n + β n T n y n,
Strong convergence for a countable family of nonexpansive mappings 2167 where {α n } and {β n } are real in [0,1] and {T n } is a sequence of nonexpansive mappings with some conditions. Then we prove that the sequence {x n } defined by (1.4) converges strongly to a fixed point of {T n }. The result of this result extends and improves the corresponding results of Yonghong Yoa,et al.,[14]. 2 Preliminary Notes Let H be a real Hilbert space and let C be a closed convex subset of H.Then, for any x H, there exists a unique nearest point u C such that x u x y, y C. We denote u by P C (x), where P C is called the metric projection of H onto C. It is well known that P C is nonexpansive. Furthermore, for x H and u C, u = P C (x) x u, u y 0, y C Lemma 2.1. ([12]) Let C be a nonempty closed convex of a real Hilbert space H. Let T : C C be a nonexpansive mapping. Then I T is demi-closed at zero, i.e., if x n x C and x n Tx n 0, then x = Tx Lemma 2.2. ([8]) Let {x n } and {y n } be bounded sequences in a Banach space X and let {β n } be a sequence in [0, 1] with 0 < lim inf n β n lim sup n β n < 1. Suppose that x n+1 =(1 β n )y n +β n x n for all integer n 0 and lim sup n ( y n+1 y n x n+1 x n ) 0. Then lim n y n x n =0. Lemma 2.3. ([12]).Let {a n } be a sequence of nonnegative real numbers such that a n+1 (1 γ n )a n + γ n δ n, for all n 0 where {γ n } be a sequence in (0,1) and {δ n } is a sequence in R such that (i) n=10 γ n =, (ii) lim sup n δ n 0 or n=10 δ nγ n <. Then lim n a n =0. Lemma 2.4..Let H be a real Hilbert space.then for allx, y H,the following hold; (i) x + y 2 = x 2 +2 y, x + y, (ii) x + y 2 = x 2 +2 y, x
2168 J. Joomwong Throughout the rest of this paper, Let each t (0, 1), we consider the following mapping T t given by T t x = TP C [(1 t)x], x C. It is easy to check that T t x T t y (1 t) x y which implies that T t is a contraction. Using the Banach contraction principle, there exists a unique fixed point x t of T t in C, i.e., x t = TP C [(1 t)x t ]. (2.1) Lemma 2.5. ([14]). Let C be a nonempty bounded closed convex subset of Hilbert space H. LetT : C C be a nonexpansive mapping with F (T ). For each t (0, 1), let the net {x t } be generated by (2.1). Then, as t 0, the net {x t } converges strongly to a fixed point of T. Lemma 2.6. ([11]). Let C be a nonempty bounded closed convex subset of Hilbert space H and {T n } be a sequence of mappings of C into itself. Suppose that lim k,l ρk l =0, (2.2) where ρ k l = sup{ T k z T l z : z C} <, for all k, l N. Then for each x C, {T n x} converges strongly to some point of C. Moreover, let T be a mapping from C into itself defined by Tx = lim T n x, for all x C. Then lim sup { Tz T n z : z C} =0 3 Main results In this section, we prove the strong convergence theorems for a countable family of nonexpansive mappings in a real Hilbert space. Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let {T n } be a sequence of nonexpansive mapping of H into itself such that n=1 F (T n) is nonempty. Let {α n }, {β n } are the sequences in (0, 1). For given x 0 C arbitrarily, let the sequence {x n }, n 0. be generated by (1.4). Suppose the following conditions are satisfied: (i) lim α n =0 and n=10 α n = ; (ii) n=10 α n+1 α n < and n=10 β n+1 β n < ; (iii) 0 < lim inf n β n lim sup n β n < 1.
Strong convergence for a countable family of nonexpansive mappings 2169 Suppose that for any CofH, the sequence {T n } satisfied condition (2.2) in lemma (2.6). Let T be a mapping of H into itself defined by Ty = lim T n y for all y H. If F (T )= n=1 F (T n) then {x n } converges strongly to a fixed point z in F (T ). Proof. First, we observed that {x n } is bounded. Indeed, pick any p n=1 F (T n)=f (T ) to obtain, x n+1 p = (1 β n )x n + β n T n y n p (1 β n ) x n p + β n T n y n p (1 β n ) x n p + β n y n p (1 β n ) x n p + β n (1 α n )x n p (1 β n ) x n p + β n [(1 α n ) x n p + α n p ] (1 α n β n ) x n p + α n β n p max{ x n p, p }. Hence, {x n } is bounded and so is {T n x n }. Let B = {y H : y p K} where K = max{ x n p, p }, n 0. Clearly, B is bounded closed convex subset of H, T (B) B, {x n } B, and {T n x n } B. Set z n = T n y n, n 0. It follows that z n+1 z n T n+1 y n+1 T n+1 y n + T n+1 y n T n y n (1 α n+1 )x n+1 (1 α n )x n + T n+1 y n T n y n x n+1 x n + α n+1 x n+1 + α n x n + T n+1 (1 α n+1 )x n+1 T n (1 α n )x n x n+1 x n + α n+1 x n+1 + α n x n + sup T n+1 s T n s. s B Then, we obtained z n+1 z n x n+1 x n = α n+1 x n+1 + α n x n + sup T n+1 s T n s. s B By our assumptions, we get lim sup( z n+1 z n x n+1 x n ) 0. This together with lemma (2.2) imply that lim z n x n =0. Therefore, lim n+1 x n = lim (1 β n )x n + β n z n x n = lim β n z n x n =0
2170 J. Joomwong Now, we observe that x n T n x n x n x n+1 + x n+1 T n x n x n x n+1 + (1 β n )x n β n T n y n T n x n x n x n+1 +(1 β n ) x n T n x n + β n y n x n x n x n+1 +(1 β n ) x n T n x n + β n (1 α n )x n x n x n x n+1 +(1 β n ) x n T n x n + α n x n That is x n T n x n 1 β n { x n x n+1 + α n x n } 0 as n. Let the net {x t } defined by (2.1). By lemma (2.5), we have x t z as t 0 Next,we prove that lim sup z, z x n 0. Indeed, we calculate x t x n 2 = (x t T n x n )+(T n x n x n ) 2 = x t T n x n 2 +2 x t T n x n,t n x n x n + T n x n x n 2 x t T n x n 2 +2 x t x n,t n x n x n 2 T n x n x n,t n x n x n + T n x n x n 2 T n P C [(1 t)x t T n x n 2 +2 x t x n T n x n x n T n x n x n 2 (1 t)x t x n 2 +2 x t x n T n x n x n x t x n 2 2t x t,x t x n + t 2 x t 2 +2 x t x n T n x n x n x t x n 2 2t x t,x t x n + t 2 M + M T n x n x n where M>0such that sup{ x t 2, 2 x t x n,t (0, 1),n 1} M. It follows that x t,x t x n t M + M x 2 2t n T n x n. Therefore, We note that lim sup t 0 lim sup x t,x t x n 0. (3.1) z, z x n = z, z x t + z x t,x t x n + x t,x t x n z, z x t + z x t x t x n + x t,x t x n z, z x t + z x t M + x t,x t x n. This together with x t z and (3.1) implies that lim sup z, z x n 0
Strong convergence for a countable family of nonexpansive mappings 2171 Finally, we show that x n z. From (1.4), we have x n+1 z 2 = (1 β n )x n + β n T n y n z 2 (1 β n ) x n z 2 + β n T n y n z 2 (1 β n ) x n z 2 + β n y n z 2 (1 β n ) x n z 2 + β n (1 α n )(x n z) α n z 2 (1 β n ) x n z 2 + β n [(1 α n ) x n z 2 2α n (1 α n ) z, x n z + α 2 n z 2 ] (1 α n β n ) x n z 2 + α n β n [2(1 α n ) z, z x n + α n β n z 2 ]. By lemma (2.3), we obtained x n z as n. This completes the proof. Setting T n T in Theorem 3.1, we have the following result. Corollary 3.2. [14, Theorem 3.2] Let C be a nonempty closed convex subset of a real Hilbert space H. LetT : C C be a nonexpansive mapping such that F (T ) is nonempty. Let {α n }, {β n } are the sequences in (0, 1). For given x 0 C arbitrarily, let the sequence {x n }, n 0. be generated by (1.3). Suppose the following conditions are satisfied: (i) lim α n =0 and n=10 α n = ; (ii) 0 < lim inf n β n lim sup n β n < 1. Then {x n } converges strongly to a fixed point of T. ACKNOWLEDGEMENTS. The author like to thank the referees for Their helpful comments and suggestions, which improve the presentation of this paper. References [1] K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of countable family of nonexpansive mapping in Banach space, Nonlinear Analysis,67 (2007)2350-2360 [2] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20(2004) 103-120. [3] P.L. Combettes, On the numerical robustness of the parallel projection method in signal synthesis,ieee signal process. Lett.,8(2001) 45-47.
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