WEAK AND STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR NONEXPANSIVE MAPPINGS IN HILBERT SPACES

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1 Applicable Analysis and Discrete Mathematics available online at Appl. Anal. Discrete Math. 2 (2008), doi: /aadm m WEAK AND STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR NONEXPANSIVE MAPPINGS IN HILBERT SPACES Yu Miao, Junen Li In a real Hilbert space H, startin rom an arbitrary initial point x 0 H, an iterative process is deined as ollows: x n+1 = a nx n +(1 a n)t λ n+1 y n, y n = b nx n + (1 b n)t x n, n 0, where T λ n+1 x = Tx λ n+1µ (Tx), T x = Tx β nµ (Tx), ( x H), T : H H a nonexpansive mappin with F(T) and (resp. ) : H H an η (resp. η )-stronly monotone and k (resp. k )-Lipschitzian mappin, {a n} (0,1), {b n} (0, 1) and {λ n} [0, 1), {β n} [0, 1). Under some suitable conditions, several converence results o the sequence {x n} are shown. 1. INTRODUCTION Let H be a Hilbert space with inner product, and norm. A mappin T : H H is said to be nonexpansive i Tx Ty x y or any x, y H. A mappin : H H is said to be η-stronly monotone i there exists constant η > 0 such that x y, x y η x y 2 or any x, y H. : H H is said to be k-lipschitzian i there exists constant k > 0 such that x y k x y or any x, y H. The interest and importance o construction o ixed points o nonexpansive mappins stem mainly rom the act that it may be applied in many areas, such as imae recovery and sinal processin (see, e.., [1, 2, 12]), solvin convex minimization problems (see, e.., [3, 16 19]). Iterative techniques or approximatin ixed points o nonexpansive mappins have been studied by various authors (see, e.., [1, 6 10, 13, 14], etc.), usin amous Mann iteration method, Ishikawa iteration method, and many other iteration methods such as, viscosity approximation method [7] and CQ method [8] Mathematics Subject Classiication. 47H09, 47H10. Keywords and Phrases. Nonexpansive mappin, iterative process, weak converence, stron converence. 197

2 198 Yu Miao, Junen Li Let : H H be a nonlinear mappin and K a nonempty closed convex subset o H. The variational inequality problem is ormulated as indin a point u K such that (1.1) (V I(, K)) (u ), ν u 0, ν K. The variational inequalities were initially studied by Kinderlehrer and Stampacchia [5], and ever since have been widely studied. It is well known that the V I(, K) is equivalent to the ixed point equation (1.2) u = P K (u µ(u )), where P K is the projection rom H onto K and µ is an arbitrarily ixed constant. In act, when is an η-stronly monotone and Lipschitzian mappin on K and µ > 0 small enouh, then the mappin deined by the riht-hand side o (1.2) is a contraction. For reducin the complexity o computation caused by the projection P K, Yamada [18] proposed an iteration method to solve the variational inequalities V I(, K). For arbitrary u 0 H, (1.3) u n+1 = Tu n λ n+1 µ(t(u n )), n 0, where T is a nonexpansive mappin rom H into itsel, K is the ixed point set o T, is an η-stronly monotone and k-lipschitzian mappin on K, {λ n } is a real sequence in [0, 1), and 0 < µ < 2η/k 2. Then Yamada [18] proved that {u n } converes stronly to the unique solution o the V I(, K) as {λ n } satisies the ollowin conditions: (1) lim λ n = 0; (2) + n=0 λ n = ; (3) lim (λ n λ n+1 )/λ 2 n+1 = 0. Based on the idea o iterative process (1.3), recently, Wan [15] discussed the more eneral Mann iteration scheme and ave the ollowin results: Let H be a Hilbert space, T : H H a nonexpansive mappin with F(T) := {x H, Tx = x}, and : H H an η-stronly monotone and k-lipschitzian mappin. For any x 0 H, {x n } is deined by (1.4) x n+1 = a n x n + (1 a n )T λn+1 x n, n 0, where (1.5) T λ x = Tx λµ(tx), x H, where {a n } (0, 1) and {λ n } [0, 1), then under some suitable conditions, the sequence {x n } is shown to convere stronly to a ixed point o T and the necessary and suicient conditions that {x n } converes stronly to a ixed point o T are obtained. Motivated by the above works, we will eneralize the scheme (1.4) as ollows.

3 Weak and stron converence o an iterative method 199 Let H be a Hilbert space, T : H H a nonexpansive mappin with F(T) and (resp. ) : H H an η (resp. η )-stronly monotone and k (resp. k )- Lipschitzian mappin. For any x 0 H, {x n } is deined by (1.6) x n+1 = a n x n + (1 a n )T λn+1 y n, y n = b n x n + (1 b n )T x n, n 0, where (1.7) T λn+1 x = Tx λ n+1 µ (Tx), x H, T x = Tx β nµ (Tx), x H, and {a n } (0, 1), {b n } (0, 1) and {λ n } [0, 1), {β n } [0, 1) satisy the ollowin conditions: (i) α a n 1 α, β b n 1 β or some α, β (0, 1/2); (ii) + λ n < +, + β n < + ; (iii) 0 < µ < 2η /k 2, 0 < µ < 2η /k PRELIMINARIES In this section we will state some useul notations and lemmas. A Banach space E is said to satisy Opial s condition i or any sequence {x n } in E, x n x implies that limsup x n x < limsup x n y or all y E with y x, where x n x denotes that {x n } converes weakly to x. It is well known that every Hilbert space satisies Opial s condition. A mappin T with domain D(T) and rane R(T) in E is said to be demiclosed at p; i whenever {x n } is a sequence in D(T) such that {x n } converes weakly to x D(T) and {Tx n } converes stronly to p, then Tx = p. A mappin T : K E is said to be demicompact i, or any sequence {x n } in K such that x n Tx n 0 (n ), there exists subsequence {x nj } o {x n } such that {x nj } converes stronly to x K. Lemma 2.1. [18]. Let T λ x = Tx λµ(tx), where T : H H is a nonexpansive mappin rom H into itsel and is an η-stronly monotone and k-lipschitzian mappin rom H into itsel. I 0 λ < 1 and 0 < µ < 2η/k 2, then T λ is a contraction and satisies (2.1) T λ x T λ y (1 λτ) x y, x, y H, where τ = 1 1 µ(2η µk 2 ).

4 200 Yu Miao, Junen Li Lemma 2.2. [14]. Let {s n }, {t n } be two nonneative sequence satisyin or some real number N 0 1, s n+1 s n + t n n N 0. I + t n < +, then lim s n exists. Lemma 2.3. [4]. Let K be a nonempty closed convex subset o a real Hilbert space H and T a nonexpansive mappin rom K into itsel. I T has a ixed point, then I T is demiclosed at zero, where I is the identity mappin o H. 3. MAIN RESULTS First we ive the ollowin key lemma. Lemma 3.1. For the iterative process (1.6), we have (1) lim x n p exists or each p F(T); (2) lim x n Tx n = 0. Proo. At irst we recall the well known identity in Hilbert space H: or any x, y H and t [0, 1], (3.1) tx + (1 t)y 2 = t x 2 + (1 t) y 2 t(1 t) x y 2. For any p F(T), rom Lemma 2.1, we have (3.2) T x n p = T x n T p + T p p (1 β n ) x n p + β n µ (p), and (3.3) where T λn+1 y n p = T λn+1 T λn+1 y n T λn+1 y n T λn+1 p + T λn+1 p p p + T λn+1 p p (1 λ n+1 τ ) y n p + λ n+1 µ (p), = 1 1 µ (2η µ k 2), τ = 1 1 µ (2η µ k 2). Furthermore, by the elementary inequality, 2ab ta 2 + (1/t)b 2, or any a, b R, t > 0,

5 Weak and stron converence o an iterative method 201 we obtain T x n p 2 (1 + β n 1 β n )(1 β n ) 2 x n p 2 (3.4) and + (1 + 1 β n ) 2 β n τ µ2 (p) 2 = (1 β n ) x n p 2 + β nµ 2 (p) 2 (3.5) T λn+1 y n p 2 (1 λ n+1 τ ) y n p 2 + λ n+1µ 2 (p) 2. τ From (3.1) and (3.4), it ollows (3.6) y n p 2 = b n (x n p) + (1 b n )(T x n p) 2 Thus by (3.5) and (3.6), we have (3.7) x n+1 p 2 which implies = b n x n p 2 + (1 b n ) T x n p 2 b n (1 b n ) T x n x n 2 [b n + (1 b n )(1 β n )] x n p 2 + (1 b n ) β nµ 2 (p) 2 = a n (x n p) + (1 a n )(T λn+1 y n p) 2 b n (1 b n ) T x n x n 2. = a n x n p 2 + (1 a n ) T λn+1 y n p 2 a n (1 a n ) T λn+1 y n x n 2 {a n + (1 a n )(1 λ n+1 τ )[b n + (1 b n )(1 β n )]} x n p 2 + (1 a n )(1 λ n+1 τ )(1 b n ) β nµ 2 (p) 2 + (1 a n ) λ n+1µ 2 (p) 2 τ (1 a n )(1 λ n+1 τ )b n (1 b n ) T x n x n 2 x n+1 p 2 x n p 2 + β nµ 2 a n (1 a n ) T λn+1 y n x n 2, (p) 2 + λ n+1µ 2 τ (p) 2. From Lemma 2.2 and the conditions: + λ n < +, + β n < +, it ollows that lim x n p exists or each q F(T). It ollows that {x n } is bounded. From the iterative process (1.6) we have (3.8) x n+1 x n = (1 a n ) T λn+1 y n x n

6 202 Yu Miao, Junen Li and (3.9) y n x n = (1 b n ) T x n x n. By (3.7), (3.8) and the condition a n [α, 1 α], it ollows that x n+1 p 2 x n p 2 + β nµ 2 that is to say that (p) 2 + λ n+1µ 2 τ (p) 2 α 1 α x n+1 x n 2, α 1 α x n+1 x n 2 x n p 2 x n+1 p 2 + β nµ 2 (p) 2 + λ n+1µ 2 (p) 2 τ which implies that (3.10) lim x n+1 x n = 0. In addition, rom (3.8), we know that λn+1 (3.11) lim T y n x n = 0. From (3.9), (3.7) and similar proo as (3.10) and (3.11), we have (3.12) lim y n x n = 0 and (3.13) lim T x n x n = 0. Thus (3.14) x n Tx n = x n T x n + T x n Tx n x n T x n + β n µ (Tx n ). Since {x n } is bounded, then {Tx n } and {(Tx n )} are bounded as well. Thereore lim n x n Tx n = 0. Theorem 3.2. The iterative process {x n }, which is taken as in (1.6), converes weakly to a ixed point o T. Proo. The proo is normal. It ollows rom Lemma 3.1 that lim x n p exists and {x n } is bounded. Now we prove that {x n } has a unique weak subsequential limit in F(T). To prove this, let p 1 and p 2 be weak limits o subsequences {x nk } and {x nj } o {x n }, respectively. It ollows rom Lemma 2.3 and Lemma 3.1 that lim x n Tx n = 0 and I T is demiclosed with respect to zero, thereore

7 Weak and stron converence o an iterative method 203 we obtain Tp 1 = p 1. Similarly Tp 2 = p 2, i.e., p 1, p 2 F(T). Next we prove the uniqueness. For this purpose that p 1 p 2. then by Opial s condition, we obtain lim x n p 1 = lim k + x n k p 1 < lim k + x n k p 2 = lim x n p 2 = lim j + x n j p 2 < lim j + x n j p 1 = lim x n p 1. This is a contradiction. Hence {x n } converes weakly to a point in F(T). Theorem 3.3. Let T be completely continuous, then the iterative process {x n }, which is taken as in (1.6), converes stronly to a ixed point o T. Proo. By Lemma 3.1, {x n } is bounded and lim x n Tx n = 0, then {Tx n } is also bounded. Since T is completely continuous, there exists subsequence {Tx nj } o {Tx n } and p H, such that Tx nj p 0 as n j +. It ollows rom Lemma 3.1 that lim x n n j Tx nj = 0. So by the continuity o T and Lemma j + 3.1, we have lim x n j p = 0 and p F(T). Furthermore by Lemma 3.1 n j + aain, we et that lim x n p exists. Thus lim x n p = 0 which implies the desired result. Theorem 3.4. Let T be demicompact, then the iterative process {x n }, which is taken as in (1.6), converes stronly to a ixed point o T. Proo. Since T is demicompact, {x n } is bounded and lim x n Tx n = 0, then there exists subsequence {x nj } o {x n } such that {x nj } converes stronly to p H. It ollows rom Lemma 2.3 that p F(T). Since the subsequence {x nj } o {x n } such that {x nj } converes stronly to p and lim x n p exists or all p F(T) by Lemma 3.1, then {x n } converes stronly to the common ixed point p F(T). The proo is completed. REFERENCES 1. F. E. Browder, W. V. Petryshyn: Construction o ixed points o nonlinear mappins Hilbert space. J. Math. Anal. Appl., 20 (2) (1967), C. Byrne: A uniied treatment o some iterative alorithms in sinal processin and imae reconstruction. Inverse Problems, 20 (1) (2004), F. Deutsch, I. Yamada: Minimizin certain convex unctions over the intersection o the ixed point sets o nonexpansive mappins. Numer. Funct. Anal. Optim., 19 (1998), K. Geobel, W. A. Kirk: Topics in Metric Fixed Point Theory. Cambride Stud. Adv. Math., vol. 28, Cambride Univ. Press, 1990.

8 204 Yu Miao, Junen Li 5. D. Kinderlehrer, G. Stampacchia: An Introduction to Variational Inequalities and Their Applications, vol. 88 o Pure and Applied Mathematics, Academic Press, New York, NY, USA, M. Maiti, M. K. Ghosh: Approximatin ixed points by Ishikawa iterates. Bull. Austral. Math. Soc., 40 (1) (1989), G. Marino, H. K. Xu: A eneral iterative method or nonexpansive mappins in Hilbert spaces. J. Math. Anal. Appl., 318 (2006), A. Moudai: Viscosity approximation methods or ixed-points problems. J. Math. Anal. Appl., 241 (1) (2000), K. Nakajo, K. Shimoji, W. Takahashi: Stron converence theorems by the hybrid method or amilies o nonexpansive mappins in Hilbert spaces. Taiwanese J. Math., 10 (2) (2006), K. Nakajo, W. Takahashi: Stron converence theorems or nonexpansive mappins and nonexpansive semiroups. J. Math. Anal. Appl., 279 (2) (2003), M. O. Osilike, S. C. Aniabosor, B. G. Akuchu: Fixed points o asymptotically demicontractive mappins in arbitrary Banach spaces. Panamer. Math. J., 12 (2) (2002), C. I. Podilchuk, R. J. Mammone: Imae recovery by convex projections usin a least-squares constraint. J. Optical Society o America. A, 7 (3) (1990), H. F. Senter, W. G. Dotson Jr.: Approximatin ixed points o nonexpansive mappins. Proc. Amer. Math. Soc., 44 (2) (1974), K. K. Tan, H. K. Xu: Approximatin ixed points o nonexpansive mappins by the Ishikawa iteration process. J. Math. Anal. Appl., 178 (2) (1993), L. Wan: An Iteration Method or Nonexpansive Mappins in Hilbert Spaces, Fixed Point Theory Appl., (2007), Art. ID H. K. Xu: Iterative alorithms or nonlinear operators. J. London Math. Soc., 66 (2002), H. K. Xu: An iterative approach to quadratic optimization. J. Optim. Theory Appl., 116 (2003), I. Yamada: The hybrid steepest descent method or the variational inequality problem o the intersection o ixed point sets o nonexpansive mappins, in: D. Butnariu, Y. Censor, S. Reich (Eds.): Inherently Parallel Alorithm or Feasibility and Optimization. Elsevier, 2001, pp I. Yamada, N. Oura, Y. Yamashita, K. Sakaniwa: Quadratic approximation o ixed points o nonexpansive mappins in Hilbert spaces. Numer. Funct. Anal. Optim., 19 (1998), Collee o Mathematics and Inormation Science, (Received February 26, 2008) Henan Normal University, (Revised July 24, 2008) Henan, China yumiao728@yahoo.com.cn junen li@yahoo.com.cn