Strong convergence theorems by hybrid method for asymptotically k -strict pseudo-contractive mapping in Hilbert space

Nonlinear Analysis: Hybrid Systems 3 (2009) 380–385

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Strong convergence theorems by hybrid method for asymptotically k-strict pseudo-contractive mapping in Hilbert space Issara Inchan a , Kamonrat Nammanee b,∗ a

Department of Mathematics and Computer, Uttaradit Rajabhat University, Uttaradit 53000, Thailand

b

Department of Mathematics, School of Science and Technology, Naresuan University at Phayao, Phayao 56000, Thailand

article

abstract

info

Article history: Received 18 November 2008 Accepted 18 February 2009 Keywords: Asymptotically k-strict pseudo-contractive mapping Mann’s iteration method Opial’s condition

In this paper, we introduce the hybrid method of modified Mann’s iteration for an asymptotically k-strict pseudo-contractive mapping. Then we prove that such a sequence converges strongly to PF (T ) x0 . This main theorem improves the result of Issara Inchan [I. Inchan, Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces, Int. J. Math. Anal. 2 (23) (2008) 1135–1145] and concerns the result of Takahashi et al. [W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 341 (2008) 276–286], and many others. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Let H be a real Hilbert space, C a nonempty closed convex subset of H and T : C → C a mapping. Recall that T is nonexpansive if kTx − Tyk ≤ kx − yk for all x, y ∈ C . A point x ∈ C is a fixed point of T provided Tx = x. Denote by F (T ) the set of fixed points of T ; that is, F (T ) = {x ∈ C : Tx = x}. We know that a Hilbert space H satisfies Opial’s condition [1], that is, for any sequence {xn } ⊂ H with xn * x, the inequality lim inf kxn − xk < lim inf kxn − yk n→∞

n→∞

holds for every y ∈ H with y 6= x. Recall that a mapping T : C → C is said to be a strict pseudo-contractive mapping [2] if there exists a constant 0 ≤ k < 1 such that

kTx − Tyk2 ≤ kx − yk2 + kk(I − T )x − (I − T )yk2 ,

(1.1)

for all x, y ∈ C . (If (1.1) holds, we also say that T is a k-strict pseudo-contraction.) It is known that if T is a 0-strict pseudo-contractive mapping, T is a nonexpansive mapping. In this paper we will consider an iteration method of modified Mann for asymptotically k-strict pseudo-contractive mapping. We say that T : C → C is an asymptotically k-strict pseudo-contractive mapping if there exists a constant 0 ≤ k < 1 satisfying

kT n x − T n yk2 ≤ (1 + γn )kx − yk2 + kk(I − T n )x − (I − T n )yk2 , ∗

Corresponding author. E-mail addresses: [email protected] (I. Inchan), [email protected] (K. Nammanee).

1751-570X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2009.02.002

(1.2)

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381

for all x, y ∈ C and for all n ∈ N where γn ≥ 0 for all n such that limn→∞ γn = 0. We see that if k = 0, then T is an asymptotically nonexpansive mapping. By Goebel and Kirk [3], T is an asymptotically nonexpansive mapping if there exists a sequence {γn } of nonnegative numbers with limn→∞ γn = 0 and such that

kT n x − T n yk2 ≤ (1 + γn )kx − yk2 ,

(1.3)

for all x, y ∈ C and all integers n ≥ 1. Fixed point iteration processes for nonexpansive mappings and asymptotically nonexpansive mappings in Hilbert spaces and Banach spaces including Mann and Ishikawa iteration processes have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities: see [4–7]. However, Mann and Ishikawa iteration processes have only weak convergence even in Hilbert space: see [8,7]. Our iteration method for finding a fixed point of an asymptotically k-strict pseudo-contractive mapping T is the modified Mann’s iteration method studied in [9–12] which generates a sequence {xn } via xn+1 = αn xn + (1 − αn )T n xn ,

n ≥ 0,

(1.4)

where the initial guess x0 ∈ C is arbitrary and the sequence lies in the interval (0, 1). In 2007, Takahashi, Takeuchi and Kubota [7] introduced the modification of the Mann iteration method for a family of nonexpansive mappings {Tn }. Let x0 ∈ H. For C1 = C and u1 = PC1 x0 , define a sequence {un } of C as follows:

{αn }∞ n=0

yn = αn un + (1 − αn )Tn un , Cn+1 = {z ∈ Cn : kyn − z k ≤ kun − z k}, un+1 = PCn+1 x0 , n ∈ N,

(

(1.5)

where 0 ≤ αn ≤ a < 1 for all n ∈ N. Then we prove that the sequence {un } converges strongly to z0 = PF (T ) x0 . In 2008, Inchan [13], introduced the modified Mann iteration processes for an asymptotically nonexpansive mapping. Let C be a closed bounded convex subset of a Hilbert space H, T be an asymptotically nonexpansive mapping of C into itself and let x0 ∈ C . For C1 = C and x1 = PC1 (x0 ), define {xn } as follows:

 yn = αn xn + (1 − αn )T n xn , C = {z ∈ Cn : kyn − z k2 ≤ kxn − z k2 + θn }, x n+1 = P n ∈ N, n+1 Cn+1 x0 ,

(1.6)

where θn = (1 − αn )(k2n − 1)(diam C )2 → 0 as n → ∞ and 0 ≤ αn ≤ a < 1 for all n ∈ N. Then he proves that {xn } converges strongly to z0 = PF (T ) x0 . Inspired and motivated by these facts, it is the purpose of this paper to introduce the modified Mann iteration processes for an asymptotically k-strict pseudo-contractive mapping by the idea in (1.6). Let C be a closed convex subset of a Hilbert space H, T be an asymptotically k-strict pseudo-contractive mapping of C into itself and let x0 ∈ C . For C1 = C and x1 = PC1 (x0 ), define {xn } as follows:

 yn = αn xn + (1 − αn )T n xn , C = {z ∈ Cn : kyn − z k2 ≤ kxn − z k2 + [k − αn (1 − αn )]kxn − T n xn k + θn }, x n+1 = P n ∈ N, n +1 Cn+1 x0 ,

(1.7)

where θn = (diam C )2 (1 − αn )γn → 0, (n → ∞). We shall prove that the iteration generated by (1.7) converges strongly to z0 = PF (T ) x0 . 2. Preliminaries Let H be a real Hilbert space with norm k · k and inner product h·, ·i and let C be a closed convex subset of H. For every point x ∈ H, there exists a unique nearest point in C , denoted by PC x, such that

kx − PC xk ≤ kx − yk,

for all y ∈ C .

PC is called the metric projection of H onto C . It is well known that PC is a nonexpansive mapping of H onto C . We collect some lemmas which will be used in the proof of the main result. Lemma 2.1 ([14]). The following identities hold in a Hilbert space H: (i) kx + yk2 = kxk2 + kyk2 + 2hx, yi, ∀x, y ∈ H . (ii) kλx + (1 − λ)yk2 = λkxk2 + (1 − λ)kyk2 − λ(1 − λ)kx − yk2 for all x, y ∈ H and λ ∈ [0, 1]. Lemma 2.2 ([15]). Let T be an asymptotically k-strict pseudo-contractive mapping defined on a bounded closed convex subset C of a Hilbert space H. Assume that {xn } is a sequence in C with the properties (i) xn * z and (ii) Txn − xn → 0. Then (I − T )z = 0.

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Lemma 2.3 ([16]). Let C be a closed convex subset of a real Hilbert space H. Given x ∈ H and y ∈ C , then y = PC x if and only if the following inequality holds

hx − y, y − z i ≥ 0,

∀z ∈ C .

Lemma 2.4 ([15]). Assume that C is a closed convex subset of a Hilbert space H and let T : C → C be an asymptotically k-strict pseudo-contraction. Then for each n ≥ 1, T n satisfies the Lipschitz condition:

kT n x − T y k ≤ Ln kx − yk for all x, y ∈ C , where Ln =

√

k+ 1+γn (1−k) . 1−k

3. Main results In this section, we prove strong convergence theorems by hybrid methods for asymptotically k-strict pseudo-contractive mappings in Hilbert spaces. Theorem 3.1. Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let T be an asymptotically k-strict pseudo-contractive mapping of C into itself such that F (T ) 6= ∅ and let x0 ∈ C . For C1 = C and x1 = PC1 x0 , assume that the control sequence {αn }∞ n=1 is chosen such that lim supn→∞ αn < 1 − k. Then {xn } generated by (1.7) converges strongly to z0 = PF (T ) x0 . Proof. We first show that F (T ) ⊂ Cn for all n ∈ N, by induction. For any z ∈ F (T ) we have z ∈ C = C1 hence F (T ) ⊂ C1 . Let F (T ) ⊂ Cm for each m ∈ N. Then we have, for u ∈ F (T ) ⊂ Cm

kym − uk2 = = = ≤ = ≤ ≤

kαm xm + (1 − αm )T m xm − uk2 kαm (xm − u) + (1 − αm )(T m xm − u)k2 αm kxm − uk2 + (1 − αm )kT m xm − uk2 − αm (1 − αm )kxm − T m xm k2 αm kxm − uk2 + (1 − αm )[(1 + γm )kxm − uk2 + kkxm − T m xm k2 ] − αm (1 − αm )kxm − T m xm k2 (1 + (1 − αm )γm )kxm − uk2 + [k − αm (1 − αm )]kxm − T m xm k2 kxm − uk2 + [k − αm (1 − αm )]kxm − T m xm k2 + (1 − αm )γm kxm − uk2 kxm − uk2 + [k − αm (1 − αm )]kxm − T m xm k2 + θm .

It follows that u ∈ Cm+1 and F (T ) ⊂ Cm+1 , hence F (T ) ⊂ Cn for all n ∈ N. Next, we show that Cn is closed and convex for all n ∈ N. It obviously follows that C1 = C is closed and convex. Suppose that Cm is closed and convex for each m ∈ N. Let zj ∈ Cm+1 ⊂ Cm with zj → z. Since Cm is closed, z ∈ Cm and kym − zj k2 ≤ kzj − xm k2 + [k − αm (1 − αm )]kxm − T m xm k2 + θm . Then

kym − z k2 = kym − zj + zj − z k2 = kym − zj k2 + kzj − z k2 + 2hym − zj , zj − z i ≤ kzj − xm k2 + [k − αm (1 − αm )]kxm − T m xm k2 + θm + kzj − z k2 + 2kym − zj kkzj − z k. Taking j → ∞,

kym − z k2 ≤ kz − xm k2 + [k − αm (1 − αm )]kxm − T m xm k2 + θm . Hence z ∈ Cm+1 . Let x, y ∈ Cm+1 ⊂ Cm with z = α x + (1 − α)y where α ∈ [0, 1]. Since Cm is convex, z ∈ Cm and kym − xk2 ≤ kx − xm k2 +[k −αm (1 −αm )]kxm − T m xm k2 +θm , kym − yk2 ≤ ky − xm k2 +[k −αm (1 −αm )]kxm − T m xm k2 +θm , we have

kym − (α x + (1 − α)y)k2 kα(ym − x) + (1 − α)(ym − y)k2 αkym − xk2 + (1 − α)kym − yk2 − α(1 − α)k(ym − x) − (ym − y)k2 α(kx − xm k2 + [k − αm (1 − αm )]kxm − T m xm k2 + θm ) + (1 − α)(ky − xm k2 + [k − αm (1 − αm )]kxm − T m xm k2 + θm ) − α(1 − α)ky − xk2 = αkx − xm k2 + (1 − α)ky − xm k2 − α(1 − α)k(xm − x) − (xm − y)k2 + [k − αm (1 − αm )]kxm − T m xm k2 + θm = kα(xm − x) + (1 − α)(xm − y)k2 + [k − αm (1 − αm )]kxm − T m xm k2 + θm = kxm − z k2 + [k − αm (1 − αm )]kxm − T m xm k2 + θm .

kym − z k2 = = = ≤

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Then z ∈ Cm+1 , it follows that Cm+1 is closed and convex. Hence Cn is closed and convex for all n ∈ N. This implies that {xn } is well defined. From xn = PCn x0 , we have

hx0 − xn , xn − yi ≥ 0,

for all y ∈ Cn .

Since F (T ) ⊂ Cn , we have

hx0 − xn , xn − ui ≥ 0 for all u ∈ F (T ) and n ∈ N.

(3.1)

So, for u ∈ F (T ), we have 0 ≤ hx0 − xn , xn − ui = hx0 − xn , xn − x0 + x0 − ui

= −hxn − x0 , xn − x0 i + hx0 − xn , x0 − ui ≤ −kxn − x0 k2 + kx0 − xn kkx0 − uk. This implies that

kx0 − xn k2 ≤ kx0 − xn kkx0 − uk, hence

kx0 − xn k ≤ kx0 − uk for all u ∈ F (T ) and n ∈ N.

(3.2)

From xn = PCn x0 and xn+1 = PCn+1 x0 ∈ Cn+1 ⊂ Cn , we also have

hx0 − xn , xn − xn+1 i ≥ 0 for all n ∈ N.

(3.3)

So, for xn+1 ∈ Cn , we have, for n ∈ N 0 ≤ hx0 − xn , xn − xn+1 i = hx0 − xn , xn − x0 + x0 − xn+1 i

= −hxn − x0 , xn − x0 i + hx0 − xn , x0 − xn+1 i ≤ −kxn − x0 k2 + kx0 − xn kkx0 − xn+1 k. This implies that

kx0 − xn k2 ≤ kx0 − xn kkx0 − xn+1 k, hence

kx0 − xn k ≤ kx0 − xn+1 k for all n ∈ N.

(3.4)

From (3.2) we have {xn } is bounded, limn→∞ kxn − x0 k exists. Next, we show that kxn − xn+1 k → 0. In fact, from (3.3) we have

k x n − x n +1 k 2 = = = = ≤ =

k(xn − x0 ) + (x0 − xn+1 )k2 kxn − x0 k2 + 2hxn − x0 , x0 − xn+1 i + kx0 − xn+1 k2 kxn − x0 k2 + 2hxn − x0 , x0 − xn + xn − xn+1 i + kx0 − xn+1 k2 kxn − x0 k2 − 2hx0 − xn , x0 − xn i − 2hx0 − xn , xn − xn+1 i + kx0 − xn+1 k2 kxn − x0 k2 − 2kxn − x0 k2 + kx0 − xn+1 k2 −kxn − x0 k2 + kx0 − xn+1 k2 .

Since limn→∞ kxn − x0 k exists, we have that lim kxn − xn+1 k = 0.

n→∞

(3.5)

On the other hand, xn+1 ∈ Cn+1 ⊂ Cn implies that

kyn − xn+1 k2 ≤ kxn − xn+1 k2 + [k − αn (1 − αn )]kxn − T n xn k2 + θn , By the definition of yn , we have

kyn − xn k = kαn xn + (1 − αn )T n xn − xn k = (1 − αn )kT n xn − xn k. From (3.6), we have

(1 − αn )2 kT n xn − xn k2 = kyn − xn k2 = kyn − xn+1 + xn+1 − xn k2

(3.6)

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≤ kyn − xn+1 k2 + kxn+1 − xn k2 + 2kyn − xn+1 kkxn+1 − xn k ≤ kxn − xn+1 k2 + [k − αn (1 − αn )]kxn − T n xn k2 + θn + kxn+1 − xn k2 + 2kyn − xn+1 kkxn+1 − xn k = [k − αn (1 − αn )]kxn − T n xn k2 + 2kxn+1 − xn k(kxn+1 − xn k + kyn − xn+1 k) + θn . It follows that

((1 − αn )2 − (k − αn (1 − αn )))kxn − T n xn k2 ≤ 2kxn+1 − xn k(kxn+1 − xn k + kyn − xn+1 k) + θn . Hence

(1 − k − αn )kT n xn − xn k ≤ 2kxn+1 − xn k(kxn+1 − xn k + kyn − xn+1 k) + θn .

(3.7)

From lim supn→∞ αn < 1 − k, we can choose  > 0 such that αn ≤ 1 − k −  for large enough n. From (3.5) and (3.7), we have lim kT n xn − xn k = 0.

(3.8)

n→∞

Next, we show that limn→∞ kTxn − xn k = 0. From Lemma 2.4, we have

kTxn − xn k ≤ kTxn − T n+1 xn k + kT n+1 xn − T n+1 xn+1 k + kT n+1 xn+1 − xn+1 k + kxn+1 − xn k ≤ L1 kxn − T n xn k + kT n+1 xn+1 − xn+1 k + (1 + Ln+1 )kxn − xn+1 k.

(3.9)

From (3.5) and (3.8), we have lim kTxn − xn k = 0.

(3.10)

n→∞

By (3.9), Lemma 2.2 and boundedness of {xn } we obtain ∅ 6= ωw (xn ) ⊂ F (T ). By the fact that kxn − x0 k ≤ kz0 − x0 k for all n ≥ 0 where z0 = PF (T ) (x0 ) and the weak lower semi-continuity of the norm, we have

kx0 − z0 k ≤ kx0 − wk ≤ lim inf kx0 − xn k n→∞

≤ lim sup kx0 − xn k ≤ kx0 − z0 k, n→∞

for all w ∈ ωw (xn ). However, since ωw (xn ) ⊂ F (T ), we must have w = z0 for all w ∈ ωw (xn ). Thus ωw (xn ) = {z0 } and then xn * z0 . Hence, xn → z0 = PF (T ) (x0 ) by

kxn − z0 k2 = kxn − x0 k2 + 2hxn − x0 , x0 − z0 i + kx0 − z0 k2 ≤ 2(kz0 − x0 k2 + hxn − x0 , x0 − z0 i) → 0 as n → ∞. This completes the proof.



Using this Theorem 3.1, we have the following corollaries. Corollary 3.2. Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let T be a k-strict pseudo-contractive mapping of C into itself for some 0 ≤ k < 1 such that F (T ) 6= ∅ and let x0 ∈ C . For C1 = C and x1 = PC1 x0 , define {xn } as follows;

 yn = αn xn + (1 − αn )Txn , C = {z ∈ Cn : kyn − z k2 ≤ kxn − z k2 },  x n +1 = P n +1 Cn+1 x0 ,

(3.11)

for all n ∈ N, where {αn } ⊂ [α, β] for some α, β ∈ [k, 1). Then {xn } generated by (3.11) converges strongly to z0 = PF (T ) x0 . Corollary 3.3 ([13]). Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let T be an asymptotically nonexpansive mapping of C into itself such that F (T ) 6= ∅ and let x0 ∈ C . For C1 = C and x1 = PC1 x0 , define {xn } as follows;

 yn = αn xn + (1 − αn )T n xn , C = {z ∈ Cn : kyn − z k2 ≤ kxn − z k2 + θn },  x n +1 = P n ∈ N, n +1 Cn+1 x0 ,

(3.12)

where θn = (1 − αn )(k2n − 1)(diam C )2 → 0 as n → ∞ and 0 ≤ αn ≤ a < 1 for all n ∈ N. Then {xn } generated by (3.12) converges strongly to z0 = PF (T ) x0 .

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Corollary 3.4 ([7, Theorem 4.1]). Let H be a Hilbert space and C be a nonempty closed convex subset of H. Let T be a nonexpansive mapping of C into H such that F (T ) 6= ∅ and let x0 ∈ H. For C1 = C and u1 = PC1 x0 , define a sequence {un } of C as follows: yn = αn un + (1 − αn )Tun , Cn+1 = {z ∈ Cn : kyn − z k ≤ kun − z k}, un+1 = PCn+1 x0 , n ∈ N,

(

(3.13)

where 0 ≤ αn ≤ a < 1 for all n ∈ N. Then {un } converges strongly to z0 = PF (T ) x0 . Acknowledgements The first author was supported by The Thailand Research Fund and the Commission on Higher Education under grant MRG5180026. The second author was supported by The Thailand Research Fund and the Commission on Higher Education under grant MRG5180011. Moreover, we would like to thank Prof. Dr. Somyot Plubiteng for providing valuable suggestions and we would also like to thank the referee for the comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591–597. F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967) 197–228. K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972) 171–174. S. Ishikawa, Fixed point theorems for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974) 147–150. W.A. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–510. J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Amer. Math. Soc. 43 (1991) 153–159. W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341 (2008) 276–286. T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear. Anal. 64 (2006) 1140–1152. T.C. Lim, H.K. Xu, Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal. 22 (1994) 1345–1355. J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991) 407–413. J. Schu, Approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 112 (1991) 143–151. K.K. Tan, H.K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122 (1994) 733–739. I. Inchan, Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces, Int. J. Math. Anal. 2 (23) (2008) 1135–1145. G. Marino, H.K. Xu, Weak and strong convergence theorems for k-strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–349. T.H. Kim, H.K. Xu, Convergence of the modified Mann’s iterations method for asymptotically strict pseudo-contractions, Nonlinear. Anal. 68 (2008) 2828–2836. S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math Anal. Appl. 336 (2007) 455–469.