STRONG CONVERGENCE THEOREMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS AND ASYMPTOTICALLY NONEXPANSIVE SEMIGROUPS YONGFU SU AND XIAOLONG QIN Received 22 April 2006; Accepted 14 July 2006 Strong convergence theorems are obtained from modified Halpern iterative scheme for asymptotically nonexpansive mappings and asymptotically nonexpansive semig roups, re- spectively. Our results extend and improve the recent ones announced by Nakajo, Taka- hashi, Kim, Xu, and some others. Copyright © 2006 Y. Su and X. Qin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and preliminary Let H be a real Hilbert space, C a nonempty closed convex subset of H,andT : C → C a mapping. Recall that T is nonexpansive if Tx− Ty≤x − y∀x, y ∈ C, (1.1) and T is asymptotically nonexpansive if there exists a sequence {k n } of positive real num- bers with lim n→∞ k n = 1 and such that   T n x − T n y   ≤ k n x − y∀n ≥ 1, x, y ∈ C. (1.2) Apointx ∈ 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}. Also, recall that a family S ={T(s) | 0 ≤ s<∞} of mappings from C into itself is called an asymptotically nonexpansive semigroup on C if it satisfies the following conditions: (i) T(0)x = x for all x ∈ C; (ii) T(s +t) = T(s)T(t)foralls,t ≥ 0; (iii) there exists a positive valued function L :[0, ∞) → [1,∞) such that lim s→∞ L s = 1 and T(s)x − T(s)y≤L s x − y for all x, y ∈ C and s ≥ 0; (iv) for all x ∈ C, s → T(s)x is continuous. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 96215, Pages 1–11 DOI 10.1155/FPTA/2006/96215 2 Nonexpansive mapping We denote by F(S) the set of all common fixed points of S, that is, F(S) =  0≤s<∞ F(T(s)). It is known that F(S) is closed and convex. Construction of fixed point of non- expansive mapping is an important subject in the theory of nonexpansive mappings and finds applications in a number of applied areas, in particular, in image recovery and signal processing (see, e.g., [14, 15]). However, the sequence {T n x} ∞ n=0 of iterates of the map- ping T at a point x ∈ C may not converge in the weak topology. Thus averaged iterations prevail. In fact, Mann’s iterations do have weak convergence. More precisely, Mann’s iter- ation procedure is a sequence {x n } defined by x n+1 = α n x n +  1 − α n  Tx n , (1.3) where the initial guess x 0 ∈ C is chosen arbitrarily. Reich [9]provedthatifE is a uniformly convex Banach space with a Fr ´ echet differen- tiable norm and if {α n } is chosen such that  ∞ n=1 α n (1 − α n ) =∞, then the sequence {x n } defined by (1.3) converges weakly to a fixed point of T. However we note that Mann’s iterations have only weak convergence even in a Hilbert space [1]. Recently many authors want to modify the Mann iteration method (1.3)sothatstrong convergence is guaranteed have recently been made. Nakajo and Takahashi [8]proposed the following modification of the Mann iteration (1.3) for a single nonexpansive mapping T in a Hilber t space: x 0 ∈ C arbitrarily, y n = α n x n +  1 − α n  Tx n , C n =  z ∈ C :   y n − z   ≤   x n − z    , Q n =  z ∈ C :  x 0 − x n ,x n − z  ≥ 0  , x n+1 = P C n ∩Q n x 0 , (1.4) where P K denotes the metric projection from H onto a closed convex subset K of H and proved that sequence {x n } converges strongly to P F(T) x 0 . They also proposed the following iteration process for a nonexpansive semigroup S = { T(s)|0 ≤ s<∞} in a Hilbert space H: x 0 ∈ C arbitrarily, y n = α n x n +  1 − α n  1 t n  t n 0 T(s)x n ds, C n =  z ∈ C :   y n − z   ≤   x n − z    , Q n =  z ∈ C :  x 0 − x n ,x n − z  ≥ 0  , x n+1 = P C n ∩Q n x 0 . (1.5) They proved that if the sequence {α n } is bounded from one and if {t n } is a positive real divergent sequence, then the sequence {x n } generated by (1.5) converges strongly to P F(S) x 0 . Y. Su and X. Qi n 3 Halpern [3] firstly studied iteration scheme as follows: x n+1 = α n u +  1 − α n  Tx n , n ≥ 0, (1.6) where u,x 0 ∈ C are arbitrary (but fixed) and {α n }⊂(0,1). He pointed out that the con- ditions lim n→∞ α n = 0and  ∞ n=1 α n =∞are necessary in the sense that if the iteration scheme (1.6) converges to a fixed point of T, then these conditions must be satisfied. Ten years later, Lions [6] investigated the general case in Hilbert space under the conditions lim n→∞ α n = 0, ∞  n=1 α n =∞,lim n→∞ α n − α n+1 α 2 n+1 = 0 (1.7) on the parameters. However, Lions’ conditions on the parameters were more restric- tive and did not include the natural candidate {α n = 1/n}.Reich[10] gave the iteration scheme (1.6) in the case when E is uniformly smooth and α n = n −δ with 0 <δ<1. Wittmann [13] studied the iteration scheme (1.6) in the case when E is a Hilbert space and {α n } satisfies lim n→∞ α n = 0, ∞  n=1 α n =∞, ∞  n=1   α n+1 − α n   < ∞. (1.8) Reich [11] obtained a strong convergence of the iterates (1.6) with two necessary and decreasing conditions on parameters for convergence in the case when E is uniformly smooth with a weakly continuous duality mapping. Recently, Martinez-Yanes and Xu [7]adaptedtheiteration(1.6) in Hilbert space as follows: x 0 ∈ C arbitrarily, y n = α n x 0 +  1 − α n  Tx n , C n =  z ∈ C :   y n − z   2 ≤   x n − z   2 + α n    x 0   2 +2  x n − x 0 ,z  , Q n =  z ∈ C :  x 0 − x n ,x n − z  ≥ 0  , x n+1 = P C n ∩Q n x 0 . (1.9) More precisely, they prove the following theorem. Theorem 1.1 (Martinez-Yanes and Xu [7]). Let H be a real Hilbert space, C aclosedconvex subset of H,andT : C → C a nonexpansive mapping such that F(T) =∅. Assume that {α n }⊂(0,1) is such that lim n→∞ α n = 0.Thenthesequence{x n } defined by (1.9)converges strongly to P F(T) x 0 . The purpose of this paper is to employ Nakajo and Takahashi’s [8]ideatomodifypro- cess (1.6) for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroup to have strong convergence theorem in Hilbert space. In the sequel, we need the following lemmas for the proof of our main results. 4 Nonexpansive mapping Lemma 1.2. Let K beaclosedconvexsubsetofrealHilbertspaceH and le t P K be the metric projection from H onto K (i.e., for x ∈ H, P k is the only point in K such that x − P k x= inf{x − z : z ∈ K}). Given x ∈ H and z ∈ K. Then z = P K x if and only if there holds the relations x − z, y − z≤0 ∀y ∈ K. (1.10) Lemma 1.3(Linetal.[5]). Let T be an asymptotically nonexpansive mapping defined on a bounded closed convex subset C of a Hilbert space H. Assume that {x n } is a sequence in C with the properties (i) x n  p and (ii) Tx n − x n → 0. Then p ∈ F(T). Lemma 1.4 (Kim and Xu [4]). Let C be a nonexpansive bounded closed convex subset of H and let S ={T(t):0≤ t<∞} be an asymptotically nonexpansive semigroup on C.Thenit holds that limsup s→∞ limsup n→∞ sup x∈C     T(s)  1 t  t 0 T(u)x n du  − 1 t  t 0 T(u)x n du     = 0. (1.11) Lemma 1.5. Let C be a nonex pansive bounded closed convex subset of H and let S ={T(s): 0 ≤ s<∞} be an asymptotically nonexpansive semigroup o n C.If{x n } isasequencein C satisfying the properties (i) x n  z; (ii) limsup s→∞ limsup n→∞ T(s)x n − x n =0, then z ∈ F(S). Proof. This lemma is the continuous version of [12, Lemma 2.3]. The proof given in [12] is easily extended to the continuous case.  2. Main results In this section we propose a modification of the Halpern iteration method to have strong convergence for asymptotically nonexpansive mappings and asymptotically nonexpan- sive semigroup in Hilbert space. Theorem 2.1. Let C be a bounded closed convex subset of a Hilbert space H and let T : C → C be an asymptotically nonexpansive mapping with sequence {k n }. Assume that {α n } ∞ n=0 and {β n } ∞ n=0 are sequences in (0,1) such that lim n→∞ α n = 0, lim n→∞ β n = 1,andM is an appropriate constant such that M ≥x 0 − v 2 ,forallv ∈ C.Defineasequence{x n } in C by the following algorithm: x 0 ∈ C chosen arbitrarily, z n = β n x n +  1 − β n  T n x n , y n = α n x 0 +  1 − α n  T n z n , C n =  v ∈ C :   y n − v   2 ≤   x n − v   2 +   z n   2 −   x n   2 +2  x n − z n ,v  + α n M  , Q n =  v ∈ C :  x 0 − x n ,x n − v  ≥ 0  , x n+1 = P C n ∩Q n x 0 . (2.1) Then {x n } converges to P F(T) x 0 ,providedk 2 n (1 − α n ) − 1 ≤ 0. Y. Su and X. Qi n 5 Proof. From [2]weknowthatT has a fixed point in C. That is, F(T) =∅.Itisobviously that C n is closed and Q n is closed and convex for each n ≥ 0. Next observe that C is convex. For v 1 ,v 2 ∈ C n and t ∈ (0,1), putting v = tv 1 +(1− t)v 2 .Itissufficient to show that v ∈ C n . Indeed, the defining inequality in C n is equivalent to the inequality 2  z n − y n , v  ≤   z n   2 −   y n   2 + α n M. (2.2) Therefore, we have 2  z n − y n ,v  = 2  z n − y n ,tv 1 +(1− t)v 2  = 2t  z n − y n ,v 1  +2(1− t)  z n − y n ,v 2  ≤   z n   2 −   y n   2 + α n M, (2.3) which implies that C is convex. Next, we show that F(T) ⊂ C n for all n. Indeed, for each p ∈ F(T),   y n − p   2 =   α n x 0 − p +  1 − α n  T n z n − p    2 ≤ α n   x 0 − p   2 +  1 − α n  k 2 n   z n − p   2 ≤   x n − p   2 −   x n − p   2 + α n   x 0 − p   2 +  1 − α n  k 2 n   z n − p   2 ≤   x n − p   2 +    z n − p   2 −   x n − p   2  + α n   x 0 − p   2 ≤   x n − p   2 +    z n   2 −   x n   2 +2  x n − z n , p   + α n M. (2.4) Therefore, p ∈ C n for each n ≥ 1, which implies that F(T) ⊂ C n . Next we show that F(T) ⊂ Q n ∀n ≥ 0. (2.5) We prove this by induction. For n = 0, we have F(T) ⊂ C = Q 0 . Assume that F(T) ⊂ Q n . Since x n+1 is the projection of x 0 onto C n ∩ Q n ,byLemma 1.2 we have  x 0 − x n+1 ,x n+1 − z  ≥ 0 ∀z ∈ C n ∩ Q n . (2.6) As F(T) ⊂ C n ∩ Q n by the induction assumptions, the last inequality holds, in particu- lar, for all z ∈ F(T). This to gether with the definition of Q n+1 implies that F(T) ⊂ Q n+1 . Hence (2.5)holdsforalln ≥ 0. In order to prove lim n→∞ x n+1 − x n =0, from the def- inition of Q n we have x n = P Q n x 0 which together with the fact that x n+1 ∈ C n ∩ Q n ⊂ Q n implies that   x 0 − x n   ≤   x 0 − x n+1   . (2.7) This shows that the sequence {x n − x 0 } is nondecreasing. Since C is bounded. We ob- tain that lim n→∞ x n − x 0  exists. Notice again that x n = P Q n x 0 and x n+1 ∈ Q n which give 6 Nonexpansive mapping x n+1 − x n ,x n − x 0 ≥0. Therefore, we have   x n+1 − x n   2 =    x n+1 − x 0  −  x n − x 0    2 ≤   x n+1 − x 0   2 −   x n − x 0   2 − 2  x n+1 − x n ,x n − x 0  ≤   x n+1 − x 0   2 −   x n − x 0   2 . (2.8) It follows that lim n→∞   x n − x n+1   = 0. (2.9) On the other hand, It follows from x n+1 ∈ C n that   y n − x n+1   2 ≤   x n − x n+1   2 +   z n   2 −   x n   2 +2  x n − z n ,x n+1  + α n M. (2.10) It follows from (2.1)andlim n→∞ β n = 1that   z n − x n   =  1 − β n    x n − T n x n   −→ 0. (2.11) Next, we consider   z n   2 −   x n   2 +2  x n − z n ,x n+1  =   z n   2 +   x n   2 − 2  z n ,x n  +2  x n − z n ,x n+1  − 2   x n   2 +2  z n ,x n  =   z n − x n   2 +2  x n − z n ,x n+1  − 2   x n   2 +2  z n ,x n  =   z n − x n   2 +2  z n ,x n − x n+1  − 2   x n   2 +2  x n ,x n+1  . (2.12) Therefore, it follows from (2.9)and(2.11)that   z n   2 −   x n   2 +2  x n − z n ,x n+1  −→ 0. (2.13) Furthermore, from (2.9), (2.13), and lim n→∞ α n = 0, we obtain lim n→∞   y n − x n+1   = 0. (2.14) On the other hand, we consider   y n − T n x n   ≤   y n − T n z n   +   T n z n − T n x n   ≤ α n   x 0 − T n z n   + k n   z n − x n   = α n   x 0 − T n z n   + k n  1 − β n    x n − T n x n   . (2.15) Y. Su and X. Qi n 7 Therefore, it follows that   x n − T n x n   ≤   x n − x n+1   +   x n+1 − y n   +   y n − T n x n   ≤   x n − x n+1   +   x n+1 − y n   + α n   x 0 − T n z n   + k n  1 − β n    x n − T n x n   . (2.16) That is,  1 − k n  1 − β n    x n − T n x n   ≤   x n − x n+1   +   x n+1 − y n   + α n   x 0 − T n z n   . (2.17) It follows from lim n→∞ β n = 1, lim n→∞ α n = 0, (2.9), and (2.14)that lim n→∞   x n − T n x n   −→ 0. (2.18) Putting k = sup{k n : n ≥ 1} < ∞,weobtain   Tx n − x n   ≤   Tx n − T n+1 x n   +   T n+1 x n − T n+1 x n+1   +   T n+1 x n+1 − x n+1   +   x n+1 − x n   ≤ k   x n − T n x n   +  1+k    x n − x n+1   +   T n+1 x n+1 − x n+1   , (2.19) which implies that   Tx n − x n   −→ 0. (2.20) Assume that {x n i } is a subsequence of {x n } such that x n i  x.ByLemma 1.3 we hav e x ∈ F(T). Next we show that x = P F(T) x 0 and the convergence is strong. Put x = P F(T) x 0 and consider the sequence {x 0 − x n i }.Thenwehavex 0 − x n i  x 0 − x and by the weak lower semicontinuity of the norm and by the fact that x 0 − x n+1 ≤x 0 − x for all n ≥ 0 which is implied by the fact that x n+1 = P C n ∩Q n x 0 ,wehave   x 0 − x   ≤   x 0 − x   ≤ liminf i→∞   x 0 − x n i   ≤ limsup i→∞   x 0 − x n i   ≤   x 0 − x   . (2.21) This gives   x 0 − x   =   x 0 − x   ,   x 0 − x n i   −→   x 0 − x   . (2.22) It follows that x 0 − x n i → x 0 − x;hence,x n i → x.Since{x n i } is an arbitrary subsequence of {x n },weconcludethatx n → x. The proof is completed.  8 Nonexpansive mapping Theorem 2.2. Let C be a nonempty bounded closed convex subset of H and let S ={T(s): 0 ≤ s<∞} be an asymptotically nonexpansive semigroup on C. Assume that {α n } ∞ n=0 and {β n } ∞ n=0 are seque nces in (0,1) such that lim n→∞ α n = 0 and lim n→∞ β n = 1. {t n } is a positive real divergent sequence and M is an appropriate constant such that M ≥x 0 − v for all v ∈ C. Define a sequence {x n } in C by the following algori thm: x 0 ∈ C chosen arbitrarily, z n = β n x n +  1 − β n  1 t n  t n 0 T(s)x n ds, y n = α n x 0 +  1 − α n  1 t n  t n 0 T(s)z n ds, C n =  v ∈ C :   y n − v   2 ≤ α n   x n − v   2 +   z n   2 −   x n   2 +2  x n − z n ,v  + α n M  , Q n =  v ∈ C :  x 0 − x n ,x n − z  ≥ 0  , x n+1 = P C n ∩Q n x 0 . (2.23) Then {x n } converges to P F(S) x 0 ,provided((1/t n )  t n 0 L s dt) 2 (1 − α n ) − 1 ≤ 0. Proof. We only conclude the difference. First we show F(S) ⊂ C n .ItfollowsfromC is bounded, we obtain that F( S) =∅(see [12]). Taking p ∈ F(S), we have   y n − p   2 ≤ α n   x 0 − p   2 +  1 − α n      1 t n  t n 0 T(s)z n ds− p     2 ≤ α n   x 0 − p   2 +  1 − α n   1 t n  t n 0   T(s)z n − p   ds  2 ≤ α n   x 0 − p   2 +  1 − α n   1 t n  t n 0 L s ds  2   z n − p   2 =   x n − p   2 +    z n − p   2 −   x n − p   2  + α n   x 0 − p   2 ≤   x n − p   2 +    z n   2 −   x n   2 +2  x n − z n , p   + α n M. (2.24) It follows that F(S) ⊂ C n for each n ≥ 0. From the proof of Theorem 2.1 we have the se- quence {x n } is well defined and F(S) ⊂ C n ∩ Q n for each n ≥ 0. Similarly to the argument of Theorem 2.1 and noticing q = P F(S) x 0 ,wehavex n+1 − x 0 ≤q − x 0  for each n ≥ 0 and x n+1 − x n →0. Next, we assume that a subsequence {x n i } of {x n } converges weakly Y. Su and X. Qi n 9 to q. It follows that   T(s)x n − x n   ≤     T(s)x n − T(s)  1 t n  t n 0 T(s)x n ds      +     T(s)  1 t n  t n 0 T(s)x n ds  − 1 t n  t n 0 T(s)x n ds     +     1 t n  t n 0 T(s)x n ds− x n     ≤ 2     1 t n  t n 0 T(s)x n ds− x n     +     T(s)  1 t n  t n 0 T(s)x n ds  − 1 t n  t n 0 T(s)x n ds     , (2.25) for each n ≥ 0. It follows from (2.23)that     y n − 1 t n  t n 0 T(s)z n ds     ≤ α n     x 0 − 1 t n  t n 0 T(s)z n ds     . (2.26) Therefore, we obtain     y n − 1 t n  t n 0 T(s)z n ds     −→ 0. (2.27) Next, we consider the first term on the right-hand side of (2.25)     1 t n  t n 0 T(s)x n ds− x n     ≤   x n − x n+1   +   x n+1 − y n   +     y n − 1 t n  t n 0 T(s)x n ds     . (2.28) Since x n+1 ∈ C n ,wehave   y n − x n+1   2 ≤ α n   x n − x n+1   2 +   z n   2 −   x n   2 +2  x n − z n ,x n+1  + α n M. (2.29) Similar to the proof of Theorem 2.1,wehave lim n→∞   y n − x n+1   = 0, (2.30) and hence     y n − 1 t n  t n 0 T(s)x n ds     ≤     y n − 1 t n  t n 0 T(s)z n ds     +     1 t n  t n 0 T(s)z n ds− 1 t n  t n 0 T(s)x n ds     ≤     y n − 1 t n  t n 0 T(s)z n ds     + 1 t n  t n 0   T(s)z n − T(s)x n   ds ≤     y n − 1 t n  t n 0 T(s)z n ds     +  1 t n  t n 0 L s ds    z n − x n   2 . (2.31) 10 Nonexpansive mapping Since lim n→∞ β n = 1, we have   z n − x n   =  1 − β n      x n − 1 t n  t n 0 T(s)x n ds     −→ 0. (2.32) It follows from (2.27)and(2.32)that     y n − 1 t n  t n 0 T(s)x n ds     −→ 0. (2.33) It follows from (2.30)and(2.33)that     x n − 1 t n  t n 0 T(s)x n ds     −→ 0. (2.34) On the other hand, by using Lemma 1.4 we obtain limsup s→∞ limsup n→∞     T(s)  1 t n  t n 0 T(s)x n ds  − 1 t n  t n 0 T(s)x n ds     = 0. (2.35) It follows from (2.34)and(2.35)that limsup s→∞ limsup n→∞   T(s)x n − x n   = 0. (2.36) Assume that a {x n i } is a subsequence of {x n } such that {x n i }  q ∈ C,thenq ∈ F(S) (by Lemma 1.5). Next we show that q = Π F(S) x 0 and the convergence is strong. Put q  = Π F(S) x 0 ,fromx n+1 = Π C n ∩Q n x 0 and q  ∈ F(S) ⊂ C n ∩ Q n ,wehavex n+1 − x 0 ≤q  − x 0 . On the other hand, from weakly lower semicontinuity of the norm, we obtain   q  − x 0   ≤   x 0 − q   ≤ liminf i→∞   x 0 − x n i   ≤ limsup i→∞   x 0 − x n i   ≤   q − x 0   . (2.37) It follows from definition of Π F(S) x 0 that we obtain q = Π F(S) x 0 and hence   q  − x 0   =   q − x 0   . (2.38) It follows that x n i → q  .Since{x n i } is an arbitrarily weakly convergent sequence of {x n }, we can conclude that {x n } convergesstronglytoonepointofΠ F(S) x 0 . This completes the proof.  References [1] A. Genel and J. Lindenstrauss, An example concerning fixed points, Israel Journal of Mathematics 22 (1975), no. 1, 81–86. [2] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings,Pro- ceedings of the American Mathematical Society 35 (1972), no. 1, 171–174. [...]... 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Our results extend and improve. paper is to employ Nakajo and Takahashi’s [8]ideatomodifypro- cess (1.6) for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroup to have strong convergence theorem in