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STRONG CONVERGENCE THEOREMS FOR INFINITE FAMILIES OF NONEXPANSIVE MAPPINGS IN GENERAL BANACH SPACES TOMONARI SUZUKI Received 2 June 2004 In 1979, Ishikawa proved a strong convergence theorem for finite families of nonexpan- sive mappings in general Banach spaces. Motivated by Ishikawa’s result, we prove strong convergence theorems for infinite families of nonexpansive mappings. 1. Introduction Throughout this paper, we denote by N the set of positive integers and by R the set of real numbers. For an arbitrary set A, we also denote by A the cardinal number of A. Let C be a closed convex subset of a Banach space E.LetT be a nonexpansive mapping on C, that is, Tx−Ty≤x − y (1.1) for all x, y ∈ C. We denote by F(T) the set of fixed points of T.WeknowF(T) = ∅ in the case that E is uniformly convex and C is bounded; see Browder [2], G ¨ ohde [9], and Kirk [13]. Common fixed point theorems for families of nonexpansive mappings are proved in [2, 4, 5], and other references. Many convergence theorems for nonexpansive mappings and families of n onexpansive mappings have been studied; see [1, 3, 6, 7, 10, 11, 12, 14, 15, 17, 18, 19, 20, 21] and others. For example, in 1979, Ishikawa proved the following theorem. Theorem 1.1 [12]. Let C beacompactconvexsubsetofaBanachspaceE.Let {T 1 ,T 2 , ,T k } be a finite family of commuting nonexpansive mappings on C.Let{β i } k i=1 be a finite sequence in (0,1) and put S i x = β i T i x +(1−β i )x for x ∈C and i = 1,2, ,k.Letx 1 ∈ C and define asequence{x n } in C by x n+1 =  n  n k−1 =1  S k n k−1  n k−2 =1  S k−1 ···  S 3 n 2  n 1 =1  S 2 n 1  n 0 =1 S 1  ···  x 1 (1.2) for n ∈ N. Then {x n } converges strongly to a common fixed point of {T 1 ,T 2 , ,T k }. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:1 (2005) 103–123 DOI: 10.1155/FPTA.2005.103 104 Convergence to common fixed point The author thinks this theorem is one of the most interesting convergence theorems for families of nonexpansive mappings. In the case of k = 4, this iteration scheme is as follows: x 2 = S 4 S 3 S 2 S 1 x 1 , x 3 = S 4 S 3 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 x 2 , x 4 = S 4 S 3 S 2 S 1 S 1 S 1 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 x 3 , x 5 = S 4 S 3 S 2 S 1 S 1 S 1 S 1 S 2 S 1 S 1 S 1 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 S 1 S 1 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 x 4 , x 6 = S 4 S 3 S 2 S 1 S 1 S 1 S 1 S 1 S 2 S 1 S 1 S 1 S 1 S 2 S 1 S 1 S 1 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 S 1 S 1 S 1 S 2 S 1 S 1 S 1 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 S 1 S 1 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 x 5 , x 7 = S 4 S 3 S 2 S 1 S 1 S 1 S 1 S 1 S 1 S 2 S 1 S 1 S 1 S 1 S 1 S 2 S 1 S 1 S 1 S 1 S 2 S 1 S 1 S 1 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 S 1 S 1 S 1 S 1 S 2 S 1 S 1 S 1 S 1 S 2 S 1 S 1 S 1 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 S 1 S 1 S 1 S 2 S 1 S 1 S 1 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 S 1 S 1 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 S 1 S 2 S 1 S 3 S 2 S 1 x 6 . (1.3) We remar k that S i S j = S j S i does not hold in general. Very recently, in 2002, the following theorem was proved in [19]. Theorem 1.2 [19]. Let C be a compact convex subset of a Banach space E and let S and T be nonexpansive mappings on C with ST = TS.Letx 1 ∈ C and de fine a sequence {x n } in C by x n+1 = α n n 2 n  i=1 n  j=1 S i T j x n +  1 −α n  x n (1.4) for n ∈ N,where{α n } is a sequence in [0,1] such that 0 < liminf n α n ≤ limsup n α n < 1. Then {x n } converges strongly to a common fixed point z 0 of S and T. This theorem is simpler than Theorem 1.1. However, this is not a convergence theorem for infinite families of nonexpansive mappings. Under the assumption of the strict convexity of the Banach space, convergence theo- rems for infinite families of nonexpansive mappings were proved. In 1972, Linhart [15] proved the following; see also [20]. Theorem 1.3 [15]. Let C beacompactconvexsubsetofastrictlyconvexBanachspaceE. Let {T n : n ∈N} be an infinite family of commuting nonexpansive mappings on C.Let{β n } beasequencein(0,1).PutS i x =β i T i x +(1−β i )x for i ∈ N and x ∈ C.Let f be a mapping on N satisfying ( f −1 (i)) =∞for all i ∈N. Define a seque nce {x n } in C by x 1 ∈ C and x n+1 = S f (n) ◦S f (n−1) ◦···◦S f (1) x 1 (1.5) for n ∈ N. Then {x n } converges strongly to a common fixed point of {T n : n ∈N}. Tomonari Suzuki 105 The following mapping f on N satisfies the assumption in Theorem 1.3:ifn ∈N sat- isfies k−1  j=1 j<n≤ k  j=1 j (1.6) for some k ∈ N,thenput f (n) =n − k−1  j=1 j. (1.7) That is, f (1) = 1, f (2) =1, f (3) =2, f (4) =1, f (5) =2, f (6) =3, f (7) =1, f (8) =2, f (9) =3, f (10) =4, f (11) =1, f (12) =2, f (13) =3, f (14) =4, f (15) =5, f (16) =1, f (17) =2, (1.8) It is a natural problem whether or not there exists an iteration to find a common fixed point for infinite families of commuting nonexpansive mappings without assuming the strict convexity of the Banach space. This problem has not been solved for twenty-five years. In this paper, we give such iteration. That is, our answer of this problem is positive. 2. Lemmas In this section, we prove some lemmas. The following lemma is connected with Kras- nosel’ski ˘ ı and Mann’s type sequences [14, 16]. This is a generalization of [19, Lemma 1]. See also [8, 20]. Lemma 2.1. Let {z n } and {w n } be sequences in a Banach space E and let {α n } beasequence in [0,1] with limsup n α n < 1.Put d =limsup n→∞   w n −z n   or d =liminf n→∞   w n −z n   . (2.1) Suppose that z n+1 = α n w n +(1−α n )z n for all n ∈N, limsup n→∞    w n+1 −w n   −   z n+1 −z n    ≤ 0, (2.2) and d<∞. Then liminf n→∞     w n+k −z n   −  1+α n + α n+1 + ···+ α n+k−1  d   = 0 (2.3) hold for all k ∈ N. 106 Convergence to common fixed point Proof. Since   w n+1 −z n+1   −   w n −z n   ≤   w n+1 −w n   +   w n −z n+1   −   w n −z n   =   w n+1 −w n   −   z n+1 −z n   , (2.4) we have limsup n→∞    w n+ j −z n+ j   −   w n −z n    = limsup n→∞ j−1  i=0    w n+i+1 −z n+i+1   −   w n+i −z n+i    ≤ limsup n→∞ j−1  i=0    w n+i+1 −w n+i   −   z n+i+1 −z n+i    ≤ j−1  i=0 limsup n→∞    w n+i+1 −w n+i   −   z n+i+1 −z n+i    ≤ 0 (2.5) for j ∈ N.Puta = (1 −limsup n α n )/2. We note that 0 <a<1. Fix k, ∈ N and ε>0. Then there exists m  ≥  such that a ≤1 −α n , w n+1 −w n −z n+1 −z n ≤ε,andw n+ j − z n+ j −w n −z n ≤ε/2foralln ≥ m  and j = 1,2, ,k. In the case of d = limsup n w n − z n ,wechoosem ≥m  satisfying   w m+k −z m+k   ≥ d − ε 2 (2.6) and w n −z n ≤d + ε for all n ≥ m. We note that   w m+ j −z m+ j   ≥   w m+k −z m+k   − ε 2 ≥ d −ε (2.7) for j =0,1, ,k −1. In the case of d =liminf n w n −z n ,wechoosem ≥m  satisfying   w m −z m   ≤ d + ε 2 (2.8) and w n −z n ≥d −ε for all n ≥ m. We note that   w m+ j −z m+ j   ≤   w m −z m   + ε 2 ≤ d + ε (2.9) for j = 1, 2, , k. In both cases, such m satisfies that m ≥ , a ≤ 1 −α n ≤ 1, w n+1 −w n − z n+1 −z n ≤ε for all n ≥m,and d −ε ≤   w m+ j −z m+ j   ≤ d + ε (2.10) for j =0,1, ,k. We next show   w m+k −z m+ j   ≥  1+α m+ j + α m+ j+1 + ···+ α m+k−1  d − (k − j)(2k +1) a k−j ε (2.11) Tomonari Suzuki 107 for j =0,1, ,k −1. Since d −ε ≤   w m+k −z m+k   =   w m+k −α m+k−1 w m+k−1 −  1 −α m+k−1  z m+k−1   ≤ α m+k−1   w m+k −w m+k−1   +  1 −α m+k−1    w m+k −z m+k−1   ≤ α m+k−1   z m+k −z m+k−1   + ε +  1 −α m+k−1    w m+k −z m+k−1   = α 2 m+k−1   w m+k−1 −z m+k−1   + ε +  1 −α m+k−1    w m+k −z m+k−1   ≤ α 2 m+k−1 d +2ε +  1 −α m+k−1    w m+k −z m+k−1   , (2.12) we obtain   w m+k −z m+k−1   ≥  1 −α 2 m+k−1  d −3ε 1 −α m+k−1 ≥  1+α m+k−1  d − 2k +1 a ε. (2.13) Hence (2.11)holdsfor j = k −1. We assume (2.11)holdsforsomej ∈{1,2, ,k −1}. Then since  1+ k−1  i=j α m+i  d − (k − j)(2k +1) a k−j ε ≤   w m+k −z m+ j   =   w m+k −α m+ j−1 w m+ j−1 −  1 −α m+ j−1  z m+ j−1   ≤ α m+ j−1   w m+k −w m+ j−1   +  1 −α m+ j−1    w m+k −z m+ j−1   ≤ α m+ j−1 k−1  i=j−1   w m+i+1 −w m+i   +  1 −α m+ j−1    w m+k −z m+ j−1   ≤ α m+ j−1 k−1  i=j−1    z m+i+1 −z m+i   + ε  +  1 −α m+ j−1    w m+k −z m+ j−1   ≤ α m+ j−1 k−1  i=j−1   z m+i+1 −z m+i   + kε +  1 −α m+ j−1    w m+k −z m+ j−1   = α m+ j−1 k−1  i=j−1 α m+i   w m+i −z m+i   + kε +  1 −α m+ j−1    w m+k −z m+ j−1   ≤ α m+ j−1 k−1  i=j−1 α m+i (d + ε)+kε +  1 −α m+ j−1    w m+k −z m+ j−1   ≤ α m+ j−1 k−1  i=j−1 α m+i d +2kε+  1 −α m+ j−1    w m+k −z m+ j−1   , (2.14) 108 Convergence to common fixed point we obtain   w m+k −z m+ j−1   ≥ 1+  k−1 i=j α m+i −α m+ j−1  k−1 i=j−1 α m+i 1 −α m+ j−1 d − (k − j)(2k +1)/a k−j +2k 1 −α m+ j−1 ε ≥  1+ k−1  i=j−1 α m+i  d − (k − j + 1)(2k +1) a k−j+1 ε. (2.15) Hence (2.11)holdsfor j := j −1. Therefore (2.11)holdsforallj = 0, 1, , k −1. Spe- cially, we have   w m+k −z m   ≥  1+α m + α m+1 + ···+ α m+k−1  d − k(2k +1) a k ε. (2.16) On the other hand, we have   w m+k −z m   ≤   w m+k −z m+k   + k−1  i=0   z m+i+1 −z m+i   =   w m+k −z m+k   + k−1  i=0 α m+i   w m+i −z m+i   ≤ d + ε + k−1  i=0 α m+i (d + ε) ≤ d + k−1  i=0 α m+i d +(k +1)ε. (2.17) From (2.16)and(2.17), we obtain     w m+k −z m   −  1+α m + α m+1 + ···+ α m+k−1  d   ≤ k(2k +1) a k ε. (2.18) Since  ∈ N and ε>0 are arbitrary, we obtain the desired result.  By using Lemma 2.1, we obtain the following useful lemma, which is a generalization of [19, Lemma 2] and [20, Lemma 6]. Lemma 2.2. Let {z n } and {w n } be bounded sequences in a Banach space E and let {α n } be asequencein[0, 1] with 0 < liminf n α n ≤ lim sup n α n < 1.Supposethatz n+1 = α n w n +(1− α n )z n for all n ∈N and limsup n→∞    w n+1 −w n   −   z n+1 −z n    ≤ 0. (2.19) Then lim n w n −z n =0. Tomonari Suzuki 109 Proof. We put a = liminf n α n > 0, M = 2sup{z n  + w n  : n ∈ N} < ∞,andd = limsup n w n −z n  < ∞.Weassumed>0. Then fix k ∈ N with (1 + ka)d>M.ByLemma 2.1,wehave liminf n→∞     w n+k −z n   −  1+α n + α n+1 + ···+ α n+k−1  d   = 0. (2.20) Thus, there exists a subsequence {n i } of a sequence {n} in N such that lim i→∞    w n i +k −z n i   −  1+α n i + α n i +1 + ···+ α n i +k−1  d  = 0, (2.21) the limit of {w n i +k −z n i }exists, and the limits of {α n i + j }exist for all j ∈{0,1, ,k −1}. Put β j = lim i α n i + j for j ∈{0,1,···,k −1}. It is obvious that β j ≥ a for all j ∈{0,1, , k −1}.Wehave M<(1 + ka)d ≤  1+β 0 + β 1 + ···+ β k−1  d = lim i→∞  1+α n i + α n i +1 + ···+ α n i +k−1  d = lim i→∞   w n i +k −z n i   ≤ limsup n→∞   w n+k −z n   ≤ M. (2.22) This is a contradiction. Therefore d = 0.  We prove the following lemmas, which are connected with real numbers. Lemma 2.3. Let {α n } be a real sequence with lim n (α n+1 −α n ) =0.Theneveryt ∈ R with liminf n α n <t<limsup n α n is a cluster point of {α n }. Proof. We assume that there exists t ∈ (liminf n α n ,limsup n α n )suchthatt is not a cluster point of {α n }. Then there exist ε>0andn 1 ∈ N such that liminf n→∞ α n <t−ε<t<t+ ε<limsup n→∞ α n , α n ∈ (−∞, t −ε] ∪[t + ε,∞), (2.23) for all n ≥n 1 .Wechoosen 2 ≥ n 1 such that |α n+1 −α n | <εfor all n ≥ n 2 . Then there exist n 3 ,n 4 ∈ N such that n 4 ≥ n 3 ≥ n 2 , α n 3 ∈ (−∞, t −ε], α n 4 ∈ [t + ε,∞). (2.24) 110 Convergence to common fixed point We put n 5 = max  n : n<n 4 , α n ≤ t −ε  ≥ n 3 . (2.25) Then we have α n 5 ≤ t −ε<t+ ε ≤ α n 5 +1 (2.26) and hence ε ≤2ε ≤α n 5 +1 −α n 5 =   α n 5 +1 −α n 5   <ε. (2.27) This is a contradiction. Therefore we obtain the desired result.  Lemma 2.4. For α,β ∈ (0,1/2) and n ∈N,   α n −β n   ≤|α −β|, ∞  k=1   α k −β k   ≤ 4|α −β| (2.28) hold. Proof. We assume that n ≥2 because the conclusion is obvious in the case of n =1. Since α n −β n = (α −β) n−1  k=0 α n−1−k β k , (2.29) we have   α n −β n   =| α −β| n−1  k=0 α n−1−k β k ≤|α −β| n−1  k=0 1 2 n−1 =|α −β| n 2 n−1 ≤|α −β|. (2.30) We also have ∞  k=1   α k −β k   =      ∞  k=1  α k −β k       =     α 1 −α − β 1 −β     =     α −β (1 −α)(1 −β)     ≤ 4|α −β|. (2.31) This completes the proof.  Tomonari Suzuki 111 We know the following. Lemma 2.5. Let C be a subset of a B anach space E and let {V n } beasequenceofnonex- pansive mappings on C w ith a common fixed point w ∈C.Letx 1 ∈ C and define a sequence {x n } in C by x n+1 = V n x n for n ∈ N. Then {x n −w} is a nonincreasing sequence in R. Proof. We have x n+1 −w=V n x n −V n w≤x n −w for all n ∈N.  3. Three nonexpansive mappings In this section, we prove a convergence theorem for three nonexpansive mappings. The purpose for this is that we give the idea of our results. Lemma 3.1. Let C be a closed convex subset of a Banach space E.LetT 1 and T 2 be nonex- pansive mappings on C with T 1 ◦T 2 = T 2 ◦T 1 .Let{t n } beasequencein(0,1) converging to 0 and let {z n } beasequenceinC such that {z n } converges strongly to some w ∈ C and lim n→∞    1 −t n  T 1 z n + t n T 2 z n −z n   t n = 0. (3.1) Then w is a common fixed point of T 1 and T 2 . Proof. It is obvious that sup m,n∈N   T 1 z m −T 1 z n   ≤ sup m,n∈N   z m −z n   . (3.2) So {T 1 z n } is bounded because {z n } is bounded. Similarly, we have that {T 2 z n } is also bounded. Since lim n→∞    1 −t n  T 1 z n + t n T 2 z n −z n   = 0, (3.3) we have   T 1 w −w   ≤ limsup n→∞    T 1 w −T 1 z n   +   T 1 z n −  1 −t n  T 1 z n −t n T 2 z n   +    1 −t n  T 1 z n + t n T 2 z n −z n   +   z n −w    ≤ limsup n→∞  2   w −z n   + t n   T 1 z n −T 2 z n   +    1 −t n  T 1 z n + t n T 2 z n −z n    = 0 (3.4) and hence w is a fixed point of T 1 . We note that T 1 ◦T 2 w =T 2 ◦T 1 w =T 2 w. (3.5) 112 Convergence to common fixed point We assume that w is not a fixed point of T 2 .Put ε =   T 2 w −w   3 > 0. (3.6) Then there exists m ∈ N such that   z m −w   <ε,    1 −t m  T 1 z m + t m T 2 z m −z m   t m <ε. (3.7) Since 3ε =   T 2 w −w   ≤   T 2 w −z m   +   z m −w   <   T 2 w −z m   + ε, (3.8) we have 2ε<   T 2 w −z m   . (3.9) So, we obtain   T 2 w −z m   ≤   T 2 w −  1 −t m  T 1 z m −t m T 2 z m   +    1 −t m  T 1 z m + t m T 2 z m −z m   ≤  1 −t m    T 2 w −T 1 z m   + t m   T 2 w −T 2 z m   +    1 −t m  T 1 z m + t m T 2 z m −z m   =  1 −t m    T 1 ◦T 2 w −T 1 z m   + t m   T 2 w −T 2 z m   +    1 −t m  T 1 z m + t m T 2 z m −z m   ≤  1 −t m    T 2 w −z m   + t m   w −z m   +    1 −t m  T 1 z m + t m T 2 z m −z m   <  1 −t m    T 2 w −z m   +2t m ε <  1 −t m    T 2 w −z m   + t m   T 2 w −z m   =   T 2 w −z m   . (3.10) This is a contradiction. Hence, w is a common fixed point of T 1 and T 2 .  [...]... Browder, Nonexpansive nonlinear operators in a Banach space, Proc Nat Acad Sci U.S.A 54 (1965), 1041–1044 , Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces, Arch Rational Mech Anal 24 (1967), 82–90 R E Bruck Jr., A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific J Math 53 (1974), 59–71 R DeMarr, Common fixed points for commuting... one-parameter continuous semigroup of mappings √ is F(T(1)) ∩ F(T( 2)), submitted to Proc Amer Math Soc , Strong convergence theorem to common fixed points of two nonexpansive mappings in general Banach spaces, J Nonlinear Convex Anal 3 (2002), no 3, 381–391 , Convergence theorems to common fixed points for in nite families of nonexpansive mappings in strictly convex Banach spaces, Nihonkai Math J 14 (2003),... 4.1, we have that w is a common fixed point of T1 ,T2 , ,T Since ∈ N is arbitrary, we obtain that w is a common fixed point of {Tn : n ∈ N} This completes the proof Theorem 4.3 Let C be a compact convex subset of a Banach space E Let {Tn : n ∈ N} be an in nite family of commuting nonexpansive mappings on C Fix λ ∈ (0,1) Let {αn } be a sequence in [0,1/2] satisfying liminf αn = 0, lim αn+1 − αn = 0 limsup... point of {Tn : n ∈ N} We note that w is a cluster point of {xn } because so are ztn for all n ∈ N Hence, liminf n xn − w = 0 We also have that { xn − w } is nonincreasing by Lemma 2.5 Thus, limn xn − w = 0 This completes the proof Similarly, we can prove the following Theorem 4.5 Let C be a compact convex subset of a Banach space E Let {Tn : n ∈ N} be an in nite family of commuting nonexpansive mappings. .. points and iteration of a nonexpansive mapping in a Banach space, Proc Amer Math Soc 59 (1976), no 1, 65–71 , Common fixed points and iteration of commuting nonexpansive mappings, Pacific J Math 80 (1979), no 2, 493–501 W A Kirk, A fixed point theorem for mappings which do not increase distances, Amer Math Monthly 72 (1965), 1004–1006 M A Krasnosel’ski˘, Two remarks on the method of successive approximations,... fixed point of T1 ,T2 , ,T By induction, we obtain the desired result Lemma 4.2 Let C be a bounded closed convex subset of a Banach space E Let {Tn : n ∈ N} be an in nite family of commuting nonexpansive mappings on C Let {tn } be a sequence in (0,1/2) converging to 0 and let {zn } be a sequence in C such that {zn } converges strongly to some w ∈ C and ∞ 1− k=1 ∞ k tn T1 zn + k=2 k tn−1 Tk zn = zn for. .. J Linhart, Beitr¨ ge zur Fixpunkttheorie nichtexpandierender Operatoren, Monatsh Math 76 a (1972), 239–249 (German) W R Mann, Mean value methods in iteration, Proc Amer Math Soc 4 (1953), 506–510 S Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J Math Anal Appl 67 (1979), no 2, 274–276 T Suzuki, The set of common fixed points of a one-parameter continuous semigroup of mappings. .. convex subset of a Banach space E Let ∈ N with ≥ 2 and let {T1 ,T2 , ,T } be a finite family of commuting nonexpansive mappings on C Let {αn } be a sequence in [0,1/2] satisfying liminf αn = 0, n→∞ limsup αn > 0, n→∞ lim αn+1 − αn = 0 n→∞ (4.24) Define a sequence {xn } in C by x1 ∈ C and −1 xn+1 = 1 1 1 − αk T1 xn + n 2 2 k=1 1 αk−1 Tk xn + xn n 2 k=2 (4.25) for n ∈ N Then {xn } converges strongly to a... C be a closed convex subset of a Banach space E Let T1 , T2 , and T3 be commuting nonexpansive mappings on C Let {tn } be a sequence in (0,1/2) converging to 0 and let {zn } be a sequence in C such that {zn } converges strongly to some w ∈ C and lim n→∞ 2 2 1 − tn − tn T1 zn + tn T2 zn + tn T3 zn − zn = 0 2 tn (3.11) Then w is a common fixed point of T1 , T2 , and T3 Proof We note that {T1 zn }, {T2... fixed point of T1 , T2 , and T3 Theorem 3.3 Let C be a compact convex subset of a Banach space E Let T1 , T2 , and T3 be commuting nonexpansive mappings on C Fix λ ∈ (0,1) Let {αn } be a sequence in [0,1/2] satisfying liminf αn = 0, lim αn+1 − αn = 0 (3.19) xn+1 = λ 1 − αn − α2 T1 xn + λαn T2 xn + λα2 T3 xn + (1 − λ)xn n n (3.20) limsup αn > 0, n→∞ n→∞ n→∞ Define a sequence {xn } in C by x1 ∈ C and for . STRONG CONVERGENCE THEOREMS FOR INFINITE FAMILIES OF NONEXPANSIVE MAPPINGS IN GENERAL BANACH SPACES TOMONARI SUZUKI Received 2 June 2004 In 1979, Ishikawa proved a strong convergence. theorem for finite families of nonexpan- sive mappings in general Banach spaces. Motivated by Ishikawa’s result, we prove strong convergence theorems for in nite families of nonexpansive mappings. 1 10.1155/FPTA.2005.103 104 Convergence to common fixed point The author thinks this theorem is one of the most interesting convergence theorems for families of nonexpansive mappings. In the case of k = 4, this