BROWDER’S TYPE STRONG CONVERGENCE THEOREMS FOR INFINITE FAMILIES OF NONEXPANSIVE MAPPINGS IN BANACH SPACES TOMONARI SUZUKI Received 19 August 2005; Revised 24 February 2006; Accepted 26 February 2006 We prove Browder’s type strong convergence theorems for infinite families of nonexpan- sive mappings. One of our main results is the following: let C be a bounded closed convex subset of a uniformly smooth Banach space E.Let {T n : n ∈ N} be an infinite family of commuting nonexpansive mappings on C.Let {α n } and {t n } be sequences in (0,1/2) satisfying lim n t n = lim n α n /t n = 0for ∈ N.Fixu ∈ C and define a sequence {u n } in C by u n = (1 −α n )((1 − n k =1 t k n )T 1 u n + n k =1 t k n T k+1 u n )+α n u for n ∈ N.Then{u n } con- verges strongly to Pu,whereP is the unique sunny nonexpansive retraction from C onto ∞ n=1 F(T n ). Copyright © 2006 Tomonari Suzuki. 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 Let C beaclosedconvexsubsetofaBanachspaceE.AmappingT on C is called a non- expansive mapping if Tx−Ty≤x − y for all x, y ∈ C. We denote by F(T) the set of fixed points of T.WeknowthatF(T) is nonempty in the case that E is uniformly smooth and C is bounded; see Baillon [1]. When E has the Opial property and C is weakly com- pact, F(T)isalsononempty;see[11, 13]. See also [4, 5, 10] and others. Fix u ∈ C.Then for each α ∈ (0,1), there exists a unique point x α in C satisfying x α = (1 −α)Tx α + αu be- cause the mapping x → (1 −α)Tx+ αu is contractive; see [2]. In 1967 Browder [6]proved the following strong convergence theorem. Theorem 1.1 (Browder [6]). Let C beaboundedclosedconvexsubsetofaHilbertspaceE and let T be a nonexpansive mapping on C.Let {α n } beasequencein(0,1) converging to 0. Fix u ∈ C and define a sequence {u n } in C by u n = 1 −α n Tu n + α n u (1.1) for n ∈ N. Then {u n } converges strongly to the element of F(T) nearest to u. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 59692, Pages 1–16 DOI 10.1155/FPTA/2006/59692 2 Infinite families of nonexpansive mappings Reich extended this theorem to uniformly smooth Banach spaces in [17]. Using the notion of Bochner integral and (invariant) mean, Shioji and Takahashi in [18]proved Browder’s type strong convergence theorems for families of nonexpansive mappings Very recently, the author proved the following Browder’s type strong convergence the- orem for one-parameter nonexpansive semigroups. This is a generalization of the results in [19, 25]. We remark that we do not use the notion of Bochner integral. Theorem 1.2 [24]. Let C be a weakly compact convex subset of a Banach space E.Assume that either of the following holds: (i) E is uniformly convex with uniformly G ˆ ateaux differentiable norm; (ii) E is uniformly smooth; or (iii) E is a smooth Banach space with the Opial property and the duality mapping J of E is weakly sequentially continuous at zero. Let {T(t):t ≥0} be a one-parameter nonexpansive semigroup on C.Letτ be a nonnegative real number. Let {α n } and {t n } be sequences of real numbers satisfying 0 <α n < 1, 0 <τ+ t n and t n = 0 for n ∈N,andlim n t n = lim n α n /t n = 0.Fixu ∈ C and define a sequence {u n } in C by u n = 1 −α n T τ + t n u n + α n u (1.2) for n ∈ N. Then {u n } converges strongly to Pu,whereP is the unique sunny nonexpansive retraction from C onto t≥0 F(T(t)). Also, ver y recently, the author proved Krasnoselskii and Mann’s type convergence the- orems for infinite families of nonexpansive mappings in [21]. See also [20]. In this paper, using the idea in [21], we prove Browder’s type strong convergence theorems for infinite families of nonexpansive mappings without assuming the strict convexity of the Banach space. We remark that if we assume the strict convexity, its proof is very easy because the set of common fixed points of countable families of nonexpansive mappings is the set of fixed points of some single nonexpansive mapping; see Bruck [8]. We also remark that we do not use the notion of (invariant) mean. 2. Preliminaries Throughout this paper, we denote by N, Z, Q,andR the set of all positive integers, all integers, all rational numbers, and all real numbers, respectively. Let {x n } be a sequence in a topological space X. By the axiom of choice, there exist a directed set (D, ≤)andauniversal subnet {x f (ν) : ν ∈D} of {x n }, that is, (i) f is a mapping from D into N such that for each n ∈ N there exists ν 0 ∈ D such that ν ≥ ν 0 implies f (ν) ≥n; (ii) for each subset A of X, there exists ν 0 ∈ D such that either {x f (ν) : ν ≥ν 0 }⊂A or {x f (ν) : ν ≥ν 0 }⊂X \A holds. In this paper, we often use {x ν : ν ∈D} instead of {x f (ν) : ν ∈D}, for short. We know that if a net {x ν } is universal and g is a mapping from X into an arbitrary set Y,then{g(x ν )} is also universal. We also know that if X is compact, then a universal net {x ν } always converges. See [12] for details. Tomonari Suzuki 3 Let E be a real Banach space. We denote by E ∗ the dual of E. E is called uniformly convex if for each ε>0, there exists δ>0suchthat x + y/2 < 1 − δ for all x, y ∈ E with x=y=1andx − y≥ε. E is said to be smooth or said to have a G ˆ ateaux differentiable norm if the limit lim t→0 x + ty−x t (2.1) exists for each x, y ∈ E with x=y=1. E is said to have a uniformly G ˆ ateaux differ- entiable norm if for each y ∈ E with y=1, the limit is attained uniformly in x ∈ E with x=1. E is said to be uniformly smooth or said to have a uniformly Fr ´ echet differentiable norm if the limit is attained uniformly in x, y ∈ E with x=y=1. E is said to have the Opial property [14] if for each weakly convergent sequence {x n } in E with weak limit x 0 , liminf n→∞ x n −x 0 < liminf n→∞ x n −x (2.2) holds for all x ∈ E with x = x 0 .Weremarkthatwemayreplace“liminf”by“limsup.” That is, E has the Opial property if and only if for each weakly convergent sequence {x n } in E with weak limit x 0 , limsup n→∞ x n −x 0 < limsup n→∞ x n −x (2.3) holds for all x ∈ E with x = x 0 . Let E be a smooth Banach space. The duality mapping J from E into E ∗ is defined by x, J(x) = x 2 = J(x) 2 (2.4) for all x ∈ E. J is said to be weakly sequentially continuous at zero if for every sequence {x n } in E which converges weakly to 0 ∈ E, {J(x n )} converges weakly ∗ to 0 ∈E ∗ . AconvexsubsetC of a Banach space E is said to have normal structure [3]ifforevery bounded convex subset K of C which contains more than one point, there exists z ∈ K such that sup x∈K x −z< sup x,y∈K x − y. (2.5) We know that compact convex subsets of any Banach spaces and closed convex subsets of uniformly convex Banach spaces have normal structure. Turett [27]provedthatuni- formly smooth Banach spaces have normal structure. Also, Gossez and Lami Dozo [11] proved that every weakly compact convex subset of a Banach space with the Opial prop- erty has normal str ucture. We recall that a closed convex subset C ofaBanachspaceE is said to have the fixed point property for nonexpansive mappings (FPP, for short) if for every bounded closed convex subset K of C, every nonexpansive mapping on K has a fixed point. So, by Kirk’s fixed point theorem [13], every weakly compact convex subset with normal structure has FPP. 4 Infinite families of nonexpansive mappings Let C and K be subsets of a Banach space E.AmappingP from C into K is called sunny [7]if P Px +t(x −Px) = Px (2.6) for x ∈ C with Px+ t (x −Px) ∈ C and t ≥ 0. The following is proved in [15]. Lemma 2.1 (Reich [15]). Let E be a smooth Banach space and let C be a convex subset of E.LetK beasubsetofC and let P bearetractionfromC onto K. Then the following are equivalent: (i) x −Px, J(Px − y)≥0 for all x ∈ C and y ∈ K; (ii) P is both sunny and nonexpansive. Hence, there is at most one sunny nonexpansive retraction from C onto K. The following lemma is proved in [24]. However, it is essentially proved in [16]. See also [26]. Lemma 2.2 (Reich [16]). Let C be a nonempt y closed convex subset of a Banach space E with a uniformly G ˆ ateaux differentiable norm. Let {x α : α ∈D} be a net in E and let z ∈ C. Suppose that the limits of {x α − y} exist for all y ∈C. Then the following are equivalent: (i) lim α∈D x α −z=min y∈C lim α∈D x α − y; (ii) limsup α∈D y −z, J(x α −z)≤0 for all y ∈ C; (iii) liminf α∈D y −z, J(x α −z)≤0 for all y ∈C. The following lemma is well known. Lemma 2.3. Let {u n } beasequenceinaBanachspaceE and let z belong to E.Assume that every subsequence {u n i } of {u n } has a subsequence converging to z. Then {u n } itself converges to z. From Lemma 2.3,weobtainthefollowing. Lemma 2.4. Let {u n } be a sequence in a Banach space E. Assume that {u n } hasatmostone cluster point, and every subsequence of {u n } has a cluster point. Then {u n } converges. Proof. Since {u n } is a subsequence of {u n }, {u n } has a cluster point z ∈ E.Let{u n i } be an arbitrary subsequence of {u n }. Then by assumption {u n i } has a cluster point w ∈ E. Since w is also a cluster point of {u n },wehavew = z.Hence,{u n i } has a cluster point z ∈ E. That is, there exists a subsequence of {u n i } converging to z.So,byLemma 2.3, {u n } converges to z. This completes the proof. 3. Fixed point theorem The following theorem is one of the most famous fixed point theorems for families of nonexpansive mappings. Theorem 3.1 (Bruck [9]). Suppose a closed convex subse t C of a Banach space E has the fixed point property for nonexpansive mappings, and C is either weakly compact, or bounded and separable. Then for any commuting family S of nonexpansive mappings on C, the set of common fixed points of S is a nonempty nonexpansive retract of C. Tomonari Suzuki 5 Using Theorem 3.1, we prove the following fixed point theorem. Theorem 3.2. Let C be a closed convex subset of a Banach space E.LetA beaweakly compact c onvex subset of C. Assume that A has the fixed point property for nonexpansive mappings. Let {T n : n ∈ N} be an infinite family of commuting nonexpansive mappings on C such that T 1 (A) ⊂ A, T +1 A ∩ k=1 F T k ⊂ A (3.1) for all ∈ N. The n there exists a common fixed point z 0 ∈ A of {T n : n ∈N}. Proof. We put B := A ∩( k =1 F(T k )) for ∈ N. We first show B is nonempty and there exists a nonexpansive retraction P from A onto B for all ∈N. From the assumption of T 1 (A) ⊂A, there exists a fixed point z 1 ∈ A of T 1 , that is, B 1 = ∅.ByTheorem 3.1,there exists a nonexpansive retraction P 1 from A onto B 1 .WeassumeB is nonempty and there exists a nonexpansive retraction P from A onto B for some ∈ N.Fromtheassumption of T +1 (B ) ⊂ A,wehavethatT +1 ◦P is a nonexpansive mapping on A. We note that B +1 = F(T +1 ◦P ). Indeed, B +1 ⊂ F(T +1 ◦P )isobvious.Conversely,weassumez 2 ∈ A satisfies T +1 ◦P z 2 = z 2 .Fork ∈ N with k ≤ ,wehave T k z 2 = T k ◦T +1 ◦P z 2 = T +1 ◦T k ◦P z 2 = T +1 ◦P z 2 = z 2 , (3.2) that is, z 2 ∈ B and hence P z 2 = z 2 . Thus, we also have T +1 z 2 = T +1 ◦P z 2 = z 2 . (3.3) Therefore z 2 ∈ B +1 and hence B +1 ⊃ F(T +1 ◦P ). We have shown B +1 = F(T +1 ◦P ). Since A has the fixed point property, we have B +1 = F T +1 ◦P = ∅. (3.4) By Theorem 3.1 again, there exists a nonexpansive retraction P +1 from A onto B +1 .So, by induction, we have shown that B is nonempty and there exists a nonexpansive retrac- tion P from A onto B for all ∈ N. Define a sequence {Q n : n ∈ N} of nonexpansive mappings on A by Q n := P n ◦P n−1 ◦···◦P 2 ◦P 1 (3.5) for n ∈ N.SinceP m x =P n ◦P m x for x ∈ A, m,n ∈N with m ≥n,wehave Q m ◦Q n = P max{m,n} ◦P max{m,n}−1 ◦···◦P 2 ◦P 1 (3.6) for all m,n ∈ N and hence Q m ◦Q n = Q n ◦Q m for all m,n ∈N.So,byTheorem 3.1,there exists a common fixed point z 0 ∈ A of {Q n : n ∈N}.Letusprovethatz 0 is also a common fixed point of {T n : n ∈N}.Since P 1 z 0 = Q 1 z 0 = z 0 , (3.7) 6 Infinite families of nonexpansive mappings we have z 0 ∈ B 1 , that is, T 1 z 0 = z 0 .Weassume T 1 z 0 = T 2 z 0 =···=T z 0 = z 0 (3.8) for some ∈ N.Then z 0 = Q +1 z 0 = P +1 ◦P ◦···◦P 2 ◦P 1 z 0 = P +1 ◦P ◦···◦P 2 z 0 =···=P +1 ◦P z 0 = P +1 z 0 (3.9) and hence z 0 ∈ B +1 , that is, T +1 z 0 = z 0 . So, by induction, z 0 is a common fixed point of {T n : n ∈N}. This completes the proof. 4. Lemmas In this section, we prove some lemmas which are used in the proofs of our main results. Lemma 4.1. Let C be a closed convex subset of a Banach space E.Let {T n : n ∈ N} be an infinite family of commuting nonexpansive mappings on C with a common fixed point. Let {α n } and {t n } be sequences in (0,1/2) satisfying lim n t n = lim n α n /t n = 0 for ∈ N.Let{I n } beasequenceofnonemptysubsetsofN such that I n ⊂ I n+1 for n ∈N,and ∞ n=1 I n = N .For I ⊂ N and t ∈ (0,1/2) with I = ∅, define nonexpansive mappings S(I, t) on C by S(I, t)x : = 1 − k∈I t k T 1 x + k∈I t k T k+1 x (4.1) for x ∈ C.Fixu ∈ C and define a s equence {u n } in C by u n = (1 −α n )S(I n ,t n )u n + α n u (4.2) for n ∈ N.Let{u n β : β ∈ D} beasubnetof{u n }. Then the following hold. (i) limsup β u n β −T 1 x≤limsup β u n β −x for x ∈C. (ii) If x ∈ C satisfies T 1 x =x, then limsup β u n β −T 2 x≤lim sup β u n β −x. (iii) If x ∈ C satisfies T 1 x = T 2 x =···=T −1 x = x for some ∈ N with ≥ 3, then limsup β u n β −T x≤lim sup β u n β −x. Proof. Let v be a common fixed point of {T n : n ∈N}. It is obvious that S(I,t)v = v for all I ⊂ N and t ∈ (0,1/2) w ith I = ∅.Forx ∈C and k ∈N,wehave T k x ≤ T k x −v + v= T k x −T k v + v≤x −v+ v. (4.3) Hence, {T k x : k ∈ N}is bounded for every x ∈ C. Therefore S(I,t)iswelldefinedforevery I ⊂ N and t ∈(0,1/2) with I = ∅. It is obvious that S(I,t) is a nonexpansive mapping on C for every I and t.Since u n −v = 1 −α n S I n ,t n u n + α n u −v ≤ 1 −α n S I n ,t n u n −v + α n u −v ≤ 1 −α n u n −v + α n u −v, (4.4) Tomonari Suzuki 7 we have u n −v≤u −v for n ∈ N. Therefore {u n } is bounded. Since T k u n ≤ T k u n −v + v≤ u n −v + v≤ u n +2v (4.5) for all n,k ∈ N, {T k u n : n,k ∈ N} is also bounded. We fix x ∈ C and we put M : = max u,v,sup n∈N u n ,sup n,k∈N T k u n ,x,sup k∈N T k x < ∞. (4.6) It is obvious that S(I, t)u n ≤M and S(I,t)x≤M for all n ∈N, I ⊂N and t ∈(0,1/2) with I = ∅.Fromtheassumption,wehave S I n ,t n u n −u n = α n S I n ,t n u n −u (4.7) for n ∈ N.Wehave u n β −T 1 x ≤ u n β −S I n β ,t n β u n β + S I n β ,t n β u n β −S I n β ,t n β x + S I n β ,t n β x −T 1 x ≤ α n β S I n β ,t n β u n β −u + u n β −x + − k∈I n β t k n β T 1 x + k∈I n β t k n β T k+1 x ≤ 2Mα n β + u n β −x +2M k∈I n β t k n β ≤ 2Mα n β + u n β −x +2M t n β 1 −t n β (4.8) for β ∈ D and hence limsup β∈D u n β −T 1 x ≤ limsup β∈D u n β −x . (4.9) This is (i). We next show (ii). We assume that T 1 x = x.ThenT 1 ◦T 2 x = T 2 ◦T 1 x = T 2 x. For β ∈ D with 1,2 ∈I n β ,wehave u n β −T 2 x ≤ u n β −S I n β ,t n β u n β + S I n β ,t n β u n β −T 2 x ≤ α n β S I n β ,t n β u n β −u + 1 − k∈I n β t k n β T 1 u n β −T 2 x + t n β T 2 u n β −T 2 x + k∈I n β \{1} t k n β T k+1 u n β −T 2 x ≤ 2Mα n β + 1 −t n β T 1 u n β −T 2 x + t n β u n β −x +2M k∈I n β \{1} t k n β ≤ 2Mα n β + 1 −t n β T 1 u n β −T 1 ◦T 2 x + t n β u n β −x +2M t 2 n β 1 −t n β ≤ 2Mα n β + 1 −t n β u n β −T 2 x + t n β u n β −x +2M t 2 n β 1 −t n β (4.10) 8 Infinite families of nonexpansive mappings and hence u n β −T 2 x ≤ 2M α n β t n β + u n β −x +2M t n β 1 −t n β . (4.11) Therefore we obtain limsup β∈D u n β −T 2 x ≤ limsup β∈D u n β −x . (4.12) Let us prove (iii). We assume T 1 x = T 2 x =···=T −1 x = x for some ∈ N with ≥ 3. Then T m ◦T x = T ◦ T m x = T x for every m ∈ N with 1 ≤ m<.Forβ ∈ D with 1,2, , −1 ∈I n β ,wehave u n β −T x ≤ u n β −S I n β ,t n β u n β + S I n β ,t n β u n β −T x ≤ α n β S I n β ,t n β u n β −u + 1 − k∈I n β t k n β T 1 u n β −T x + −2 m=1 t m n β T m+1 u n β −T x + t −1 n β T u n β −T x + k∈I n β \{1,2, ,−1} t k n β T k+1 u n β −T x ≤ 2Mα n β + 1 − −1 m=1 t m n β T 1 u n β −T x + −2 m=1 t m n β T m+1 u n β −T x + t −1 n β u n β −x +2M k∈I n β \{1,2, ,−1} t k n β ≤ 2Mα n β + 1 − −1 m=1 t m n β T 1 u n β −T 1 ◦T x + −2 m=1 t m n β T m+1 u n β −T m+1 ◦T x + t −1 n β u n β −x +2M t n β 1 −t n β ≤ 2Mα n β + 1 − −1 m=1 t m n β u n β −T x + −2 m=1 t m n β u n β −T x + t −1 n β u n β −x +2M t n β 1 −t n β = 2Mα n β + 1 −t −1 n β u n β −T x+ t −1 n β u n β −x +2M t n β 1 −t n β (4.13) Tomonari Suzuki 9 and hence u n β −T x ≤ 2M α n β t −1 n β + u n β −x +2M t n β 1 −t n β . (4.14) Therefore we obtain limsup β∈D u n β −T x ≤ limsup β∈D u n β −x . (4.15) This completes the proof. Remark 4.2. Let g be a strictly increasing mapping on N. Then it is obvious that lim n t g(n) = lim n α g(n) /t g(n) = 0forall ∈ N, I g(n) ⊂ I g(n+1) for n ∈ N,and ∞ n=1 I g(n) = N .Thus,the same conclusions of Lemmas 4.3–4.6 also hold for {u g(n) }. Lemma 4.3. Let E, C, {T n }, {α n }, {t n }, {I n }, u,and{u n } be as in Lemma 4.1. Assume that {u n }converges strongly to some point x ∈ C. Then x is a common fixed point of {T n : n ∈N}. Proof. From Lemma 4.1(i), we have limsup n→∞ u n −T 1 x ≤ lim n→∞ u n −x = 0. (4.16) This means {u n } converges to T 1 x and hence T 1 x = x. We assume that T 1 x =···= T −1 x =x for some ∈N with ≥ 2. Then from Lemma 4.1(ii) and (iii), we have limsup n→∞ u n −T x ≤ lim n→∞ u n −x = 0. (4.17) This means {u n }converges to T x and hence T x =x. So, by induction, we obtain T n x =x for all n ∈ N. This completes the proof. Lemma 4.4. Let E, C, {T n }, {α n }, {t n }, {I n }, u,and{u n } be as in Lemma 4.1. Assume that E is smooth and z ∈ C is a common fixed point of {T n : n ∈N}. Then u n −u,J u n −z ≤ 0 (4.18) for all n ∈ N. Proof. Since α n (u n −u) =(1 −α n )(S(I n ,t n )u n −u n ), we have α n 1 −α n u n −u, J u n −z = S I n ,t n u n −u n ,J u n −z = S I n ,t n u n −z,J u n −z + z −u n , J u n −z = S I n ,t n u n −S I n ,t n z, J u n −z − u n −z 2 ≤ S I n ,t n u n −S I n ,t n z u n −z − u n −z 2 ≤ u n −z 2 − u n −z 2 = 0. (4.19) 10 Infinite families of nonexpansive mappings Thus we obtain u n −u, J u n −z ≤ 0 (4.20) for all n ∈ N. Lemma 4.5. Let E, C, {T n }, {α n }, {t n }, {I n }, u,and{u n } be as in Lemma 4.1. Assume that E is smooth. Then {u n } hasatmostoneclusterpoint. Proof. We assume that a subsequence {u n i } of {u n } converges strongly to x, and that another subsequence {u n j } of {u n } converges strongly to y.ApplyingLemma 4.3 to the subsequences {u n i } and {u n j },wehavethatx and y are common fixed points of {T n : n ∈ N} .So,byLemma 4.4,wehave u n i −u, J u n i − y ≤ 0 (4.21) for all i ∈ N. Therefore we obtain x −u, J( x − y) ≤ 0. (4.22) Similarly we can prove y −u, J(y −x) ≤ 0. (4.23) So we obtain x − y 2 = x − y, J(x − y) = x −u, J(x − y) + u − y, J(x − y) = x −u, J(x − y) + y −u, J(y −x) ≤ 0. (4.24) This implies x = y. This completes the proof. Lemma 4.6. Let E be a reflexive Banach space with uniformly G ˆ ateaux differentiable norm and let C be a closed convex subset of E with the fixed point property for nonexpansive map- pings. Let {T n }, {α n }, {t n }, {I n }, u,and{u n } be as in Lemma 4.1. Then {u n } has a cluster point which is a common fixed point of {T n : n ∈N}. Proof. From the proof of Lemma 4.1,wehavethat {u n } is bounded. Take a universal subnet {u ν : ν ∈D} of {u n }. Define a continuous convex function f from C into [0,∞) by f (x): = lim ν∈D u ν −x (4.25) for all x ∈ C. We note that f is well defined because {u ν −x} is a universal net in some compactsubsetof R for each x ∈ C.FromthereflexivityofE and lim x→∞ f (x) =∞,we can put r : = min x∈C f (x) and define a nonempty weakly compact convex subset A of C by A : = x ∈C : f (x) =r . (4.26) [...]... 227–239 , Strong convergence theorems for in nite families of nonexpansive mappings in general [21] Banach spaces, Fixed Point Theory and Applications 2005 (2005), no 1, 103–123 , The set of common fixed points of an n-parameter continuous semigroup of mappings, [22] Nonlinear Analysis 63 (2005), no 8, 1180–1190 , The set of common fixed points of a one-parameter continuous semigroup of mappings [23]... 2)), Proceedings of the American Mathematical Society 134 (2006), no 3, 673–681 , Browder’s type convergence theorems for one-parameter semigroups of nonexpansive map[24] pings in Banach spaces, to appear in Israel Journal of Mathematics , Browder’s type convergence theorems for one-parameter semigroups of nonexpansive map[25] pings in Hilbert spaces, to appear in Proceedings of the Fourth International... Jr., Nonexpansive retracts of Banach spaces, Bulletin of the American Mathematical Society 76 (1970), 384–386 , Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Transactions of [8] the American Mathematical Society 179 (1973), 251–262 , A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific [9] Journal of Mathematics 53 (1974), 59–71 16 In nite families. .. Proceedings of the National Academy of Sciences of the United States of America 53 (1965), 1272–1276 , Nonexpansive nonlinear operators in a Banach space, Proceedings of the National Acad[5] emy of Sciences of the United States of America 54 (1965), 1041–1044 , Convergence of approximants to fixed points of nonexpansive non-linear mappings in Ba[6] nach spaces, Archive for Rational Mechanics and Analysis... mapping J of E is weakly sequentially continuous at zero Let {Tn : n ∈ N} be an in nite family of commuting nonexpansive mappings on C Let {αn } and {tn } be sequences in (0,1/2) satisfying limn tn = limn αn /tn = 0 for ∈ N Let {In } be a sequence of nonempty subsets of N such that In ⊂ In+ 1 for n ∈ N, and ∞ 1 In = N Fix u ∈ C n= and define a sequence {un } in C by un = 1 − αn 1− k In k tn T1 un + k In. .. A Kirk, A fixed point theorem for mappings which do not increase distances, The American Mathematical Monthly 72 (1965), 1004–1006 [14] Z Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bulletin of the American Mathematical Society 73 (1967), 591–597 [15] S Reich, Asymptotic behavior of contractions in Banach spaces, Journal of Mathematical Analysis and... Product formulas, nonlinear semigroups, and accretive operators, Journal of Functional [16] Analysis 36 (1980), no 2, 147–168 , Strong convergence theorems for resolvents of accretive operators in Banach spaces, Journal [17] of Mathematical Analysis and Applications 75 (1980), no 1, 287–292 [18] N Shioji and W Takahashi, Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert... Hilbert spaces, Nonlinear Analysis 34 (1998), no 1, 87–99 [19] T Suzuki, On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proceedings of the American Mathematical Society 131 (2003), no 7, 2133–2136 , Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonex[20] pansive semigroups without Bochner integrals, Journal of Mathematical Analysis... } in C by n un = 1 − αn 1− k =1 vn = (1 − αn ) ∞ 1− k =1 k tn T1 un + k tn T1 vn + n k =1 ∞ k =1 for n ∈ N Then {un } and {vn } converge strongly to Pu k tn Tk+1 un + αn u, k tn Tk+1 vn (5.5) + αn u 14 In nite families of nonexpansive mappings From the proofs of lemmas in Section 4, we also obtain the following Theorem 5.8 Let E and C be as in Theorem 5.4 Let {Tn : n = 1,2, , } be a finite family of. .. -parameter nonexpansive semigroups We recall that a family of mappings {T(p) : p ∈ [0, ∞) } is said to be an -parameter nonexpansive semigroup on a closed convex subset C of a Banach space E if the following are satisfied (i) For each p ∈ [0, ∞) , T(p) is a nonexpansive mapping on C (ii) T(p + q) = T(p) ◦ T(q) for all p, q ∈ [0, ∞) (iii) For each x ∈ C, the mapping p → T(p)x from [0, ∞) into C is continuous . families of nonexpansive mappings in [21]. See also [20]. In this paper, using the idea in [21], we prove Browder’s type strong convergence theorems for in nite families of nonexpansive mappings. Takahashi in [18]proved Browder’s type strong convergence theorems for families of nonexpansive mappings Very recently, the author proved the following Browder’s type strong convergence the- orem for. BROWDER’S TYPE STRONG CONVERGENCE THEOREMS FOR INFINITE FAMILIES OF NONEXPANSIVE MAPPINGS IN BANACH SPACES TOMONARI SUZUKI Received 19 August