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Abstract and Applied Analysis Nonlinear ergodic theorems for a semitopological semigroup of nonLipschitzian mappings without convexity
Nonlinear ergodic theorems for a semitopological semigroup of nonLipschitzian mappings without convexity
Li, G., Kim, J. K.Quanto ti piace questo libro?
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Volume:
4
Anno:
1999
Lingua:
english
Rivista:
Abstract and Applied Analysis
DOI:
10.1155/s1085337599000056
File:
PDF, 1,84 MB
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NONLINEAR ERGODIC THEOREMS FOR A SEMITOPOLOGICAL SEMIGROUP OF NONLIPSCHITZIAN MAPPINGS WITHOUT CONVEXITY G. LI AND J. K. KIM Received 2 February 1999 Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H , and = {Tt : t ∈ G} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x) inf t∈G Tt x −z} = {z ∈ H : inf s∈G supt∈G Tts x −z = for each x ∈ C and L() = x∈C L(x). In this paper, we prove that s∈G conv{Tts x : t ∈ G} L() is nonempty for each x ∈ C if and only if there exists a unique nonexpansive retraction P of C into L() such that P Ts = P for all s ∈ G and P (x) ∈ conv{Ts x : s ∈ G} for every x ∈ C. Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of nonLipschitzian mappings without convexity. 1. Introduction and preliminaries Let H be a Hilbert space with norm · and inner product (·, ·). Let G be a semitopological semigroup, that is, a semigroup with a Hausdorff topology such that for each s ∈ G the mappings s → s · t and s → t · s of G into itself are continuous. Let C be a nonempty subset of H and let = {Tt : t ∈ G} be a semigroup on C, that is, Tst (x) = Ts Tt (x) for all s, t ∈ G and x ∈ C. Recall that a semigroup is said to be (a) nonexpansive if Tt x − Tt y ≤ x − y for x, y ∈ C and t ∈ G. (b) asymptotically nonexpansive [6] if there exists a function k : G → [0, ∞) with inf s∈G supt∈G kts ≤ 1 such that Tt x − Tt y ≤ kt x − y for x, y ∈ C and t ∈ G. (c) of asymptotically nonexpansive type [6] if for each x in C, there is a function r(·, x) : G → [0, ∞) with inf s∈G supt∈G r(ts, x) = 0 such that Tt x − Tt y ≤ x − y + r(t, x) for all y ∈ C and t ∈ G. It is easily seen that (a)⇒(b)⇒(c) and that both the inclusions are proper (cf. [6, page 112]). Baillon [1] proved the first nonlinear mean ergodic theorem for nonexpansive mappings in a Hilbert space: let C be a nonempty closed convex subset of a Hilbert space H and T a nonexpansive mapping of C into itself. If the ; set F (T ) of fixed points of T Copyright © 1999 Hindawi Publishing Corporation Abstract and Applied Analysis 4:1 (1999) 49–59 1991 Mathematics Subject Classification: 47H09, 47H10, 47H20 URL: http://aaa.hindawi.com/volume4/S1085337599000056.html 50 Nonlinear ergodic theorems is nonempty, then for each x ∈ C, the Cesáro means n−1 Sn (x) = 1 k T x n (1.1) k=0 converge weakly as n → ∞ to a point of F (T ). In this case, putting y = P x for each x ∈ C, P is a nonexpansive retraction of C onto F (T ) such that P T = T P = P and P x ∈ conv{T n x : n = 0, 1, 2, . . .} for each x ∈ C, where convA is the closure of the convex hull of A. The analogous results are given for nonexpansive semigroups on C by Baillon [2] and BreźisBrowder [3]. In [10], MizoguchiTakahashi proved a nonlinear ergodic retraction theorem for Lipschitzian semigroups by using the notion of submean. Recently, Li and Ma [8, 9] proved the nonlinear ergodic retraction theorems for nonLipschitzian semigroups in a Banach space without using the notion of submean. Also, in 1992, Takahashi [13] proved the ergodic theorem for nonexpansive semigroups on condition that s∈G conv{Tst x : t ∈ G} ⊂ C for some x ∈ C. In this paper, without using the concept of submean, we prove nonlinear ergodic theorem for semitopological semigroup of nonLipschitzian mappings without convexity in a Hilbert space. We first prove that if C is a nonempty subset of a Hilbert space H, G a semitopological semigroup, and = {Tt : t ∈ G} a representation of G as asymptotically nonexpansive type mappings of C into itself, then conv{T ts x : s∈G t ∈ G} L() is nonempty for each x ∈ C if and only if there exists a unique nonexpansive retraction P of C into L() such that P Ts = P for all s ∈ G and P x is in the closed convex hull of {Ts x : s ∈ G}, where L(x) = {z : inf s∈G supt∈G Tts x − z = inf t∈G Tt x − z} and L() = x∈C L(x). By using this result, we also prove the ergodic convergence theorem for semitopological semigroup of nonLipschitzian mapping without convexity. Our results are generalizations and improvements of the previously known results of BrézisBrowder [3], HiranoTakahashi [4], MizoguchiTakahashi [10], TakahashiZhang [14], and Takahashi [11, 12, 13] in many directions. Further, it is safe to say that in the results [1, 2, 3, 4, 5, 7, 10, 11, 12, 13, 14], many key conditions are not necessary. 2. Ergodic convergence theorems Throughout this paper, we assume that C is a nonempty subset of a real Hilbert space H, G a semitopological semigroup, and = {Tt : t ∈ G} an asymptotically nonexpansive type semigroup on C. For each x ∈ C, define L(x) and L() by L(x) = z : inf sup Tts x − z = inf Tt x − z , L() = L(x), (2.1) s∈G t∈G t∈G x∈C respectively. We denote F () by the set {x ∈ C : Ts (x) = x for all s ∈ G} of common fixed point of . We begin with the following lemma. Lemma 2.1. Let C be a nonempty subset of a Hilbert space H and = {Tt : t ∈ G} an asymptotically nonexpansive type semigroup on C. Then F () ⊂ L(). G. Li and J. K. Kim 51 Proof. Let x ∈ C and f ∈ F (). Since is asymptotically nonexpansive type, for an arbitrary ε > 0, there exists s0 ∈ G such that for all t ∈ G r ts0 , f < ε. (2.2) Hence, for each a ∈ G, inf sup Tts x − f ≤ sup Tts0 a x − f ≤ sup Ta x − f + r ts0 , f s∈G t∈G t∈G t∈G ≤ Ta x − f + ε. (2.3) Since ε > 0 is arbitrary, we have inf s∈G supt∈G Tts x − f ≤ inf t∈G Tt x − f . Therefore, f ∈ L(x). This completes the proof. Remark 2.2. It is not easy to prove that F () is nonempty when C is not a convex subset. However, we can show that L() is nonempty under some conditions and it is important for the ergodic convergence theorem. The following proposition plays a crucial role in the proof of our main theorems in this paper. Proposition 2.3. Let G be a semitopological semigroup, C a nonempty subset of a Hilbert space H , and = {Tt : t ∈ G} an asymptotically nonexpansive type semigroup on C. Then, for every x ∈ C, the set L(x), (2.4) conv Tts x : t ∈ G s∈G consists of at most one point. Proof. Let u, v ∈ sume that s∈G conv{Tts x : t ∈ G} L(x), without loss of generality, we as 2 2 inf Tt x − u ≤ inf Tt x − v . t∈G (2.5) t∈G Now, for each t, s ∈ G, since 2 2 u − v2 + 2 Tts x − u, u − v = Tts x − v − Tts x − u , (2.6) we have 2 2 u − v2 + 2 inf Tts x − u, u − v ≥ inf Tts x − v − sup Tts x − u t∈G t∈G t∈G 2 2 ≥ inf Tt x − v − sup Tts x − u . t∈G t∈G From u ∈ L(x), we have 2 2 u − v2 + 2 sup inf Tts x − u, u − v ≥ inf Tt x − v − inf sup Tts x − u s∈G t∈G (2.7) t∈G s∈G t∈G t∈G t∈G (2.8) 2 2 = inf Tt x − v − inf Tt x − u ≥ 0. 52 Nonlinear ergodic theorems Therefore, for ε > 0 there is an s1 ∈ G such that u − v2 + 2 Tts1 x − u, u − v > −ε ∀t ∈ G. (2.9) From v ∈ conv{Tts1 x : t ∈ G}, we have u − v2 + 2(v − u, u − v) ≥ −ε. (2.10) This inequality implies that u−v2 ≤ ε. Since ε > 0 is arbitrary, we have u = v. This completes the proof. Remark 2.4. In the TakahashiZhang’s result [14], it is assumed that C is a closed convex subset, G a reversible semigroup, and an asymptotically nonexpansive semigroup. Proposition 2.3 shows those key conditions are not necessary. Let m(G) be the Banach space of all bounded realvalued functions on a semitopological semigroup G with the supremum norm and let X be a subspace of m(G) containing constants. Then, an element µ of X ∗ (the dual space of X) is called a mean on X if µ = µ(1) = 1. Let µ be a mean on X and f ∈ X. Then, according to time and circumstances, we use µt (f (t)) instead of µ(f ). For each s ∈ G and f ∈ m(G), we define elements ls f and rs f in m(G) given by (ls f )(t) = f (st) and (rs f )(t) = f (ts) for all t ∈ G, respectively. Throughout the rest of this section, let X be a subspace of m(G) containing constants invariant under ls and rs for each s ∈ G. Furthermore, suppose that for each x ∈ C and y ∈ H, a function f (t) = Tt x −y2 is in X. For µ ∈ X ∗ , we define the value µt (Tt x, y) of µ at this function. By Riesz theorem, there exists a unique element µ x in X such that µt Tt x, y = µ x, y ∀y ∈ H. (2.11) Lemma 2.5. Suppose that X has an invariant mean µ. Then we have conv Tts x : t ∈ G L(x) = µ x for every x ∈ C. s∈G Further, if Tt is continuous for each t ∈ G and x ∈ C, then µ x ∈ F (). s∈G conv{Tst x (2.12) : t ∈ G} ⊂ C for some Proof. Since µ is an invariant mean, it is easy to show that µ x ∈ s∈G conv{Tts x : t ∈ G} for each x ∈ C. By Proposition 2.3, it is enough to prove that µ x ∈ L(x) for each x ∈ C. To this end, let ε > 0, since is an asymptotically nonexpansive type semigroup, for each t ∈ G there is an ht ∈ G such that for each h ∈ G, (2.13) r hht , Tt x < ε. Put M = supt,s∈G Tt x − Ts x, then we have Thh t x − µ x 2 − Tt x − µ x 2 = µs Thh t x − Ts x 2 − Tt x − Ts x 2 t t 2 2 = µs Thht t x − Thht s x − Tt x − Ts x ≤ 2Mε for each h ∈ G. (2.14) G. Li and J. K. Kim 53 Hence, we have 2 2 inf sup Ths x − µ x ≤ Tt x − µ x + 2Mε s∈G h∈G ∀t ∈ G. (2.15) Since ε > 0 is arbitrary, we have µ x ∈ L(x). Finally, suppose that s∈G conv{Tst x : t ∈ G} ⊂ Cand each Tt is continuous from C into itself. Then, we can easily prove that µ x ∈ s∈G conv{Tst x : t ∈ G} and hence we have µ x ∈ C. For each h ∈ G and ε ∈ (0, 1), there exists 0 < δ < ε such that Th y − Th µ x < ε whenever y ∈ C and y − µ x ≤ δ. Since is an asymptotically nonexpansive type semigroup, there is s0 ∈ G such that 1 δ 2 ∀t ∈ G, (2.16) r ts0 , µ x < 2 M1 + 1 where M1 = supt∈G Tt x − µ x. Then for each t, s ∈ G, we have Tss µ x − µ x 2 + 2 Tt x − µ x, µ x − Tss µ x 0 0 2 2 = Tt x − Tss0 µ x − Tt x − µ x 2 2 2 2 = Tss0 t x − Tss0 µ x − Tt x − µ x − Tss0 t x − Tss0 µ x + Tt x − Tss0 µ x 2 2 ≤ δ 2 − Tss0 t x − Tss0 µ x + Tt x − Tss0 µ x . (2.17) It follows that Tss µ x − µ x ≤ δ 0 ∀s ∈ G. (2.18) This implies that Th µ x − µ x ≤ Th µ x − Th Tss µ x + Thss µ x − µ x < 2ε. 0 0 (2.19) Since ε > 0 is arbitrary, we have Th µ x = µ x. This completes the proof. Now, we prove a nonlinear ergodic theorem for asymptotically nonexpansive type semigroups without convexity. Before doing this, we give a definition concerning means. Let {µα : α ∈ A} be a net of means on X, where A is a directed set. Then {µα : α ∈ A} is said to be asymptotically invariant if for each f ∈ X and s ∈ G, µα (f ) − µα ls f −→ 0, µα (f ) − µα rs f −→ 0. (2.20) Theorem 2.6. Let C be a nonempty subset of a Hilbert space H , X an invariant subspace of m(G) containing constants, and = {Tt : t ∈ G} an asymptotically nonexpansive type semigroup on C. If for each x ∈ C and y ∈ H , the function f on G defined by f (t) = Tt x − y2 belong to X, then for an asymptotically invariant net {µα : α ∈ A} on X, the net {µα x}α∈A converges weakly to an element x0 ∈ L(x). 54 Nonlinear ergodic theorems Further, if Tt is continuous for each t ∈ G and x0 ∈ F (). s∈G conv{Tst x : t ∈ G} ⊂ C, then Proof. Let W be the set of all weak limit points of subnet of the net {µα x : α ∈ A}. By Proposition 2.3, it is enough to prove that W⊂ conv Tts x : t ∈ G L(x). (2.21) s∈G To show this, let z ∈ W and let {uαβ x} be a subnet of {µα x} such that {µαβ x} converges weakly to z. Now, without loss of generality, we can suppose that {µαβ x} converges weakly* to µ ∈ X ∗ . It is easily seen that µ is an invariant mean on X and then Lemma 2.5 implies that z = µ x ∈ s∈G conv{Tts x : t ∈ G} L(x). This completes the proof. Let C(G) be the Banach space of all bounded continuous realvalued functions on G and let RUC(G) be the space of all bounded right uniformly continuous functions on G, that is, all f ∈ C(G) such that the mapping s → rs f is continuous. Then RUC(G) is a closed subalgebra of C(G) containing constants and invariant under ls and rs . As a direct consequence of Theorem 2.6, we obtain the following corollary. Corollafry 2.7 (see [13]). Let C be a nonempty subset of a Hilbert space H and let G be a semitopological semigroup such that RUC(G) has an invariant mean. Let = {T t : t ∈ G} be a nonexpansive semigroup on C such that {Tt x : t ∈ G} is bounded and s∈G conv{Tst x : t ∈ G} ⊂ C for some x ∈ C. Then, F () = ∅. Further, for an asymptotically invariant net {µα }α∈A of means on RUC(G), the net {µα }α∈A , converges weakly to an element x0 ∈ F (). Remark 2.8. For the proof of Corollary 2.7, Takahashi [13] used the condition s∈G conv{Tst x : t ∈ G} ⊂ C. But, from Theorem 2.6, we can prove the result without this condition except proving the fact that the weak limit of {µα x} is in F (). 3. Nonexpansive retractions In this section, we prove an ergodic retraction theorem for a semitopological semigroup of asymptotically nonexpansive type mappings without convexity. Theorem 3.1. Let C be a nonempty subset of a Hilbert space H and let = {Tt : t ∈ G} be a semitopological semigroup of asymptotically nonexpansive type mappings on C such that statements are equivalent: L() = ∅. Then the following (a) s∈G conv{Tts x : t ∈ G} L() = ∅ for each x ∈ C. (b) There is a unique nonexpansive retraction P of C into L() such that P Tt = P for every t ∈ G and P x ∈ conv{Tt x : t ∈ G} for every x ∈ C. Proof. (b)⇒(a). Let x ∈ C, then P x ∈ L(). Also P x ∈ s∈G conv{Tts x : t ∈ G}. In fact, for each s ∈ G, P x = P Ts x ∈ conv{Tt Ts x : t ∈ G} = conv{Tts x : t ∈ G}. G. Li and J. K. Kim 55 (a)⇒(b). Let x ∈ C. Then by Proposition 2.3, s∈G conv{Tts x : t ∈ G} L() contains exactly one point P x. For each a ∈ G, we have {P Ta x} = conv Ttsa x : t ∈ G L() s∈G ⊇ conv Tts x : t ∈ G (3.1) L() = {P x} s∈G and hence we have P Ta = P for every a ∈ G. Finally, we have to show that P is nonexpansive. Let x, y ∈ C and 0 < λ < 1. Then for any ε > 0, there exists s1 ∈ G such that (3.2) sup Tts1 x − P y ≤ inf Tt x − P y + ε, t∈G t∈G from P y ∈ L(). Hence, we have λTtss x + (1 − λ)P x − P y 2 1 2 = λ Ttss1 x − P y + (1 − λ)(P x − P y) 2 2 = λTtss1 x − P y + (1 − λ)P x − P y2 − λ(1 − λ)Ttss1 x − P x 2 2 ≤ λ Tab x − P y + ε + (1 − λ)P x − P y2 − λ(1 − λ) inf Tt x − P x , t∈G (3.3) for each t, s, a, b ∈ G. Since ε > 0 is arbitrary, this implies 2 inf sup λTts x + (1 − λ)P x − P y s∈G t∈G 2 2 ≤ λTab x − P y + (1 − λ)P x − P y2 − λ(1 − λ) inf Tt x − P x t∈G 2 2 2 = λTab x+(1−λ)P x−P y + λ(1 − λ) Tab x − P x − λ(1 − λ) inf Tt x − P x . t∈G (3.4) Then it is easily seen that 2 2 inf sup λTts x + (1 − λ)P x − P y − λ(1 − λ) inf sup Tab x − P x s∈G t∈G b∈G a∈G 2 2 ≤ sup inf λTab x + (1 − λ)P x − P y − λ(1 − λ) inf Tt x − P x . t∈G b∈G a∈G Since P x ∈ L(), we have 2 2 inf sup λTts x + (1 − λ)P x − P y ≤ sup inf λTts x + (1 − λ)P x − P y . s∈G t∈G Let (3.5) s∈G t∈G 2 h(λ) = inf sup λTts x + (1 − λ)P x − P y . s∈G t∈G (3.6) (3.7) 56 Nonlinear ergodic theorems Then for any ε > 0, there exists s2 ∈ G such that for all t ∈ G, λTts x + (1 − λ)P x − P y 2 ≤ h(λ) + ε 2 and hence 1/2 λTts2 x + (1 − λ)P x − P y, P x − P y ≤ h(λ) + ε P x − P y (3.8) ∀t ∈ G. From P x ∈ conv{Tts2 x : t ∈ G}, we have 1/2 λP x + (1 − λ)P x − P y, P x − P y ≤ h(λ) + ε P x − P y. (3.9) (3.10) Since ε > 0 is arbitrary, this yields that That is, P x − P y2 ≤ h(λ). (3.11) 2 P x − P y2 ≤ inf sup λTts x + (1 − λ)P x − P y . (3.12) s∈G t∈G Now, one can choose an s3 ∈ G such that Tts3 x − P x ≤ M for all t ∈ G, where M = 1 + inf t∈G Tt x − P x. Then, we have λTtss x + (1 − λ)P x − P y 2 3 2 = λ Ttss3 x − P x + (P x − P y) (3.13) 2 = λ2 Ttss3 x − P x + P x − P y2 + 2λ Ttss3 x − P x, P x − P y ≤ M 2 λ2 + P x − P y2 + 2λ Ttss3 x − P x, P x − P y . It then follows from (3.6) and (3.12) that 2λ sup inf Tts x − P x, P x − P y s∈G t∈G ≥ 2λ sup inf Ttss3 x − P x, P x − P y s∈G t∈G 2 ≥ sup inf λTtss3 x + (1 − λ)P x − P y − P x − P y2 − M 2 λ2 s∈G t∈G 2 = sup inf λTts Ts3 x + (1 − λ)P Ts3 x − P y − P x − P y2 − M 2 λ2 (3.14) s∈G t∈G 2 ≥ P Ts3 x − P y − P x − P y2 − M 2 λ2 = −M 2 λ2 . Hence, we have 1 sup inf Tts x − P x, P x − P y ≥ − M 2 λ. 2 s∈G t∈G (3.15) Letting λ → 0, then we have sup inf Tts x − P x, P x − P y ≥ 0. s∈G t∈G (3.16) G. Li and J. K. Kim Let ε > 0, then there is s4 ∈ G such that r ts4 , x < ε ∀t ∈ G. 57 (3.17) For such an s4 ∈ G, from (3.16), we have sup inf Tts Ts4 x − P Ts4 x, P Ts4 x − P y ≥ 0 (3.18) and hence there is s5 ∈ G such that inf Tts5 Ts4 x − P Ts4 x, P Ts4 x − P y > −ε. (3.19) Then, from P Ts4 x = P x, we have inf Tts5 s4 x − P x, P x − P y > −ε. (3.20) Similarly, from (3.16), we also have sup inf Tts Ts5 s4 y − P Ts5 s4 y, P Ts5 s4 y − P x ≥ 0, (3.21) and there exists s6 ∈ G such that inf Tts6 s5 s4 y − P Ts5 s4 y, P Ts5 s4 y − P x ≥ −ε, (3.22) s∈G t∈G t∈G t∈G s∈G t∈G t∈G that is, inf P y − Tts6 s5 s4 y, P x − P y ≥ −ε. (3.23) On the other hand, from (3.20) inf Tts6 s5 s4 x − P x, P x − P y > −ε. (3.24) t∈G t∈G Combining (3.23) and (3.24), we have −2ε < Tts6 s5 s4 x − Tts6 s5 s4 y, P x − P y − P x − P y2 ≤ Tts6 s5 s4 x − Tts6 s5 s4 y · P x − P y − P x − P y2 ≤ r ts6 s5 s4 , x) + x − y · P x − P y − P x − P y2 ≤ ε + x − y · P x − P y − P x − P y2 . (3.25) Since ε > 0 is arbitrary, this implies P x −P y ≤ x −y. The proof is completed. Using Lemma 2.1, we have the following ergodic retraction theorem for asymptotically nonexpansive type semigroups. Theorem 3.2. Let C be a nonempty subset of a real Hilbert space H and let = {Tt : t ∈ G} be a semitopological semigroup of asymptotically nonexpansive type mappings on C such that F () = ∅.Then the following statements are equivalent: (a) s∈G conv{Tts x : t ∈ G} F () = ∅ for each x ∈ C. (b) There is a unique nonexpansive retraction P of C onto F () such that P Tt = Tt P = P for every t ∈ G and P x ∈ conv{Tt x : t ∈ G} for every x ∈ C. 58 Nonlinear ergodic theorems We denote by B(G) the Banach space of all bounded realvalued functions on G with supremum norm. Let X be a subspace of B(G) containing constants. Then, according to MizoguchiTakahashi [10], a realvalued function µ on X is called a submean on X if the following conditions are satisfied: (1) µ(f + g) ≤ µ(f ) + µ(g) for every f, g ∈ X; (2) µ(αf ) = αµ(f ) for every f ∈ X and α ≥ 0; (3) for f, g ∈ X, f ≤ g implies µ(f ) ≤ µ(g); (4) µ(c) = c for every constant c. The following corollaries are immediately deduced from Theorem 3.2. Corollafry 3.3 (see [10]). Let C be a closed convex subset of a Hilbert space H and let X be an rs invariant subspace of B(G) containing constants which has a right invariant submean. Let = {Tt : t ∈ G} be a Lipschitzian semigroup on C with 2 ≤ 1 and F () = ∅, where k is the Lipschitzian constants. If for each inf s supt kts t x, y ∈ C, the function f on G defined by 2 f (t) = Tt x − y ∀t ∈ G (3.26) and the function g on G defined by g(t) = kt2 ∀t ∈ G (3.27) belongto X, then the following statements are equivalent: (a) s∈G conv{Tts x : t ∈ G} F () = ∅ for each x ∈ C. (b) There is a nonexpansive retraction P of C onto F () such that P Tt = Tt P = P for every t ∈ G and P x ∈ conv{Tt x : t ∈ G} for every x ∈ C. Corollafry 3.4 (see [7]). Let C be a nonempty closed convex subset of a Hilbert space H and let = {Tt : t ∈ G} be a continuous representation of a semitopological semigroup as nonexpansive mappings from C into itself. If for each x ∈ C, the set conv{T x : t ∈ G} F () = ∅, then there exists a nonexpansive retraction P of ts s∈G C onto F () such that P Tt = Tt P = P for every t ∈ G and P x ∈ conv{Tt x : t ∈ G} for every x ∈ C. Remark 3.5. By Theorem 3.2, many key conditions, in Corollaries 3.3 and 3.4, such as C is convex closed subset and is continuous Lipschitzian semigroup, are not necessary. Acknowledgement The authors wish to acknowledge the financial support of the Korea Research Foundation made in the program year of 1998. G. Li and J. K. Kim 59 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] J.B. Baillon, Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér. AB 280 (1975), no. 22, 1511–1514 (French). MR 51#11205. Zbl 307.47006. , Quelques propriétés de convergence asymptotique pour les semigroupes de contractions impaires, C. R. Acad. Sci. Paris Sér. AB 283 (1976), no. 3, 75–78 (French). MR 54#13655. Zbl 339.47028. H. Brézis and F. E. Browder, Remarks on nonlinear ergodic theory, Advances in Math. 25 (1977), no. 2, 165–177. MR 57#1218. Zbl 399.47058. N. Hirano and W. Takahashi, Nonlinear ergodic theorems for nonexpansive mappings in Hilbert spaces, Kodai Math. J. 2 (1979), no. 1, 11–25 (English). MR 80j:47064. Zbl 404.47031. H. Ishihara and W. Takahashi, A nonlinear ergodic theorem for a reversible semigroup of Lipschitzian mappings in a Hilbert space, Proc. Amer. Math. Soc. 104 (1988), no. 2, 431–436 (English). MR 90g:47120. Zbl 692.47010. W. A. Kirk and R. Torrejón, Asymptotically nonexpansive semigroups in Banach spaces, Nonlinear Anal. 3 (1979), no. 1, 111–121 (English). MR 82a:47062. Zbl 411.47035. A. T. M. Lau, K. Nishiura, and W. Takahashi, Nonlinear ergodic theorems for semigroups of nonexpansive mappings and left ideals, Nonlinear Anal. 26 (1996), no. 8, 1411–1427 (English). MR 97b:47074. Zbl 880.47048. G. Li, Weak convergence and nonlinear ergodic theorems for reversible semigroups of nonLipschitzian mappings, J. Math. Anal. Appl. 206 (1997), no. 2, 451–464 (English). MR 98k:47139. Zbl 888.47046. G. Li and J. Ma, Nonlinear ergodic theorem for semitopological semigroups of nonLipschitzian mappings in Banach spaces, Chinese Sci. Bull. 42 (1997), no. 1, 8–11 (English). MR 98e:47110. Zbl 904.47063. N. Mizoguchi and W. Takahashi, On the existence of fixed points and ergodic retractions for Lipschitzian semigroups in Hilbert spaces, Nonlinear Anal. 14 (1990), no. 1, 69–80 (English). MR 91h:47071. Zbl 695.47063. W. Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981), no. 2, 253–256 (English). MR 82f:47079. Zbl 456.47054. , A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 97 (1986), no. 1, 55–58. MR 88f:47051. , Fixed point theorem and nonlinear ergodic theorem for nonexpansive semigroups without convexity, Canad. J. Math. 44 (1992), no. 4, 880–887 (English). MR 93j:47091. Zbl 786.47047. W. Takahashi and P.J. Zhang, Asymptotic behavior of almostorbits of reversible semigroups of Lipschitzian mappings, J. Math. Anal. Appl. 142 (1989), no. 1, 242–249 (English). MR 90g:47121. Zbl 695.47062. G. Li: Department of Mathematics, Yangzhou University, Yangzhou 225002, China Email address: ligang@cims1.yzu.edu.cn J. K. 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