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Abstract and Applied Analysis Uniform asymptotic normal structure, the uniform semiOpial property, and fixed points of asymptotically..
Uniform asymptotic normal structure, the uniform semiOpial property, and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part II
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3
年:
1998
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english
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Abstract and Applied Analysis
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10.1155/s1085337598000554
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UNIFORM ASYMPTOTIC NORMAL STRUCTURE, THE UNIFORM SEMIOPIAL PROPERTY, AND FIXED POINTS OF ASYMPTOTICALLY REGULAR UNIFORMLY LIPSCHITZIAN SEMIGROUPS. PART II MONIKA BUDZYŃSKA, TADEUSZ KUCZUMOW AND SIMEON REICH Abstract. In this part of our paper we present several new theorems concerning the existence of common ﬁxed points of asymptotically regular uniformly lipschitzian semigroups. 1. Introduction Let (X, ·) be a Banach space and C a subset of X. A mapping T : C → C is said to be uniformly klipschitzian if for each x, y ∈ C and every natural number n, T n x − T n y ≤ k x − y. If k = 1, then the mapping T is called nonexpansive. These deﬁnitions can also be introduced in metric spaces. The class of uniformly lipschitzian mappings on C is completely characterized as the class of those mappings on C which are nonexpansive with respect to some metric on C which is equivalent to the norm [16]. In this part of our paper we use the new geometric coeﬃcients introduced in its ﬁrst part to study the existence of (common) ﬁxed points for this class of mappings. 2. Basic notations and facts Throughout this paper we will use the notations from the ﬁrst part of our paper [6]. However, before we recall several known ﬁxed point theorems we need a few additional notations. 1991 Mathematics Subject Classiﬁcation. 47H10, 47H20. Key words and phrases. The uniform semiOpial property, asymptotically regular and uniformly lipschitzian semigroups, ﬁxed points. Received: September 15, 1997. c 1996 Mancorp Publishing, Inc. 247 248 MONIKA BUDZYŃSKA, TADEUSZ KUCZUMOW AND SIMEON REICH Let (X, ·) denote a Banach space. For x ∈ X and a bounded sequence {xn } the asymptotic radius of {xn } at x is the number ra (x, {xn }) = lim supn x − xn . Now for a nonempty closed convex subset C of X the asymptotic radius of {xn } in C is the number r (C, {xn }) = inf {ra (x, {xn }) : x ∈ C}. The asymptotic center of {xn } in C [13] is the set Ac (C, {xn }) = {x ∈ C : ra (x, {xn }) = r (C, {xn })} . For more details see [1], [16] ; and [17]. Let B (0, 1) be the closed unit ball in X. The modulus of convexity of X is the function δ : [0, 2] → [0, 1] deﬁned by δ () = inf 1 − x + y ; x, y ∈ B (0, 1) , x − y ≥ 2 [8]. The characteristic of convexity of X is the number 0 (X) = sup { : δ () = 0} [16]. When 0 (X) = 0 X is called a uniformly convex space [8]. The Lifshitz characteristic κ (M ) of a metric space (M, ρ) is the supremum of all positive real numbers b such that there exists a > 1 such that for each x, y ∈ M and r > 0 with ρ (x, y) > r there exists z ∈ M satisfying B (x, br) ∩ B (y, ar) ⊂ B (z, r) [23]. It is obvious that κ (M ) ≥ 1. In a Banach space (X, ·) we denote by κ0 (X) the inﬁmum of the numbers κ (C) where C is a closed, convex, bounded and nonempty subset of X. It is known [2] , [16] that (1) 1 ≤ κ0 (X) ≤ N (X) 1 − δ (1) √ and 0 (X) < 1 if and only κ0 (X) > 1 [12]. Therefore κ0 (X) ≤ 2 [2]. Unfortunately, we know the exact value of κ0 (X) or some lower bounds for κ0 (X) in special spaces only [2]. Therefore it is convenient to introduce a new coeﬃcient which plays a role similar to the one of κ0 (X). In [10] T. Domı́nguez Benavides and H.K. Xu introduced such a new constant κω (X) in Banach spaces. We give here a slightly diﬀerent deﬁnition of κω (X) from the one given in [10]. Namely, if (X, ·) is a Banach space and C is a nonempty bounded closed convex subset of X, then (a) a number b ≥ 0 has property (Pω ) with respect to C if there exists some a > 1 such that for all x, y ∈ C and r > 0 with x − y ≥ r and each weakly convergent sequence {zn } with elements in C such that lim supn x − zn ≤ ar and lim supn y − zn ≤ br, there exists z ∈ C such that lim inf n z − zn ≤ r; (b) κω (C) = sup {b > 0 : b has property (Pω ) with respect to C} ; (c) κω (X) = inf {κω (C) : C is a nonempty bounded closed convex subset of X}. It is clear that κω (C) ≥ κ (C) for all nonempty bounded closed convex subsets C ⊂ X. Next let us observe that κω (X) = inf {κω (C) : C is a convex weakly compact subset of X} . Hence we get that κω (X) ≤ W CS (X) [2]. UNIFORM ASYMPTOTIC NORMAL STRUCTURE 249 Let (M, ρ) be a metric space, where M is not a singleton and T : M → M . Then we will use the symbol T  to denote the exact Lipschitz constant of T , i.e., ρ (T x, T y) : x, y ∈ M, x = y . T  = sup ρ (x, y) Let X be a Banach space, C a nonempty bounded closed convex subset of X , G an unbounded subset of [0, ∞) such that (2) t + h ∈ G for all t, h ∈ G, (3) t − h ∈ G for all t, h ∈ G with t ≥ h, and T = {Tt : t ∈ G} a family of selfmappings on C . T is called a semigroup of mappings on C if (i) Ts+t x = Ts Tt x for all s, t ∈ G and x ∈ C, (ii) for each x ∈ C, the mapping t → Tt x from G into C is continuous when G has the relative topology of [0, ∞). Let us observe that in the particular case G = N we get the semigroups of iterates T = {Tt : t ∈ G} = {Tn1 : n ∈ N} . If T satisﬁes i.  ii. and in addition there exists k > 0 such that Tt x − Tt y ≤ k x − y for all x, y ∈ C and t in G, then we say that T is a uniformly lipschitzian (klipschitzian) semigroup of mappings on C. If T satisﬁes i.  ii. and for each x ∈ C, h ∈ G, lim Tt+h x − Tt x = 0, t→∞ then T is said to be asymptotically regular. The concept of asymptotic regularity is due to F.E. Browder and W.V. Petryshyn [5]. Let us observe that the notions of the asymptotic radius and the asymptotic center can be formulated in an obvious way for {xt }t∈G , where G satisﬁes (2) and (3). The ﬁrst positive result about ﬁxed points of uniformly lipschitzian mappings is due to K. Goebel and W.A. Kirk. Theorem 2.1. [14] Let X be a Banach space with 0 (X) < 1 and let C be a nonempty bounded closed convex subset of X. Suppose T : C → C is uniformly lipschitzian with a constant k < γ, where γ > 1 satisﬁes the equation 1 γ 1−δ (4) = 1. γ Then T has a ﬁxed point in C. By (1) it is obvious that the constant γ from (4) is strictly less than κ0 (X) [23]. In [30] K.K. Tan and H.K. Xu proved the following theorem which is formulated in the same spirit as the previous one. 250 MONIKA BUDZYŃSKA, TADEUSZ KUCZUMOW AND SIMEON REICH Theorem 2.2. [30] Let X be a uniformly convex Banach space, C a nonempty bounded closed convex subset of X and T : C → C a klipschitzian mapping with k < γ1 , where γ1 > 1 is the solution of the equation N (X) (5) = 1. γ1 1 − δ γ12 Then T has a ﬁxed point in C. The constant γ1 given by formula (5) is always bigger than the constant γ deﬁned by (4). In [23] Lifshitz extended the result of Goebel and Kirk in the following way: Theorem 2.3. [23] Let (M, ρ) be a complete metric space and T : M → M a uniformly lipschitzian mapping with constant k < κ (M ). If there exists x0 ∈ M such that the orbit {T n x0 } is bounded, then T has a ﬁxed point in M. Here we must note that in the case of a Banach space we do not know in general how to compare the constant γ1 given by (5) to the constant κ (X) or κ0 (X). In each Banach space we have κ0 (X) ≤ N (X) (see (1)) but in particular cases we can have κ0 (X) < N (X) [7]. Therefore the following result is important. Theorem 2.4. [7] Let X be a Banach space X with uniform normal structure and C a nonempty bounded closed convex subset of X. If T : C → C is a uniformly klipschitzian mapping with k < N (X), then T has a ﬁxed point. Now we recall a ﬁxed point theorem in which the coeﬃcients κ0 (X) and N (X) appear simultaneously. Theorem 2.5. [9] Let X be a Banach space, C a nonempty bounded closed convex subset of X and T : C → C. If n lim inf T  < n then T has a ﬁxed point. 1+ 1 + 4 · N (X) · (κ0 (X) − 1) , 2 For a discussion of this theorem see [9]. For asymptotically regular mappings we have the following results. Theorem 2.6. [20] Let X be an inﬁnite dimensional uniformly convex Banach space, C a nonempty bounded closed convex separable subset of X and T : C → C an asymptotically regular mapping with lim inf T n  < γ2 , where n γ2 > 1 is the solution of the equation W CS (X) (6) = 1. γ2 1 − δ γ22 Then T has a ﬁxed point in C. UNIFORM ASYMPTOTIC NORMAL STRUCTURE 251 Theorem 2.7. [2, 10] Suppose X is a Banach space. Suppose also C is a convex weakly compact subset of X and T : C → C is asymptotically regular. If lim inf T n  < κω (C), then T has a ﬁxed point. n Theorem 2.8. [2, 10] Suppose X is a Banach space such that W CS (X) > 1, C is a convex weakly compact subset of X and T : C → C is asymptotically n regular. If lim inf T  < W CS (X), then T has a ﬁxed point. n Theorem 2.9. [9] Let X be a reﬂexive Banach space, C a nonempty bounded closed convex subset of X and T : C → C an asymptotically regular mapping. If 1 + 1 + 4 · W CS (X) · (κω (X) − 1) n lim inf T  < , n 2 then T has a ﬁxed point. For a discussion of the connections among the above results see [9, 10]. In [11], [20], [21], [30], [31] and [32] some of the above results were reformulated in terms of semigroups. It is also worthwhile to see [4], [15], [18], [19], [22], [24], [25], [26], [27], [28], [29], [33], [34]. 3. Existence of fixed points of lipschitzian semigroups of mappings We begin with a generalization of Theorem 2.2. Theorem 3.1. Let (X, ·) be a Banach space with 0 < 1, C a nonempty bounded closed convex subset of X and T = {Tt : t ∈ G} a uniformly lipschitzian semigroup of mappings on C with supt Tt  = k < γ, where γ > 1 is the solution of the equation N (X) = 1. γ Then there exists a common ﬁxed point of T. (7) γ 1−δ Proof. Without loss of generality we can assume that (X, ·) is uniformly convex and k > 1. If 0 < 0 < 1 we need only make minor changes in the following proof. Let us denote N (X) by N . For each x ∈ C let z be the unique element of Ac (C, {Tt x}). Let us assume that d (x) diama {Tt x} = ≥ r = r (C, {Tt x}) > 0 N N and d (z) = diama {Tt z} > 0. Then we can ﬁnd sequences {sn } and {tn } such that lim sn = lim tn = ∞ (8) and Next we have n n lim Ttn z − Tsn z = diama {Tt z} = d (z) . n ra (Tsn z, {Tt x}) ≤ k · r 252 MONIKA BUDZYŃSKA, TADEUSZ KUCZUMOW AND SIMEON REICH and ra (Ttn z, {Tt x}) ≤ k · r. Hence r ≤ lim inf ra n 1≤ 1−δ and ﬁnally d (z) k·r ·k ≤ 1−δ 1 k · δ −1 1 − d (z) ≤ N k (9) d (z) Tsn z + Ttn z , {Tt x} ≤ 1 − δ 2 k·r N · d (z) d (x) · k · k · r, · k, · d (x) . Let us denote Nk · δ −1 1 − k1 by a. Directly from our assumption we get a < 1. Now we deﬁne a sequence {xn }∞ n=0 in C in the following way: x0 is an arbitrarily chosen element of C and xn+1 is the unique element of Ac (C, {Tt xn }) for n = 0, 1, 2, ... . By (9) we obtain d (xn ) ≤ an · d (x0 ) , and the inequalities xn − xn+1 ≤ xn − Tj xn−1 + Tj xn−1 − Ti xn + Ti xn − xn+1 ≤ xn − Tj xn−1 + k Tj−i xn−1 − xn + Ti xn − xn+1 , which are valid for i < j, lead to xn − xn+1 ≤ (1 + k) d (xn−1 ) + d (xn ) ≤ (1 + k) an−1 + an · d (x0 ) . This, in turn, yields the conclusion that {xn }∞ n=0 is norm Cauchy and hence strongly convergent. Let x = limn xn . Then by (8) for each s ∈ G we have x − Ts x ≤ lim x − xn+1 + lim sup lim sup xn+1 − Tt+s xn n n t + lim sup lim sup Tt+s xn − Ts xn+1 + lim Ts xn+1 − Ts x n t n d (xn ) = 0. ≤ lim (1 + k) x − xn+1 + n N This completes the proof. Remark 3.1. The constant γ given by (7) is bigger than the constant γ deﬁned by (5). For our next results we need the following simple fact. Lemma 3.1. Let (X, ·) be a Banach space. 1) For each 0 < 1θ < wSOC (X) and every asymptotically regular sequence {xn } with a weakly compact conv {xn } there exists a weakly convergent to w subsequence {xni } such that (i) r (w, {xni }) ≤ θ · diama ({xn }), (ii) w − y ≤ ra (y, {xni }) for every y ∈ X. UNIFORM ASYMPTOTIC NORMAL STRUCTURE 253 2) For each 0 < 1θ < wAN (X) and every asymptotically regular sequence {xn } with a weakly compact conv {xn } there exists a subsequence {xni } and a point w ∈ conv {xn } such that r (w, {xni }) ≤ θ · diama ({xn }) . Proof. 1) ii. It is suﬃcient to apply the lower semicontinuity of · with respect to the weak topology because we have ra (y, {xni }) = lim sup y − xni i and y − xni * y − w. We will use the above lemma in the proofs of the next three theorems. Theorem 3.2. Let (X, ·) be an inﬁnite dimensional Banach space with 0 < 1, C a nonempty bounded closed convex subset of X and T = {Tt : t ∈ G} a uniformly klipschitzian asymptotically regular semigroup of mappings on C with k < γ, where γ satisﬁes wAN (X) γ > max 1, 2 and is the solution of the equation 2 (10) γ 1−δ wAN (X) 2 (γ) = 1. Then there exists in C a common ﬁxed point of T. Proof. Without loss of generality we can assume that (X, ·) is uniformly (X) convex, k 2 > wAN and k > 1. If 0 < 0 < 1 we need only make minor 2 changes in our proof. First we choose θ such that 1 < 1θ < wAN (X) and 1 < 1. θk 2 Next we ﬁx t0 , h ∈ G with h > 0 and deﬁne tn = t0 + nh for n = 1, 2, ... Let us observe that for each x ∈ C the sequence {Ttn x} is asymptotically regular and by Lemma 3.1 for each x ∈ C there exists a weakly convergent subsequence Ttni x with w ∈ conv {Ttn x} such that k 1−δ (11) ra w, Ttni x ≤ θ · diama ({Ttn x}) . In other words, for each x ∈ C we can ﬁnd a subsequence Ttni x and w (x) ∈ C which satisfy the inequality (11). Hence, if z (x) is the unique element of Ac C, Ttni x , then we have (12) r (x) = ra z (x) , Ttni x ≤ ra w (x) , Ttni x ≤ θ · diama ({Ttn x}) . 254 MONIKA BUDZYŃSKA, TADEUSZ KUCZUMOW AND SIMEON REICH The asymptotic regularity of T leads to ra Ttm z (x) , Ttni x ≤ ra Ttm z (x) , Ttni +tm x + lim sup Ttni +tm x − Ttni x i (13) = ra Ttm z (x) , Ttni +tm x ≤ k · ra z (x) , Ttni x for every m. Now we set R (x) = ra (x, {Ttn x}) = lim sup x − Ttn x . n We observe that by the asymptotic regularity of T and since tl = t0 + lh we have for l = 1, 2, ..., diama ({Ttn x}) = lim sup Ttm x − Ttn x = lim sup m̃ m̃<m<n max Ttm x − Ttm+n x , sup Ttm x − Ttm+n0 +l x 1≤n≤n0 m̃ m̃<m = lim sup m̃ m̃<m ≤ lim sup m̃ l>0 sup Ttm x − Ttm+n0 +l x l>0 sup Ttm x − Tmh x m̃<m sup Tmh x − Ttm+n0 +l x + lim sup sup m̃ m̃<m l>0 ≤ k · sup x − Ttn0 +l x = k · sup x − Ttn x for n0 ≥ 1. Hence (14) n>n0 l>0 diama ({Ttn x}) ≤ k · lim sup x − Ttn x = k · R (x) . n We construct the sequence {xm } in the following way: x0 ∈ C is arbitrary and xm+1 = z (xm ) for m = 0, 1, ... . By (12) and (14) we get (15) rm = r (xm ) ≤ θ · k · R (xm ) . Now we have to consider two cases. If for some m0 we have rm0 = 0, then we get R (z (xm0 )) = lim sup z (xm0 ) − Ttn z (xm0 ) n ≤ lim sup lim sup z (xm0 ) − Ttni xm0 n i + Ttni xm0 − Ttn +tni xm0 + Ttn +tni xm0 − Ttn z (xm0 ) = 0. UNIFORM ASYMPTOTIC NORMAL STRUCTURE 255 In the second case we have rm > 0 for all m ≥ 0. For each m we ﬁrst choose an arbitrary 0 < < R (xm+1 ) and then j such that Ttj xm+1 − xm+1 ≥ R (xm+1 ) − . Now we ﬁnd i0 such that for all i ≥ i0 we have Ttni xm − xm+1 ≤ rm + 2 (the subsequence {tni } depends on xm here) and (see (13)) Ttni xm − Ttj xm+1 ≤ k · (rm + ) . It follows that Tt xm − 1 xm+1 + Tt xm+1 j ni 2 R (xm+1 ) − . k · (rm + ) ≤ k · (rm + ) · 1 − δ Letting i tend to inﬁnity we obtain rm ≤ k · (rm + ) · 1 − δ Taking now to 0 we get R (xm+1 ) − . k · (rm + ) . . R (xm+1 ) r m ≤ k · rm · 1 − δ k · rm which after applying the inequality (15) implies that 1 R (xm+1 ) ≤ θ · k 2 · δ −1 1 − · R (xm ) . k Let us observe that 1 0 < α = θ · k 2 · δ −1 1 − <1 k and therefore R (xm ) ≤ αm · R (x0 ) (16) for m = 1, 2, ... . Hence by (15) and (16) we deduce that xm+1 − xm ≤ lim sup xm+1 − Ttni xm + lim sup Ttni xm − xm i ≤ ra xm+1 , Ttni xm i + R (xm ) = rm + R (xm ) ≤ (θ · k + 1) · R (xm ) ≤ (θ · k + 1) · αm · R (x0 ) (the subsequence {tni } depends on xm here) and therefore the sequence {xn } is strongly convergent to some x ∈ C. By the inequality R (x) − R (y) ≤ (1 + k) · x − y , which is valid for all x, y ∈ C, we have R (x) = 0. Thus in both cases we can ﬁnd y ∈ C with R (y) = 0. This means that y = lim Ttn y n 256 MONIKA BUDZYŃSKA, TADEUSZ KUCZUMOW AND SIMEON REICH and by the asymptotic regularity of T we have Tt y = lim Tt+tn y = lim Ttn y = y n n for each t ∈ G. This completes the proof. Theorem 3.3. Let C be a convex weakly compact subset of a Banach space X with wSOC (X) > 1. Then every asymptotically regular uniformly klipschitzian semigroup T = {Tt : t ∈ G} of mappings on C with k < 1 [wSOC (X)] 2 has a common ﬁxed point. Proof. Let k ≥ 1 and let us ﬁx θ such that 1 k 2 < < wSOC (X) . θ We choose t0 , h ∈ G with h > 0 and consider the sequence {tn } = {t0 + nh}. For each x ∈ C we deﬁne R (x) by R (x) = ra (x, {Ttn x}) = lim sup x − Ttn x . n Let us observe that for each x ∈ C the sequence {Ttn x} is asymptotically regular and by Lemma 3.1 for each x ∈ C there is a weakly convergent to w subsequence Ttni x such that (i) ra w, Ttni x ≤ θ · diama ({Ttn x}), (ii) y − w ≤ ra y, Ttni x for every y ∈ X. By i., ii., (14) and the asymptotic regularity of T we obtain ra Ttm w, Ttni x ≤ ra Ttm w, Ttni +tm x ≤ k · ra w, Ttni x + lim sup Ttni +tm x − Ttni x i ≤ k · θ · diama ({Ttn x}) ≤ θ · k 2 · R (x) , Ttm w − w ≤ ra Ttm w, Ttni x ≤ θ · k 2 · R (x) for each m, and ﬁnally R (w) ≤ θ · k 2 · R (x) = α · R (x) , where 0 ≤ α = θ · k 2 < 1. Consequently, for each x ∈ C there is w (x) such that Ttni x * w (x) , ra w (x) , Ttni x ≤ θ · k · R (x) and R (w) ≤ α · R (x) . This allows us to construct a sequence {xm } which is convergent to a ﬁxed point of T. We simply choose the ﬁrst element x1 arbitrarily and next we set xm = w (xm−1 ) for m = 2, 3, ... . Now it is suﬃcient to observe that R (xm ) ≤ αm−1 · R (x1 ) UNIFORM ASYMPTOTIC NORMAL STRUCTURE 257 and repeat the arguments from the end of the proof of Theorem 3.2 to ﬁnish the present proof. In the next theorem we employ κω (X) and wSOC (X). Theorem 3.4. Let (X, ·) be a Banach space with wSOC (X) > 1, C a convex weakly compact subset of X and T = {Tt : t ∈ G} an asymptotically regular uniformly klipschitzian semigroup of mappings on C. If 1+ 1 + 4 · [wSOC (X)] · (κω (X) − 1) , 2 then T has a common ﬁxed point. (17) k< Proof. The proof is based on ideas presented in [9]. Let us denote S = wSOC (X) and κ = κω (X). Without loss of generality we may assume that S < +∞ (see Theorem 3.2 in [6]) and k > 1. The inequality (17) and k > 1 imply κ > 1. Next we observe that 1+ 1 + 4 · S · (κ − 1) ≤ S < +∞, 2 because (see Section 2) κ ≤ W CS(X) ≤ S < +∞. The inequality k< implies the inequality 1+ 1 + 4 · S · (κ − 1) 2 k κ−1 < . S k−1 Directly from the deﬁnition of κ we ﬁnd a > 1 and 1 < b < κ such that b −1 k < a , S k−1 (18) and for all x, y ∈ C and r > 0 with x − y ≥ r and each weakly convergent sequence {zn } in C for which lim supn x − zn ≤ a·r and lim supn y − zn ≤ b · r there exists some z ∈ C such that lim inf n z − zn ≤ r. Next we choose > 0 such that 1 + 2 = α < 1. a Similarly as in the proofs of the previous theorems we consider the sequence {tn } = {t0 + nh}, where t0 , h ∈ G and h > 0. For x ∈ C we deﬁne R (x) as follows R (x) = inf r > 0 : ∃y∈C lim inf x − Ttn y ≤ r . n ) = 0 for some x ∈ C. To this end we take an First we will show that R (x arbitrary x ∈ C. Assume that R (x) > 0. Then we can ﬁnd y ∈ C with lim inf x − Ttn y < R (x) · (1 + ) . n 258 MONIKA BUDZYŃSKA, TADEUSZ KUCZUMOW AND SIMEON REICH There are two possibilities; either S · R (x) · (1 + ) k·a sup x − Ttn (x) ≤ (19) n or S · R (x) · (1 + ) x − Ttj (x) > (20) k·a for some j. Let us take a look at the ﬁrst case. By (14) and (19) we get S · R (x) · (1 + ) a n and after applying the deﬁnition of wSOC (X) we obtain diama ({Ttn x}) ≤ k · lim sup x − Ttn (x) ≤ S · inf ra w, Ttni x This implies that : Ttni x is weakly convergent to w S · R (x) · (1 + ) . a ≤ inf ra w, Ttni x : Ttni x is weakly convergent to w ≤ R (x) · (1 + ) . a Hence there exists a subsequence Ttni x which weakly converges to w ∈ C such that R (x) · (1 + 2) (21) = αR (x) lim w − Ttni x < i a and therefore (22) R (w) < αR (x) . Next we see that w − x ≤ w − Ttni x + x − Ttni x and by (19) and (21) this yields w − x ≤ lim w − Ttni x + lim sup x − Ttni x i ≤ α · R (x) + i S · R (x) · (1 + ) k·a S · α · R (x) S =α· 1+ · R (x) . k k Let us now consider the second case. By (20) we have (23) ≤ α · R (x) + S · R (x) · (1 + ) x − Ttj x > k·a for some j. Let us choose a weakly convergent Ttni y such that x − Ttni y < R (x) · 1 + 2 UNIFORM ASYMPTOTIC NORMAL STRUCTURE and 259 tj Ttni +m−1 y − Ttni +m y < R (x) · 2 m=1 for each i. This implies (24) tj x − Ttni +tj y < x − Ttni y + Ttni +m−1 y − Ttni +m y < R (x) · (1 + ) m=1 and Ttj x − Ttni +tj y ≤ k · R (x) · (1 + ) (25) for every i. By (18) we can choose λ such that b −1 k <λ< a . S k−1 Then by (24), (25) and (26) we have (26) λTtj x + (1 − λ) x − Ttni +tj y ≤ λ Ttj x − Ttni +tj y + (1 − λ) x − Ttni +tj y ≤ λ · k · R (x) · (1 + ) + (1 − λ) · R (x) · (1 + ) (27) = [λ (k − 1) + 1] · R (x) · (1 + ) < b · and next by (20) and (26) , R (x) · (1 + ) a λTtj x + (1 − λ) x − x = λ Ttj x − x S · R (x) · (1 + ) R (x) · (1 + ) > . k·a a Directly from the deﬁnition of b and by (24), (27) and (28) there exists w ∈ C such that R (x) · (1 + ) (29) ≤ α · R (x) w − Ttni +tj y ≤ a for every i. Therefore by the asymptotic regularity of T and (29) we get (28) >λ· R (w) ≤ lim sup w − Ttni y i ≤ lim sup w − Ttni +tj y + lim sup Ttni +tj y − Ttni y i (30) i = lim sup w − Ttni +tj y ≤ α · R (x) , i and by (24) and (29) we obtain w − x ≤ w − Ttni +tj y + x − Ttni +tj y (31) ≤ α · R (x) + R (x) · (1 + ) = (1 + + α) · R (x) . 260 MONIKA BUDZYŃSKA, TADEUSZ KUCZUMOW AND SIMEON REICH Thus in both cases for every x ∈ C we can ﬁnd w ∈ C for which the following inequalities are valid: (32) R (w) ≤ α · R (x) and (33) w − x ≤ max 1 + + α, α · 1 + S k · R (x) = M · R (x) (by (22), (23), (30) and (31)). This allows us to deﬁne a function f : C → C by f (x) = w, where w satisﬁes (32) and (33). We introduce a sequence {xn } as follows: x0 is chosen in an arbitrary way and next xn = f (xn−1 ) for n = 1, 2, ... . The sequence {xn } is a Cauchy sequence. Indeed, the inequalities (32) and (33) lead to xn − xn−1 ≤ M · R (xn−1 ) ≤ M · αn−1 · R (x0 ) . as the limit of {xn } and applying the inequality Setting x R (u) − R (v) ≤ u − v , ) = 0. which is valid for all u, v ∈ C, we get R (x Now we prove that R (x) = 0 implies that x is a common ﬁxed point of T. To prove this fact it is suﬃcient to observe that for every y ∈ C, t ∈ G and every natural n we have − Tt x ≤ x − Ttn y + Ttn y − Ttn +t y + Ttn +t y − Tt x x − Ttn y + Ttn y − Ttn +t y . ≤ (1 + k) x Applying the deﬁnition of R and the asymptotic regularity of T we complete the proof. 4. Examples Example 4.1. Let us consider the space Xβp [6], where 1 < p < ∞ and 1 1 1 1 < β < 4 p . Then for 2 p ≤ β < 4 p we have κω Xβp N Xβp = 1 [2], but wAN Xβp = wSOC Xβp = 1 4p β = W CS Xβp > 1 [6]. Example 4.2. Let us observe that for the space Xβ2 with 1 < β < √ 2 β Xβ2 = max 1, [2, 3], wAN get W CS and κω (X) ≥ κ0 (X) [10], where κ0 Xβ2 Xβ2 = wSOC Xβ2 1 1 2 = 1 + 2 − 2 β2 − 1 β β 2 [9] (see also [35]). Hence for β suﬃciently close to 1 the constants 1+ 1 + 4 · W CS Xβ2 · κ0 Xβ2 − 1 2 = √ = 5 2 √ , we 2 [6] UNIFORM ASYMPTOTIC NORMAL STRUCTURE < 1+ 1 + 4 · wSOC Xβ2 that [16] wSOC Xβ2 0 Xβ2 = · κ0 Xβ2 − 1 2 are strictly bigger than 261 2 β2 − 1 2 √ 1 2 1 = 2 4 . Let us observe in addition √ for 1√< β ≤ 2 for 2 < β < 2 and therefore for 1 < β < 25 the constant γ given by (10) is strictly bigger than the constant γ2 deﬁned by (6). 5. Acknowledgments The third author was partially supported by the Fund for the Promotion of Research at the Technion. Part of the work on this paper was done when the third author visited the Institute of Mathematics at UMCS. He thanks the Institute and its members for their hospitality. All the authors thank the referee for several useful comments and corrections. 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