CUBO A Mathematical Journal Vol.10, N o ¯ 04, (101–108). December 2008 Browder Convergence and Mosco Convergence for Families of Nonexpansive Mappings Tomonari Suzuki* Department of Mathematics, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan email: suzuki-t@mns.kyutech.ac.jp ABSTRACT We study the relationship between Browder’s strong convergence and Mosco conver- gence of fixed-point set for families of nonexpansive mappings. RESUMEN Estudiamos la relación entre la convergencia fuerte de Browder y la convergencia de Mosco del conjunto de puntos fijos para familias de aplicaciones no espansivas. Key words and phrases: Nonexpansive mapping, nonexpansive semigroup, fixed point, Browder convergence, Mosco convergence. Math. Subj. Class.: 47H10, 47H09, 47H20. *The author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Educa- tion, Culture, Sports, Science and Technology. 102 Tomonari Suzuki CUBO 10, 4 (2008) 1. Introduction Let C be a subset of a Banach space E. A mapping T on C is called a nonexpansive mapping if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x,y ∈ C. We denote by F(T ) the set of fixed points of T . Using the results in Gossez and Lami Dozo [6] and Kirk [8], we can prove that F(T ) is nonempty in the case where C is weakly compact, convex and has the Opial property. See also [1, 5, 7] and others. In 1967, Browder [2] proved the following strong convergence theorem, Theorem 1 (Browder [2]). Let C be a bounded closed convex subset of a Hilbert space E and let T be a nonexpansive mapping on C. Let {αn} be a sequence in (0, 1) converging to 0. Fix u ∈ C and define a sequence {xn} in C by xn = (1 − αn) Txn + αn u for n ∈ N. Then {xn} converges strongly to Pu, where P is the metric projection from C onto F(T ). A family of mappings {T (t) : t ≥ 0} is called a one-parameter strongly continuous semigroup of nonexpansive mappings (nonexpansive semigroup, for short) on C if the following are satisfied: (NS1) For each t ≥ 0, T (t) is a nonexpansive mapping on C. (NS2) T (s + t) = T (s) ◦ T (t) for all s,t ≥ 0. (NS3) For each x ∈ C, the mapping t 7→ T (t)x from [0,∞) into C is strongly continuous. Suzuki [15] proved that ⋂ t F(T (t)) is nonempty provided C is bounded closed convex and every nonexpansive mapping on C has a fixed point. He also proved a semigroup version of Browder’s convergence theorem in [11, 17]. Theorem 2 ([11, 17]). Let E be a smooth Banach space with the Opial property such that the normalized duality mapping J of E is weakly sequentially continuous at zero. Let C be a weakly compact convex subset of E. Let {T (t) : t ≥ 0} be a nonexpansive semigroup on C. Let τ be a nonnegative real number. Let {αn} and {tn} be sequences in R satisfying 0 < αn < 1, 0 ≤ τ + tn and tn 6= 0 for n ∈ N, and limn tn = limn αn/tn = 0. Fix u ∈ C and define a sequence {xn} in C by xn = (1 − αn) T (τ + tn)xn + αn u for n ∈ N. Then {xn} converges strongly to Pu, where P is the unique sunny nonexpansive retraction from C onto ⋂ t F(T (t)). Motivated by Theorem 2, Suzuki [19] considered the Mosco convergence of {F(T (τ + tn))}. The following theorem is a corollary of the main result in [19]. Theorem 3 ([19]). Let E, C and {T (t) : t ≥ 0} be as in Theorem 2. Let τ be a nonnegative real number and let {tn} be a sequence in R satisfying 0 ≤ τ +tn and tn 6= 0 for n ∈ N, and limn tn = 0. Then {F(T (τ + tn))} converges to ⋂ t F(T (t)) in the sense of Mosco. Therefore we can guess that Browder convergence is strongly connected with Mosco conver- gence. In this paper, we study the relationship between Browder convergence and Mosco conver- gence for families of nonexpansive mappings. CUBO 10, 4 (2008) Browder Convergence and Mosco Convergence ... 103 2. Preliminaries Throughout this paper we denote by N the set of all positive integers and by R the set of all real numbers. Let E be a Banach space and let {An} be a sequence of subsets of E. Define two sets s-liminf n→∞ An and w-limsup n→∞ An as follows: x ∈ s-liminfn An if and only if there exist a sequence {xn} in E and n0 ∈ N such that {xn} converges strongly to x and xn ∈ An for n ∈ N with n ≥ n0. x ∈ w-limsupn An if and only if there exists a sequence {xn} in E such that {xn} converges weakly to x and {n ∈ N : xn ∈ An} is an infinite subset of N. It is obvious that s-liminfn An ⊂ w-limsupn An holds. We say {An} converges to a subset A of E in the sense of Mosco [9] if A = s-liminfn An = w-limsupn An. And we write A = M-lim n→∞ An. Let E be a Banach space. The normalized duality mapping J of E is defined by J(x) = {f ∈ E∗ : 〈x,f〉 = ‖x‖2 = ‖f‖2}. E is said to be smooth if and only if J(x) consists of one element for every x ∈ E. If E is smooth, then we can consider that J is a mapping from E into E∗. J is said to be weakly sequentially continuous at zero if for every sequence {xn} in E which converges weakly to 0 ∈ E, {J(xn)} converges weakly ∗ to 0 ∈ E∗. A nonempty subset C of a Banach space E is said to have the Opial property [10] if for each weakly convergent sequence {xn} in C with weak limit z0 ∈ C, lim inf n→∞ ‖xn − z0‖ < lim inf n→∞ ‖xn − z‖ holds for z ∈ C with z 6= z0. All nonempty compact subsets have the Opial property. Also, all Hilbert spaces, ℓp(1 ≤ p < ∞) and finite dimensional Banach spaces have the Opial property. A Banach space with a duality mapping which is weakly sequentially continuous also has the Opial property [6]. We know that every separable Banach space can be equivalently renormed so that it has the Opial property [4]. Let C and K be subsets of a Banach space E. A mapping P from C into K is called sunny [3] if P ( Px + t (x − Px) ) = Px for x ∈ C and t ≥ 0 with Px + t (x − Px) ∈ C. Let {Sn} be a sequence of nonexpansive mappings on a closed convex subset C of a Banach space E and let {αn} be a sequence in (0, 1] with limn αn = 0. (E,C,{Sn},{αn}) is said to have Browder’s property [16] if for each u ∈ C, a sequence {xn} defined by xn = (1 − αn) Snxn + αn u (1) 104 Tomonari Suzuki CUBO 10, 4 (2008) for n ∈ N converges strongly. We note that {xn} is well defined because x 7→ (1 − αn) Snx + αn u is contractive. We know the following. Lemma 1 ([16]). Let (E,C,{Sn},{αn}) have Browder’s property. For each u ∈ C, put Pu = lim n→∞ xn, (2) where {xn} is a sequence in C defined by (1). Then P is a nonexpansive mapping on C. Using P , we can rewrite Theorem 2 as follows. Theorem 4. Let E, C, {T (t) : t ≥ 0}, τ, {αn} and {tn} be as in Theorem 2. Then ( E,C,{T (τ + tn)},{αn} ) has Browder’s property. Moreover a mapping P defined by (2) is the unique sunny nonexpansive retraction from C onto ⋂ t F(T (t)). 3. Main results In this section, we prove our main results. Theorem 5. Let (E,C,{Sn},{αn}) satisfy Browder’s property. Assume that C has the Opial property. Define a mapping P on C by (2). Then w-limsupn F(Sn) ⊂ F(P) holds. Proof. Fix x ∈ w-limsupn F(Sn). Then there exist a subsequence {nk} of {n} and a sequence {uk} in C such that uk ∈ F(Snk ) and {uk} converges weakly to x. We note that {uk} is bounded. Define a sequence {vn} in C by vn = (1 − αn) Snvn + αn x. Then from the assumption, {vn} converges strongly to Px. We have ‖uk − vnk ‖ ≤ (1 − αnk ) ‖uk − Snkvnk ‖ + αnk ‖uk − x‖ ≤ (1 − αnk ) ‖uk − vnk ‖ + αnk ‖uk − x‖ and hence ‖uk − vnk ‖ ≤ ‖uk − x‖. So lim inf k→∞ ‖uk − Px‖ ≤ lim inf k→∞ ( ‖uk − vnk ‖ + ‖vnk − Px‖ ) ≤ lim inf k→∞ ‖uk − x‖. From the Opial property, we obtain Px = x. As a direct consequence of Theorem 5, we obtain the following. Theorem 6. Let (E,C,{Sn},{αn}) satisfy Browder’s property. Define a mapping P on C by (2). Assume that C has the Opial property and F(P) ⊂ F(Sn) for n ∈ N. Then M-limn F(Sn) = F(P) holds. CUBO 10, 4 (2008) Browder Convergence and Mosco Convergence ... 105 Proof. From the assumption, F(P) ⊂ s-liminfn F(Sn). So by Theorem 5, we obtain the desired result. Remark. Using Theorems 2 and 6, we can prove Theorem 3. We next apply Theorem 6 to infinite families of nonexpansive mappings. The following con- vergence theorem was proved in [12, 14]. Theorem 7 ([12, 14]). Let E and C be as in Theorem 2. Let {Tn : n ∈ N} be an infinite family of commuting nonexpansive mappings on C. Let {αn} and {tn} be sequences in (0, 1/2) satisfying limn tn = limn αn/tn ℓ = 0 for ℓ ∈ N. Let {In} be a sequence of nonempty subsets of N such that In ⊂ In+1 for n ∈ N, and ⋃ n In = N. Define a sequence {Sn} of nonexpansive mappings on C by Snx = (( 1 − ∑ k∈In tn k ) T1x + ∑ k∈In tn k Tk+1x ) . Then ( E,C,{Sn},{αn} ) has Browder’s property. Moreover a mapping P defined by (2) is the unique sunny nonexpansive retraction from C onto ⋂ n F(Tn). By Theorem 6, we obtain the following. Theorem 8. Let E and C be as in Theorem 2. Let {Tn : n ∈ N} be an infinite family of commuting nonexpansive mappings on C. Let {tn} be a sequence in (0, 1/2) converging to 0. Let {In} and {Sn} be as in Theorem 7. Then {F(Sn)} converges to ⋂ n F(Tn) in the sense of Mosco. Proof. Put αn = tn n . Then it is obvious that limn αn/tn ℓ = 0 holds for ℓ ∈ N. By Theorem 7, ( E,C,{Sn},{αn} ) has Browder’s property and a mapping P defined by (2) is the unique sunny nonexpansive retraction from C onto ⋂ n F(Tn). Since F(P) = ⋂ n F(Tn), we have F(P) ⊂ F(Tn). So by Theorem 6, we obtain the desired result. We recall that a family of mappings {T (p) : p ∈ [0,∞)ℓ} is said to be an ℓ-parameter nonex- pansive semigroup on a subset C of a Banach space E if the following are satisfied: (ℓNS1) For each p ∈ [0,∞)ℓ, T (p) is a nonexpansive mapping on C. (ℓNS2) T (p + q) = T (p) ◦ T (q) for all p,q ∈ [0,∞)ℓ. (ℓNS3) For each x ∈ C, the mapping p 7→ T (p)x from [0,∞)ℓ into C is continuous. We denote by Q the set of all rational numbers. Using the result in [13], we obtain the following. Theorem 9. Let E and C be as in Theorem 2. Let {T (p) : p ∈ [0,∞)ℓ} be an ℓ-parameter nonexpansive semigroup on C. Let p1,p2, · · · ,pℓ ∈ [0,∞) ℓ such that {p1,p2, · · · ,pℓ} is linearly independent in the usual sense. Let β1,β2, · · · ,βℓ ∈ R such that {1,β1,β2, · · · ,βℓ} is linearly 106 Tomonari Suzuki CUBO 10, 4 (2008) independent over Q. Suppose p0 := β1p1 + β2p2 + · · · + βℓpℓ ∈ [0,∞) ℓ. Let {tn} be a sequence in (0, 1/2) converging to 0. Define a sequence {Sn} of nonexpansive mappings on C by Snx = ( 1 − ℓ ∑ k=1 tn k ) T (p0)x + ℓ ∑ k=1 tn k T (pk)x. Then {F(Sn)} converges to ⋂ p F(T (p)) in the sense of Mosco. 4. Additional results In this section, we observe Browder’s property. Proposition 1. Let (E,C,{T},{αn}) satisfy Browder’s property. Define a mapping P on C by (2). Then P is a nonexpansive retraction from C onto F(T ). Proof. We first fix x ∈ C and define a sequence {un} in C by un = (1 − αn) Tun + αn x. Then since {un} converges strongly to Px, we obtain Px = TPx, which implies Px ∈ F(T ). We next fix y ∈ F(T ) and define a sequence {vn} in C by vn = (1 − αn) Tvn + αn y. Then since y = (1 − αn) Ty + αn y, we have vn = y and hence Py = y. This completes the proof. Remark. Though it is not interesting, we have confirmed that M-limn F(T ) = F(P) holds. There is an example such that P is not a retraction. See also [18]. Example 1. Let E be the two dimensional real Hilbert space and put C = E. For t ≥ 0, define a 2 × 2 matrices T (t) by T (t) = [ cos(t) − sin(t) sin(t) cos(t) ] . We can consider that {T (t) : t ≥ 0} is a linear nonexpansive semigroup on C. Let {αn} and {tn} be sequences in R satisfying 0 < αn < 1 and 0 < tn for n ∈ N, limn αn = limn tn = 0 and η := limn tn/αn ∈ (0,∞). Then (E,C,{T (tn)},{αn}) satisfies Browder’s property. However, a mapping P defined by (2) is not a retraction. Proof. For α ∈ (0, 1) and t ∈ (0,∞), we put a 2 × 2 matrix P(α,t) by P(α,t) = α 4 (1 − α) sin2(t/2) + α2 [ a −b b a ] , where a = α+ 2 (1−α) sin2(t/2) and b = (1−α) sin(t). It is easy to verify that for u ∈ C, P(α,t)u is the unique point satisfying x = (1 − α) T (t)x + αu. We have P := lim n→∞ P(αn, tn) = 1 η2 + 1 [ 1 −η η 1 ] = 1 √ η2 + 1 T (θ), CUBO 10, 4 (2008) Browder Convergence and Mosco Convergence ... 107 where θ := arctan(η) ∈ (0,π/2). Hence (E,C,{T (tn)},{αn}) satisfies Browder’s property. How- ever, P does not satisfy P 2 = P . We finally give an example such that M-limn F(Sn) $ F(P). Example 2. Let T be a nonexpansive mapping on a bounded closed convex subset C of a Banach space E. Assume that T is not the identity mapping on C. Define a sequence {Sn} of nonexpansive mappings on C by Snx = (1 − tn) x + tn Tx, where {tn} is a sequence in (0, 1) converging to 0. Let {αn} be a sequence in (0, 1) such that limn αn = 0 and limn αn/tn = ∞. Then (E,C,{Sn},{αn}) satisfies Browder’s property, a mapping P defined by (2) is the identity mapping on C and M-limn F(Sn) $ F(P) holds. Proof. Fix x ∈ C and define a sequence {un} in C by un = (1 − αn) Snun + αn x. We have ‖un − x‖ = (1 − αn) ‖Snun − x‖ ≤ (1 − αn) (1 − tn) ‖un − x‖ + (1 − αn) tn ‖Tun − x‖ and hence lim n→∞ ‖un − x‖ ≤ lim n→∞ (1 − αn) tn αn + tn − αn tn ‖Tun − x‖ = 0. Thus, {un} converges strongly to x. Therefore (E,C,{Sn},{αn}) satisfies Browder’s property and Px = x holds. From the assumption, F(T ) $ C = F(P). Since F(Sn) = F(T ), we have M-limn F(Sn) = F(T ) and hence M-limn F(Sn) $ F(P). Received: April 2008. Revised: April 2008. References [1] F.E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. USA, 53 (1965), 1272–1276. [2] , Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. Ration. Mech. Anal., 24 (1967), 82–90. [3] R.E. Bruck, Nonexpansive retracts of Banach spaces, Bull. Amer. Math. Soc., 76 (1970), 384–386. [4] D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc., 25 (1982), 139–144. 108 Tomonari Suzuki CUBO 10, 4 (2008) [5] J. Garćıa Falset, E. Llorens Fuster and E.M. Mazcuñán Navarro, Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings, J. Funct. Anal., 233 (2006), 494–514. [6] J.-P. Gossez and E. 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(2007), doi:10.1016/j.na.2007.04.026. N8-suzuki-accept