Fixed point properties for nonexpansive representations of topological semigroups

J. Math. Anal. Appl. 327 (2007) 246–250 www.elsevier.com/locate/jmaa

Fixed point properties for nonexpansive representations of topological semigroups E. Nazari ∗ , A.H. Riazi Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave, Tehran, Iran Received 8 September 2003 Available online 11 May 2006 Submitted by R. Triggiani

Abstract In this paper, we study a fixed point and a nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings on a nonempty subset of a Hilbert space. © 2006 Elsevier Inc. All rights reserved. Keywords: Nonexpansive mapping; Fixed point; Semitopological semigroup; Invariant means; Hilbert space

1. Introduction Let S be a semitopological semigroup with identity, i.e., S is a semigroup with a Hausdorff topology such that for each a ∈ S, the mappings s → a.s and s → s.a from S into S are continuous. Let D be a nonempty subset of a Hilbert space H and  = {Tt : t ∈ S} be a continuous representation of S as mappings from D into D, i.e., (i) Te = I ; (ii) Tst = Ts Tt for all s, t ∈ S; (iii) the mapping S × C → C defined by (s, t) → Ts x, s ∈ S, x ∈ C, is continuous when S × C has the product topology. Let  be a continuous representation of the semigroup S and each Ts , s ∈ S, be a nonexpansive self-mapping of D, i.e., Ts x − Ts y  x − y for all x, y ∈ D and s ∈ S, then  is called a continuous nonexpansive semigroup on D. Goebel and Schöneberg in [2] proved that if T be a nonexpansive self-mapping of a nonempty subset D of a Hilbert space H , then T has a fixed point in D if and only if {T n x} is bounded for some x ∈ D and for y ∈ clco{T n x: n > 0}, there is a unique p ∈ D such that y − p = infz∈D y − z. In this paper * Corresponding author.

E-mail addresses: [email protected] (E. Nazari), [email protected] (A.H. Riazi). 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.03.093

E. Nazari, A.H. Riazi / J. Math. Anal. Appl. 327 (2007) 246–250

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we study this theorem for nonexpansive continuous representation of an amenable semitopological semigroup on a nonempty subset D of a Hilbert space H . 2. Some preliminaries All topologies in this paper are assumed to be Hausdorff. Given a nonempty set S, we denote by ∞ (S) the Banach space of all bounded real-valued functions on S with supremum norm. Let S be a semigroup. Then a subspace X of ∞ (S) is left (respectively right) translation invariant if a (X) ⊆ X (respectively ra (X) ⊆ X) for all a ∈ S, where (a f )(s) = f (as) and (ra f )(s) = f (sa), s ∈ S. If S is a semitopological semigroup, we denote by CB(S) the closed subalgebra of ∞ (S) consisting of continuous functions. Let LUC(S) (respectively RUC(S)) be the space of left (respectively right) uniformly continuous functions on S, i.e., all f ∈ CB(S) such that the mapping from S into CB(S) defined by s → s f (respectively s → rs f ) is continuous when CB(S) has the sup norm topology. It is well known that LUC(S) and RUC(S) are left and right translation invariant closed subalgebras of CB(S), respectively, containing constants [1]. Note that when S is a topological group, then LUC(S) is precisely the space of left uniformly continuous functions on S defined in [3]. Now suppose X be a subspace of ∞ (S) containing constants. Then μ ∈ X ∗ is called a mean on X if μ = μ(1) = 1. Moreover, let X be rs -invariant then, a mean μ on X is right invariant if μ(rs f ) = μ(f ) for all s ∈ S and f ∈ X. Similarly we can define left invariant mean. μ is called an invariant mean if it is left and right invariant. The value of a mean μ at f ∈ X will also be denoted by μ(f ), μ, f or μt f (t). A net {μα } on RUC(S) is called asymptotically invariant [9] if for each f ∈ RUC(S) and a ∈ S, μα (ra f ) − μα (f ) −→ 0 and μα (la f ) − μα (f ) −→ 0. Let μ be a mean on X, E be a Banach space, φ : S → E be a bounded function, and K be a closed convex subset of E. Suppose for each x ∈ K, the real-valued function f on S defined by 2  fx (t) = φ(t) − x  for all t ∈ S, belongs to X. Then set r(x) = μ, fx for all x ∈ K, and define r = infx∈K r(x) and Mμ = {y ∈ K: r(y) = r}. Lemma 2.1. The nonnegative real-valued function r on K defined as above is continuous, convex and r(xn ) → ∞ as xn  → ∞. If E is reflexive or K is weakly compact, then Mμ is a nonempty closed convex subset of K. Furthermore, if E is a Hilbert space, then Mμ contains a unique element y and r + y − x  r(x) for all x ∈ K [7]. Remark 2.2. Let D be a subset of the real Hilbert space H , x ∈ H and  = {Tt : t ∈ S} be a nonexpansive semigroup on D. Suppose {Tt z: t ∈ S} is bounded for some z ∈ D. Then the functions fx (t) = Tt z − x2 and gx (t) = Tt z, x are in RUC(S) [6], see also [5]. Now if μ is a mean on RUC(S), by the Riesz representation theorem, there exists zμ ∈ H such that μ Tt z, x = zμ , x for each x ∈ H [8]. 3. Fixed point theorems Lemma 3.1. Let S be a semitopological semigroup, D be a nonempty subset of a real Hilbert space H and  = {Tt : t ∈ S} be a nonexpansive continuous representation of a semitopological

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semigroup S on D. Suppose {Tt z: t ∈ S} is bounded for some z ∈ D and RUC(S) has a left invariant mean μ. Then zμ ∈ Mμ , and Tst x − zμ   Tt x − zμ  for all x ∈ D and s, t ∈ S. Proof. Let μ be a LIM on RUC(S). By Remark 2.2 for x ∈ H , the function ft (x) = Tt z − x2 , t ∈ S, is in RUC(S). Let Mμ = {y ∈ H : r(y) = r} where r = inf{r(x): x ∈ H } and r(x) = μ, fx . By Lemma 2.1, Mμ contains a unique element y and r + y − x2  r(x) for all x ∈ H . Now let x ∈ H and t ∈ S. Then, zμ − x2 = Tt z − x2 − Tt z − zμ 2 − 2 Tt z − zμ , zμ − x . So,   0  zμ − x2 = μt Tt z − x2 − Tt z − zμ 2 − 2 Tt z − zμ , zμ − x

= μt Tt z − x2 − μt Tt z − zμ 2 − 2 zμ − zμ , zμ − x

= μt Tt z − x2 − μt Tt z − zμ 2 . This implies that Mμ consists of a single point zμ . To proof of the second part, let x ∈ D, then {Tt x: t ∈ S} is bounded because Tt x − Tt z  x − z for t ∈ S. For p = zμ ∈ Mμ and s, t, θ ∈ S, we have 2 Tt x − Tst x, Tθ z − p = Tt x − p2 − Tst x − p2 + Tst x − Tθ z2 − Tt x − Tθ z2 . Now, applying μ to both sides of the above equality with respect to θ we have 0 = 2 Tt x − Tst x, p − p

= Tt x − p2 − Tst x − p2 + μθ Tst x − Tθ z2 − μθ Tt x − Tθ z2 . On the other hand, since μθ Tst x − Tθ z2 = μθ Tst x − Tsθ z2

(by left invariance μ)

 μθ Tt x − Tθ z . 2

Therefore, Tst x − p2  Tt x − p2 + μθ Tt x − Tθ z2 − μθ Tt x − Tθ z2 . So Tst x − p  Tt x − p. This completes the proof.

2

We now state our main fixed point theorem extending Goebel–Schöneberg’s theorem [2] for nonexpansive continuous representation of an amenable semitopological semigroup on a subset D of a Hilbert space. Theorem 3.2. Let S be a semitopological semigroup with identity, D be a nonempty subset of a real Hilbert space H and  = {Tt : t ∈ S} be a nonexpansive continuous representation of S on D. Suppose RUC(S) has a left invariant mean μ. Then  has common fixed point in D if and only if {Tt x: t ∈ S} is bounded for some x ∈ D and for any y ∈ clco{Tt x: t ∈ S}, there is a unique p ∈ D such that y − p = infz∈D y − z.

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Proof. Necessity is obvious. Let us prove the sufficiency. Assume {Tt x: t ∈ S} is bounded for some x ∈ D and Mμ be as in the proof of Lemma 3.1 and c = zμ ∈ Mμ . Then by [4] we have c ∈ clco{Tt x: t ∈ S}. So there exists a unique p ∈ D such that c − p  c − z for all z ∈ D. On the other hand, by Lemma 3.1 we know Tt p − c = Tte p − c  Te p − c = p − c. Hence, we have c − p = infz∈D c − z  c − Tt p  c − p, t ∈ S, and the uniqueness of p implies that Tt p = p for all t ∈ S, i.e., p is a common fixed point for . 2 Corollary 3.3. (Goebel–Schöneberg [2]) Let T be a nonexpansive self-mapping of a nonempty subset D of a Hilbert space H . Then T has a fixed point in D if and only if {T n x} is bounded for some x ∈ D and for any y ∈ clco{T n x: n > 0} there is a unique p ∈ D such that y − p = infz∈D y − z. Proof. Let S = (N, +) and  = {T n : n ∈ N }, then since S is amenable by Theorem 3.2 the proof is compete. 2 4. Ergodic theorem We are now ready to establish our nonlinear ergodic theorem by using Lemma 3.1 for an arbitrary subset D of a Hilbert space H . Theorem 4.1. Let H , D, S,  be as in Theorem 3.2. If μ is an invariant mean on RUC(S), and {Tt z: t ∈ S} is bounded for some z ∈ D, then for any asymptotically invariant net {μα } of means on RUC(S), the net Zμα converges weakly to Zμ . In particular, if ν is another invariant mean on RUC(S), then zμ = zν . Proof. If μ is invariant mean on RUC(S), then μt Tt z − x2  inf sup Tts z − x2 s

(by right invariance of μ)

t

for each x ∈ H [10]. On the other hand, let y ∈ Mμ and s ∈ S, by Lemma 3.1 we have   inf sup y − Ttu Ts z2 − y − Ts z2  0. u

t

And hence inf sup Ttu z − y2  inf sup Ttus z − y2 = inf sup Ttu Ts z − y2  Ts z − y2 . u

u

t

t

u

t

So, we have inf sup Ttu z − y2  μs Ts z − y2 . u

t

Therefore, for each y ∈ Mμ , μt Tt z − y2 = inf sup Tts z − y2 . s

t

Hence if ν is another invariant mean on RUC(S), then by Lemma 3.1, zν ∈ Mν . Hence μt Tt z − zμ 2 = inf sup Tts z − zμ 2  μt Tt z − zν 2 s

t

 inf sup Tts z − zν 2 = νTt z − zν 2  νt Tt z − zμ 2 s

t

 inf sup Tts z − zμ2 = μt Tt z − zμ2 . s

t

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Hence μt Tt z − zμ 2 = μt Tt z − zν 2 . By uniqueness of the element in Mμ , we have zμ = zν . Now, if {μα } is an asymptotically invariant net and μ is a cluster point of {μα } in the weak topology, then μ is invariant mean on RUC(S). Hence if {zμβ } is a subnet of the net {zμα } such that zμβ converges weakly to some y in H , then, since a cluster point ν of {μβ } is also a cluster point of {μα }, ν is invariant mean. So, y = zμ = zν by the above. This implies that zμα converges weakly to zμ . 2 References [1] J.F. Berglund, H.D. Junghenn, P. Milnes, Analysis on Semigroups, Wiley, New York, 1989. [2] K. Goebel, R. Schöneberg, Moons, bridges, birds . . . and nonexpansive mappings in Hilbert space, Bull. Austral. Math. Soc. 17 (1977) 463–466. [3] E. Hewitt, K. Ross, Abstract Harmonic Analysis, I, Springer-Verlag, Berlin, 1963. [4] O. Kada, A.T. Lau, W. Takahashi, Asymptotically invariant net and fixed point set for semigroup of nonexpansive mapping, Nonlinear Anal. 29 (1997) 539–550. [5] A.T. Lau, W. Takahashi, Invariant means and fixed point properties for nonexpansive representations of topological semigroups, Topol. Methods Nonlinear Anal. 5 (1995) 39–57. [6] A.T. Lau, Semigroup of nonexpansive mappings on a Hilbert space, J. Math. Anal. Appl. 105 (1985) 514–522. [7] N. Mizoguchi, W. Takahashi, On the existence of fixed points and ergodic retractions for Lipschitzian semigroups in Hilbert spaces, Nonlinear Anal. 14 (1990) 69–80. [8] W. Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981) 253–256. [9] W. Takahashi, Fixed point theorem and nonlinear ergodic theorem for nonexpansive semigroups without convexity, Canad. J. Math. 35 (1992) 1–8. [10] W. Takahashi, Fixed point theorems for families of nonexpansive mappings on unbounded sets, J. Math. Soc. Japan 36 (1984) 543–553.